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[
"Split Learning without Local Weight Sharing To Enhance Client-side Data Privacy",
"Split Learning without Local Weight Sharing To Enhance Client-side Data Privacy"
] |
[
"Ngoc Duy Pham ",
"Tran Khoa Phan ",
"Alsharif Abuadbba ",
"Doan Nguyen ",
"Naveen Chilamkurti "
] |
[] |
[] |
Split learning (SL) aims to protect user data privacy by splitting deep models between client-server and keeping private data locally. SL has been demonstrated to achieve similar accuracy as the centralized learning model. In SL with multiple clients, the local training weights are shared between clients for local model aggregation. This paper investigates the potential of data leakage due to local weight sharing among the clients in SL by performing model inversion attacks. To mitigate the identified leakage issue, we propose and analyze privacyenhancement SL (P-SL), e.g., SL without local weight sharing, to boost client-side data privacy. We also propose paralleled P-SL to speed up the training process by employing multiple servers without accuracy reduction. Finally, we investigate P-SL with late participating clients and develop a server-based cache-based training to address the forgetting phenomenon in SL. Experimental results demonstrate that P-SL helps reduce up to 50% of client-side data leakage compared to SL. Moreover, P-SL and its cache-based version achieve comparable accuracy to SL under various data distributions with lower computation and communications costs. Also, caching in P-SL reduces the negative effect of forgetting, stabilizes the learning, and enables effective and low-complexity training in a dynamic environment with late-arriving clients.
|
10.48550/arxiv.2212.00250
|
[
"https://export.arxiv.org/pdf/2212.00250v1.pdf"
] | 254,125,105 |
2212.00250
|
a4008b5f6f882c56c20b1e54f00786718407244d
|
Split Learning without Local Weight Sharing To Enhance Client-side Data Privacy
Ngoc Duy Pham
Tran Khoa Phan
Alsharif Abuadbba
Doan Nguyen
Naveen Chilamkurti
Split Learning without Local Weight Sharing To Enhance Client-side Data Privacy
1Index Terms-Split learningprivacy preservationprivacy leakagehonest-but-curiousCNN
Split learning (SL) aims to protect user data privacy by splitting deep models between client-server and keeping private data locally. SL has been demonstrated to achieve similar accuracy as the centralized learning model. In SL with multiple clients, the local training weights are shared between clients for local model aggregation. This paper investigates the potential of data leakage due to local weight sharing among the clients in SL by performing model inversion attacks. To mitigate the identified leakage issue, we propose and analyze privacyenhancement SL (P-SL), e.g., SL without local weight sharing, to boost client-side data privacy. We also propose paralleled P-SL to speed up the training process by employing multiple servers without accuracy reduction. Finally, we investigate P-SL with late participating clients and develop a server-based cache-based training to address the forgetting phenomenon in SL. Experimental results demonstrate that P-SL helps reduce up to 50% of client-side data leakage compared to SL. Moreover, P-SL and its cache-based version achieve comparable accuracy to SL under various data distributions with lower computation and communications costs. Also, caching in P-SL reduces the negative effect of forgetting, stabilizes the learning, and enables effective and low-complexity training in a dynamic environment with late-arriving clients.
I. INTRODUCTION
Deep learning (DL), influenced by the rapid growth of data, is becoming increasingly important in our daily life. However, the privacy of data used in the model needs to be protected as required by various privacy regulations [1]. Split learning (SL) [2]- [5] is a new collaborative learning technique to protect user privacy by allowing model training without exposing users' raw private data. In a simple vanilla setting, SL divides a deep model into two parts deployed between a client (data owner) and a server (computing service), where only intermediate data (output of raw data fed through the local part) is exposed for collaborative training with the server part [2]. Compared to federated learning (FL) [6], SL is suitable for DL applications on resource-constrained devices (e.g. IoT, mobile) because the clients only need to run the first few layers of the deep model while the server conducts the rest layers, which are the most critical computation part. With the growing availability of different sources of data, SL has been extended to process learning on multiple clients such Pham as [2], [7]- [9]. In [10], [11], the authors have conducted a comprehensive study of SL on various scenarios from a low to a high number of clients, from balanced to imbalanced to extreme data distribution, etc., to provide a thorough insight into SL. Regarding SL on multiple data sources, the clients typically share their local weights with other clients to aggregate the learnt knowledge from different data sources, such as sequentially passing weights to the next client [2] or averaging all local weights at the client side [9]. In these settings, it is assumed that only the server is semi-trust (honest-but-curious [12]) while all clients trust each other. However, in case a client is malicious and colludes with the server, local weight sharing would lead to the potential of raw data leakage. Fig. 1 demonstrates an example of data leakage we identify in the original SL [2] with two clients, C 1 and C 2 , where C 2 uses the model inversion attack [13] to reconstruct the raw data of C 1 . Specifically, using the C 1 's shared local weights, C 2 could train a decoder [13], [14] on its local data. Then the decoder, acting as an inversed model of C 1 's model, is used to reconstruct raw data from C 1 's intermediate data exposed during training or inference. In addition, a decoder trained on C 2 's local model could be used to attack the next client that receives the C 2 's local weights to update its model. In order to better preserve data privacy at the client side, a research question (RQ) is raised: How to develop novel effective SLbased training to minimize data leakage in multi-client SL? As the answer, we propose a non-sharing local weights method among the clients during SL training dubbed a privacyenhancement SL (P-SL). Compared to the original SL, which provides client-server privacy preservation, P-SL additionally preserves data privacy at the clients. Our work is the first to identify data leakage among SL clients and the first to investigate SL performance without local weight sharing under various data distributions.
Furthermore, in SL, in addition to the data leakage issue among clients, the commitment to participate by all clients in the training process at the same time is a significant challenge [11]. Due to various network, energy, and resource problems, some devices might not be active during the whole training process or even join the training process later -when the collaborative training has been finished. Considering a new client joining the learning process after the completion of training the existing clients (called the first training), newcomer training would be challenging. This raises another RQ, How to ensure stable, low complexity, and high accuracy P-SL in dynamic environments where more clients join later? As the first training cycle has been completed, the learning of the new clients would impact the learnt knowledge of existing clients due to the forgetting phenomena [15] as discussed in [11].
In this paper, we propose a novel cache-based approach to deal with the forgetting phenomena to enhance the learning performance of P-SL. In summary, the contributions of this paper are:
• We define a novel honest-but-curious participants (clients and server) threat model, then identify the client-side data leakage of the original SL and its variants based on this threat model using model inversion attacks. • We propose P-SL to protect clients' data privacy under the threat model, which shows a reduction of up to 50% leakage as compared to the original SL. In P-SL, while the clients do not share local weights, they are still able to learn collaboratively at the server side to exploit local knowledge to improve the training effectiveness. • We present a comprehensive empirical evaluation with different datasets and distributions to demonstrate that the learning performance of P-SL is comparable to the original SL. We also propose a parallelizing P-SL to perform clients' learning simultaneously which reduce the time for training without accuracy reduction. • We propose a server-side caching approach for P-SL to enable only training the newly arriving clients to address the forgetting phenomenon of existing clients. Existing clients will also benefit from the training of new clients with new data without the need for re-training, hence reducing the training overhead. Experiment results show the advantages of caching in SL, especially in dynamic training environment.
The remainder of this paper is organized as follows: Section II provides background information about SL with variants, data distributions, and current research on privacy preservation in SL. Section III presents the identified threat model underlying the proposed P-SL with privacy measurement and performance evaluation, followed by parallelizing P-SL for speedup training time given in Section IV. Section V presents the proposed cache-based P-SL to deal with newly arriving clients in order to ensure reliable P-SL. We conclude the paper with some proposed future directions in Section VI.
II. BACKGROUND
This section presents the background information of SL with its variants for multiple clients, and distribution of user data.
A. Vanilla split learning
A deep model is a hypothesis h θ : X → Y that maps an input x ∈ X to an output y ∈ Y. Model training is finding the parameters (weights) θ that can accurately reflect the relationship between X and Y. In order to preserve user data privacy during model training, SL [2] splits layers of the entire deep model into multiple parts. In a simple vanilla setting, the model is divided into two parts h θ = f u · g w , where the localized f u contains the first few layers while the rest, g w , is deployed at the server, which is the most computationally complex part. While training, the client processes forward propagation on its local data batch and sends the output (called intermediate data or smashed data) together with the associated ground-truth labels f u (x batch ), y batch to the server. The server continues to process forward propagation on the received intermediate data to calculate the loss between y batch and g w (f u (x batch )). The gradients of the loss function are back-propagated at the server until the split layer (at which the deep model is cut/split), which is then sent to the client to complete back-propagation to the first layer locally. Then, based on the computed gradients, the client and the server update their corresponding weights, u and w. This process in the simple vanilla SL is the core mechanism for many other variants, including SL with multiple clients and our proposal.
B. SL with multiple clients
SL can be extended to train a deep model on N ≥ 2 clients. The deep model is also split into two, f θ = f u ·g w , where f u is distributed to all clients (f ui to client C i ) while g w is deployed at the central server. The training procedure uses data from multiple clients in a round-robin fashion. In general, when the training process of C i−1 is finished, C i receives the weights u i−1 of C i−1 to initialize its weights u i . Then C i continues the training on its data (collaboratively with the server following the vanilla setting) before sharing its trained weights, u i , to the next client C i+1 [2]. The weights u N trained by the last client C N are the model weights to be passed back to all clients for inference.
The model training of SL is processed sequentially among the clients which increase the latency; therefore, there are several studies to increase the training speed. In [8], the authors set up the mini-batch of each client proportional to its local data size to process the training model in parallel. All clients are initialised with the same weights and all gradients are averaged before being updated to clients in each iteration. The synchronization strategy results in all clients having the same model weights, which is a potential vulnerability to model inversion attacks.
SplitFed learning (SFL) [9] is a new approach that combines the strengths of FL and SL. In SFL, clients perform forward propagation in parallel on their respective data and send intermediate data to the central server. On receiving gradients from the server, the clients perform the back-propagation step and then send the updated weights to a Fed server. The Fed server averages (Avg(·)) all updates and sends out a single update to all clients. Similar to [8], after each global epoch, clients update the same weights for their models, which leads to the white-box case for model inversion attacks also.
C. Privacy-enhancing SL approaches
Critical privacy vulnerabilities of SL are based on the fact that a neural network is naturally predisposed to be functionally inverted [16]. That is, the intermediate data exposed by clients may be exploited to recover the raw input data. Therefore, SL privacy protection techniques typically aim to minimize data leakage from the intermediate data. For example, noise defend [17], [18] is an approach to apply additive Laplacian noise to the intermediate data to be sent to the server. Because of the added noise, the target's model is no longer a one-to-one function; therefore, it is harder for an attacker to learn the mapping from intermediate data to input data. Instead of direct noise, latent noise can be added via binarization [19] to reduce the correlation between intermediate and raw input data. However, these mechanisms require efforts to alleviate the impact of noise perturbation on model accuracy [20].
The work [21] proposed to reduce raw data leakage by adding an additional distance correlation-based loss term to the loss function. The distance correlation loss is minimized between the raw and intermediate data to help ensure the intermediate data has minimal information with regard to reconstructing the raw data while still being valuable enough to achieve model utility. In [19], the additional loss term is general that could be any leakage metric without being limited to the distance correlation. However, applying an extra loss may still result in privacy leakage because the intermediate data exposes too much information to be protected by a single leakage metric in the loss function [16]. To overcome the limitation caused by a single global loss function, in [22], clientbased privacy protection is proposed, which uses two different loss functions computed on the client and server sides. Along this direction, the work in [23] designs a framework that comprises two steps -a pre-training step that builds up a feature extractor with strong model-inversion resistance and a follow-up resistance transfer step that initializes the client-side models using the feature extractor. This framework requires sufficient computation resources to perform the pre-training on a source task and can be vulnerable during the early training epoch. To preserve both data privacy and label information, the authors from [24] employ sample diversity to mix the clientside models' intermediate data and creates obfuscated labels before sending them from clients to the server. The mixed intermediate data preserve a low distance correlation with raw data, thus preventing private data from being reconstructed separately. However, this mixing technique would not reduce data leakage as intended when conducting inference on a single data sample.
D. SL under diverse data distributions
In general, data is often distributed among clients in an imbalanced manner, e.g., some sensors are more active than others, hence having more data, or big healthcare institutions typically have more patient data [11]. Regarding a classification task, each client holds samples from all classes with similar quantities under balanced data distribution. In imbalanced data distribution, each client still has samples from all classes but with an imbalanced total number of samples. Note that the ratio of samples between classes at each client is similar to the ratio of the whole dataset. In [11], the authors investigated three different distributions for user data, including balanced, imbalanced, and non-IID (nonindependent identically distributed) data. Their findings reveal that SL performs well (compared to FL) under balanced and imbalanced data while being very sensitive to non-IID data. Therefore, this work investigates SL under balanced and imbalanced data settings. More specifically, we consider experiments with classification tasks on small-scale image datasets using 2D-CNN-based deep models. Table I summarizes the information of the two selected datasets with the corresponding Very Deep Convolutional Networks (VGG) [25] based deep models. The deep models are split into two parts deployed at the clients (the first two convolutions) and the server (the rest of the model), respectively. In this work, we have selected two datasets, Fashion [26] and CIFAR10 [27], which has 10 classes each with separate train and test sets. We distribute 60k samples of the train set to N clients and use the same test set to verify the learning performance of each client. For the imbalanced data distribution, the number of samples per client follows a half bell curve of the standard normal distribution; see Table II, which shows the total number of data samples per N = 6 clients.
Client index C 1 C 2 C 3 C 4 C 5 C 6 Splitting ratio 1% 3% 9% 19% 30% 38% No. of samples 600 1.8k 5.4k 11.4k 18k 22.8k
III. PRIVACY-ENHANCEMENT SPLIT LEARNING
We define a threat model as the underlying context for the proposed P-SL and the analysis of data leakage.
A. Threat model, model inversion attack, and data leakage
In traditional SL, the server is assumed to be honestbut-curious [28], e.g., it follows the training procedure but is curious about the raw data from clients. Note that the above reviewed works on SL privacy protection techniques assume this threat model. In our study, we further assume that the clients are also honest-but-curious. To the best of our knowledge, our work is the first study that assumes both honest-but-curious clients and server in SL.
In the new threat model, we define the leakage of user data as the difference between the reconstructed data compared to the private raw data of a client. A model inversion attack [13] is deployed by a malicious client to reconstruct the raw data from the intermediate data exposed by neighbor clients. The quality of the reconstruction can be measured using a variety of metrics such as mean squared error (MSE), structural similarity index measure (SSIM), peak signal-to-noise ratio (PNSR), Kullback-Leibler divergence, etc. [13], [19], [29], [30]. In this work, we use SSIM as data leakage metric.
B. Privacy-enhancement SL (P-SL) algorithm
In order to protect the local model from inversion attacks, we propose a non-sharing local weights method at the client side. We present the proposed P-SL algorithm in Alg. 1, followed by the performance analysis and evaluation.
In P-SL we deliver the same f u of a deep model h θ = f u ·g w to all clients to collaboratively conduct training with a shared server, g w . Fig. 2 depicts the architecture of P-SL together with the highlighted differences between SL and SFL. The proposed P-SL is based on SL with multiple clients connecting to a central server where there is no communication between the clients (for sharing snapshot [2] -local weights) or the usage of a Fed server (for local model aggregation [9]). Alg. 1 presents the collaborative training procedure between clients and server in the proposed P-SL. In the initial phase, clients and server receive their corresponding parts, C i ← f ui and S ← g w , from a split model, h θ = f u · g w , and then initialize their model weights too, u i and w. During a global epoch, following round-robin manner, each client C i starts its training with the server and the inner while loop (lines 2 − 10) demonstrates the exact training procedure of a simple vanilla SL. Note that the box (lines 5 − 8) wraps the executes at the server, and the data transmission between clients -server (e.g. transferring intermediate data, labels, gradients, etc.) is done via network connection. When the training of N clients is completed, we would have N different local models combined with the server model to form N different deep models (i.e. while Client i has data to train with Server do 3:
h θi = f ui ·g w where 1 ≤ i ≤ N ).
Client i does forward propagation on its data 4: Client i sends intermediate data and labels to Server
5:
Server propagates incoming data on its layers 6: Server computes errors based on the labels 7:
Server back-propagates gradients until its first layer 8: Server sends gradients of split layer to Client i 9:
Client i back-propagates the received gradients 10: Client i and Server update their model weights performs inference on its live data using its local private model combined with the shared server model. Compared to SL and SFL, in P-SL, we maintain the client-server collaboration but prevent the weight exchanges among the clients. This approach would reduce clients' computation and affect model accuracy, which will be analysed and discussed further.
C. Computation analysis
For simplicity, we analyze the total computation and communication costs of P-SL compared to SL and SFL under a balanced data distribution. Assume that N is the number of clients, |X | is the total number of dataset items, S is the size of the split layer (the last layer of f u ), C P is the computation cost for processing one forward and backward propagation on f u with one data item, C U is the cost for updating a client's local weights from the received weights from previous client or the Fed server, |U | is the size of the local model f u . Table III shows that the computation and communication costs at the client side in P-SL are less than SL and SFL due to no local weight sharing; the cost reduction depends on the size of the local model. The factor 2, in communication costs, means uploading intermediate data and downloading corresponding gradients (2 |X | N S) or uploading and downloading local models (2|U |) at the client side.
N C P + C U 2 |X | N S + 2|U | SFL |X | N C P + C U 2 |X | N S + 2|U | P-SL |X | N C P 2 |X | N S
D. Performance evaluation 1) Training accuracy: Using the selected datasets and local data distributions described in the previous section, we implement P-SL with N = 6 clients and a central server. After training, we measure the learning performance of each client when performing inference on a test set collaboratively with the server (h θi = f ui · g w ). We compare the results with multiple SL -named mSL -where we set up N different SL processes between N pairs client-server (h θi = f ui · g wi ). Note that, with mSL we have 6 different server instances, while P-SL use one shared instance of the server model. We also present the results of SL and SFL, which consider a more simplified threat model of only honest-but-curious server (see Table IV). The training accuracy with Fashion dataset of each client is presented in Table V, while the results with CIFAR10 dataset are visualized in Fig. 3. With mSL, the accuracy of each client depends on the amount of data samples held by that client as expected. Therefore, these clients have similar accuracy under balanced data (around 90% with Fashion) while their accuracy ranges from lower (C 1 with 78.6%) to higher (C 6 with 92.1%) values (see Table V). We can visually observe similar results in Fig. 3, which shows the results using CIFAR10 -the more complex/difficult dataset causing a higher accuracy difference between the clients with fewer and higher data samples. In the proposed P-SL, even though separate training of local models, the learning performance is better than mSL due to the shared server model, which aggregates the knowledge from all clients. P-SL achieves 3% with Fashion and 12% with CIFAR10 higher accuracy than mSL under balanced data distribution. Under imbalanced distribution, the results are more impressive because we can observe the high accuracy improvement of clients with fewer data such as C 1 , C 2 , etc. (see Fig. 3b). This result demonstrates the benefit of P-SL in collaborative learning even though sharing clients' weights is not allowed. We also compare our results with SL and SFL, which achieve state-of-the-art collaborative learning performance. By sharing local models among the clients, knowledge is aggregated at both client and server sides; therefore, the accuracy of SL and SFL is higher than P-SL, which only aggregates knowledge at the server. In summary, our experiments demonstrate that without local weight sharing, P-SL still benefits the collaborative learning between multiple clients and a central server. Under imbalanced data distribution, clients with fewer data can learn more by participating in learning with clients having more data.
In our experiments, we fixed the training order of clients, that is, in sequence C 1 , C 2 , · · · C 6 , e.g., clients with least training data are trained first. The client training order would affect the accuracy performance because the learning with more data could be different to with fewer data clients (under imbalanced distribution). Further investigation on the training order is deferred to the next section.
2) Privacy preservation at client side: We now conduct experiments to evaluate the privacy preservation of P-SL. While training, the model inversion attack is used to reconstruct raw data of a client from the intermediate data that the client has sent out to the server. Specifically, we conducted experiments with 6 clients under balanced data distribution. While all clients are performing the training, using model inversion attack [13], we train a decoder [14] using the local weights and data of a client (could be any client; however, in our experiments, we select C 1 as a malicious client who is curious about data of other clients). Then, using the decoder to reconstruct raw data from intermediate data that any client sent to the server as a natural process of SL. The exposed intermediate data is considered public because it leaves the client and could be touched by the server or any man-in-the-middle who controls the networking infrastructure. Based on this setup, privacy preservation is evaluated by measuring the amount of leakage from clients' raw data. Data leakage is measured using Structural Similarity Index Measure [31], a perceptual metric that quantifies image quality degradation, which is also used to measure data leakage (similarity) between the raw and reconstructed images in [13], [30], [32]. SSIM provides a more intuitive and interpretable metric compared to other commonly used metrics like MSE or PNSR [32]. SSIM is a single value between 0 and 1, where 0 represents least similarity, and 1 indicates most similarity -most leakage. Fig. 4. Data leakage at client side in P-SL: raw private image (leftmost) and the reconstructed ones using intermediate data from C 1 , C 2 , and C 3 , respectively. Fig. 4 demonstrates the reconstructed images from other clients on assuming that C 1 is malicious. Because the decoder is trained using C 1 local model and its raw data, the reconstruction from C 1 's intermediate data is very clear. This also demonstrates the quality of the decoder. However, the quality of reconstruction is dropped significantly when applying the decoder to other clients' intermediate data. The reconstructed images from C 2 and C 3 in Fig. 4 are vague with high noise compared to the raw image. Numerical results presented in Table VI reveals that the reconstruction quality (SSIM value) of all clients (except C 1 ) in P-SL is only about 0.5. With SL and SFL, as partially visualized in Fig. 1, C 1 almost entirely reconstructs the raw data of all other clients (SSIM values are more than 0.95), similar to C 1 self-reconstructing its data. Attacker We also conducted experiments with imbalanced data and obtained similar results presented in Table VII. From the experimental results, we can conclude that P-SL outperforms SL and SFL in preserving data privacy at the client side. However, the SSIM values between reconstructed and raw images of P-SL are more significant than 0, which means leakage still happens. This leakage can be explained by the query-free attack in [13], where the attacker does not require target model information nor be able to query the target model. The only assumption for this type of attack is the same data distribution between adversary and victim. In our experiments, data is distributed uniformly to all clients with equal or different numbers of samples. Therefore, the model of C 1 acts as a shadow model for the query-free attack which causes partial data leakage. An insight observed from Table VII is that attackers with more data (e.g. C 6 ) could reconstruct better quality compared to attackers with fewer data (e.g. C 1 ).
C 1 C 2 C 3 C 4 C 5 C 6 C 1 0.
IV. PARALLELIZING P-SL WITH MULTIPLE SERVERS
In the original SL, the training is conducted sequentially by collaboratively training a client with the server part before sending the local weights to the next client. Therefore, during training, there is only one client is active, which causes very high latency and large idle time at the client side. SFL and especially the generalized SFL (SFLG - [11]) have addressed this latency problem by proposing client-side training following the FL manner. Specifically, the client conduct training in parallel and their local weights are aggregated using a local Fed server as per SFL architecture. In this case, depending on the number of server instances would affect the number of clients conducting their training simultaneously. If there are N clients, then SFLv1 is the case of deploying N server instances, SFLv2 is the case of using only 1 server instance, and SFLG allows m instances of the server where 1 ≤ m ≤ N [11].
Regarding the proposed P-SL, each client conducts collaborative training with the server separately. Therefore, we can process clients' training simultaneously if multiple server instances are available. To ease the understanding of parallelizing P-SL, we use an example shown in Fig. 5 for description. Parallel P-SL follows the steps below in each round: 1) During the setup phase, all clients receive the same model f u , and the server starts with model g w . The server setups a pool of m instances (this example has two). 2) On training, all clients connect to the server and are associated with available instances. Then they perform propagation on their local models using their local data in parallel and independently. After that, they send their intermediate data to the server. 3) The corresponding server instances conduct forwardbackward operations on the received intermediate data from the client and send back the computed gradients. 4) The collaborative training between a client and the corresponding server instance is indicated by label 1 .
On completing the training, resulting in a pair clientserver model, f ui · g wj , the server instance becomes available and waits in the pool for the next client to connect (label 3 ). 5) When a server instance is available after training, a snapshot of server model weights, w j , is recorded (label 2 ). After a period of time or a number of snapshots is recorded, the server aggregates (i.e. Avg(.)) all snapshots to form a new version of server model weights, w * . Then all server instances, g wj , update the new weights, w * , for the next round of training.
It should be noted that the aggregation of the server models, g wj , is performed asynchronously and the degree of parallelization depends on the number of server instances. Table VIII presents the experimental results of parallelizing P-SL with 6 clients and 2 server instances. Each client is randomly associated with a server instance to perform the training. Due to having two servers, two groups of clients can process training in parallel, which theoretically speeds up the training two times. Based on the reported results, we can see that parallelized P-SL achieves similar results with sequential P-SL, where we sequentially train each client with a single server. Therefore, parallelized P-SL can be 'scalable' to speed up the training without affecting the model's accuracy.
V. CACHE-BASED SPLIT LEARNING A. SL with newly participating clients: A case study
In the previous section, we have demonstrated that P-SL could enable collaborative learning among distributed clients. Especially with imbalanced data, during training, one client's participation contributes to the accuracy of others, and clients with fewer data get the most benefit. However, setting up the training when all clients simultaneously participate is challenging due to the unstable status of IoT/mobile devices. Besides the study of offline clients during training in [11], there is no study about learning for newcomers. This is a real-world situation where a new client with its data also wants to join the training to share the knowledge from other clients that have learnt before. In order to study this situation, we conducted experiments with 6 clients with a similar data distribution described in Table II. Firstly, we let 4 clients (C 1 , C 3 , C 4 , C 6 ) collaboratively learn their models using P-SL, called the first training. Then, C 2 and C 5 come later to join, called the second training. C 2 and C 5 also have their data and want to learn their models with the knowledge from other clients' data. There are two solutions for the second training: 1. train all clients, which causes extra overhead for existing clients, and 2. train the new arriving clients only to reduce the training complexity. There is a hybrid approach where we could train new clients for some epochs and then train existing ones together, but we focus on the extreme case (train all or train new) to study the effect of learning new things to the existing knowledge.
We continue to conduct experiments on both Fashion and CIFAR10 datasets. Table IX shows the experimental results of the first training (without C 2 and C 5 ), the second training with solution one (train all clients) and with solution two (train new clients only). Note that all training is done using P-SL. From the obtained results, we can observe that training all clients helps new clients learn their deep models while slightly improving the accuracy of existing clients (e.g. C 1 and C 6 ) because of reinforcement learning from the newcomers' data. In contrast, training only new clients leads to the server forgetting the knowledge learnt from existing clients, consequently reducing the learning performance of the new joining clients (C 2 and C 5 ). Of course, the accuracy of the existing clients is also reduced due to the updating server model when training with the new clients. We can observe a similar effect in Fig. 6, which visualizes the results on CIFAR10. After the second training with the new clients only, the accuracy of the existing clients dropped 10% − 20% with both balanced and imbalanced data, demonstrating the forgetting phenomenon of deep learning. Therefore, training all clients when there are newcomers is a suitable approach to maintain the benefit of collaborative learning. However, training the existing clients again causes network and computation overhead, which is the limitation of low-end devices. The results of the second full training are comparable to the results in Section III -learning performance with 6 clients -but not the same due to the randomness and the starting point (the second training starts based on the first training while we start training from scratch in the above experiments). In summary, our experiments on P-SL with some clients joining after the first learning has demonstrated that retraining the whole network would benefit newcomers and reinforce the performance of the existing clients. Besides, retraining would increase the extra cost of existing clients, which is a disadvantage of low-end devices.
B. Cached-based P-SL algorithm
To deal with the forgetting phenomenon, we design an enhanced method for training only newcomers to reduce the extra cost of retraining while maintaining the learnt knowledge of the existing clients. Specifically, we propose caching the intermediate data sent from the clients to the server during training to improve the server model's learning. Due to the catastrophic forgetting phenomenon when a model is serially trained among clients, caching would help review the previous knowledge while learning new ones. We present the detailed cached-based algorithm in Alg. 2 followed by the computation and privacy aspects analysis.
We modify the execution of the server part in P-SL as presented from line 5 to line 8 in Alg. 1 Server concatenates the cached labels into Client i 's labels 6: Server propagates the concatenated data on its layers 7: Server computes errors based on the concatenated labels 8: Server back-propagates the gradients until its first layer 9: Server slices the gradients based on the split layer's size 10: Server sends the sliced gradients of split layer to Client i For each iteration of client C i 's training, on receiving intermediate data, z i = f ui (x train i ), with corresponding labels, y train i , the server stores them into a cache pool (line 1). Before processing the forward propagation, the server concatenates the cached data randomly selected from the cache pool, (z cache , y cache ), to the incoming intermediate data and labels of C i (lines 3 − 5). Then, the server processes the forward and backward passes as usual (lines 6 − 9). Let L denote the loss function applied to measure the distance between the ground-truth labels and the model's predicted outputs, then the gradients at the server's last layer are computed as follows:
∇L(outputs, labels) = ∇ ui,u k ,w L g w (z i ), y train i + L g w (z cache ), y cache
Remember that the computed gradients for the split layer have the size of the concatenated features instead of z i 's size. Therefore, the server needs to slice the gradients to fit the size of z i before sending them to the client (line 9). The execution in clients is kept the same as P-SL in Alg. 1. From the above equation, the gradients are not only computed based on the errors from training with C 1 data but also from other clients' data (cached intermediate data and labels). Therefore, by updating g w using these gradients, the server can both be able to learn new knowledge from C 1 's data and review knowledge that has been learnt with other clients' data. Analysis and evaluation for the proposed cached-based P-SL will be presented and discussed following.
C. Computation and Privacy Analysis
In the proposed cache-based P-SL, we only modify the procedure in the server so that the cost at the client side is the same as P-SL. At the server, the extra costs for storing cached data and processing concatenation are acceptable because the server is assumed to have enough computing resources to provide to multiple clients. In addition, we can control the size of cached data to adjust the server's performance. Therefore, cache-based P-SL does not increase the cost at the client side, which keeps the benefit when applied to IoT/mobile environment.
Regarding data privacy, in P-SL, we do not allow the clients to share their local weights, so the risk of being attacked by model inversion is reduced. Besides, we cache all clients' intermediate data during the training of cache-based P-SL. This caching approach also does not violate any privacy issue because those data is public as default in SL, and clients are willing to share it with the server to obtain the model utility of the learning. In summary, cache-based P-SL does not increase the cost at the client side nor violate the privacy of local private data. However, there is extra overhead for computing and storage resources at the server, which is more acceptable than with the low-end devices at the client side. To comprehensively evaluate the learning performance of the proposed scheme, we conduct experiments and present results in the following.
D. Performance Evaluation 1) Training newcomers: When a new client joins later, the better approach is to retrain the whole network to learn a deep model for the new client and to update the existing clients if possible. However, retraining would lead to extra computation costs and wasting network bandwidth resources. Therefore, we experiment again, training only newcomers using cached-based P-SL to evaluate the proposed scheme. Due to knowledge being reviewed through cached data while training new clients, the performance of cached-based P-SL is more stable and achieves higher accuracy for both new and existing clients. Fig. 7 shows the learning performance of newcomers (C 2 and C 5 ) and existing clients (C 1 , C 3 , C 4 and C 6 ) using P-SL (left column) and cached-based P-SL (right column) on both Fashion (top half) and CIFAR10 (bottom half) datasets. By using caching, learning with only newcomers of P-SL is more stable, and the accuracy results are comparable with retraining the whole network. Therefore, we can train the newcomers only using cached-based P-SL to save the extra cost of retraining with a slight reduction in the accuracy of existing clients.
2) Order of clients in training: In the previous section, we questioned how the order of clients participating in the training would affect the final accuracy result, especially under imbalanced data distribution where some clients have more data than others. We conduct experiments with P-SL where each epoch, we randomly select the order of clients to participate in the training with the server. Then we compare the learning performance to the fixed order (i.e., starting from C 1 and ending at C 6 each epoch) to evaluate the effect of the order of clients. The experimental results reveal no difference between training with fixed or random order under balanced data distribution. Due to the similar quantity and distribution of data, the learning performance of the server with C 1 is also similar to that of another client. However, under imbalanced data distribution, the learning performance of server with more data clients should be different with fewer data clients. We plot the learning performance of P-SL in Fig. 8a, where we can observe that the achieved accuracy is not stable. However, the experiment with cache-based P-SL (shown in Fig. 8b) demonstrates the effect of caching approach to stabilize the learning curve.
From the above experiments, we can conclude that caching supports to stabilizing and maintaining the learning performance of P-SL. Regarding parallelization to speed up training, this caching approach can be extended to the server with multiple instances that share a cache pool. Furthermore, the strategy for caching, such as which and how much data to be cached, should be the next question we leave for future work.
VI. CONCLUSION This paper identifies the possible data leakage between clients in the traditional SL and other variants due to sharing local weights during their training. We propose and analyse SL without local weight sharing to enhance privacy preservation for user data. The experimental results under various data distributions demonstrate that the proposed P-SL enables collaborative learning from distributed clients, which has comparable accuracy to SL and SFL while reducing half of the data leakage at the client side. Additionally, P-SL can be paralleled to speed up client training without sacrificing the model accuracy. We also study P-SL in a dynamic environment where newly arriving clients participate in the training, whose learning could affect the existing clients due to the forgetting phenomenon. We then propose a server-caching mechanism for P-SL that helps review the learnt knowledge while conducting training with newcomers. The experiment results demonstrate that cached-based P-SL stablizes the learning performance and enables training only for late-arriving clients to reduce client-side overhead and address the server-side forgetting. Further investigation on strategy for caching would be a future direction to push forward the proposed P-SL on preserving user data privacy in collaboratively distributed learning for IoT/mobile devices in a real-world dynamic environment.
Fig. 1 .
1Demonstration of data leakage at the client side of SL: raw private image (left) is reconstructed (right) by a malicious client using model inversion attack.
Fig. 2 .
2P-SL architecture with differences from original SL and SFL.
Fig. 3 .
3Accuracy (%) results with CIFAR10 dataset.
Fig. 5 .
5Parallelizing P-SL with two server instances.
Fig. 6 .
6Accuracy (%) results of 6 clients, with 2 joinings late on CIFAR10 dataset.
Fig. 7 .
7Learning performance comparison between non-cached (left column) and cached-based (right column) P-SL with Fashion (top half) and CIFAR10 (bottom half) when training newcomers, C 2 and C 5 (second training).
Fig. 8 .
8Learning performance of P-SL on imbalanced CIFAR10 where the order of clients participating in the training each epoch is random.
, Phan, and Chilamkurti are with School of Computing, Engineering, and Mathematical Sciences, La Trobe University, Victoria, Australia. Email: {ngocduy.pham,k.phan,o.nguyen,n.chilamkurti}@latrobe.edu.au. Abuadbba is with CSIRO's Data61 & Cybersecurity CRC, Australia. Email: [email protected]. Corresponding authors: T. K. Phan and N. Chilamkurti
TABLE I DATASETS
IAND CORRESPONDING DEEP LEARNING MODELSDataset
Input size
Samples
Deep model architecture
Client side
Server side
Fashion
1 × 28 × 28
60, 000
2conv
4conv+1dense
CIFAR10
3 × 32 × 32
60, 000
2conv
8conv+1dense
TABLE II IMBALANCED
IIDATA DISTRIBUTION FOR 6 CLIENTS
After the training, each client Algorithm 1 Procedure for one global epoch of P-SL.Initialize:Clients and Server receive their model parts Clients and Server initialize their model weights1: for each Client i among all the Clients do2:
TABLE III COMPUTATION
IIIAND COMMUNICATION COSTS AT A CLIENT OF SL, SFL, AND P-SL DURING ONE GLOBAL EPOCH.Scheme
Computation
Communication
SL
|X |
TABLE IV BENCHMARKING
IVACCURACY (%) OF SL AND SFLDataset
Balanced data Imbalanced data
SL
SFL
SL
SFL
Fashion
93.7
93.1
93.7
93.4
CIFAR10
85.6
84.2
85.4
84.6
TABLE V
ACCURACY (%) RESULTS WITH FASHION DATASET
Client
C 1
C 2
C 3
C 4
C 5
C 6
Scheme
With balanced data
mSL
89.5
90.1
89.8
90.0
89.9
90.2
P-SL
92.5
92.5
92.4
92.6
92.6
92.6
Scheme
With imbalanced data
mSL
78.6
84.9
88.2
90.4
91.1
92.1
P-SL
88.8
91.1
92.1
92.8
92.9
92.9
TABLE VI DATA
VILEAKAGE (SSIM) COMPARISON BETWEEN P-SL TO SL AND SFL WHEN C 1 TRAIN A DECODER USING ITS LOCAL MODEL TO RECONSTRUCT RAW DATA OF OTHER CLIENTS WHEN THEIR INTERMEDIATE DATA IS EXPOSED.Scheme
C 1
C 2
C 3
C 4
C 5
C 6
SL
0.97
0.96
0.95
0.95
0.95
0.95
SFL
0.97
0.97
0.97
0.97
0.97
0.97
P-SL
0.97
0.53
0.49
0.51
0.53
0.56
TABLE VII DATA
VIILEAKAGE (SSIM) OF P-SL UNDER IMBALANCED DATA WITH DIFFERENT ATTACKERS.
TABLE VIII
VIIIACCURACY (%) RESULTS WHEN PARALLELIZING P-SL WITH TWO SERVER
INSTANCES.
Client
C 1
C 2
C 3
C 4
C 5
C 6
Data dist.
Fashion dataset
Balance
92.1
92.3
92.4
92.2
92.1
92.1
Imbalance
90.0
91.0
92.2
92.6
92.8
92.7
CIFAR10 dataset
Balance
81.0
81.6
81.8
81.4
81.4
81.5
Imbalance
59.1
74.6
81.1
83.1
84.1
83.8
TABLE IX ACCURACY
IX(%) RESULTS OF 6 CLIENTS, WITH 2 JOININGS LATE ON FASHION DATASET.Client
C 1
C 2
C 3
C 4
C 5
C 6
Training stage
Balanced data distribution
1 st w. 4 clients
91.4
91.5
91.6
91.2
2 nd w. ALL clients
92.6
92.6
92.6
92.4
92.3
92.3
2 nd w. NEW clients
91.0
91.4
90.9
90.6
91.6
91.4
Training stage
Imbalanced data distribution
1 st w. 4 clients
88.7
91.0
91.6
92.1
2 nd w. ALL clients
90.5
90.2
92.3
92.6
92.6
93.2
2 nd w. NEW clients
87.5
89.8
91.0
91.2
92.1
92.1
(inside the box) to provide caching intermediate data from all clients. Then this cached data can be combined with the incoming intermediate data in the next client's training to 'review' the previous knowledge for the server. The modification is presented in Alg. 2.Algorithm 2 Server executes in cache-based P-SL.
Input: Intermediate data and labels from Client i
Output: Gradients of the split layer to Client i
1: Server caches Client i 's intermediate data and labels
2: if cache pool is not empty then
3:
Server randomly selects some intermediate data and
labels from cache pool
4:
Server concatenates the cached intermediate data into
Client i 's intermediate data
5:
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|
[] |
[
"Logic and operator algebras",
"Logic and operator algebras"
] |
[
"Ilijas Farah "
] |
[] |
[] |
The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.(2010). 03C20, 03C98, 03E15, 03E75, 46L05, 46L10.Mathematics Subject Classification
| null |
[
"https://arxiv.org/pdf/1404.4978v3.pdf"
] | 18,372,997 |
1404.4978
|
5020cc309440f785f18d5ce9194238bcbea25dd2
|
Logic and operator algebras
2 Jun 2014
Ilijas Farah
Logic and operator algebras
2 Jun 2014Classification of C*-algebrastracial von Neumann algebraslogic of metric structuresBorel reducibilityultraproducts
The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.(2010). 03C20, 03C98, 03E15, 03E75, 46L05, 46L10.Mathematics Subject Classification
Introduction
The connection between logic and operator algebras in the past century was sparse albeit fruitful. Dramatic progress has brought set theory and operator algebras closer together over the last decade. A number of long-standing problems in the theory of C*-algebras were solved by using set-theoretic methods, and solutions to some of them were even shown to be independent from ZFC. There is much to be said about these developments (as witnessed in three almost disjoint recent survey papers [96], [45], [30]), but that is not what this paper is about. New applications of logic to operator algebras are being found at such a pace that any survey is bound to become obsolete within a couple of years. Instead of presenting an encyclopaedic survey, I shall proceed to describe the current developments (many of them from the unpublished joint work, [33], [34]) and outline some possible directions of research. The choice of the material reflects my interests and no attempt at completeness has been made. Several results proved by operator algebraists without using logic that have logical content are also included.
'Logic' in the title refers to model theory and (mostly descriptive) set theory, with a dash of recursion theory in a crucial place.
I am indebted to Bradd Hart, Isaac Goldbring and Aaron Tikuisis for a number of remarks on the first draft of the present paper that considerably improved it in many ways.
Operator algebras
Let B(H) denote the Banach algebra of bounded linear operators on a complex Hilbert space H equipped with the operation * of taking the adjoint. A C*algebra is a Banach algebra with involution which is *-isomorphic to a subalgebra of B(H) for some H. Notably, all algebraic isomorphisms between C*-algebras are isometries. All C*-algebras considered here will be unital, unless otherwise specified. A von Neumann algebra is a unital subalgebra of B(H) which is closed in the weak operator topology. An algebra isomorphic to a von Neumann algebra is called a W*-algebra. Standard terminology from operator theory is imported into operator algebras, and in particular positivity of self-adjoint operators plays an important role.
I only have something to say about those von Neumann algebras that have a trace. A normalized trace (on a von Neumann algebra or a unital C*-algebra) is a unital positive functional such that τ (ab) = τ (ba) for all a and b. We shall only consider unital algebras and normalized traces. A trace on a von Neumann algebra is automatically continuous in the weak operator topology. A tracial infinitedimensional von Neumann algebra with a trivial center is a II 1 factor. The terminology comes from von Neumann's type classification, in which the unique I n factor is M n (C); we shall not consider other types of factors.
If τ is a trace on an operator algebra A then the ℓ 2 -norm a 2 = τ (a * a) 1/2 turns A into a pre-Hilbert space. The algebra A is represented on this space by the left multiplication; this is the GNS representation corresponding to τ . If A is a C*-algebra, then the weak closure of the image of A is a tracial von Neumann algebra. If A is simple and infinite-dimensional, this algebra is a II 1 factor. A GNS representation can be associated to an arbitrary positive unital functional (state).
The category of abelian C*-algebras is equivalent to the category of locally compact Hausdorff spaces and the category of abelian von Neumann algebras with a distinguished trace is equivalent to the category of measure algebras. Because of this, these two subjects are considered to be noncommutative (or quantized) topology and measure theory, respectively.
There is only one (obvious, spatial) way to define the tensor product of von Neumann algebras. A C*-algebra A is nuclear if for every C*-algebra B there is a unique C*-norm on the algebraic tensor product of A and B. The importance of this notion is evident from a variety of its equivalent characterizations (see [11]), one of them being Banach-algebraic amenability. Although by a result of Junge and Pisier (see [11]) there is finite subset F ⋐ B(H) such that no nuclear C*-algebra includes F , these algebras are ubiquitous in a number of applications.
For more on C*-algebras and von Neumann algebras see [8], [55], and [11].
Intertwining.
A metric structure is a complete metric space (A, d) equipped with functions f : A n → A and predicates p : A n → R, all of which are assumed to be uniformly continuous on d-bounded sets. Consider two separable complete metric structures A and B. Assume we have partial isometric homomorphisms Φ n : F n → G n , Ψ n : G n → F n+1 for n ∈ N such that F n ⊆ F n+1 ⊆ A and G n ⊆ G n+1 ⊆ B for n ∈ N and n F n and n G n are dense in A and B respectively. Furthermore assume that in the following diagram
F 1 F 2 F 3 F 4 . . . A G 1 G 2 G 3 G 4 . . . B Φ 1 Φ 2 Φ 3 Φ 4 Ψ 1 Ψ 2 Ψ 3
the n-th triangle commutes up to 2 −n . Then Φ : n F n → B defined by Φ(a) = lim n Φ n (a) and Ψ : n G n → A defined by Ψ(b) = lim n Ψ n (b) are well-defined isometric homomorphisms. Their continuous extensions to A and B are respectively an isomorphism from A onto B and vice versa.
Variations of this method for constructing isomorphisms between C*-algebras comprise Elliott's intertwining argument. In Elliott's program for classification of separable, nuclear, unital and simple C*-algebras maps Φ n and Ψ n are obtained by lifting morphism between the K-theoretic invariants (so-called Elliott invariants) of A and B. The first result along these lines was the Elliott-Bratteli classification of separable AF algebras (i.e., direct limits of finite-dimensional C*-algebras) by the ordered K 0 . Remarkably, for A and B belonging to a rather large class of nuclear C*-algebras this method shows that any morphism between Elliott invariants lifts to a morphism between the algebras. Elliott conjectured that the separable, nuclear, unital and simple algebras are classified by K-theoretic invariant known as the Elliott invariant. This bold conjecture was partially confirmed in many instances. See [77] for more on the early history of this fascinating subject.
Examples of separable, nuclear, unital and simple C*-algebras that limit the extent of Elliott's classification program were given in [78] and [90]. Algebras defined in [90] have a remarkable additional property. Not only do the nonisomorphic algebras A and B have the same Elliott invariant, but in addition they cannot be distinguished by any homotopy-invariant continuous functor. We shall return to these examples in §4.3. The revised Elliott program is still one of the core subjects in the study of C*-algebras (see [25]).
2.2.
Strongly self-absorbing (s.s.a.) algebras. An infinite-dimensional C*-algebra is UHF (uniformly hyperfinite) if it is an infinite tensor product of full matrix algebras M n (C). If A is UHF, then every two unital copies of M n (C) in it are unitarily conjugate and therefore every endomormphism of A is a pointnorm limit of inner automorphisms. The generalized natural number of A has as its 'divisors' all n such that M n (C) embeds unitally into A. Glimm proved that this is a complete isomorphism invariant for the separable UHF algebras.
If A is UHF then it has a unique trace τ . The tracial von Neumann algebra corresponding to the τ -GNS representation of A ( §2) is the hyperfinite II 1 factor, R, and it does not depend on the choice of A. It is the only injective II 1 factor and it has played a key role in the classification of injective factors ( [19]).
Two *-homomorphisms Φ and Ψ from A into B are approximately unitarily equivalent if there is a net of inner automorphisms α λ , for λ ∈ Λ, of B such that
lim λ α λ • Φ(a) = Ψ(a)
for all a ∈ A (convergence is taken in the operator norm for C*-algebras and in the ℓ 2 -norm for tracial von Neumann algebras). If A ⊗ B ∼ = A we say that A is B-absorbing and if A ⊗ A ∼ = A then we say that A is self-absorbing.
Here and in what follows, we will often be providing two definitions at once, one for von Neumann algebras and another for C*-algebras. The difference comes in the interpretation of ⊗, either as the von Neumann (spatial) tensor product⊗ or as the C*-algebra minimal (spatial) tensor product ⊗. McDuff factors are the Rabsorbing II 1 factors. A separable C*-algebra D is strongly self-absorbing (s.s.a.) ( [92]) if there is an isomorphism Φ : D → D ⊗ D and map id ⊗1 D : D → D ⊗ D is approximately unitarily equivalent to Φ. The definition of strongly self-absorbing is modified to II 1 factors following the convention stated above, by replacing · with · 2 and ⊗ with⊗.
The hyperfinite factor R is the only s.s.a. tracial von Neumann algebra with separable predual (Stefaan Vaes pointed out that this was essentially proved in [19,Theorem 5.1(3)]). A UHF algebra A is s.s.a. if and only if it is self-absorbing. However, the latter notion is in general much stronger. For any unital C*-algebra A the infinite tensor product N A is self-absorbing but not necessarily s.s.a. Every s.s.a. C*-algebra D is simple, nuclear and unital ( [23]). Three s.s.a. algebras are particularly important. The Jiang-Su algebra Z is an infinite-dimensional C*-algebra which is indistinguishable from C by its Elliott invariant. Conjecturally, Z-absorbing infinite-dimensional separable, nuclear, unital and simple algebras are classifiable by their Elliott invariant. The Cuntz algebra O 2 is the universal algebra generated by two partial isometries with complementary ranges. The Cuntz algebra O ∞ is the universal unital C*-algebra generated by partial isometries v n , for n ∈ N, with orthogonal ranges. The first step in the Kirchberg-Phillips classification of purely infinite separable, nuclear, unital and simple algebras was Kirchberg's result that every such algebra is O ∞ -absorbing and that O 2 is A-absorbing for every separable, nuclear, unital and simple algebra (see [77]).
Abstract classification
A Polish space is a separable, completely metrizable topological space. A subset of a Polish space is analytic if it is a continuous image of some Polish space. Essentially all classical classification problems in mathematics (outside of subjects with a strong set-theoretic flavour) can be modelled by an analytic equivalence relation on a Polish space. Moreover, the space of classifying invariants is also of this form, and computation of the invariant is usually given by a Borel measurable map. This is indeed the case with C*-algebras and the Elliott invariant ( [43]).
If E and F are equivalence relations on Polish spaces, E is Borel-reducible to F , E ≤ B F , if there exists a Borel-measurable f : X → Y such that x E y if and only if f (x) E f (y). One can interpret this as stating that the classification problem for E is not more difficult than the classification problem for F . Following Mackey, an equivalence relation E Borel-reducible to the equality relation on some Polish space is said to be smooth. By the Glimm-Effros dichotomy the class of non-smooth Borel-equivalence relations has an initial object ( [52]), denoted E 0 . It is the tail equality relation on {0, 1} N . While the Glimm-Effros dichotomy was proved by using sophisticated tools from effective descriptive set theory, the combinatorial core of the proof can be traced back to work of Glimm and Effros on representations of locally compact groups and separable C*-algebras. See [57], [47] for more on (invariant) descriptive set theory.
When is an equivalence relation classifiable? Many non-smooth equivalence relations are considered to be satisfactorily classified. An example from the operator algebras is the Elliott-Bratteli classification of separable AF algebras by countable abelian ordered groups. A rather generous notion is being 'classifiable by countable structures.' Hjorth's theory of turbulence ( [54]) provides a powerful tool for proving that an orbit equivalence relation is not classifiable by countable structures.
Sasyk and Törnquist have proved that every class of injective factors that was not already satisfactorily classified is not classifiable by countable structures ( [80], [81]). By combining results of [44], [71], [48], [24] and [79], one proves that the following isomorphism relations are Borel-equireducible. Each of these equivalence relations (as well as the isometry of a class of separable metric structures of any given signature) is Borel-reducible to an orbit equivalence relation of a Polish group action ( [24]).
Being Borel-reducible to an orbit equivalence relation is, arguably, the most generous definition of being concretely classifiable. Conjecturally, E 1 , the tailequivalence relation on [0, 1] N , is an initial object among Borel equivalence relations not Borel-reducible to an orbit equivalence relation ( [58]). Notably, the isomorphism of separable Banach spaces is the ≤ B -terminal object among analytic equivalence relations ( [46]).
The answer to the question 'When is an equivalence relation classifiable' is frequently of somewhat sociological nature. It is notable that the isomorphism relation of abelian unital C*-algebras (generally considered intractable) is Borelreducible to the isomorphism relation of Elliott-classifiable AI algebras (for which there is a satisfactory classification relation). Also, as pointed out by David Fremlin, most analysts find that normal operators are satisfactorily classified up to conjugacy by the spectral theorem, although they are not classifiable by countable structures.
Nevertheless, the theory of Borel-reducibility is a great example of a situation in which logic provides concrete obstructions to sweeping conjectures. For example, the classification of countable abelian torsion free groups of rank n + 1 is strictly more complicated than the classification of countable abelian torsion free groups of rank n for every n ( [89]). (Notably, the proof of this result uses Popa superrigidity of II 1 factors, [75].) This theory was recently successfully applied to (non)classification of automorphisms of group actions on operator algebras ( [59], automorphisms of C*-algebras ( [64], [60]) and subfactors ( [9]).
A partial Borel-reducibility diagram of classification problems in operator algebras is given below. For an explanation of terminology see [30, §9]. I am indebted to Marcin Sabok for pointing out that the isomorphism of countable structures of any signature is Borel-reducible to the isomorphism relation of separable AF algebras ( [14] Borel-reduction of equivalence relations as defined above does not take into the account the functorial nature of the classification of C*-algebras. Some preliminary results on Borel functorial classification were obtained by Lupini.
Model-theoretic methods
Until recently there was not much interaction between model theory and operator algebras (although model theory was fruitfully applied to the geometry of Banach spaces, see [53]). Recent emergence of the logic of metric structures ( [5]), originally introduced only for bounded metric structures, created new opportunities for such interactions. It was modified to allow operator algebras in [37].
Logic of metric structures.
Model theory can roughly be described as the study of axiomatizable classes of structures and sets definable in them. Axiomatizable properties can be expressed in syntactic terms, but they are also characterized by preservation under ultraproduts and ultraroots (see §6). A category C is axiomatizable if there exists a first-order theory T such that the category M(C) of all models of T is equivalent to the original category. Classical model theory deals with discrete structures, and its variant suitable for metric structures as defined in §2.1 was introduced in [5]. In this logic interpretations of formulas are real-valued, propositional connectives are real-valued functions, and quantifiers are sup x and inf x . Each function and predicate symbol is equipped with a modulus of uniform continuity. This modulus is a part of the language. If the diameter of the metric structures is fixed, then every formula has its own modulus of uniform continuity, respected in all relevant metric structures. Formulas form a real vector space equipped with a seminorm, φ(x) = | sup φ(ā) A | where the supremum is taken over all metric structures A of the given language and all tuplesā in A of the appropriate type. Formulas are usually required to have range in [0, ∞) (or [0, 1] in the bounded case) but allowing negative values results in equivalent logic; see also [4]. The theory of a model is the kernel of the functional φ → φ A , where φ ranges over all sentences (i.e., formulas with no free variables) of the language. This kernel uniquely defines the functional, which can alternatively be identified with the theory. The weak*-topology on this space is also known as the logic topology. If the language is countable then the space of formulas is separable and the spaces of theories and types (see §4.4) are equipped with compact metric topology.
Two metric structures are elementarily equivalent if their theories coincide.
A formula is existential if it is of the form infx φ(x) for some quantifier-free for- mula φ(x). The existential theory of A is Th ∃ (A) = {ψ ∈ Th(A) : ψ is existential}.
There are several equivalent ways to adapt the logic of metric structures to operator algebras and to unbounded metric structures in general ( [37], [3]). Axiomatizability is defined via equivalence of categories as above, but model M (A) associated with A has more (albeit artificial) structure. It is equipped with domains of quantification, bounded subsets of A on which all functions and predicates are uniformly continuous (with a fixed modulus of uniform continuity) and over which quantification is allowed. It is the existence of category M(C), and not its particular choice, that matters.
In the simplest version of M (A) quantification is allowed only over the (operator norm) n-balls of the algebra. The notion of sorts over which one can quantify corresponds to those functors from the model category into metric spaces with uniformly continuous functions that commute with ultraproducts (see [37,2]). For example, M (A) can be taken to consist of all matrix algebras M n (A) for n ∈ N, as well as completely positive, contractive maps between them and finite-dimensional algebras. This is important because nuclearity is equivalently characterized as the CPAP, the completely positive approximation property (see [11] and [12]). C*-algebras are axiomatized as Banach algebras with an involution that satisfy the C*-equality, aa * = a 2 , by the Gelfand-Naimark and Segal (GNS mentioned earlier) theorem. Abelian C*-algebras are obviously axiomatized by sup x,y xy − yx and non-abelian C*-algebras are slightly less obviously axiomatized by inf x ≤1 |1 − x | + x 2 (a C*-algebra is nonabelian if and only if it contains a nilpotent element).
The proof that the tracial von Neumann algebras with a distinguished trace are also axiomatizable ( [37], first proved in [6]) goes deeper and uses Kaplansky's Density Theorem. Again, quantification is allowed over the (operator norm) unit ball and the metric is the ℓ 2 metric a 2 = τ (a * a) 1/2 . The operator norm is not continuous with respect to the ℓ 2 metric and it therefore cannot be added to II 1 factors as a predicate.
There are elementarily equivalent but nonisomorphic separable unital AF algebras. This is proved by using descriptive set theory. The association A → Th(A) is Borel, and hence the relation of elementary equivalence is smooth ( §3). The category of AF algebras is equivalent to the category of their ordered K 0 groups. By the Borel version of this result and the fact that the isomorphism of dimension groups is not smooth the conclusion follows.
The following proposition is taken from ( [34]).
Proposition 4.1.
(1) For every separable, nuclear, unital and simple C*-algebra there exists an elementarily equivalent, separable, non-nuclear, C*-algebra.
(2) The reduced group C*-algebra of the free group with infinitely many generators C * r (F ∞ ) is not elementarily equivalent to a nuclear C*-algebra. Instead of providing a genuine obstruction, this proposition precipitated some of the most interesting progress in the field.
Here is a simple but amusing observation. The Kadison-Kastler distance between subalgebras of B(H) is the Hausdorff (norm) distance between their unit balls. For every sentence φ the map A → φ A is continuous with respect to this metric. Therefore the negation of an axiomatizable property is stable under small perturbations of an algebra (see [18] and references thereof for more on perturbations of C*-algebras).
Elementary submodels. If
A is a submodel of B, it is said to be an elementary submodel if φ A = φ B ↾ A n for every n and every n-ary formula φ. The Downwards Löwenheim-Skolem theorem implies that every model has a separable elementary submodel. Its C*-algebraic variant is known as 'Blackadar's method' and is used to provide separable examples from known nonseparable examples (see [8,II.8.5] and [72]).
Proposition 4.2. Assume
A is a C*-algebra and B is its elementary submodel. Then B is a C*-algebra with the following properties.
(1) Every trace of B extends to a trace of A.
(2) Every ideal of B is of the form I ∩ B for some ideal I of A.
(3) Every character of B extends to a character of A.
(4) If A is nuclear so is B.
In particular, B is monotracial and/or simple if and only if A has these properties. It should be noted that neither of these properties is axiomatizable, because neither of them is preserved under taking ultrapowers (see [76] for the nonaxiomatizability of having a unique trace and §6 for the ultrapowers).
A drastic example of a property that does not persist to elementary submodels is given in Theorem 8.1.
Intertwining again.
We return to Elliott's intertwining argument ( §2.1):
A 1 A 2 A 3 A 4 . . . A = lim n A n B 1 B 2 B 3 B 4 . . . B = lim n B n Φ 1 Φ 2 Φ 3 Φ 4 Ψ 1 Ψ 2 Ψ 3
If the maps Φ n are expected to converge to an isomorphism, it is necessary that they approximate elementary maps. For a formula φ(x) and a tupleā in the domain of Φ n one must have φ(ā) A = lim n φ(Φ n (ā)) B . Even more elementarily, the algebras A and B ought to be elementarily equivalent (no pun intended). Every known counterexample to Elliott's program involves separable, nuclear, unital and simple algebras with the same Elliott invariant, but different theories. For example, the radius of comparison was used in [91] to distinguish between continuum many nonisomorphis separable, nuclear, unital and simple algebras with the same Elliott invariant, and it can be read off from the theory of an algebra ( [34]). This motivates an outrageous conjecture, that the following question has a positive answer. Since being Z-stable is axiomatizable (see §6.1), the revised Elliott conjecture that all Z-stable separable, nuclear, unital and simple algebras are classified by their Elliott invariant is a special case of a positive answer to Question 4.3. All known nuclear C*-algebras belong to the so-called bootstrap class, obtained by closing the class of type I algebras under operations known to preserve nuclearity (see [8]). An (expected) negative answer to 4.4. Omitting types. Let F n be the set of formulas whose free variables are included in {x 1 , . . . , x n }. An n-type is a subset t of F n such that for every finite t 0 ⋐ t and for every ε > 0 there are a C*-algebra A and n-tupleā in the unit ball of A such that |φ(ā)| < ε for all φ ∈ t 0 . By applying functional calculus one sees that this definition is equivalent to the apparently more general standard definition ( [5]) in which types consist of arbitrary closed conditions. An n-type t is realized by n-tupleā in the unit ball of A if φ(ā) A = 0 for all φ in t. It is omitted in A if it is not realized by any n-tuple in A 1 . Loś's theorem implies that every type is realized in an ultraproduct; we shall return to this in §6.0.1 but presently we are concerned with omitting types.
The omitting types theorem of classical ('discrete') model theory ( [67]) provides a simple condition for omitting a type in a model of a given theory. A predicate p is definable if for every ε > 0 there exists a formula φ(x) which up to ε approximates the value of p. A type is definable if the distance function to its realization in a saturated model ( §6.0.1, §7.1) is definable. By the omitting types theorem of [5] a type is omissible if and only if it is not definable, with the additional stipulation that it be complete (i.e., maximal under the inclusion). While a definable type is never omissible even if it is incomplete, Ben Yaacov has isolated types that are neither definable nor omissible. His example was simplified by T. Bice. (1) There is a theory T in a separable language such that the set of types omissible in some model of T is a complete Σ 1 2 set. (2) There are a complete theory T in a separable language and a countable set P of types such that for every finite P 0 ⋐ P there exists a model M of T that omits all types in P 0 , but no model of T omits all types in P.
Therefore the question of whether a type is omissible in a model of a given metric theory is by (1) far from being Borel or even analytic and therefore intractable, and by (2) separately omissible types over a complete theory are not necessarily jointly omissible. Both results stand in stark contrast to the situation in classical model theory.
The idea that the omitting types theorem can be used in the study of C*algebras emerged independently in [83] and [15]. A sequence t n , for n ∈ N, of m-types is uniform if there are formulas φ j (x) for j ∈ N with the same modulus of uniform continuity such that t n = {φ j (x) ≥ 2 −n : j ∈ N} for every n. In this situation, the interpretation of the infinitary formula φ(x) = inf j φ j (x) is uniformly continuous in every model (with a fixed modulus of uniform continuity) and moreover supx φ(x) A = 0 if and only if A omits all t n .
Nuclearity, simplicity, as well as many other important non-axiomatizable properties of C*-algebras (including nuclear dimension or decomposition rank ≤ n; see [99]) are characterized by omitting a uniform sequence of types. The classical theory of omitting types applies to such types unchanged: a uniform sequence of types is omissible in a model of a complete theory T if and only if none of the types is isolated ( [39]). As an extra, this characterization shows that one can find a separable elementary submodel of a nonnuclear algebra that is itself nonnuclear by assuring that it includes a tuple that realizes the relevant type 4.5. Strongly self-absorbing algebras II. These algebras have remarkable model-theoretic properties. Every s.s.a. algebra D is a prime model of its theory (it elementarily embeds into every other model of its theory) and every unital morphism of D into another model of its theory is elementary ( §4.2). (1) E is D-absorbing: E ⊗ D ∼ = E.
(2) D is isomorphic to a subalgebra of E.
(3) Th ∃ (D) ⊆ Th ∃ (E).
The implications from (1) to (2) and from (2) to (3) are always true, but both converses fail in general. S.s.a. algebras are as rare as they are important and the following diagram represents all known s.s.a. algebras, given in the order defined by either clause of Proposition 4.5.
O 2 O ∞ ⊗ UHF O ∞ UHF Z
Finding an s.s.a. algebra other than the ones in the diagram would refute the revised Elliott program.
Tracial von Neumann algebras
Many of the pathologies that plague (or enrich, depending on the point of view) the theory of C*-algebras are not present in von Neumann algebras.
By a result of McDuff, the relative commutant of a II 1 factor in its ultrapower is trivial, nontrivial and abelian, or the factor tensorially absorbs R (see Proposition 6.1). Each of these three classes is nonempty, and there is presently no other known method for distinguishing theories of II 1 factors (see [36]).
The hyperfinite II 1 factor R is a canonical object and every embedding of R into a model of its theory is elementary ( §4.5). However, there are embeddings between models of the theory of R that are not elementary (i.e., the theory of R is not model-complete), and in particular this theory does not allow the elimination of quantifiers ( [31], [51]). This may be an indication that we do not have the right language for the theory of II 1 factors. The obstruction for the elimination of quantifiers extracted in [31] from [56] is removed by adding a predicate for the unitary conjugacy relation. As this is a definable relation, adding such predicate affects only syntactical structure of the language. It is not clear whether adding finitely, or even countably, many such predicates could make the theory of R model-complete. This may suggest that the theory of R is as complicated as the first-order arithmetic or ZFC.
Given a II 1 factor M and a projection p in M , are M and its corner pM p elementarily equivalent? By the Keisler-Shelah theorem, this is equivalent to asking whether these algebras have isomorphic ultrapowers. A positive answer would imply that all free group factors L(F n ), for n ≥ 2, are elementarily equivalent, giving a 'poor man's' solution to the well-known problem whether the free group factors are isomorphic (see [22]). On the other hand, a negative answer would provide a continuum of distinct theories of II 1 factors that are corners of L(F 2 ) . A deeper analysis of the model theory of II 1 factors will nceessarily involve Voiculescu's free probability.
In recent years theories of C*-algebras and von Neumann algebras are increasingly considered as inseparable. Some of the most exciting progress on understanding tracial C*-algebras was initiated in [68]. We shall return to this in §6.1, but see also [12].
Massive algebras I: Ultraproducts
We now consider algebras that are rarely nuclear and never separable, but are nevertheless indispensable tools in the study of separable nuclear algebras.
Ultraproducts emerged independently in logic and in functional analysis (more precisely, in the theory of II 1 factors) in the 1950's (see the introduction to [88]). If (A n , d n ), for n ∈ N, are bounded metric structures of the same signature and U is an ultrafilter on N, then the ultraproduct U A n is defined as follows. On the product structure n A n consider the quasi-metric d U ((a n ), (b n )) = lim n→U d n (a n , b n ).
Since every function symbol f has a fixed modulus of uniform continuity, it defines a uniformly continuous function on the quotient metric structure n A n / ∼ dU . This structure is the ultraproduct of A n , for n ∈ N, associated to the ultrafilter U. It is denoted by U A n .
In the not necessarily bounded case one replaces n A n with {(a n ) ∈ n A n : a n belong to the same domain of quantification}. With our conventions, in the operator algebra case this is the ℓ ∞ -product usually denoted n A n . The nontrivial fact that an ultrapower of tracial von Neumann algebras is a tracial von Neumann algebra is an immediate consequence of the axiomatizability.
The usefulness of ultraproducts draws its strength largely from two basic principles. The first one is Loś's theorem, stating that for any formula φ(x) we have
φ(ā) U An = lim n→U φ(ā n ) An .
This in particular implies that the diagonal embedding of A into its ultrapower is elementary ( §4.2), and therefore the theory is preserved by taking ultrapowers. The second principle will be discussed in §6.0.1.
This may be a good place to note two results in abstract model theory that carry over to the metric case ( [5]). A category K with an appropriately defined ultraproduct construction is closed under the elementary equivalence if and only if it is closed under isomorphisms, ultraproducts, and ultraroots (i.e., A U ∈ K implies A ∈ K). By the Keisler-Shelah theorem, two models are elementarily equivalent if and only if they have isomorphic ultrapowers. Both results require considering ultrafilters on arbitrarily large sets (see [86]).
The fact that it is easier to prove that an ultraproduct of C*-algebras is a C*algebras than that an ultraproduct of tracial von Neumann algebras is a tracial von Neumann algebra is reflected in the fact that it is easier to prove that the C*-algebras are axiomatizable than that the tracial von Neumann algebras are axiomatizable.
All ultrafilters considered here concentrate on N and are nonprincipal. It is not possible to construct such an ultrafilter in ZF alone, as a (rather weak) form of the Axiom of Choice is required for its construction. However, results about separable C*-algebras and separably acting II 1 factors proved using ultrafilters can be proved without appealing to the Axiom of Choice, by standard absoluteness arguments.
An ultrapower of an infinite-dimensional, simple, unital C*-algebra is by Loś's theorem unital. It is, however, nonseparable, not nuclear, and it is simple only under exceptional circumstances. This shows that none of these three properties is axiomatizable (cf. Proposition 4.1). Nevertheless, separable, nuclear, unital and simple C*-algebras can be constructed by using the Henkin construction and omitting types theorem ( [39], see §4.4).
6.0.1. Countable saturation. We define the second important property of massive algebras. If a type (see §4.4) is allowed to contain formulas with parameters from an algebra A we say that it is a type over A.
An algebra A is countably saturated if every countable type t(x) over A is realized in A if and only if it is consistent. (These algebras are sometimes said to be ℵ 1 -saturated. The latter terminology is more conveniently extended to higher cardinalities.) Every ultrapower associated to a nonprincipal ultrafilter on N is countably saturated. A weakening of countable saturation suffices for many purposes (see §7), and we shall return to full saturation in §7.1.
Relative commutants.
In the theory of operator algebras even more important than the ultrapower itself is the relative commutant of the algebra inside the ultrapower,
A ′ ∩ A U = {b ∈ A U : ab = ba for all a ∈ A}.
The current prominence of ultrapowers as a tool for studying separable algebras can be traced back to McDuff ( [70]) and the following proposition (generalized to s.s.a. algebras in [92]). Proposition 6.1. If D is strongly self-absorbing and A is separable, then A is D-absorbing if and only if D embeds into A ′ ∩ A U .
The nontrivial, converse, implication uses the following (a lemma in model theory that I learned from Wilhelm Winter) proved using the intertwining argument. Noting that all nonprincipal ultrafilters on N 'look the same' and in particular that the choice of U in Proposition 6.1 is irrelevant as long as it is a nonprincipal ultrafilter on N, one may ask the following. If M is a Hilbert space or a measure algebra, then a simple argument (using Maharam's theorem in the latter case) gives a negative answer. Also, Continuum Hypothesis (CH) implies negative answer to both questions for an arbitrary separable M (see §7.1). Therefore, the question is whether CH can be removed from this proof. Question 6.3 for relative commutants was asked by McDuff ( [70]) and Kirchberg ([61]) in the case of McDuff factors and C*-algebras, respectively. In [49] it was proved that, under some additional assumptions on M , CH is equivalent to the positive answer to either of these questions! This was achieved by using only results from classical ('discrete') model theory. By using the logic of metric structures and Shelah's non-structure theory, the full result was proved in [35] and [41]. Theorem 6.4. Assume CH fails. If M is a separable C*-algebra or a McDuff factor with a separable predual, then M has 2 c nonisomorphic ultrapowers and 2 c nonisomorphic relative commutants associated to nonprincipal ultrafilters on N.
Let's zoom out a bit. A complete first-order theory T has the order property if there exist n ≥ 1 and a 2n-ary formula φ(x,ȳ) such that for every m there is a model M of T which has a 'φ-chain' of length at least m. A φ-chain is a sequencē
x i ,ȳ i , for i ≤ m, such that φ(x i ,ȳ j ) = 0 if i ≤ j and φ(x i ,ȳ j ) = 1 if i > j.
This is the metric version of one of the important non-structural properties of theories in Shelah's stability theory ([85] and [35]). The theory of any infinitedimensional C*-algebra and of any II 1 factor has the order property. This is proved by continuous functional calculus and by utilizing noncommutativity, respectively. However, the theories of abelian tracial von Neumann algebras do not have the order property, essentially by applying Maharam's theorem on measure algebras. Theorem 6.5. Suppose that A is a separable structure in a separable language.
(1) If the theory of A does not have the order property then all of its ultrapowers associated to nonprincipal ultrafilters on N are isomorphic.
(2) If the theory of A has the order property then the following are equivalent:
(a) A has fewer than 2 2 ℵ 0 nonisomorphic ultrapowers associated with nonprincipal ultrafilters on N. While the structure of relative commutants of II 1 factors in their ultrapowers provides the only known method for distinguishing their theories, every infinitedimensional separable C*-algebra has a nontrivial relative commutant in its ultrapower ( [61], also [35]). The relative commutant of the Calkin algebra ( §7) in its ultrapower is trivial ( [61]) and the relative commutant of B(H) may or may not be trivial, depending on the choice of the ultrafilter ( [40]).
It is not difficult to see that the existential theory of A ′ ∩ A U depends only on the theory of A. However, a result of [61] implies that there is a separable C*-algebra A elementarily equivalent to O 2 such that A ′ ∩ A U and O 2 ∩ O U 2 have different ∀∃-theories. (An ∀∃-sentence is one of the form supx infȳ φ(x,ȳ) where φ is quantifier-free.) In the following all ultrafilters are nonprincipal ultrafilters on N. Proposition 6.6. Assume A is a separable C*-algebra.
(1) For all U and V, the algebras A ′ ∩A U and A ′ ∩A V are elementarily equivalent.
(2) For every separable C ⊆ A ′ ∩A U we have Th ∃ (A ′ ∩C ′ ∩A U ) = Th ∃ (A ′ ∩A U ). (3) If D is a separable unital subalgebra of A ′ ∩ A U then there are ℵ 1 commuting copies of D inside A ′ ∩ A U .
An entertaining proof of (1) can be given by using basic set theory. Collapse 2 ℵ0 to ℵ 1 without adding reals. Then U and V are still ultrafilters on N and one can use saturation to find an isomorphism between the ultrapowers that sends A to itself. The theories of two algebras are unchanged, and therefore by absoluteness the result follows. Clause (3) is an immediate consequence of (2) and it is a minor strengthening of a result in [61].
When A is not Z-stable, the relative commutant of A can have characters even if it is simple ( [62]). In the case when algebra A is nuclear and Z-stable, A ′ ∩ A U inherits some properties from A. For example, each of the traces on A ′ ∩ A U extends to a trace on A U by [68] (cf. Proposition 4.2). The relative commutants of s.s.a. algebras are well-understood; the following was proved in [33]. Proposition 6.7. If D is a s.s.a. algebra and U is a nonprincipal ultrafilter on N, then D ′ ∩ D U is an elementary submodel of D U . Moreover, CH implies that these two algebras are isomorphic.
6.3. Expansions and traces. If a metric structure A is expanded by adding a new predicate τ , its ultrapower A U expands to the ultrapower of the expanded structure (A, τ ) U which still satisfies Loś's theorem and is countably saturated.
If A is a unital tracial C*-algebra then its traces form a weak*-compact convex subset T (A) of the dual unit ball. For τ ∈ T (A) denote the tracial von Neumann algebra associated with the τ -GNS representation ( §2) by N τ . If A is simple and infinite-dimensional and τ is an extremal trace then N τ is a factor, and if A is in addition nuclear and separable then N τ is isomorphic to the hyperfinite factor R. This is because A is nuclear if and only if its weak closure in every representation is an injective von Neumann algebra, and R is the only injective II 1 factor with a separable predual. The following was proved in [68] and improved to the present form in [62].
Proposition 6.8. If A is separable and τ ∈ T (A), then the quotient map from
A ′ ∩ A U to N ′ τ ∩ (N τ ) U is surjective.
If b ∈ A U is such that its image is in the commutant of N ′ τ , then by countable saturation one finds a positive element c of norm 1 such that τ (c) = 0 and c(a n b − ba n ) = (a n b − ba n )c = 0 for all a n in a fixed countable dense subset of A. The fact that the type of such c is consistent follows from the fact that the image of b is in N ′ τ . Then (1 − c)b(1 − c) is in A ′ ∩ A U and it has the same image under the quotient map as b.
Proposition 6.8 precipitated remarkable progress on understanding tracial C*algebras, the most recent results of which are [69] and [82].
Massive algebras II: Coronas
Another class of massive C*-algebras (with no analogue in von Neumann algebras) has special relevance to the study of separable algebras. If A is a non-unital C*algebra, the multiplier algebra of A, M (A), is the noncommutative analogue of thě Cech-Stone compactification of a locally compact Hausdroff space. It is the surjectively universal unital algebra containing A as an essential ideal. The corona (or outer multiplier ) algebra of A is the quotient M (A)/A. Some examples of coronas are the Calkin algebra Q(H) (the corona of the algebra of compact operators) and the asymptotic sequence algebra ℓ ∞ (A)/c 0 (A) for a unital A. The latter algebra, as well as the associated central sequence algebra A ′ ∩ ℓ ∞ (A)/c 0 (A) are sometimes used in classification of C*-algebras instead of the metamathematically heavier ultrapowers and the corresponding relative commutants. While Loś's theorem miserably fails for the asymptotic sequence algebra, all coronas and corresponding relative commutants share some properties of countably saturated algebras. The simplest of these properties is being SAW*: for any two orthogonal separable subalgebras A and B of a corona there exists a positive element c such that ca = a for all a ∈ A and cb = 0 for all b ∈ B.
7.0.1. Quantifier-free saturation. An algebra C is quantifier-free saturated if every countable type over C consisting only of quantifier-free formulas is consistent if and only if it is realized in C. An algebra C is countably degree-1 saturated if every countable type over C consisting only of formulas of the form p , where p is a *-polynomial of degree 1, is consistent if and only if it is realized in C. A dummy variable argument shows that the degree-2 saturation is equivalent to quantifierfree saturation. By refining an argument introduced by Higson, the following was proved in [32].
Theorem 7.1. If A is a corona of a separable non-unital C*-algebra, or a relative commutant of a separable subalgebra of such corona, then A is countably degree-1 saturated.
A very interesting class of countable degree-1 saturated C*-algebras was isolated in [94].
A sampler of properties of countable degree-1 saturated algebras.
Assume C is countably degree-1 saturated (the results below also apply to tracial von Neumann algebras, and in this case (1), (3) and (5) do not even require countable degree-1 saturation).
(1) C has SAW* as well as every other known countable separation property ( [32]).
(2) A separable algebra A is isomorphic to a unital subalgebra of C if and only if Th ∃ (A) ⊆ Th ∃ (C).
(3) A representation of a group Γ in A is a homomorphism π : Γ → (GL(A), ·). It is unitarizable if there is an invertible h ∈ A such that h −1 π(g)h is a unitary for all g ∈ Γ. Conjecturally unitarizability of all uniformly bounded representations of a group Γ on B(H) is equivalent to the amenability of Γ (see [74]). If Γ is a countable amenable group, then every uniformly bounded representation π of Γ in C is unitarizable ( [17]).
(4) C is not isomorphic to the tensor product of two infinite-dimensional algebras ( [26] for the ultraproducts of II 1 factors and [50] for the general result). Therefore an ultrapower or a corona is never isomorphic to a nontrivial tensor product and the separability assumption is needed in Proposition 6.1.
(5) ('Discontinuous functional calculus.') If a is a normal operator, then by the continuous functional calculus for every continuous complex-valued function g on the spectrum, sp(a), of a the naturally defined g(a) belongs to the abelian algebra generated by a. If moreover g is real-valued then c can be chosen to be self-adjoint.
The 'Second Splitting Lemma' ( [10,Lemma 7.3]) is a special case of the above when C is the Calkin algebra, a = h 0 is self-adjoint, and the range of g is {0, 1}. 7.0.3. Failure of saturation. While the asymptotic sequence algebras, as well as some abelian coronas, are fully countably saturated ( [42]), this is not true for sufficiently noncommutative coronas. By a K-theoretic argument N. C. Phillips constructed two unital embeddings of the CAR algebra into the Calkin algebra Q(H) that are approximately unitarily equivalent, but not conjugate by a unitary ( [32, §4]). This gives a countable quantifier-free type over Q(H) that is consistent but not realized. Even coronas of separable abelian C*-algebras provide a range of different saturation properties (see [42]).
Automorphisms. A metric model
A is saturated if every type over A whose cardinality is smaller than the density character χ(A) of A (i.e., the smallest cardinality of a dense subset) which is consistent is realized in A. The Continuum Hypothesis (CH) implies that all countably saturated models of cardinality 2 ℵ0 are saturated. A transfinite back-and-forth argument shows that any two elementarily equivalent saturated models of the same density character are isomorphic and that a saturated model A has 2 χ(A) automorphisms. By a counting argument, most of these automorphisms are outer and moreover nontrivial when 'trivial automorphism' is defined in any reasonable way; see [20] for a (lengthy) discussion. This explains the effectiveness of CH as a tool for resolving problems of a certain form. A deeper explanation is given in Woodin's celebrated Σ 2 1 -absoluteness theorem (see [100]).
By the above, CH implies that an ultrapower A U of a separable, infinitedimensional algebra has automorphisms that do not lift to automorphisms of ℓ ∞ (A). Much deeper is a complementary series of results of Shelah, to the effect that if ZFC is consistent then so is the assertion that any isomorphism between ultraproducts of models with the strong independence property lifts to an isomorphism of the products of these models ( [87]). No continuous version of this result is known. One difficulty in taming ultrapowers is that the ultrafilter is not a definable object; in particular Shelah's results apply only to a carefully constructed ultrafilter in a specific model of ZFC.
Motivated by work on extension theory and a very concrete question about the unilateral shift, in [10] it was asked whether the Calkin algebra has outer automorphisms. Since the Calkin algebra is not countably saturated ( §7.0.3) it took some time before such an automorphism was constructed using CH ( [73]). This is one of the most complicated known CH constructions, involving an intricate use of EEtheory to extend isomorphisms of direct limits of separable subalgebras. A simpler proof was given in [28, §1], and the method was further refined in [20]. Instead of following the usual back-and-forth construction in which isomorphisms between separable subalgebras are recursively extended, one uses CH to embed the first derived limit of an inverse system of abelian groups into the outer automorphism group.
Forcing axioms imply that the Calkin algebra has only inner automorphisms ( [28]). Conjecturally, for every non-unital separable C*-algebra the assertion that its corona has only (appropriately defined) 'trivial' automorphisms is independent of ZFC (see [20]). Even the abelian case of this conjecture is wide open ( [42]).
The 'very concrete question' of Brown-Douglas-Fillmore alluded to two paragraphs ago is still wide open: Is there an automorphism of Q(H) that sends the image of the unilateral shiftṡ to its adjoint? Fredholm index obstruction shows that such an automorphism cannot be inner. Since the nonexistence of outer automorphisms of Q(H) is relatively consistent with ZFC, so is a negative answer to the BDF question. Every known automorphism α of Q(H) in every model of ZFC has the property that its restriction to any separable subalgebra is implemented by a unitary. Bothṡ andṡ * are unitaries with full spectrum and no nontrivial roots. It is, however, not even known whetherṡ andṡ * have the same (parameter-free) type in Q(H); a positive answer would provide a strong motivation for the question of whether Q(H) is countably homogeneous.
Gaps.
A gap in a semilattice B is a pair A, B such that a ∧ b = 0 for all a ∈ A and all b ∈ B but there is no c such that c ∧ a = a and c ∧ b = 0 for all a ∈ A and b ∈ B. There are no countable gaps in a countably saturated Boolean algebra such as P(N)/ Fin, the quotient of P(N) over the ideal Fin of finite sets. In 1908 Hausdorff constructed a gap in P(N)/ Fin with both of its sides of cardinality ℵ 1 . Later Luzin constructed a family of ℵ 1 orthogonal elements in P(N)/ Fin such that any two of its disjoint uncountable subsets form a gap. It should be emphasized that both results were proved without using CH or any other additional set-theoretic axioms.
Hausdorff's and Luzin's results show that P(N)/ Fin is not more than countably saturated. In particular, if the Continuum Hypothesis fails then the obvious back-and-forth method for constructing automorphisms of P(N)/ Fin runs into difficulties after the first ℵ 1 stages. In one form or another, gaps were used as an obstruction to the existence of morphisms in several consistency results in analysis, notably as obstructions to extending a partial isomorphism ( [84, §V], [21], [28]).
Two subalgebras A and B of an ambient algebra C form a gap if ab = 0 for all a ∈ A and b ∈ B, but there is no positive element c such that ca = a and cb = 0 for all a ∈ A and all b ∈ B. The gap structure of P(N)/ Fin can be imported into the Calkin algebra, but the gap structure of the latter is also much richer ( [101]).
However, the failure of higher saturation in coronas is also manifested in a genuinely noncommutative fashion. A countable family of commuting operators in a corona of a separable algebra can be lifted to a family of commuting operators if and only if this is true for each one of its finite subsets. Proposition 7.3. In M 2 (ℓ ∞ /c 0 ) there exists a family of ℵ 1 orthogonal projections such that none of its uncountable subsets can be lifted to a commuting family of projections in M 2 (ℓ ∞ ).
This was stated in [45] for the Calkin algebra in place of (barely noncommutative) M 2 (ℓ ∞ /c 0 ), but the proof given there clearly gives the stronger result. The combinatorial essence for the proof of Proposition 7.3 echoes Luzin's original idea. One recursively constructs projections p γ in M 2 (ℓ ∞ ) so that p γ p γ ′ is compact but [p γ , p γ ′ ] > 1/4 for all γ = γ ′ . Then the image this family in the corona is as required, as a counting argument shows that no uncountable subfamily can be simultaneously diagonalized.
Recall that every uniformly bounded representation of a countable amenable group in a countably degree-1 saturated algebra is unitarizable (Proposition 7.2). This is false for uncountable groups. This was proved in [17] and improved to the present form in [93] using Luzin's gap.
Proposition 7.4. There is a uniformly bounded representation π of ℵ1 Z/2Z on M 2 (ℓ ∞ /c 0 ) such that the restriction of π to a subgroup is unitarizable if and only if the subgroup is countable.
The construction of Kadison-Kastler-near, but not isomorphic, nonseparable algebras in [16] involves what at the hindsight can be considered as a gap. It is not known whether there is a separable example (see [18] for several partial positive results).
Nonseparable algebras
Not surprisingly, the theory of nonseparable algebras hides surprises and problems not present in the separable case; see [95].
8.1. Nonseparable UHF algebras. Uniformly hyperfinite (UHF) algebras are defined as tensor products of full matrix algebras ( §2.2). However, there are two other natural ways to define uniformly hyperfinite: as (i) an inductive limit of a net of full matrix algebras, or (ii) as an algebra in which every finite subset can be arbitrarily well approximated by a full matrix subalgebra. These three notions, given in the order of decreasing strength, coincide in the separable unital case. Dixmier asked whether separability is needed for this conclusion. The answer is that in every uncountable density character, UHF and (i) differ, but that one needs an algebra of density character ℵ 2 in order to distinguish between (i) and (ii) ( [38]). An extension of methods of [38] resulted in a nuclear, simple C*-algebra that has irreducible representations on both separable and nonseparable Hilbert space ( [27]). This is in contrast with the transitivity of the space of irreducible representations of a separable simple C*-algebra ( [63]).
Representation theory. Representation theory of separable algebras
has deeply affected development of the classical descriptive set theory, as evident from the terminology of both subjects (terms 'smooth' and 'analytic' have the same, albeit nonstandard in other areas of mathematics, meaning). Extension of the work of Glimm and Effros on representation theory combined with methods from logic initiated the abstract classification theory ( §3). The representation theory of nonseparable algebras was largely abandoned because some of the central problems proved to be intractable (see the introduction to [1]). One of these stumbling blocks, Naimark's problem, was partially solved in [1] (see also [96]). By using a strengthening of CH (Jensen's ♦ ℵ1 principle) and a deep result on representation theory of separable C*-algebras (an extension of [63] mentioned above), Akemann and Weaver constructed a C*-algebra that has a unique (up to spatial equivalence) irreducible representation on a Hilbert space, but is not isomorphic to the algebra of compact operators on any Hilbert space. An extension of [1] shows that ♦ ℵ1 implies the existence of a simple C*-algebra with exactly m inequivalent irreducible representations. By a classical result of Glimm (closely related to the Glimm-Effros dichotomy), a simple separable C*-algebra with two inequivalent representations has 2 ℵ0 inequivalent representations. It is not known whether a counterexample to Naimark's problem can be found in ZFC alone or by using an axiom other than ♦ ℵ1 (such as ♦ κ for κ > ℵ 1 ). The fact that every forcing notion that adds a new real number destroys all ground-model examples is a bit of an annoying teaser.
Cyclic representations of C*-algebras are, via the GNS construction ( §2), in a natural bijective correspondence with their states (i.e., positive unital functionals). Pure (i.e., extremal) states are noncommutative versions of ultrafilters. The space of nonprincipal ultrafilters on N, (along with the associated quotient structure P(N)/ Fin) is arguably the most important set-theoretically malleable object known to man. The study of pure states on B(H) (i.e., 'quantized ultrafilters') has already produced some surprising results ([2], [7]; also see [66]).
Amenable operator algebras.
A prominent open problem in the theory of operator algebras is whether every algebra of operators on a Hilbert space which is amenable is isomorphic to a C*-algebra. By using Proposition 7.4, one obtains the following ( [17], [93]).
Theorem 8.1. There exists a nonseparable amenable subalgebra of M 2 (ℓ ∞ ) which is not isomorphic to a C*-algebra. None of its nonseparable amenable subalgebras is isomorphic to a C*-algebras, yet it is an inductive limit of separable subalgebras (even elementary submodels) each of which is isomorphic to a C*-algebra. Moreover, for every ε > 0 such an algebra can be found in an ε-Kadison-Kastler neighbourhood of a C*-algebra.
The question whether there exists a separable counterexample remains open; see [65].
Concluding remarks
The most recent wave of applications of logic to operator algebras started by work of Nik Weaver and his coauthors, in which several long-standing problems were solved by using additional set-theoretic axioms (see [96]). Although we now know that the answers to some of those problems (such as the existence of outer automorphisms of the Calkin algebra) are independent from ZFC, statements of many prominent open problems in operator algebras are absolute between models of ZFC and therefore unlikely to be independent (see the appendix to [29] for a discussion).
Nevertheless, operator algebras do mix very well with logic. Jon Barwise said "As logicians, we do our subject a disservice by convincing others that the logic is first-order and then convincing them that almost none of the concepts of modern mathematics can really be captured in first-order logic." Remarkably, some of the deepest results on the structure of C*-algebras have equivalent formulation in the language of (metric) first-order logic (this applies e.g., to [97] and [98]).
In many of the developments presented here methods from logic were blended with highly nontrivial operator-algebraic methods. Good examples are the proof that the theory of R does not allow elimination of quantifiers ( [51]) the key component of which comes from [13], the already mentioned use of [56], and blending of ♦ ℵ1 with the transitivity of pure state space of separable simple algebras ( [63]) in [1].
Finally, some results in pure logic were motivated by work on operator algebras. Examples are Theorem 6.5, which is new even for discrete structures, and negative and positive results on omitting types ( §4.4).
(a) Isomorphism relation of separable C*-algebras. (b) Isomorphism relation of Elliott-classifiable separable, nuclear, unital and simple algebras. (c) Isometry relation of separable Banach spaces. (d) Affine homeomorphism relation of metrizable Choquet simplices. (e) Isometry relation of Polish spaces.
Question 4 . 3 .
43Assume that separable, nuclear, unital and simple algebras A and B have the same Elliott invariant and are elementarily equivalent. Are A and B necessarily isomorphic?
Question 4.3 would require new examples of separable, nuclear, unital and simple algebras. Can model-theoretic methods provide such examples?
Proposition 4. 5 .
5If D and E are s.s.a. algebras then the following are equivalent.
Lemma 6 . 2 .
62If A ⊆ B are separable metric structures and B U has a sequence of isometric automorphisms α n such that lim n α n (a) = a for all a ∈ A and lim n dist(α n (b), A) = 0 for all b ∈ B, then A and B are isometrically isomorphic.
Question 6 . 3 .
63If M is a separable metric structure, does the isomorphism type of M U (and M ′ ∩ M U , if M is a Banach algebra) depend on U at all?
(b) all ultrapowers of A associated to nonprincipal ultrafilters on N are isomorphic. (c) the Continuum Hypothesis holds. 6.2. Model theory of the relative commutant. The notion of a relative commutant does not seem to have a useful generalization in the abstract model theory and its model-theoretic properties are still poorly understood.
Proposition 7 . 2 .
72Assume C is countably degree-1 saturated and B ⊆ {a} ′ ∩ C is separable, U ⊆ sp(a)is open, and g : U → C is a bounded continuous function. Then there exists c ∈ C ∩ C * (B, a) ′ such that for every f ∈ C 0 (sp(a)) we have cf (a) = (gf )(a).
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|
[] |
[
"AdaBin: Improving Binary Neural Networks with Adaptive Binary Sets",
"AdaBin: Improving Binary Neural Networks with Adaptive Binary Sets"
] |
[
"Zhijun Tu [email protected] \nInstitute of Artificial Intelligence and Robotics\nJiaotong University\nXi'an\n\nHuawei Noah's Ark Lab\n\n",
"Xinghao Chen [email protected] \nHuawei Noah's Ark Lab\n\n",
"Pengju Ren [email protected] \nInstitute of Artificial Intelligence and Robotics\nJiaotong University\nXi'an\n",
"Yunhe Wang [email protected] \nHuawei Noah's Ark Lab\n\n"
] |
[
"Institute of Artificial Intelligence and Robotics\nJiaotong University\nXi'an",
"Huawei Noah's Ark Lab\n",
"Huawei Noah's Ark Lab\n",
"Institute of Artificial Intelligence and Robotics\nJiaotong University\nXi'an",
"Huawei Noah's Ark Lab\n"
] |
[] |
This paper studies the Binary Neural Networks (BNNs) in which weights and activations are both binarized into 1-bit values, thus greatly reducing the memory usage and computational complexity. Since the modern deep neural networks are of sophisticated design with complex architecture for the accuracy reason, the diversity on distributions of weights and activations is very high. Therefore, the conventional sign function cannot be well used for effectively binarizing fullprecision values in BNNs. To this end, we present a simple yet effective approach called AdaBin to adaptively obtain the optimal binary sets {b1, b2} (b1, b2 ∈ R) of weights and activations for each layer instead of a fixed set (i.e., {−1, +1}). In this way, the proposed method can better fit different distributions and increase the representation ability of binarized features. In practice, we use the center position and distance of 1-bit values to define a new binary quantization function. For the weights, we propose an equalization method to align the symmetrical center of binary distribution to real-valued distribution, and minimize the Kullback-Leibler divergence of them. Meanwhile, we introduce a gradient-based optimization method to get these two parameters for activations, which are jointly trained in an end-to-end manner. Experimental results on benchmark models and datasets demonstrate that the proposed AdaBin is able to achieve state-of-the-art performance. For instance, we obtain a 66.4% Top-1 accuracy on the ImageNet using ResNet-18 architecture, and a 69.4 mAP on PASCAL VOC using SSD300. The PyTorch code is available at https://github.com/huawei-noah/ Efficient-Computing/tree/master/BinaryNetworks/AdaBin and the MindSpore code is available at https://gitee.com/mindspore/models/ tree/master/research/cv/AdaBin.
|
10.48550/arxiv.2208.08084
|
[
"https://export.arxiv.org/pdf/2208.08084v2.pdf"
] | 251,623,074 |
2208.08084
|
eacbc5a026a966beb4e648bb0cfeca299c73c5af
|
AdaBin: Improving Binary Neural Networks with Adaptive Binary Sets
Zhijun Tu [email protected]
Institute of Artificial Intelligence and Robotics
Jiaotong University
Xi'an
Huawei Noah's Ark Lab
Xinghao Chen [email protected]
Huawei Noah's Ark Lab
Pengju Ren [email protected]
Institute of Artificial Intelligence and Robotics
Jiaotong University
Xi'an
Yunhe Wang [email protected]
Huawei Noah's Ark Lab
AdaBin: Improving Binary Neural Networks with Adaptive Binary Sets
Binary Neural Networks, Adaptive Binary Sets
This paper studies the Binary Neural Networks (BNNs) in which weights and activations are both binarized into 1-bit values, thus greatly reducing the memory usage and computational complexity. Since the modern deep neural networks are of sophisticated design with complex architecture for the accuracy reason, the diversity on distributions of weights and activations is very high. Therefore, the conventional sign function cannot be well used for effectively binarizing fullprecision values in BNNs. To this end, we present a simple yet effective approach called AdaBin to adaptively obtain the optimal binary sets {b1, b2} (b1, b2 ∈ R) of weights and activations for each layer instead of a fixed set (i.e., {−1, +1}). In this way, the proposed method can better fit different distributions and increase the representation ability of binarized features. In practice, we use the center position and distance of 1-bit values to define a new binary quantization function. For the weights, we propose an equalization method to align the symmetrical center of binary distribution to real-valued distribution, and minimize the Kullback-Leibler divergence of them. Meanwhile, we introduce a gradient-based optimization method to get these two parameters for activations, which are jointly trained in an end-to-end manner. Experimental results on benchmark models and datasets demonstrate that the proposed AdaBin is able to achieve state-of-the-art performance. For instance, we obtain a 66.4% Top-1 accuracy on the ImageNet using ResNet-18 architecture, and a 69.4 mAP on PASCAL VOC using SSD300. The PyTorch code is available at https://github.com/huawei-noah/ Efficient-Computing/tree/master/BinaryNetworks/AdaBin and the MindSpore code is available at https://gitee.com/mindspore/models/ tree/master/research/cv/AdaBin.
Introduction
Deep Neural Networks (DNNs) have demonstrated powerful learning capacity, and are widely applied in various tasks such as computer vision [28], natural [5] and ReActNet [34]. (b) Visualization for activations of 2 nd layer in ResNet-18 on ImageNet. Real denotes real-valued activations, Sign and AdaBin denote the binary methods of previous BNNs and ours. language processing [3] and speech recognition [22]. However, the growing complexity of DNNs requires significant storage and computational resources, which makes the deployment of these deep models on embedded devices extremely difficult. Various approaches have been proposed to compress and accelerate DNNs, including low-rank factorization [46], pruning [10,19], quantization [12], knowledge distillation [11,23] and energy-efficient architecture design [9], etc. Among these approaches, quantization has attracted great research interests for decades, since the quantized networks with less bit-width require smaller memory footprint, lower energy consumption and shorter calculation delay. Binary Neural Networks (BNNs) are the extreme cases of quantized networks and could obtain the largest compression rate by quantizing the weights and activations into 1-bit values [18,39,43,45]. Different from the floating point matrix operation in traditional DNNs, BNNs replace the multiplication and accumulation with bit-wise operation XNOR and BitCount, which can obtain an about 64× acceleration and 32× memory saving [39]. However, the main drawback of BNNs is the severe accuracy degradation compared to the full-precision model, which also limits its application to more complex tasks, such as detection, segmentation and tracking.
According to the IEEE-754 standard, a 32-bit floating point number has 6.8 × 10 38 unique states [1]. In contrast, a 1-bit value only has 2 states {b 1 , b 2 }, whose representation ability is very weak compared with that of the full-precision values, since there are only two kinds of the multiplication results of binary values as shown in Table 1a. To achieve a very efficient hardware implementation, the conventional BNN method [13] binarizes both the weights and the activations to either +1 or -1 with sign function. The follow-up approaches on BNNs have w a -1 +1
-1 +1 -1 +1 -1 +1
(a) BNN [13] w a 0 +1
-1 0 -1 +1 0 +1 (b) SiBNN [40] w a -1 +1 0 0 0 +1 -1 +1 (c) SiMaN [31]
w a a b1 a b2 w b1 a b1 w b1 a b2 w b1 w b2 a b1 w b2 a b2 w b2 (d) AdaBin (Ours) Based on the above observation and analysis, we propose an Adaptive Binary method (AdaBin) to redefine the binary values (b 1 , b 2 ∈ R) with their center position and distance, which aims to obtain the optimal binary set that best matches the real-valued distribution. We propose two corresponding optimization strategies for weights and activations. On one hand, we introduce an equalization method for the weights based on statistical analysis. By aligning the symmetrical center of binary distribution to real-valued distribution and minimizing the Kullback-Leibler divergence (KLD) of them, we can obtain the analytic solutions of center and distance, which makes the weight distribution much balanced. On the other hand, we introduce a gradient-based optimization method for the activations with a loss-aware center and distance, which are initialized in the form of sign function and trained in an end-to-end manner. As shown in Table 1, we present the truth tables of the multiplication results for binary values in different BNNs. Most previous BNNs binarize both the weights and activations into {−1, +1} as shown in Table 1a. A few other methods [31,40] attempt to binarize weights and activations into {0, +1}, as shown in Table 1b and Table 1c. These methods result in 2 or 3 kinds of output representations. Table 1d illustrates the results of our proposed AdaBin method. The activations and weights are not fixed and could provide 4 kinds of output results, which significantly enhances the feature representation of binary networks as shown in Fig. 1b. Meanwhile, we can find that previous binary methods are the special cases of our AdaBin and we extend the binary values from ±1 to the whole real number domain.
Furthermore, we demonstrate that the proposed AdaBin can also be efficiently implemented by XNOR and BitCount operations with negligible extra calculations and parameters, which could achieve 60.85× acceleration and 31× memory saving in theory. With only minor extra computation, our proposed Ad-aBin outperforms state-of-the-art methods for various architectures, as shown in Fig. 1a. The contributions of this paper are summarize as follow:
(1) We rethink the limitation of {−1, +1} in previous BNNs and propose a simple yet effective binary method called AdaBin, which could seek suitable binary sets by adaptively adjusting the center and distance of 1-bit values.
(2) Two novel strategies are proposed to obtain the optimal binary sets of weights and activations for each layer, which can further close the performance gap between binary neural networks and full-precision networks.
(3) Extensive experiments on CIFAR-10 and ImageNet demonstrate the superior performance of our proposed AdaBin over state-of-the-art methods. Besides, though not tailored for object detection task, AdaBin also outperforms prior task-specific BNN methods by 1.9 mAP on PASCAL VOC dataset.
Related Work
Binary neural network was firstly introduced by [13]. They creatively proposed to binarize weights and activations with sign function and replace most arithmetic operations of deep neural networks with bit-wise operations. To reduce the quantization error, XNOR-Net [39] proposed a channel-wise scaling factor to reconstruct the binarized weights, which also becomes one of the most important components of the subsequent BNNs. ABC-Net [33] approximated full-precision weights with the linear combination of multiple binary weight bases and employed multiple binary activations to alleviate information loss. Inspired by the structures of ResNet [21] and DenseNet [25], Bi-Real Net [35] proposed to add shortcuts to minimize the performance gap between the 1-bit and real-valued CNN models, and BinaryDenseNet [6] improved the accuracy of BNNs by increasing the number of concatenate shortcut. IR-Net [38] proposed the Libra-PB, which can minimize the information loss in forward propagation by maximizing the information entropy of the quantized parameters and minimizing the quantization error with the constraint {−1, +1}. ReActNet [34] proposed to generalize the traditional sign and PReLU functions, denoted as RSign and RPReLU for the respective generalized functions, to enable explicit learning of the distribution reshape and shift at near-zero extra cost.
Binarization with Adaptive Binary Sets
In this section, we focus on how to binarize weights and activations respectively, and introduce a new non-linear module to enhance the capacity of BNNs.
We first give a brief introduction on the general binary neural networks. Given an input a ∈ R c×h×w and weight w ∈ R n×c×k×k , then we can get the output y ∈ R n×h ′ ×w ′ by convolution operation as Eq. 1.
y = Conv(a, w).
(1) To accelerate the inference process, previous BNNs always partition the input and weight into two clusters, −1 and +1 with sign function as Eq. 2.
Sign(x) = b 1 = −1, x < 0 b 2 = +1, x ≥ 0 .(2)
Then the floating-point multiplication and accumulation could be replaced by bit-wise operation XNOR (denoted as ⊙) and BitCount as Eq. 3, which will result in much less overhead and latency.
y = BitCount(a b ⊙ w b ).(3)
In our method, we do not constrain the binarized values to a fixed set like {−1, +1}. Instead we release b 1 and b 2 to the whole real number domain and utilize the proposed AdaBin quantizer, which could adjust the center position and distance of the two clusters adaptively as Eq. 4. In this way, the binarized distribution can best match the real-valued distribution:
B(x) = b 1 = β − α, x < β b 2 = β + α, x ≥ β ,(4)
where the α and β are the half-distance and center of the binary values b 1 and b 2 . Fig. 2 shows the binarization of AdaBin, as we can see that, the data on the left of the center will be clustered into b 1 and the data on the right of the center will be clustered into b 2 . The distance α and center β will change with different distributions, which help partition the floating point data into two optimal clusters adaptively. For the binarization of weights and activations, we exploit the same form of AdaBin but different optimization strategies.
Weight Equalization
Low-bit quantization greatly weaken the feature extraction ability of filter weights, especially for 1-bit case. Previous BNNs exploit different methods to optimize the binarized weights. XNOR-Net [39] minimizes the mean squared error (MSE) by multiplying a scale factor, and IR-Net [38] obtains the maximum information entropy by weight reshaping and then conduct the same operation as XNOR-Net. However, these methods can not get accurate quantization error between binarized data and real-valued data due to the following limitations. Firstly, the center position of previous binarized values {−1, +1} is always 0, which is not aligned with the center of original real-valued weights. Secondly, MSE is a simple metric to evaluate the quantization error but do not consider the distribution similarity between binarized data and real-valued data. On the contrary, the Kullback-Leibler divergence (KLD) is a measure on probability distributions [29] and is more accurate to evaluate the information loss than MSE. Therefore, we propose to minimize the KLD to achieve a better distributionmatch. We apply the AdaBin for weights binarization as Eq. 5:
w b = B(w) = w b1 = β w − α w , w < β w w b2 = β w + α w , w ≥ β w ,(5)
where α w and β w are distance and center of binarized weights, the binary elements of w b in the forward is β w − α w and β w + α w . And the KLD of real-valued distribution and binary distribution can be represented as Eq. 6.
D KL (P r ||P b ) = x∈w&w b P r (x)log P r (x) P b (x) dx,(6)
where the P r (x) and P b (x) denote the distribution probability of real-valued weights and binarized weights. In order to make the binary distribution more balanced, we need to align its symmetrical center (position of mean value) to the real-valued distribution, so that Eq. 7 can be obtained.
β w = E(w) ≈ 1 c × k × k c−1 m=0 k−1 j=0 k−1 i=0 w m,j,i .(7)
Therefore, we can further infer that P b (w b1 ) = P b (w b2 ) = 0.5. Since there is no convinced formula of weight distribution for neural networks, it is difficult to calculate the Kullback-Leibler divergence explicitly. However, the weights in such networks typically assume a bell-shaped distribution with tails [2,4,47], and the both sides are symmetrical on the center, then we can get the α w as Eq. 8, the detailed proof is in the supplementary.
α w = ∥w − β w ∥ 2 √ c × k × k ,(8)
where ∥ · ∥ 2 denotes the ℓ 2 -norm. In our method, the distance α w and center β w are channel-wise parameters for weight binarization, and updated along the real-valued weights during the training process. Without distribution reshaping and the constraint that the center of binary values is 0 as IR-Net and XNOR-Net, AdaBin could equalize the weights to make the binarized distribution best match the real-valued distribution. During the inference, we can decompose the binary weights matrix into 1-bit storage format as following:
w b = α w b w + β w , b w ∈ {−1, +1}.(9)
So that the same as the previous BNNs, our method can also achieve about 32× memory saving.
Gradient-based Activation Binarization
Activation quantization is a challenging task with low bit-width, and has much more impacts to the final performance than weight. HWGQ [8] proposed to address this challenge by applying a half-wave Gaussian quantization method, based on the observation that activation after Batch Normalization tends to have a symmetric, non-sparse distribution, that is close to Gaussian and ReLU is a half-wave rectifier. However, recent BNNs [36] proposed to replace the ReLU with PReLU [20], which could facilitate the training of binary networks. So that HWGQ can not be further applied because of this limitation. Besides, the distribution of real-valued activations is not as stable as weights, which keeps changing for different inputs. Therefore we can not extract the center and distance from the activations as Eq. 7 and Eq. 8, which brings extra cost to calculate them and will greatly weaken the hardware efficiency of binary neural networks during inference. In order to get the optimal binary activation during training, we propose a gradient-based optimization method to minimize the accuracy degradation arising from activation binarization. Firstly, we apply the AdaBin quantizer to activations as Eq. 10.
a b = B(a) = a b1 = β a − α a , a < β a a b2 = β a + α a , a ≥ β a ,(10)
where α a and β a are the distance and center of binarized activations, and the binary set of a b in the forward is {β a − α a , β a + α a }. To make the binary activations adapt to the dataset as much as possible during the training process, we set α a and β a as learnable variables, which could be optimized via backward gradient propagation as total loss decreases. In order to ensure that the training process can converge, we need to clip out the gradient of large activation values in the backward as Eq. 11.
∂L ∂a = ∂L ∂a b * 1 | a−βa αa |≤1 ,(11)
where L denotes the output loss, a is the real-valued activation and a b is the binarized activation, 1 |x|≤1 denotes the indicator function that equals to 1 if |x| ≤ 1 is true and 0 otherwise. This functionality can be achieved by a composite function of hard tanh and sign, thus we rewrite the Eq. 10 as following:
a b = α a × Sign(Htanh( a − β a α a )) + β a .(12)
• Binarizing Weight For simplicity, we denote g(x) = Sign(Htanh(x)), then we can get the gradient of α a and β a as Eq. 13 in the backward:
∂L ∂αa = ∂L ∂a b ∂a b ∂αa = ∂L ∂a b (g( a − βa αa ) − a αa g ′ ( a − βa αa )), ∂L ∂βa = ∂L ∂a b ∂a b ∂βa = ∂L ∂a b (1 − g ′ ( a − βa αa )),(13)
where g ′ (x) is the derivative of g(x). We set the initial values of center position β a and distance α a to 0 and 1, so that the initial effect of our binary quantizer is equivalent to the sign function [13,35,38]. Then these two parameters of different layers are dynamically updated via gradient descent-based training, and converge to the optimal center and distance values, which is much different from the unified usage of the sign function in the previous BNNs. During inference, the α a and β a of all the layers are fixed, then we can binarize the floating point activations into 1-bit as followings:
a b = α a b a + β a , b a ∈ {−1, +1},(14)
where the b a is the 1-bit storage form and obtained online with input data. Compared with the sign function of previous BNNs, AdaBin will take a little overhead but could significantly improve the feature capacity of activations with the adaptive binary sets for each layer.
Non-linearity
Prior methods [36] propose to use Parametric Rectified Linear Unit (PReLU) [20] as it is known to facilitate the training of binary networks. PReLU adds an [39] 89.8 BNN [13] 89.9 RAD [15] 90.0 IR-Net [38] 90.4 RBNN [32] 91.3 SLB [45] 92.0 AdaBin (Ours) 92.3 Table 2: Comparisons with state-of-the-art methods on CIFAR-10. W/A denotes the bit width of weights and activations.
adaptively learnable scaling factor in the negative part and remain unchanged in the positive part. However, we empirically found that the binary values with our proposed AdaBin are almost all positive in very few layers, which invalidate the non-linearity of PReLU. Therefore, to further enhance the representation of feature maps, we propose to utilize Maxout [17] for the stronger non-linearity in our AdaBin, which is defined as Eq. 15.
f c (x) = γ + c ReLU(x) − γ − c ReLU(−x),(15)
where x is the input of the Maxout function, γ + c and γ − c are the learnable coefficient for the positive part and negative part of the c-th channel, respectively. Following the setting of PReLU, the initialization of γ + c and γ − c are 1 and 0.25.
Binary Convolution for AdaBin
The goal of BNNs is to replace the computationally expensive multiplication and accumulation with XNOR and BitCount operations. Although the binary sets are not limited to {−1, +1}, our method can still be accelerated with bitwise operations by simple linear transformation. As shown in Fig. 3, we can binarize the weights and get the 1-bit matrix b w offline via Eq. 9, and binarize the activations to get the 1-bit activations b a online via Eq. 14, then decompose the binary convolution into three items. The first term is the same as the previous BNNs, and the second term only needs to accumulation for one output channel, which can be replaced by BitCount. The third term F (w) could be pre-computed in the inference process. For n = c = 256, k = 3, w ′ = h ′ = 14, compared with the binary convolution of IR-Net [39], our method only increases 2.74% operations and 1.37% parameters, which are negligible compared to the total complexity and could achieve 60.85× acceleration and 31× memory saving in theory, the detailed analysis is shown in the supplementary material.
Experiments
In this section, we demonstrate the effectiveness of our proposed AdaBin via comparisons with state-of-the-art methods and extensive ablation experiments. We implement the proposed method using PyTroch [37] and MindSpore [26].
Results on CIFAR-10
We train AdaBin for 400 epochs with a batch size of 256, where the initial learning rate is set to 0.1 and then decay with CosineAnnealing as IR-Net [38]. We adopt SGD optimizer with a momentum of 0.9, and use the same data augmentation and pre-processing in [21] for training and testing. We compare AdaBin with BNN [13], LAB [24], XNOR-Net [39], DoReFa [48], DSQ [16], RAD [15], IR-Net [38], RBNN [32], ReCU [44] and SLB [45]. Table 2 shows the performance of these methods on CIFAR-10. AdaBin obtains 93.1% accuracy for ResNet-18 architecture, which outperforms the ReCU by 0.3% and reduces the accuracy gap between BNNs and floating-point model to 1.7%. Besides, AdaBin obtains 0.4% accuracy improvement on ResNet-20 compared to the current best method RBNN, and gets 92.3% accuracy while binarizing the weights and activations of VGG-small into 1-bits, which outperforms SLB by 0.3%.
Results on ImageNet
We train our proposed AdaBin for 120 epochs from scratch and use SGD optimizer with a momentum of 0.9. We set the initial learning rate to 0.1 and then decay with CosineAnnealing following IR-Net [38], and utilize the same data augmentation and pre-processing in [21]. In order to demonstrate the generality of our method, we conduct experiments on two kinds of structures. The first group is the common architectures that are widely used in various computer vision tasks, such as AlexNet [28] and ResNet [21]. Another kind is the binary-specific structures such as BDenseNet [7], MeliusNet [5] and ReActNet [34], which are designed for BNNs and could significantly improve the accuracy with the same amount of parameters as common structures. Table 3: Comparison with state-of-the-art methods on ImageNet for AlexNet and ResNets. W/A denotes the bit width of weights and activations. * means using the two-step training setting as ReActNet.
Common structures. We show the ImageNet performance of AlexNet, ResNet-18 and ResNet-34 on Table 3, and compare AdaBin with recent methods like Bi-Real [35], IR-Net [38], SiBNN [40], RBNN [32], ReCU [44], Real2Bin [36] and ReActNet [34]. For AlexNet, AdaBin could greatly improve its performance on ImageNet, outperforming the current best method SiBNN by 3 Table 4 shows the performance comparison with BDenseNet, MeliusNet and ReActNet. For BDenseNet28, AdaBin could get 1.1% improvement with the same training setting, which costs negligible extra computational operations. Similarly, when AdaBin is applied to MeliusNet, an advanced version of BDenseNet, it outperforms the original networks by 1.0%, 0.7%, 0.5% and 0.6%, respectively, demonstrating that AdaBin could significantly improve the capacity and quality of binary networks. Besides, we also train the ReActNet-A structure with our AdaBin, following the same training setting with ReActNet [34]. As we can see that, AdaBin could get 1.0% performance improvement with the similar computational operations. Our method could explicitly improve the accuracy of BNNs with a little overhead compared to state-of-the-art methods, as shown in Fig. 1a. Table 5 presents the results of object detection on PASCAL VOC dataset for different binary methods. We follow the training strategy as BiDet [35]. The backbone network was pre-trained on ImageNet [14] and then we finetune the whole network for the object detection task. During training, we used the data augmentation techniques in [42], and the Adam optimizer [27] was applied. The learning rate started from 0.001 and decayed twice by multiplying 0.1 at the 160-th and 180-th epoch out of 200 epochs. Following the setting of BiDet [42], we evaluate our proposed AdaBin on both the normal structure and the structure with real-valued shortcut. We compare them with general binary methods BNN [13], XNOR-Net [39] and BiReal-Net [35], and also compare with BiDet [42] and AutoBiDet [41], which are specifically designed for high-performance binary detectors. And for reference, we also show the results of the multi-bit quantization method TWN [30] and DoReFa [48] with 4 bit weights and activations. Compared with the previous general BNNs, the proposed AdaBin improves the BNN by 22.0 mAP, XNOR by 13.8 mAP and Bi-Real Net by 5.6 mAP. Even for the task-specific optimization method BiDet, they are 11.6 mAP and 2.6 mAP lower than our method with two structures, and the improved AutoBiDet still lower than AdaBin by 10.5 mAP and 1.9 mAP. Besides, AdaBin with shortcut structure could outperform TWN and DoReFa, which demonstrates that our could significantly enable the binary neural networks to complex tasks.
Results on PASCAL VOC
Ablation Studies
Effect of AdaBin quantizer. We conduct the experiments by starting with a vanilla binary neural networks, and then add the AdaBin quantizer of weights and activations gradually. The results are shown in Table 6a, we can see that when combined with existing activation binarization by sign function, our equalization method for binarizing weights could get 0.6% accuracy improvement. Besides, when we free the α w and β w to two learnable parameters which are trained in an end-to-end manner as activation, it only get 86.7% accuracy and is much poorer than AdaBin (the last two row). We find that its Kullback-Leibler divergence is also less than AdaBin, which shows the KLD is much important to 1-bit quantization. When keeping the weight binarization as XNOR-Net [39], the proposed gradient-based optimization for binarizing activations could get 1.6% accuracy improvement, as shown in the 3 rd row. Combining the proposed weight equalization and activation optimization of AdaBin boosts the accuracy by 2% over vanilla BNN (the 1 st vs. 4 th row), which shows that AdaBin quantizer could significantly improve the capacity of BNNs. Effect of γ in Maxout. In addition, we evaluate four activation functions on ImageNet. The first is none, denoting it is an identity connection. The second is Maxout that only with γ + for positive part, the third is Maxout only with γ − for negative part and the last one is the complete Maxout as Eq. 15. As shown in Table 6b, the coefficient of γ + and γ − improve the accuracy by 9.6% and 9.7% individually. The activation function with both coefficients gets the best performance, which justifies the effectiveness of Maxout.
Conclusion
In this paper, we propose an adaptive binary method (AdaBin) to binarize weights and activations with optimal value sets, which is the first attempt to relax the constraints of the fixed binary set in prior methods. The proposed AdaBin could make the binary weights best match the real-valued weights and obtain more informative binary activations to enhance the capacity of binary networks. We demonstrate that our method could also be accelerated by XNOR and BitCount operations, achieving 60.85× acceleration and 31× memory saving in theory. Extensive experiments on CIFAR-10 and ImageNet show the superiority of our proposed AdaBin, which outperforms state-of-the-art methods on various architectures, and significantly reduce the performance gap between binary neural networks and real-valued networks. We also present extensive experiments for object detection, which demonstrates that our method can naturally be extended to more complex vision tasks.
Fig. 1 :
1(a) Comparisons with state-of-the-art methods. With a little extra computation, the proposed AdaBin achieves better results for various architectures such as ResNet, MeliusNet
Fig. 2 :
2AdaBin quantizer. The middle represents the mapping from floating point distribution f r (x) to binary distribution f b (x). b 1 and b 2 are the two clusters, α and β are the distance and center, respectively.
Fig. 3 :
3Binary convolution process. The I represents the identity matrix, and F (w) represents the extra computation with w, which could be pre-computed during the inference.
6: (a) Ablation studies of AdaBin for ResNet-20 on CIFAR-10. * means the α w and β w are learnable parameters to the binary sets. (b) The ablation studies of Maxout on ImageNet, the scale factor with γ − equals to PReLU.
Table 1 :
1The illustration on the feature representation ability of different binary schemes. The a represents the binarized input and the w represents binarized weights, respectively. a b1 , a b2 , w b1 , w b2 ∈ R, which are not restricted to fixed values for different layers.made tremendous efforts for enhancing the performance of binary network, but
still restrict the binary values to a fixed set (i.e., {−1, +1} or {0, +1}) for all the
layers. Given the fact that the feature distributions in deep neural networks are
very diverse, sign function can not provide binary diversity for these different
distributions. To this end, we have to rethink the restriction of fixed binary set
for further enhancing the capacity of BNNs.
Table 4 :
4Comparisons on ImageNet for binary-specific structures.egy, AdaBin could get 66.4% top-1 accuracy, which obtains 0.9% improvement
compared to ReActNet.
Binary-specific structures.
Table 5 :
5The comparison of different methods on PASCAL VOC for object detection. W/A denotes the bit width of weights and activations. * means the the proposed method with extra shortcut for the architectures[42].
{w b1 , w b2 } {-1, +1} PReLU 86.3 {−α, +α} {a b1 , a b2 } PReLU 87.3 {w b1 , w b2 } {a b1 , a b2 } PReLU 87.7 {w b1 , w b2 } * {a b1 , a b2 } Maxout 86.7 {w b1 , w b2 } {a b1 , a b2 }W set
A set Non-linearity Acc.(%)
{−α, +α} {-1, +1}
PReLU
85.7
Maxout
88.2
(a) Binary quantizer
Scale factors Top-1 (%) Top-5 (%)
None
53.2
77.2
γ +
62.8
83.9
γ −
62.9
84.1
γ − , γ +
63.1
84.3
(b) γ in Maxout
Table
We gratefully acknowledge the support of MindSpore, CANN(Compute Architecture for Neural Networks) and Ascend AI Processor used for this research.
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|
[
"https://github.com/huawei-noah/"
] |
[
"Global phase diagram of charge neutral graphene in the quantum Hall regime for generic interactions",
"Global phase diagram of charge neutral graphene in the quantum Hall regime for generic interactions"
] |
[
"Suman Jyoti De \nHarish-Chandra Research Institute\nA CI of Homi Bhabha National Institute\nChhatnag Road211019JhunsiPrayagrajIndia\n",
"Ankur Das \nDepartment of Condensed Matter Physics\nWeizmann Institute of Science\n76100Rehovot, Israel\n",
"Sumathi Rao \nInternational Centre for Theoretical Sciences (ICTS-TIFR)\nHesaraghatta Hobli560089Shivakote, BangaloreIndia\n",
"Ribhu K Kaul \nDepartment of Physics\nThe Pennsylvania State University\n16802University ParkPAUSA\n",
"Ganpathy Murthy \nDepartment of Physics & Astronomy\nUniversity of Kentucky\n40506LexingtonKYUSA\n"
] |
[
"Harish-Chandra Research Institute\nA CI of Homi Bhabha National Institute\nChhatnag Road211019JhunsiPrayagrajIndia",
"Department of Condensed Matter Physics\nWeizmann Institute of Science\n76100Rehovot, Israel",
"International Centre for Theoretical Sciences (ICTS-TIFR)\nHesaraghatta Hobli560089Shivakote, BangaloreIndia",
"Department of Physics\nThe Pennsylvania State University\n16802University ParkPAUSA",
"Department of Physics & Astronomy\nUniversity of Kentucky\n40506LexingtonKYUSA"
] |
[] |
Monolayer graphene at charge neutrality in a quantizing magnetic field is a quantum Hall ferromagnet. Due to the spin and valley (near) degeneracies, there is a plethora of possible ground states. Previous theoretical work, based on a stringent ultra short-range assumption on the symmetryallowed interactions, predicts a phase diagram with distinct regions of spin-polarized, canted antiferromagnetic, inter-valley coherent, and charge density wave order. While early experiments suggested that the system was in the canted antiferromagnetic phase at a perpendicular field, recent scanning tunneling studies universally find Kekulé bond order, and sometimes also charge density wave order. Recently, it was found that if one relaxes the stringent assumption mentioned above, a phase with coexisting canted antiferromagnetic and Kekulé order exists in the region of the phase diagram believed to correspond to real samples. In this work, starting from the continuum limit appropriate for experiments, we present the complete phase diagram of ν = 0 graphene in the Hartree-Fock approximation, using generic symmetry-allowed interactions, assuming translation invariant ground states up to an intervalley coherence. Allowing for a sublattice potential (valley Zeeman coupling), we find numerous phases with different types of coexisting order. We conclude with a discussion of the physical signatures of the various states. *
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10.1103/physrevb.107.125422
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[
"https://export.arxiv.org/pdf/2211.02531v2.pdf"
] | 253,370,626 |
2211.02531
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ee015b04c10cd53441415ae3c9e90607b84961eb
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Global phase diagram of charge neutral graphene in the quantum Hall regime for generic interactions
Suman Jyoti De
Harish-Chandra Research Institute
A CI of Homi Bhabha National Institute
Chhatnag Road211019JhunsiPrayagrajIndia
Ankur Das
Department of Condensed Matter Physics
Weizmann Institute of Science
76100Rehovot, Israel
Sumathi Rao
International Centre for Theoretical Sciences (ICTS-TIFR)
Hesaraghatta Hobli560089Shivakote, BangaloreIndia
Ribhu K Kaul
Department of Physics
The Pennsylvania State University
16802University ParkPAUSA
Ganpathy Murthy
Department of Physics & Astronomy
University of Kentucky
40506LexingtonKYUSA
Global phase diagram of charge neutral graphene in the quantum Hall regime for generic interactions
Monolayer graphene at charge neutrality in a quantizing magnetic field is a quantum Hall ferromagnet. Due to the spin and valley (near) degeneracies, there is a plethora of possible ground states. Previous theoretical work, based on a stringent ultra short-range assumption on the symmetryallowed interactions, predicts a phase diagram with distinct regions of spin-polarized, canted antiferromagnetic, inter-valley coherent, and charge density wave order. While early experiments suggested that the system was in the canted antiferromagnetic phase at a perpendicular field, recent scanning tunneling studies universally find Kekulé bond order, and sometimes also charge density wave order. Recently, it was found that if one relaxes the stringent assumption mentioned above, a phase with coexisting canted antiferromagnetic and Kekulé order exists in the region of the phase diagram believed to correspond to real samples. In this work, starting from the continuum limit appropriate for experiments, we present the complete phase diagram of ν = 0 graphene in the Hartree-Fock approximation, using generic symmetry-allowed interactions, assuming translation invariant ground states up to an intervalley coherence. Allowing for a sublattice potential (valley Zeeman coupling), we find numerous phases with different types of coexisting order. We conclude with a discussion of the physical signatures of the various states. *
I. INTRODUCTION
The quantum Hall effects (QHE) [1,2], discovered four decades ago [3] in semiconductor heterostructures, embody many phenomena observed there for the first time, but later found in many systems. The QHE represents the first and simplest topological insulator [4,5], as a consequence of which the electric and thermal Hall conductances are quantized. The bulk is insulating; charge and heat are carried by edge modes [6] which are robust against disorder. Due to the quantization of kinetic energy into discrete values, Landau levels are also the first example of truly flat bulk bands. As a result, the bulk physics is controlled entirely by electron-electron interactions in a partially filled Landau level. Notably, this leads to the fractional QHE (FQHE) states [7], which host excitations with fractional charge and statistics [8]. Due to spin or other internal degeneracies (such as valley or layer), ground states at some integer fillings are also controlled by interactions. Typically, interactions lead to ferromagnetism, as exemplified by the single layer ν = 1 spin ferromagnet [9] or the bilayer ν = 1 state in GaAs quantum wells [10][11][12]. Such quantum Hall ferromagnets also have interesting topological charged excitations such as skyrmions [9] or merons [11,12].
Graphene [13][14][15][16] is a single layer of Carbon atoms arranged in a honeycomb lattice with two sites (A and B) in each unit cell. Near charge neutrality, low-energy electrons in graphene occur in two valleys at the two inequivalent zone corners K and K , and obey a Dirac equation in each valley. In a quantizing perpendicular magnetic field B, the Dirac spectrum produces particlehole symmetric Landau levels n = 0, ±1, ±2 · · · with energy E ±n ∝ ± B|n| in each valley (ignoring the Zeeman splitting). Each Landau level is (almost) four-fold degenerate, with the four states being labelled by spin and valley. The n = 0 Landau level is special; states in each valley are localized on one sublattice. Furthermore, since the states are comprised of a superposition of particle-like and hole-like states of the B = 0 problem, there is valley mixing at the edge, leading to one edge mode with a particle-like dispersion and one with a hole-like dispersion [17].
It was recognized very early [18][19][20][21]that partial filling of the n = 0 manifold of Landau levels (called the zero-LLs or ZLLs) would lead to a rich set of quantum Hall ferromagnets at integer fillings, the most interesting of which is the ν = 0 case, when two of the four ZLLs are filled. The particular superpositions of the spin and valley LLs that are occupied determine the nature of the ground state.
In the non-interacting limit, the orbital part of the kinetic energy has an SU (4) spin/valley symmetry in the continuum limit. The Zeeman coupling E Z splits the spin ↑ and spin ↓ Landau levels. The ground state is fully spin-polarized [22], with the K ↑, and K ↑ Landau levels occupied. The chemical potential lies between the ↑-spin and ↓-spin sets of ZLLs. Due to the nature of the edge states of the ZLLs mentioned above [17], at each edge, the chemical potential intersects a particle-like level and a hole-like level of opposite spin. Since the system is symmetric under U (1) spin rotations around the total B field, these two modes cannot back-scatter, even if potential disorder is present. Therefore the system should be in a quantum spin Hall (QSH) phase [4,22,23]. Now let us add electron-electron interactions. The dominant long-range part of the Coulomb interaction does not discriminate between spin and valley indices, being SU (4) symmetric. The Coulomb interaction changes the single-particle spectrum at the edge, but leaves the two gapless, opposite-spin, charge-carrying edge modes intact, preserving the QSH phase [24]. In transport, such a state should show a two-terminal conductance of 2e 2 /h. Initial experiments [25][26][27][28] saw a trivial insulating state at ν = 0 without any protected edges. A seminal experiment measured the two-terminal conductance in tilted field [29], which allowed independent tuning of the Zeeman coupling. It was found that at large E Z , the two-terminal conductance does indeed tend asymptotically to 2e 2 /h. Below a critical Zeeman energy E * Z , the system remains a trivial insulator. The gap at the edge vanishes continuously as one approaches E Z → E * Z , indicating a second-order phase transition.
The fact that the ground state at purely perpendicular field is not a QSH state means that interactions beyond the SU (4)-symmetric Coulomb interactions must play an important role [18][19][20][21]. The ground state must be chosen by SU (4)-anisotropic residual interactions, arising from lattice-scale couplings. Low-energy effective symmetries inherited from the B = 0 problem can be used to deduce a U (1) valley symmetry at the four-Fermi level (reduced to a Z 3 symmetry when higher-Fermi interactions are included) [18]. One can classify the four-Fermi anisotropic residual interactions into two types; an Ising-like coupling in the valley space v z (q) and an xy-like coupling in the valley space v xy (q). See Section II A for the full definition of v z (q), v xy (q) and the interaction Hamiltonian.
Since the residual interactions arise from lattice-scale couplings, and the magnetic length = eB is two orders of magnitude larger than the lattice spacing ( a), the bare interactions are ultra-short-range (USR). The first step in obtaining the effective interactions in the ZLL manifold is to project the bare interactions to this manifold. Operationally, this assumption implies that v µ are independent of q in the ZLL. It should be noted that if one projects USR bare interactions to a N = 0 manifold in graphene, the effective interactions will not be USR [30].
Using the USR assumption for the interactions and building upon previous work [18][19][20][21], Kharitonov [31] found the phase diagram in the Hartree-Fock approximation for ν = 0 graphene. There are four phases: A fully polarized phase F , an antiferromagnetic phase AF (which becomes a canted AF, or CAF in the presence of the Zeeman coupling), a charge density wave (CDW) phase, and a phase with inter-valley coherence (IVC), sometimes also called a Kekulé distorted or bondordered (KD/BO) phase. All the phase boundaries are first-order, except for the CAF to F transition, which is second-order. Upon the addition of a valley Zeeman coupling, the CDW and bond order coexist, leading to a partially sublattice polarized (PSP) phase [32], but the transition between the PSP and the CAF phases remains first-order.
This picture indeed reproduces the phenomenology of ν = 0 graphene in tilted field [29], assuming that the anisotropic couplings are such that the system (in perpendicular field) is in the CAF phase [31]. Subsequent magnon transmission experiments [33][34][35][36] through a ν = 0 region surrounded by ferromagnetic ν = 1 regions also confirm that coherently propagating magnetic excitations are present in the ν = 0 state. In the CAF phase the natural candidate is the gapless Goldstone mode associated with the spontaneous symmetry breaking of the U (1) spin symmetry.
More recently, three scanning tunneling studies [37][38][39] on ν = 0 graphene perpendicular B field have introduced more complexity into this picture. While they cannot directly confirm or rule out CAF order, all three see Kekulé bond order, and two of the three [38,39] see CDW order as well. Partial alignment of the graphene layer with the hexagonal Boron Nitride (HBN) substrate is known [40][41][42][43] to produce a sublattice potential/valley Zeeman coupling E V , which can lead to CDW order. This origin of the CDW order was confirmed in one of the experiments [38], while it remains unclear in the other experiment [39].
It is possible that the samples used in the scanning tunneling experiments are in a different phase from those used in transport experiments. However, the most parsimonious explanation is that CAF and bond order coexist in all samples, the CAF being undetected in STM experiments and the bond order being undetected in transport.
Since there is no coexistence between CAF and BO in Kharitonov's phase diagram [31], one is led to re-examine the assumption of ultra-short-range interactions. From the point of view of an effective model in the ZLLs, there is no reason to assume any particular form for the symmetry-allowed interactions, because integrating out high energy Landau levels will lead to renormalizations in the form of the interactions [44][45][46][47][48]. In a recent work, three of us [49] used this reasoning, generalizing the symmetry-allowed interactions v z (q), v xy (q) to be arbitrary functions of q. This might seem to introduce an infinite number of new couplings. Remarkably, in the continuum Hartree-Fock (HF) approximation, assuming translation symmetry is preserved up to an inter-valley coherence, only two independent numbers per type of coupling suffice [49], namely the Hartree and the Fock couplings, defined as
g µ,H = v µ (q = 0) 2π 2 ; g µ,F = d 2 q (2π) 2 v µ (q)e −q 2 2 /2 (1)
where µ = z, xy. For ultra-short-range (USR) interactions g µ,H = g µ,F . Indeed, it was found [49] that given certain inequalities between the Hartree and Fock couplings, there was a region of couplings which showed generic coexistence between the CAF and BO orders. We will refer to this as the B/CAF phase. It should be noted that non-USR interactions have implicitly been introduced earlier by Goerbig and collaborators in the context of effective nonlinear sigma models for ν = ±1 in graphene [50,51]. In ν = −1 for example, only a single ZLL (some linear combination of the four possibilities) is occupied. As in any ferromagnet with single occupancy, the many-body wave function is completely antisymmetric in space. Thus, USR interactions cannot contribute to the energy of the state and the entire physics is controlled by the non-USR couplings [52]. The case of ν = 1 is related to that of ν = −1 by particle-hole symmetry. More recently, the connection of the parameters of the effective theory to the non-USR nature of the microscopic anisotropic interactions was made explicit by Atteia and Goerbig [52], once again in the context of ν = ±1.
There is a different line of reasoning coming from Bernal-stacked bilayer graphene (BLG) in the quantum Hall regime [53] which also leads to the non-USR condition g µ,H = g µ,F . Assuming solely nearest neighbor hoppings, the B = 0 dispersion has quadratic band touchings at K and K . However, upon including the symmetry-allowed trigonal warping (a hopping between non-Bernal stacked sites in different layers) the quadratic band touching reconstructs into four Dirac cones [54]. The inclusion of trigonal warping has a profound effect [54] on the eight-fold (nearly) degenerate manifold of states near charge neutrality in a quantizing B. The upshot is that the the symmetry-allowed interactions, when projected into the low-energy manifold, now acquire structure on the scale of and no longer satisfy the USR condition g µ,H = g µ,F . This fact is crucial in obtaining phases which show coexistence between different kinds of order in BLG at ν = 0 [53]. There is a deep analogy between MLG and BLG, as we will see; the states that we will uncover in the full phase diagram of MLG are identical to a subset of states found earlier in BLG [53].
The purpose of this paper is to find the complete HF phase diagram of monolayer graphene in the continuum approximation, assuming translation invariance up to an inter-valley coherence. Our previous work [49] was motivated by the STM experiments [37][38][39], and confined to values of couplings thought to apply to real samples. Furthermore, the valley Zeeman coupling was ignored. We will explore the full phase diagram, in the presence of nonzero E Z , E V for all possible g z,H , g z,F , g xy,H , g xy,F .
The majority of the results in the main text are for the case when the Hartree and Fock parts of a given coupling have the same sign:
g z,F g z,H > 0; g xy,F g xy,H > 0 (2)
This seems natural for weak LL-mixing, when the renormalizations from integrating out the higher energy states are expected to be small compared to the bare values of the couplings. However, for strong LL-mixing, one may well have situations when the Hartree and Fock parts of a given coupling have opposite signs. We will present some interesting results in this case as well.
To give a brief preview of our results. We find three coexistence phases: (i) The coexistence phase occurring near the BO/CAF boundary in the USR model, which was found earlier [49]. This phase, which we call the B/CAF phase, also has a spin-valley entangled order even at E V = 0, which we label as SVEY (we will explain the notation shortly in Section II B). (ii) A phase occuring near the CDW/FM boundary in the USR model. This phase displays the coexistence between CDW and FM order, mediated by a spin-valley entangled order we label SVE+ (explained in Section II B). (iii) When g z,F g z,H < 0 we find a phase where FM and SVEX/SVEY order (explained in Section II B) coexist, without any other order being present at E V = 0. We call this the FSVE phase. When E V > 0 many of these phases acquire a CDW order parameter, but remain largely unchanged otherwise. In sum, the full phase diagram of monolayer graphene for generic interactions is much richer than was previously believed.
The plan of the paper is as follows: In Section II we will briefly review the previous work on the effective model for MLG in the continuum approximation. We will generalize the interactions to be non-USR, discuss the HF approximation and find the ground state energy. Also in Section II, we present a parameterization of translationinvariant ν = 0 states [51,55,56] whose energy depends on four angles. It turns out that the states that have been found in the USR limit [31] can be characterized in terms of a single angle. Instabilities of these states, which can be computed analytically, will signal the occurrence of more complex states with coexisting order parameters. In general, the actual ground state in any region of coexistence has to be found numerically. In Section III we present our results; since there are six independent tuning parameters, we will present many two-dimensional sections through the space of coupling constants. Each two-dimensional section will satisfy different inequalities between the Hartree and Fock couplings. We end with our summary, conclusions, and open questions in Section IV. The appendices contain the details of our calculations, analytical expressions for various instabilities, and sample results for strong Landau-level mixing.
II. MODEL HAMILTONIAN, PARAMETERS, AND METHODS
We choose the primitive translation vectors for graphene as a 1 = aê x , a 2 = a(ê x 2 + √ 3êy
2 ), with the general Bravais lattice site R = n 1 a 1 + n 2 a 2 . The noninteracting Hamiltonian of graphene at zero magnetic field, suppressing the spin index for the moment, is
H 0 = −t R,j c † AR c BR+dj + h.c.(3)
where t is the nearest-neighbor hopping matrix element, c AR , c BR destroy electrons at the A and B sublattice sites of the Bravais site R, the sum on j = 1, 2, 3 with d 1 = 0, d 2 = a 1 − a 2 , d 3 = −a 2 and h.c. stands for hermitian conjugate. Note that there is no spin-orbit coupling in the Hamiltonian of Eq. (3). First-principles [57,58] and tight-binding calculations [59,60] show that the spin-orbit coupling in graphene is of the order of tens of µeV , smaller than any other energy scale in the problem. We will set the spin-orbit coupling to zero here and henceforth. Fourier transforming Eq. (3), we obtain the Bloch Hamiltonian at wave-vector k as a matrix in the sublattice space
H(k) = −t 0 f (k) f * (k) 0 (4) where f (k) = 1+e ik·(a1−a2) +e −ik·a2 .
It is easily checked that f (k) vanishes at the two inequivalent zone corners (valleys) K =ê x 4π 3a = −K indicating Dirac crossings. The low-energy effective Hamiltonian in the K valley (k = K + p, |p| 2π/a) can be obtained by expansion.
H K (p) = ta √ 3 2 0 p x − ip y p x + ip y 0(5)
The low-energy effective Hamiltonian at the K valley can be obtained by the identity H K (p) = (H K (−p)) * . In the continuum limit, we turn on a weak perpendicular magnetic field B ⊥ by allowing the Hamiltonian to act on slowly varying envelope functions. Operationally, this involves promoting p i → −i∂ i → −i∂ i + eA i (r), where the electron's charge is −e, and A(r) is the vector potential satisfying ∇×A =ê z B ⊥ . In order for the continuum limit to be justified, the magnetic length has to be much larger than the lattice spacing; = eB ⊥ a. This is extremely well-satisfied for realistic fields.
Next, one chooses Landau gauge A = B ⊥ xê y , and imposes periodic boundary conditions in the y-direction with a length L y . Let us define the Landau level wavefunctions as
x, y|n, k = e iky L y Φ n x − k 2(6)
where Φ n are the normalized harmonic oscillator wavefunctions. Note that, here and henceforth, k is a one-dimensional guiding center label, and not a twodimensional momentum. Now it is straightforward to see that the ZLL states are (0, |n = 0, k ) T in the K valley, and (|n = 0, k , 0) T in the K valley. Thus, in the ZLLs, valley and sublattice are locked together. Now we are ready to present our model Hamiltonian.
A. Hamiltonian and the Hartree-Fock Approximation
In what follows, we will index the fermion operators with a valley index α, β, which can be K ≡ 0 or K ≡ 1, and a spin index s =↑≡ 0 or s =↓≡ 1. In this notation, the non-interacting Hamiltonian of the ZLLs is
H 1b = − α,s,k (E Z (−1) s + E V (−1) α ) c † α,s,k c α,s,k (7)
which introduces the Zeeman energy E Z and the valley Zeeman/sublattice potential E V . Turning to interactions, Alicea and Fisher [18] noted that, in addition to the SU (4)-symmetric Coulomb interaction, two other types of low-energy effective interactions were allowed by SU (2) spin-rotation symmetry and momentum conservation in the B = 0 problem. We recall that the spin-orbit coupling is negligible [57][58][59][60] and has been neglected. A U (1) symmetry in the valley space (separate conservation of the number of electrons in each valley) emerges when restricting oneself to four-Fermi interactions. Upon including six-Fermi terms this is reduced to a Z 3 symmetry because 3(K − K ) is a reciprocal lattice vector. The full interaction Hamiltonian for the ZLLs in monolayer graphene, in the Landau gauge basis discussed earlier, is
H =H 1b + H int (8) H int =H Coul + H Z + H xy (9) H Z = 1 2L x L y k,k ,q v z (q)e −iqx(k−k −qy) 2 e −(q ) 2 /2 : c † k−qy τ Z c k c † k +qy τ Z c k :(10)H xy = 1 2L x L y k,k ,q v xy (q)e −iqx(k−k −qy) 2 e −(q ) 2 /2 : c † k−qy τ x c k c † k +qy τ x c k : + : c † k−qy τ y c k c † k +qy τ y c k :(11)
We have used a shorthand notation where the spin/valley indices are suppressed and summed, and τ i is a Pauli matrix in the valley space (it acts as the identity in the spin space). Furthermore, since the Coulomb interaction has no role to play in selecting the ground state, we drop it henceforth. In the generic case, when E Z , E V > 0, the Hamiltonian has a U (1) s spin-rotation symmetry generated by total σ z , a U (1) v valley-rotation symmetry generated by total τ z (which is also the difference between the number of electrons in the K and K valleys), and an entangled spin-valley U (1) sv symmetry generated by total τ z σ z . In the fine-tuned case E Z = 0 (not realizable in experimental samples) the spin-rotation symmetry is enhanced to SU (2) s . In the fine-tuned case E V = 0, the valley symmetry is enhanced to a U (1) v ⊗ Z 2v , where the Z 2v represents the symmetry exchanging the two valleys.
In the HF approximation, one looks for the single Slater determinant that has the right electron filling and minimizes the energy. Such a state, symbolically written as |HF , can be completely characterized by the set of 1-body expectation values. ∆ ss αβ (k, k ) = HF |c † α,s,k c β,s ,k |HF
We will restrict the space of HF states to those obeying translation invariance, up to an intervalley coherence. This means that the ∆ becomes diagonal in k and inde-pendent of it.
∆ ss αβ (k, k ) = δ kk ∆ ss αβ(13)
Let us make the idea of translation invariance up to an intervalley coherence more explicit. The ∆ we have assumed allows nonzero averages of the form HF |c † K,s c K s |HF . Clearly these break lattice translations since K and K are not identical up to a reciprocal lattice vector. Allowing such averages introduces a new set of reciprocal lattice vectors which are K − K and all their rotated versions, and leads to translation invariance with an enlarged unit cell of size √ 3 × √ 3 as compared to the original. Indeed, this is exactly what is seen STM experiments [37][38][39]. The ansatz of Eq. 13 makes sure that there is no translation symmetry breaking beyond the minimal one implied by intervalley coherence.
In general, the matrix ∆ is the projector on to the linear space of the occupied states. Given that two orthogonal linear combinations of the four ZLLs (call them |f 1 and |f 2 ) are occupied at ν = 0 we can write
∆ = |f 1 f 1 | + |f 2 f 2 |(14)
We can now express the HF energy of the Hamiltonian of Eq. (11) per guiding center in terms of ∆, with N φ = LxLy 2π 2 , as
E HF N φ = − E Z T r [σ Z ∆] − E V T r [τ Z ∆] + g z,H 2 (T r[τ Z ∆]) 2 − g z,F 2 T r [τ Z ∆τ Z ∆] + g xy,H 2 (T r[τ x ∆]) 2 + (T r[τ y ∆]) 2 − g xy,F 2 (T r[τ x ∆τ x ∆] + T r[τ y ∆τ y ∆])(15)
Note that g µ,h and g µ,F are defined as in Eq. (1).
B. Ansatz for States, Instabilities, and Order Parameters
We will start with an efficient parameterization [51,55,56] for the two orthogonal occupied states |f 1 and |f 2 . This parameterization has been used not only for uniform states but also for describing skyrmions [56]. (17) where n = sin θ p cos φ pêx + sin θ p sin φ pêy + cos θ pêz , and s = sin θ s cos φ sêx + sin θ s sin φ sêy + cos θ sêz are unit vec-
|f 1 = cos α 1 2 |n ⊗ |s + e iβ1 sin α 1 2 | − n ⊗ | − s (16) |f 2 = cos α 2 2 |n ⊗ | − s + e iβ2 sin α 2 2 | − n ⊗ |s
In going from |n → | − n one substitutes θ p → π − θ p and φ p → φ p + π, and likewise for s.
Given this ansatz, which depends on eight angles, we compute the HF energy.
E HF = − E Z cos θ s [cos α 1 − cos α 2 ] − E V cos θ p [cos α 1 + cos α 2 ] + g z,H 2 cos 2 θ p [cos α 1 + cos α 2 ] 2 − g z,F 16 4 cos 2 θ p (cos(2α 1 ) + cos(2α 2 )) − 8 sin 2 θ p (cos α 1 cos α 2 − sin α 1 sin α 2 cos(β 1 + β 2 )) + 8 + g xy,H 2 sin 2 θ p [cos α 1 + cos α 2 ] 2 − g xy,F 16 8 sin 2 θ p (cos α 1 cos α 2 − sin α 1 sin α 2 cos(β 1 + β 2 )) + 4 sin 2 θ p (cos(2α 1 ) + cos(2α 2 )) − 16 (cos α 1 cos α 2 − 1)(19)
There are three noteworthy features of this energy. The first is that it is independent of φ p and φ s . This results from the U (1) v and U (1) s symmetries of the Hamiltonian. Therefore we can set φ s = φ p = 0 without loss of generality. The second feature is that the dependence of the energy on β 1 , β 2 occurs only in the interacting part, and only in the combination β 1 + β 2 . The reason the one-body HF energy does not involve β 1 , β 2 is that the averages of τ z , σ z do not involve β i .
f i |τ z |f i = cos α i cos θ p (20) f i |σ z |f i = − (−1) i cos α i cos θ s(21)
The dependence of the interaction energy solely on β 1 +β 2 arises from the SU (2) spin symmetry of the interactions, which implies that the U (1) rotation |s → e iχ/2 |s ; | − s → e −iχ/2 | − s cannot change the interaction energy.
Ignoring overall phase factors, this rotation has the net effect β 1 → β 1 − χ; β 2 → β 2 + χ. This demonstrates that the energy can only depend on β 1 + β 2 . Thirdly, the dependence on β 1 + β 2 occurs via the term cos(β 1 + β 2 ), which appears linearly. Depending on the sign of its coefficient, the minimum energy will occur at cos(β 1 +
β 2 ) = ±1.
The bottom line is that the minimum of the HF energy for uniform states can be found in a subspace in which |f 1 , |f 2 can both be chosen real.
Below, we will call states which have Kekulé/BO, and/or CDW order B/CO states (because they have both bond order and/or charge order). The states originally found by Kharitonov [31] can be represented in terms of the above angles as follows (details in Appendix A),
|F M = |α 1 = 0, α 2 = π, θ p = π/2, θ s = 0, β 1 = β 2 = π (22) |CAF = |α 1 = θ CAF , α 2 = π − α 1 , θ p = π/2, θ s = 0, β 1 = β 2 = π ; g xy,F < 0 (23) |B/CO = |α 1 = α 2 = 0, θ p = θ B/CO , θ s = 0, β 1 = β 2 = 0(24)
where
θ CAF = cos −1 E Z 2|g xy,F | ; g xy,F < 0 (25) θ B/CO = cos −1 E V g V ; g V > E V(26)θ B/CO = 0; g V < E V(27)g V = 2g z,H − g z,F − 2g xy,H + g xy,F(28)
If θ B/CO = 0 the state is a pure CDW, if 0 < θ B/CO < π/2 it has coexisting BO and CDW order, and if θ B/CO = π/2 the system is in a pure Kekulé state. The reason we call these states "simple" is that they can all be described by at most a single nontrivial angle, which can be analytically computed as a function of the couplings. Generic states may depend on more than one nontrivial angle, in which case it is not possible to solve for the angles or the ground state energy analytically.
The ground state energies of the simple states are
E FM = − 2(E Z + g xy,F ) − g z,F E CAF = E 2 Z 2g xy,F − g z,F ; 0 < E Z < −2g xy,F E B/CO = − E 2 V g V − g xy,F + 2g xy,H ; 0 < E V < g V E CDW = − 2E V − g z,F + 2g z,H(29)
Our strategy is to examine the stability of these "simple" states by finding the eigenvalues of the Hessian matrix of second derivatives of the energy functional with respect to the four angles α 1 , α 2 , θ p , θ s .
E ij = ∂ 2 E HF ∂χ i ∂χ j(30)
where χ i represents all four angles. For the "simple" states one can compute the entire Hessian matrix analytically, and also obtain the eigenvalues analytically. All eigenvalues being positive means the state is stable to arbitrary small deformations. As the coupling constants are varied, a formerly positive eigenvalue may vanish, indicating an instability of the state in question. This allows us to map out the regions of stability of the "simple" states in our coupling constant space.
It can happen that when some of the angles take particular values, the projector on to the occupied subspace becomes independent of certain other angles. This occurs in the FM and B/CO phases. Consequently, certain rows and columns E ij vanish, which means that one eigenvalue always vanishes in that state independent of the coupling constants. In such cases, the instability is marked by the vanishing of an eigenvalue that does depend on coupling constants. Once the region of possible coexistence has been found, we use numerical self-consistent Hartree-Fock to obtain the ground state and confirm the coexistence predicted by the Hessian.
In preparation for showing the results, let us list all the order parameters which we will encounter and the symmetries they break. We have chosen a parameterization in which the projector matrix of the occupied states is real. Thus, out of all possible hermitian matrices that represent order parameters, only real matrices will have nonzero expectation values.
F M = σ z /2 CAF = τ z σ x /2 BO = τ x /2 CDW = τ z /2 SV EX = τ x σ x /2 SV EY = τ y σ y /2 SV E± = τ x σ x ± τ y σ y /2(31)
The SVE (spin-valley entangled) type of order parameters are so called because they break the spin and valley symmetries simultaneously in an entangled way.
Let us examine the symmetries broken by the various order parameters. In the fine-tuned case E Z = 0 (not realizable in experiment) the FM order parameter spontaneously breaks the SU (2) s symmetry. In the finetuned case E V = 0 (which is potentially realizable in experiment) the CDW order spontaneously breaks the Z 2v symmetry. In the generic case E Z , E V = 0 the FM and CDW order parameters do not break any symmetries of the Hamiltonian. The CAF order parameter breaks U (1) s and U (1) sv , but preserves U (1) v . Bond order breaks U (1) v and U (1) sv but preserves U (1) s . The SVE order parameters break U (1) s and U (1) v but preserve U (1) sv . All three U (1)s are spontaneously broken in the B/CAF phase.
We emphasize that while the U (1) s symmetry is protected by the assumed vanishing of the spin-orbit coupling in graphene (Eq. (3)), there is no such protection for U (1) v or U (1) sv . As mentioned in the introduction, since 3(K − K ) is a reciprocal lattice vector, any U (1) related to the valley space will be broken down to a Z 3 upon including six-Fermi and higher interactions. Thus, there are no Goldstone modes associated with the spontaneous breaking of the U (1) v symmetry. However, a spontaneous breaking of the U (1) s and/or the U (1) sv symmetries will lead to a phase with a Goldstone mode.
III. RESULTS
There are six coupling constants in our Hamiltonian, E Z , E V , g z,H , g z,F , g xy,H , g xy,F . Since the full sixdimensional phase diagram is impossible to visualize, we will be forced to take two-dimensional cuts.
We will often draw a correspondence between MLG and Bernal-stacked BLG [61,62]. In BLG, in addition to spin and valley, an orbital index n = 0, 1 also appears [54]. For BLG states that have no orbital mixing, and are orbitally symmetric, there is a one-to-one correspondence with states in MLG. Specifially, the role of E V in MLG is played by the perpendicular electric field D in BLG [53].
While RG arguments tell us that generic effective interactions must have nonzero range, one does not know precisely how the LL-mixing and the intergration of highenergy states affects the Hartree and Fock parts of the couplings in the ZLL manifold. For weak LL-mixing, a natural assumption is that the sign of g µ,H , g µ,F are the same, but the magnitudes could be different. The majority of the results we present in the main text assume this to be true. For strong LL-mixing, it is conceivable that the effective g µ,H and g µ,F have opposite signs. For completeness we have analyzed this case as well, though most of the details are relegated to the appendices.
In Section III A, we will present results E V = 0, restricting the Hartree and Fock parts of the couplings to have the same sign. This will allow us to examine how the original Kharitonov phase diagram [31] changes when we relax the USR assumption. We confirm the existence of the B/CAF phase found earlier [49] for g xy,F /g xy,H > 1. This phase also has the order parameter SVEY (Eq. (31)). Thus, this phase breaks U (1) s , U (1) v , and U (1) sv spontaneously. Another coexistence phase occurs near the boundary between the FM and CDW phases when g z,F /g z,H > 1. We call this the SVE+ phase because the corresponding order parameter (Eq. (31)) is nonzero in this phase. This phase breaks U (1) s and U (1) v but preserves U (1) sv . The bilayer analog of this phase was found earlier in Bernal-stacked BLG [53], although at nonzero E V .
Next, in Section III B we will keep the restriction that the Hartree and Fock parts of the couplings have the same sign, but turn on E V . Already, for USR interactions, one finds a phase with coexisting CDW and Kekulé order [56], which we call the B/CO phase. Guided by the intuition that new phases are most likely to appear near the transition lines of the original USR model, we will examine these carefully. When 0 > g xy,H > g xy,F , the B/CAF phase shrinks as E V increases (all other couplings remaining constant), and vanishes for large enough E V . For any E V > 0, both SVEX and SVEY order parameters are nonzero in the B/CAF phase. However, the symmetries that are spontaneously broken do not change in any of the phases at small E V .
Going to larger E V is even more interesting: in certain cases, even if coexistence does not occur for E V = 0, it can occur for intermediate E V , and disappear for large E V . For example, for g z,F = 0.75g z,H , g xy,F = 0.75g xy,H , both the B/CAF and SVE+ phases are absent at E V = 0. However, both phases are present for a range of intermediate E V . This is consistent with earlier results for BLG [53], which also found coexistence phases for intermediate values of the perpendicular electric field D, which plays the same role in BLG as E V plays in MLG.
From the point of view of experiment, the most interesting phases is the B/CAF phase, which occurs for 0 > g xy,H > g xy,F . At any nonzero E V , the B/CAF phase evolves into one with coexisting CAF, Kekulé, and CDW order. An analogous phase breaking all three U (1) symmetries was found earlier in BLG [53]. (Of course, the U (1) v symmetry will be reduced to a Z 3 upon the inclusion of higher-Fermi interactions, and thus does not lead to a Goldstone mode). Such a phase would be consistent with observations of magnon transmission [33,35,36] through the ν = 0 state, which implies some kind of magnetic order, and the STM experiments [37][38][39] which show Kekulé and CDW order.
Finally, in Section III C we will present some results for the signs of g µ,H and g µ,F being different for either or both of g z and g xy . As may be expected, the topology of the phase diagram can change considerably in such cases. However, a new type of coexistence phase appears, which is not present when the signs of the H and F parts of both couplings are identical. This phase, which we call the FSVE phase, has coexisting FM and SVEY order, without any other order being present at E V = 0. Such a state breaks U (1) s and U (1) v while preserving U (1) sv .
As usual, many phases acquire nonzero CDW order when E V > 0. A more detailed set of results for the H and F parts of g xy/z being of opposite signs are presented in the appendices.
A. Vanishing Valley Zeeman Coupling with gµ.F /gµ,H > 0 Throughout this section we will assume that we are "close" to the USR model in the sense that the Hartree and Fock couplings of a given type have the same sign (g µ,F /g µ,H > 0), and that there is perfect sublattice symmetry E V = 0. In order to enable comparisons with the USR model, we show the Kharitonov phase diagram [31] in Fig. 1. Here we choose E Z = 1.0 (in arbitrary units) as a fixed parameter. All the lines between phases are first-order transitions, except for transition between the CAF and FM phases, which is second-order. Our convention here and henceforth is that solid lines represent first-order phase transitions while dashed lines represent continuous phase transitions. When one relaxes the USR assumption, it turns out the inequalities g xy,F /g xy,H ≶ 1 and g z,F /g z,H ≶ 1 play a crucial role in determining whether coexistence occurs at E V = 0 (we will see later that nonzero E V overcomes this limitation). Briefly, coexistence between Kekulé and CAF order occurs only when g xy,F /g xy,H > 1, while coexistence between CDW and FM order occurs only when g z,F /g z,H > 1. This is illustrated in Figs. 2 and 3 for specific values of other couplings, and turns out to be generic as long as E V = 0. In what follows, we will fix the ratio of g µ,F to g µ,H , µ = xy, z, and plot the phase diagrams with the axes being g xy,H and g z,H . This should merely be thought of as taking a certain two-dimensional section of the full space of coupling constants, and does not represent any physical assumption about the proportionality between g µ,F and g µ,H , say, as B varies. Fig. 4 shows the phase diagram for the case g xy,F = 1.25g xy,H , g z,F = 1.25g z,H . As found earlier [49], there is a region of coexistence between Kekulé and CAF order near the first-order line of the USR model. This phase, which we call B/CAF, also has SVEY order. In addition, there is a region of coexistence between CDW and FM order (the SVE+ phase) in the neighborhood of the phase transition between the CDW and FM phases in the USR case. The transitions bordering the coexistence regions of the phase diagram are second-order, represented by dashed lines. As in Fig. 1, E Z = 1. There are two different coexistence phases: The B/CAF coexistence phase (shaded blue) was already found previously [49], and also has SVEY order. The first-order transition between the FM and CDW phases for USR interactions has broadened into the SVE+ phase (shaded green) with coexistence between CDW and FM order parameters, so called because it has a spontaneous spin-valley entangled order parameter. Fig. 5 shows the phase diagram for the case g z,F /g z,H < 1, g xy,F /g xy,H > 1. As mentioned above, this satisfies the condition for coexistence between CAF and BO order (the B/CAF phase), but fails to meet the condition for the existence of the SVE phase between the CDW and FM phases. The B/CAF phase with its attendant SVEY order is still present, though reduced in extent. Similarly, Fig. 6 has g z,F /g z,H > 1, g xy,F /g xy,H < 1, allowing the SVE+ phase to exist but forbidding coexistence between BO and CAF order. Finally, Fig. 7 has g z,F /g z,H < 1, g xy,F /g xy,H < 1, disallowing any coexistence. The reason for the difference in detail between the Kharitonov phase diagram Fig. 1 and Fig. 7 is that the boundaries between phases sometimes depend on the Hartree coupling and sometimes on the Fock coupling, as seen by the ground state energies in Eq. (29). It is of experimental interest to ask how the system evolves with increasing Zeeman coupling when all interaction parameters are fixed, and E V = 0. This corresponds to applying a tilted field to the system [29], keeping the perpendicular component of B constant. The evolution of the order parameters depends on the values chosen for g z,H/F , g xy,H/F . Clearly, if the system is in the FM phase already at E Z = 0, there will be no further evolution with increasing E Z . Similarly, if the system is in the CAF phase at E Z = 0, it can only evolve into the FM with increasing E Z . Since these possibilities have been thoroughly explored in the past literature, we will ignore them in favor of more interesting possibilities. 1, g xy,F /g xy,H > 1, which corresponds to the phase diagram of Fig. 4. Fig. 8 shows the evolution of order parameters with E Z when the system is in the CDW phase at vanishing Zeeman coupling. In addition to the FM and CDW order parameters, we also show the SVE+ order parameter. As E Z increases, the system undergoes a phase transition from the CDW phase into the SVE+ phase (coexisting FM, CDW, and SVE+ order), and then into the FM phase. Alternatively, as shown in Fig. 9 the system could start in the BO phase at E Z = 0. In this case the system first goes into the B/CAF phase (which also has SVEY order), then into the pure CAF phase, and finally into the FM phase at large E Z .
Coming next to the case of g z,F /g z,H < 1, g xy,F /g xy,H > 1, which corresponds to the phase diagram of Fig. 5, Fig. 10 shows the evolution of order parameters as a function of E Z for g z,H = −0.2, g xy,H = −1.2. The system starts in the BO phase at vanishing Zeeman coupling, undergoes a second-order transition to the B/CAF coexistence phase, undergoes yet another second-order transition to the pure CAF phase, and finally goes into the FM phase at large E Z .
Finally, we consider the case g z,F /g z,H < 1, g xy,F /g xy,H > 1, which corresponds to the phase diagram of Fig. 6. Fig. 11 shows the evolution of the order parameter as a function of E Z for g z,H = −1.5, g xy,H = −0.5. The system starts in the CDW phase at vanishing E Z , makes a second-order phase transition to the SVE+ phase at intermediate E Z , and finally goes into the FM phase via another second-order phase transition at large E Z . Fig. 6. The system is in the CDW phase at EZ = 0. As EZ increases, the system enters the SVE+ phase (coexisting CDW, FM, and SVE+ order) via a second-order transition. For larger EZ there is another second-order transition to the FM phase.
B. Nonvanishing Valley Zeeman Coupling and gµ,F /gµ,H > 0
In this subsection we consider how the phase diagrams change when E V > 0. We will still stay "close" to the USR model, assuming g µ,F /g µ,H > 0. The most obvious change is already present in the USR limit: The BO phase is replaced by the B/CO phase, in which both bond order and CDW order coexist. This is analogous to the replacement of the antiferromagnetic phase at E Z = 0 by the CAF phase at arbitrarily small E Z . Concurrently, the first-order phase transition between the CDW and BO phases in the USR phase diagram Fig. 1 is converted into a second-order transition. These are all previously known results [53,56].
The introduction of small E V does not modify the SVE+ phase (whenever it occurs). While E V > 0 does not change the symmetries broken in the B/CAF phase, it does change the phase quantitatively. Let us first look at Fig. 12, which shows the order parameters at E V = 0 along a horizontal cut across Fig. 4 at g z,H = −0.45. Recall that the B/CAF phase, at E V = 0, has BO, CAF, FM, and SVEY order. The SVEY order parameter changes discontinuously at the first-order phase transition with the SVE+ phase. All these features change when one adds a tiny E V . In Fig. 13 we show the evolution of order parameters along the same horizontal cut (g zH = −0.45) when E V = 0.01. It can be seen that the SVE+ phase does not change character qualitatively. However, the B/CAF phase now changes significantly. Firstly, it acquires a nonzero CDW order parameter, because the B/CO phase also has CDW order. More importantly, it now has both SVEX and SVEY order, and undergoes a second-order phase transition to the SVE+ phase. Thus, even a very tiny E V can affect certain phases and phase transitions significantly. Fig. 4. The first-order transition between the B/CAF and SVE+ phases has now become second-order.
Next we turn a moderate value of E V = 0.25. Fig. 14 shows the phase diagram for the ratio of Hartree and Fock parts of the couplings being g z,F /g z,H = g xy,F /g xy,H = 1.25, the same as in Fig. 4. The topology of the phase diagram is identical to that of Fig. 4, with the solitary change that the first-order transition between the B/CAF and SVE+ phases has been replaced by a second-order transition. The B/CAF region has shrinks, while the SVE+ region expands. 15 shows the phase diagram at E V = 0.5 when the ratios are g z,F = 0.75g z,H ; g xy,F = 1.25g xy,H . As in the case of the previous figure, some of the phase boundaries move, but the topology remains the same as at E V = 0. The same is true for g z,F /g z,H = 1.25; g xy,F /g xy,H = 0.75, shown in Fig. 16 and g µ,F /g µ,H = 0.75 shown in Fig. 17. More interesting phenomena occur if one increases E V to an even larger value. Recall that for g z,F /g z,H = g xy,F /g xy,H = 0.75, there is no coexistence anywhere in the phase diagram (Fig. 7) at E V = 0, or at E V = 0.5 (Fig. 17). The phase diagram for this ratio of the H and F parts of the couplings, at E V = 1.4, is shown in Fig. 18. It can be seen that a sufficiently large E V can induce coexistence, even when it does not occur at E V = 0. A similar phenomenon is seen in previous work in BLG [53]. Fig. 7 and Fig. 17. In contrast to those cases where there is no coexistence anywhere in the phase diagram, we do obtain the coexistence phases B/CAF (blue shaded region) and SVE+ (green shaded region). 19 shows a slightly different way of looking at the occurrence of the B/CAF phase as E V varies. We have fixed the Zeeman coupling at E Z = 1, and the xy interactions such that g xy,F /g xy,H < 1, implying that coexistence will not occur for E V = 0. The coupling g z,H is also fixed. We show the range of parameters in E V , g z,F where the B/CAF phase occurs. As can be seen, there is an intermediate range of E V and g z,F where the B/CAF phase appears. Once again, we choose g z,F /g z,H < 1, which implies that the SVE+ phase does not occur for E V = 0. We see that for large enough E V , the SVE+ phase is stabilized.
It is also interesting to consider the evolution of the order parameters as E Z increases in Fig. 18. If one starts in the B/CO phase, the expected sequence of phases as E Z increases for small or vanishing E V , provided the B/CAF phase occurs, is B/CO → B/CAF → CAF → FM. However, for E Z = 1.4 the sequence is different: BO → B/CAF → SVE+ → FM. This is shown in Fig. 21 It is conceivable that for strong LL-mixing, the renormalizations of the interactions could be large enough to make the signs of the Hartree and Fock parts of g z/xy opposite. For completeness we present some phase diagrams for this type of situation in this subsection.
First we consider the case g z,F /g z,H = −1, but g xy,F /g xy,H > 1. The most obvious change is in the topology of the phase diagram. The CAF phase is completely surrounded by other phases. In addition, there are two coexistence phases. The blue shaded region is the B/CAF phase familiar from the previous subsections. It has BO, CAF, and SVEY order. The brown shaded region represents a new type of coexistence phase which does not occur when g µ,F /g µ,H > 0. This phase has FM order coexisting with SVEX/SVEY order (all the ground states generated from SVEX by U (1) s ⊗ U (1) v are degenerate), with no other order parameters being present. This state spontaneously breaks the U (1) s and U (1) v symmetries, but preserves U (1) sv . Fig. 23 shows the order parameters along a horizontal cut in the phase diagram of Fig. 22 at g z,H = 1. We will call this the FSVE phase. When E V > 0, the FSVE phase acquires both SVEX and SVEY order in addition to FM and CDW order (which is natural since E V > 0). Thus, the FSVE phase remains different from the B/CAF phase (because there is no CAF order in the FSVE phase) and the SVE+ phase, which requires a precise equality of the SVEX and SVEY order parameters. More details are in the appendices.
As a second example, we show the case when g z,F /g z,H = 1.25 > 0 and g xy,F /g xy,H = −1 < 0, which is shown in Fig.24. As expected for negative ratios between the H and F parts of the coupling, the phases are drastically rearranged in the phase diagram. However, the nature of the phases remains the same as in Section III A. There is no B/CAF phase, but the SVE+ phase does occur, interpolating between the CDW and FM phases.
A more complete set of figures for other cases when one or both of the H and F coupling ratios are negative appears in the appendices.
IV. CAVEATS, CONCLUSIONS, AND OPEN QUESTIONS
Monolayer graphene at charge neutrality in the quantum Hall regime is an example of quantum Hall ferromagnetism [9][10][11][12]. While earlier theoretical [31] and experimental work [29,33] seemed to suggest a simple canted antiferromagnetic phase at perpendicular magnetic field, recent scanning tunneling observations [37][38][39] clearly show Kekulé and even charge density wave order.
A major assumption underlying most earlier theoretical work [18,20,21,31] is that the residual interactions (other than Coulomb), being descended from lattice-scale couplings, should be ultra-short-range on the scale of the magnetic length . Recently, based on renormalization group ideas, it was argued [49] that the effective lowenergy interactions will naturally acquire the length scale even if they were ultra-short-range microscopically. It should be noted that this argument does not depend on whether one considers the N = 0 manifold (the ZLLs) or some other manifold of Landau levels of graphene. For such generic interactions, coexistence between CAF and Kekulé order in the physically relevant region of the coupling constant space was shown to occur [49] in a robust regime of couplings.
Our goal in this paper was to present a more complete phase diagram of MLG at charge neutrality (and more generally, in any half-filled manifold of Landau levels). As in previous theoretical work, we assume a clean system with four-Fermi interactions only. Momentum conservation constrains the residual four-Fermi interactions to have a U (1) valley symmetry (reduced to a Z 3 upon including higher-Fermi interactions, which we ignore). This still leaves two coupling functions v z (q), v xy (q) to be determined. In the Hartree-Fock approximation, restricting to ground states respecting translation invariance up to an intervalley coherence, the energetics can be captured by a Hartree coupling and a Fock coupling for each of the two types. After the inclusion of the Zeeman and valley Zeeman couplings E Z , E V , one has six coupling constants to deal with.
Our main physically relevant finding is that coexistence between CAF and B/CO order (the bond order always coexists with CDW order when E V > 0) occurs in a large region of the coupling constant space when g xy,H , g xy,F < 0. Depending on the particular values of the interaction couplings, the valley Zeeman coupling may either enhance or suppress coexistence. In fact, in a certain range of couplings, a re-entrant transition from a pure B/CO through a coexistence phase back to a pure B/CO is possible upon increasing E V . The fact that coexistence is generic is consistent with the ubiquity of B/CO order seen in STM observations [37][38][39].
We found two other coexistence phases in regions of the phase diagram which may not be directly relevant to graphene. The first, called the SVE+ phase, interpolates between the CDW and FM phases, and also has a spin-valley entangled SVE+ order parameter. The corresponding phase in BLG was found in previous work [53]. The second one, called the FSVE phase, occurs when g z,F is allowed to be of the opposite sign as g z,H , which could conceivably occur for strong Landau-level mixing. For generic E V > 0, the FSVE phase has FM, CDW, SVEX and SVEY order, but no CAF or bond order.
Our results are complete given our assumptions, but they come with caveats. We have ignored spontaneous translation symmetry breaking beyond that required by intervalley coherence, static disorder, and quantum/thermal fluctuations. Let us consider each in turn. Intervalley coherence is allowed in our approach, and is a signal of minimal translation symmetry breaking because it implies a new reciprocal lattice vector K − K . In combination with the reduction of the U v (1) symmetry to Z 3 upon including higher-Fermi interactions, this leads to the Kekulé order, which does break lattice translation symmetry with a three-fold enlargement of the unit cell. Thus, this type of translation symmetry breaking in implicitly included in our approach. Since the STM experiments see only this minimal type of translation symmetry breaking, we believe our assumption of translation invariance up to an intervalley coherence is not a serious limitation. Next, static disorder is present in all samples. When B/CO order is present, it is expected to be pinned by the local value of disorder. In fact, since the B/CO order couples to static disorder while the CAF order does not, one expects the region of the coupling constant space where B/CO order is present to increase as disorder increases. Otherwise, static disorder is expected to have a quantitative effect on the transport gaps, but leave the nature of the state unaffected. Now we turn to quantum fluctuations. The Hartree-Fock approximation has an excellent track record in describing quantum Hall ferromagnets at zero temperature [9][10][11][12]. Near a second-order phase boundary, HF will always predict a mean-field transition. Quantum fluctuations will shift the phase boundary, and correct the critical behavior to the appropriate universality class; for example, the transition from the B/CO to the coexistence phase with the CAF order should have the universality class of the three-dimensional XY model. If a phase occurs in a very narrow sliver of coupling constant space, one might envisage quantum fluctuations making it disappear. However, since all the phases we find are robust, occurring over substantial ranges of coupling constants, we expect quantum fluctuations to alter the phase boundaries in detail, but not affect the phase diagram qualitatively. An important potential exception is the firstorder transition seen in many parameter regimes between the FM and CDW phases. When E Z , E V > 0, neither of these phases breaks any symmetry of the Hamiltonian. One cannot rule out a first-order transition without a change in symmetry between the two phases: An example is the liquid-gas transition. However, a phase transition between the FM and CDW phases is not necessary, since both have the same symmetry. Quantum fluctuations may destroy the first-order line in favor of a smooth crossover between the CDW and FM. Consider parameter regimes when the SVE+ phase intervenes between the CDW and FM mean-field phases. Since the SVE+ phase breaks U (1) s and U (1) v spontaneously, a second-order phase transition is allowed between it and the FM or CDW regions. Quantum fluctuations may change the topology of the phase diagram to make the SVE+ phase an island in the middle of the crossover between the CDW-dominated and FM-dominated regions.
Finally, we turn to T > 0 and thermal fluctuations. Recall that spontaneous intervalley coherence is subject to a Z 3 symmetry, and hence does not result in a Goldstone mode. Long-range B/CO order is expected to be survive to a critical temperature T c > 0. The U spin (1) symmetry is an almost exact symmetry, being broken only by the tiny spin-orbit coupling (≈ 10µeV ) in graphene [57][58][59][60]. Thus, setting spin-orbit coupling to zero, any spontaneous breaking of the U (1) s symmetry leads to a gapless Goldstone mode at T = 0. For T > 0 the system is in the universality class of the two-dimensional XY -model. Long-range order is absent at any nonzero T , and there should be a Berezinskii-Kosterlitz-Thouless (BKT) transition [63,64] at T KT , below which there is power-law order. T KT should vanish as the CAF order parameter vanishes, because the stiffness vanishes as well. Thus, there is an intermediate temperature regime in the B/CAF phase which is above T KT but below the T c of the Z 3 bond order.
Let us turn to experimental signatures of coexistence. STM experiments can directly measure the B/CO order; thus the key question is how to detect CAF order. Any order parameter that spontaneously breaks U (1) s will lead to a gapless Goldstone mode. Thus, the CAF phase and all three coexistence phases we have found would support a Goldstone mode at T = 0. Magnon transmission experiments [33,35,36] reveal the presence of magnetic excitations, but because the magnons are created in ferromagnetic regions with a gap of E Z , such experiments are unable to reveal whether the magnetic excitations at ν = 0 are gapless. In Bernal-stacked bilayer graphene, where a CAF state is also expected to be present, a very recent experiment has confirmed the presence of the gapless Goldstone mode [65]. If such an experiment can be carried out for MLG it would be direct confirmation of the spontaneous breaking of spin-rotation symmetry U (1) s . More broadly, in the context of transport, a measurement of the bulk thermal conductivity below T KT should reveal the Goldstone mode. Additionally, the BKT transition itself should have a signature in R xx [66].
A more indirect way to probe the CAF order parameter is to examine the detailed structure of spin/valley skyrmions, which can be induced by external charges. A thorough analysis of skyrmions in the ultra-short-range model of charge-neutral graphene was carried out very recently [56]. Using this framework, an analysis of the B/CO texture near a charge defect shows [38] that it is consistent with theory, assuming that the true ground state is pure B/CO. A phase with coexistence between B/CO and CAF order will have skyrmions that differ in detail from those of the pure B/CO phase.
There are two broad open questions. Firstly, given a microscopic model at some intermediate energy scale much larger than ω c , how does one reliably deduce the effective coupling functions in the manifold of the n = 0 Landau levels? Kharitonov [31], following earlier RG treatments [67,68], carried out just such a procedure, under the assumption that the couplings (other than Coulomb) remained ultra-short-range under RG, implying a finite number of couplings to renormalize. Based on a general fermionic RG procedure [69] which includes all low-energy interactions, there has been quite a bit of previous work attempting to integrate out higher Landau levels perturbatively [44][45][46][47][48]. While these works restricted themselves to the Coulomb interaction, it should be straightforward to extend them to include all symmetry-allowed interactions.
The second broad question is complementary to the first. Given an experimental sample, is there a complete set of measurements that could determine the cou-plings g z,H , g z,F , g xy,H , g xy,F ? Given that E Z is determined by the total field, and E V can be deduced from zero-magnetic-field gap measurements at charge neutrality, this would fully determine the effective theory at the mean-field level. One way to approach this is via a detailed investigation of skyrmions [56]. As long as the size of the skyrmions is much larger than the magnetic length, a nonlinear sigma model approach is capable of capturing their structure and energetics. The parameters that enter the nonlinear sigma model are exactly those that enter the mean-field theory, with the exception of the stiffness, which is determined by the dominant Coulomb interaction.
Last, but not least, let us briefly consider fractionally filled states in the n = 0 manifold of Landau levels in graphene. For the case of SU (4) Coulomb interactions plus ultra short range residual interactions, it is possible to construct variational states with integer and/or fractional fillings in the different flavors and compute their energies [70,71]. Determining whether this construction can be extended to generic residual interactions of arbitrary range is an important open question.
We look forward to addressing these and other questions in the near future.
ACKNOWLEDGMENTS
SJD would like to acknowledge the Infosys funding for final year students. He also wants to acknowledge ICTS for its hospitality and kind support towards academic collaboration. AD was supported by the German-Israeli Foundation Grant No. I-1505-303.10/2019, DFG MI 658/10-2, DFG RO 2247/11-1, DFG EG 96/13-1, and CRC 183 (project C01). AD also thanks the Israel planning and budgeting committee (PBC) and the Weizmann Institute of Science, the Dean of Faculty fellowship, and the Koshland Foundation for financial support. S.R. and G.M. would like to thank the VAJRA scheme of SERB, India for its support. R.K.K. was supported in part by NSF DMR-2026947. G.M. would like to acknowledge partial support from the US-Israel Binational Science Foundation (grant no. 2016130). G.M. and R.K.K. are grateful for the wonderful environment at the Aspen Center for Physics (NSF grant PHY-1607611).
Appendix A: Different Phases and Hessian
For charge-neutral graphene, assuming that interactions are ultra-short-range (USR), there are four phases, namely, the ferromagnet (FM), the canted antiferromagnet (CAF), the bond-ordered phase (BO), and the charge density wave (CDW) phase. We call these phases "simple", because they can all be described by a single nontrivial angle which is known analytically in terms of the couplings. As shown in the main text, when one removes the USR restriction on the interactions, other coexistence phases become possible. Among them is a spin-valley entangled phase SVE+, which can also be described by a single angle, and is also "simple".
A primary tool in our investigation of the phase diagram is a study of the stability of a given ground state. Recall that each candidate state is described by filling in two linear combinations of the four spin-valley degenerate states at each guiding center. We reproduce the equations from the main text here for convenience, [50][51][52]55]
|f 1 = cos α 1 2 |n ⊗ |s + e iβ1 sin α 1 2 | − n ⊗ | − s (A1) |f 2 = cos α 2 2 |n ⊗ | − s + e iβ2 sin α 2 2 | − n ⊗ |s (A2)
where n = sin θ p cos φ pêx + sin θ p sin φ pêy + cos θ pêz , and s = sin θ s cos φ sêx + sin θ s sin φ sêy + cos θ As shown in the main text, the U (1) s symmetry allows us to set φ s = 0, and the U (1) v symmetry allows us to set φ p = 0. The SU (2) s symmetry of the interactions forces the HF energy to depend only on β 1 +β 2 . The energy depends on β 1 +β 2 only via cos(β 1 +β 2 ), which appears linearly in the energy. Therefore, we can restrict consideration to the two discrete possibilities β 1 + β 2 = 0, π mod 2π. The HF energy of a given state thus depends only on the four continuously varying angles α 1 , α 2 , θ p , θ s . We now indicate how to determine the values of the four angles for the "simple" states shown in Section II B. We begin by examining the expressions for the order parameters as functions of the four angles: First we focus on the CAF phase. In this phase the BO and CDW order parameters should vanish identically, implying that α 2 = π − α 1 . Examining the FM and CAF order-parameters, we see that we can choose θ s = 0 and θ p = π/2 because cos θ s and cos θ p appear as a overall normalization factors which can be set to one. After imposing these constraints on the angles, the HF energy will depend only upon the angles α 1 , β 1 , β 2 , and can be expressed as
F M = σ z 2 = (cos[α 1 ] −E HF = 1 4 − 8E Z cos(α 1 ) + 2 sin 2 (α 1 )(g xy,F − g z,F ) cos(β 1 + β 2 ) − cos(2α 1 )(3g xy,F + g z,F ) − 5g xy,F − 3g z,F .(A5)
The CAF phase occurs for g xy,F < 0, g z,F > 0. This restricts cos(β 1 + β 2 ) = 1, allowing us to choose β 1 = β 2 = π. The CAF and FM order-parameters are
F M = cos[α 1 ] , CAF = sin[α 1 ].
Here α 1 is the single non-trivial angle which varies through the CAF phase. It has the functional dependence α 1 = cos −1 −E Z 2g xy,F , which we found by minimizing Eq. (A5). Clearly when E Z > 2|g xy,F |, the angle α 1 will be fixed to zero and this corresponds to the FM phase. The parameterization of the FM phase is subsumed in the above. Now we turn our attention to the B/CO phase which generally occurs for E V > 0. Since the CDW and BO phases are restricted versions of the B/CO phase, their parameterizations are subsumed in that of the B/CO phase. In the B/CO phase the FM and CAF order-parameters should vanish identically, allowing us to choose α 1 = α 2 = 0. The B/CO phase is a singlet, which means that the direction of s can be chosen arbitrarily, allowing us to fix θ s = 0. With this choice of angles the BO and CDW order-parameters are BO = sin[θ p ], CDW = cos[θ p ]. Thus, in this phase the angle θ p is the non-trivial angle. To find its functional dependence, we examine the HF energy with the constraints on angles α 1 = α 2 = θ s = 0, which is
E HF = − 2E V cos(θ p ) + 1 2 g V cos(2θ p ) − g xy,F 2 + g xy,H − g z,F 2 + g z,H ,(A6)
where g V = 2g z,H − g z,F − 2g xy,H + g xy,F . Minimizing Eq. (A6) leads to the functional dependence θ p =
cos −1 E V g V
for the B/CO phase. For E V > g V , the angle θ p will be fixed to zero, which corresponds to the pure CDW phase. For E V = 0, the angle θ p = π/2, which corresponds to the BO phase.
Having described how to fix the angles for the simple states, we now turn to the instabilities of these states.
Any HF state must be at least a local extremum. Thus, all the first derivatives of the energy E HF with respect to α 1 , α 2 , θ p , θ s must vanish. To look for instabilities we need to compute the second derivatives of E HF with re-spect to the four angles (the Hessian matrix).
E(α 1 , α 2 , θ p , θ s ) = ∂ 2 EHF ∂ 2 α1 ∂ 2 EHF ∂α1∂α2 ∂ 2 EHF ∂α1∂θp ∂ 2 EHF ∂α1∂θs ∂ 2 EHF ∂α2∂α1 ∂ 2 EHF ∂ 2 α2 ∂ 2 EHF ∂α2∂θp ∂ 2 EHF ∂α2∂θs ∂ 2 EHF ∂θp∂α1 ∂ 2 EHF ∂θp∂α2 ∂ 2 EHF ∂ 2 θp ∂ 2 EHF ∂θp∂θs ∂ 2 EHF ∂θs∂α1 ∂ 2 EHF ∂θs∂α2 ∂ 2 EHF ∂θs∂θp ∂ 2 EHF ∂ 2 θs . (A7)
The eigenvalues of the Hessian determine the stability of the given state. An eigenvalue crossing zero signals an instability of the given state. For "simple" states, one can analytically obtain the Hessian matrix and its eigenvalues. We will use the Hessian eigenvalues of "simple" states to map out the gross features of the phase diagram. Finer details of the phase diagrams are obtained by self-consistent iterative HF.
In the following sections we will present explicit expressions for the Hessian and its eigenvalues and attendant instabilities in the CAF (Appendix B), the B/CO (Appendix C), the FM (Appendix D), the CDW (Appendix E), and the SVE (Appendix F) phases. In Appendix G we present phase diagrams when the Hartree and Fock parts of either/both of the couplings have opposite signs, which may be relevant for very strong Landau-level mixing.
Appendix B: CAF phase
The CAF phase occurs only for g xy,F < 0 and E Z < 2|g xy,F |. It is described by the following values of the angles
α 1 = cos −1 −E Z 2g xy,F = π − α 2 (B1) θ p = π 2 , θ s = 0 (B2) β 1 =β 2 = π (B3)
The energy of the CAF phase is
E CAF = E 2 Z 2g xy,F − g z,F .(B4)
The Hessian matrix for the CAF state has the following block diagonal form:
E CAF = A 3×3 0 3×1 0 1×3 − E 2 Z g xy,F 4×4 ,(B5)−E V 1 − E 2 Z 4g 2 xy,F −E V 1 − E 2 Z 4g 2 xy,F −E V 1 − E 2 Z 4g 2 xy,F (E 2 Z −4g 2 xy,F )(gxy,F −g z,F ) 2g 2 xy,F .(B6)
Thus we find the instability equations of the CAF phase are
g xy,F = − E Z 2 (B7a) E V = √ g xy,F − g z,F E 2 Z g xy,H 2 + g 3 xy,F − g 2 xy,F (2g xy,H + g z,F ) g xy,F .(B7b)
The first instability corresponds to the second-order line between the FM and CAF phases, while the second corresponds to the instability in the B/CAF phase.
undetermined at this point. The other angles are
α 1 =α 2 = θ s = 0 (C1) θ p = cos −1 E V g V ,(C2)
with
g V = 2g z,H − g z,F − 2g xy,H + g xy,F .(C3)
The energy in the BO phase is
E BO = − E 2 V g V − g xy,F + 2g xy,H .(C4)
The Hessian matrix and its eigenvalues do depend on β 1 + β 2 , with the eigenvalues depending on cos[2(β 1 + β 2 )]. The most severe constraint on the region of stability occurs when cos[2(β 1 + β 2 )] = 1. Below, we choose β 1 = β 2 = 0, and obtain the form of the Hessian
E BO = A2×2 2 0 2×2 0 2×2 B 2×2 4×4 (C5) where A = E 2 V (g z,F −g xy,F ) (g xy,F −2g xy,H −g z,F +2g z,H ) 2 + 2E Z + g xy,F − 4g xy,H − g z,F (g xy,F − g z,F ) 1 − E 2 V (g xy,F −2g xy,H −g z,F +2g z,H ) 2 (g xy,F − g z,F ) 1 − E 2 V (g xy,F −2g xy,H −g z,F +2g z,H ) 2 E 2 V (g z,F −g xy,F ) (g xy,F −2g xy,H −g z,F +2g z,H ) 2 − 2E Z + g xy,F − 4g xy,H − g z,F ,(C6)
and
B = − 2E 2 V g xy,F −2g xy,H −g z,F +2g z,H + 2g xy,F − 4g xy,H − 2g z,F + 4g z,H 0 0 0 . (C7)
The instability lines in the BO phase are
g z,H = 1 2 (E V − g xy,F + 2g xy,H + g z,F ) (C8a) E V = (E 2 Z − 2g xy,H (−g xy,F + 2g xy,H + g z,F )) (g xy,F − 2g xy,H − g z,F + 2g z,H ) 2 2g xy,H (g xy,F − g z,F ) .(C8b)
These correspond to the instability of the CDW phase and the instability of the B/CAF phase, respectively.
Appendix D: FM phase
The FM phase is described by the angles
α 1 = 0, α 2 = π (D1a) θ p = π 2 , θ s = 0 (D1b) β 1 =β 2 = π (D1c)
The ground state energy of the FM phase is
E FM = −2(E Z + g xy,F ) − g z,F .(D2)
The form of the hessian matrix of the FM phase is
E FM = A2×2 2 0 2×2 0 2×2 B 2×2 4×4 ,(D3)
where A = 2E Z + 3g xy,F + g z,F g z,F − g xy,F g z,F − g xy,F 2E Z + 3g xy,F + g z,F ,
and
B = 0 0 0 2E Z . (D5)
The instability lines in the FM phase are
g xy,F = − E Z 2 (D6a) g z,F = − E Z − g xy,F .(D6b)
The first is the second-order line between the FM and CAF phases, while the second is the instability towards the CDW phase. Note that this is not the actual position of the dominant instability, which should depend on E V as well. The reason is that E V does not appear in the Hessian matrix for the FM phase. Note that one eigenvalue is always zero, which means one direction in angle space is flat. While we believe that some higher-order derivative of the ground state energy must reveal the instability towards the SVE+ phase, we have not pursued this issue because we can find the instability easily from the SVE+ side.
0 0 0 0 0 ,(F2)
with ξ = cos −1 E Z − E V + g xy,F + g z,H g z,F − g z,H .
Clearly, the phase exists only for the cosine argument having a magnitude smaller than unity. The angles describing this phase are
α 1 =0; α 2 = ξ (F4a) θ p =θ s = 0 (F4b) β 1 =β 2 = π (F4c)
The HF energy is E SV E+ = E 2 V − 2E V (E Z + g xy,F + g z,F ) + E 2 Z + 2E Z (g xy,F − g z,F ) + 4g z,H (E Z + g xy,F ) + g 2 xy,F + 2g z,F (g z,H − g xy,F ) − g 2 z,F 2(g z,F − g z,H ) (F5) The Hessian matrix is
E SV E+ = A 2×2 0 2×2 0 2×2 B 2×2 4×4(F6)
where, A = E V (g xy,F +g z,F )−(E Z +g xy,F +g z,F )(g xy,F −g z,F +2g z,H ) g z,F −g z,H 0 0 (E V −E Z −g xy,F −g z,F )(E V −E Z −g xy,F +g z,F −2g z,H ) g z,F −g z,H
and
B = C 0 0 D (F8)
with C = − (E V −E Z −g xy,F −g z,F )(E V (g xy,F −g xy,H )+E Z (−g xy,F +g xy,H +g z,F −g z,H )−g 2 xy,F +g xy,F g xy,H +g xy,F g z,F −2g xy,F g z,H +g xy,H g z,F ) (g z,F −g z,H ) 2 and D = E Z (E V −E Z −g xy,F +g z,F −2g z,H ) g z,F −g z,H
.
The instabilities of SVE+ phase are given by
g z,F =E V − E Z − g xy,F (F9a) g z,H = 1 2 (E V − E Z − g xy,F + g z,F ) (F9b)
g z,H = (g xy,F − g xy,H )(E V − E Z − g xy,F ) + g z,F (E Z + g xy,F + g xy,H ) E Z + 2g xy,F (F9c) E V = (E Z + g xy,F + g z,F )(g xy,F − g z,F + 2g z,H ) g xy,F + g z,F .
The first corresponds to the instability towards the FM phase, while the second corresponds to the instability towards the CDW phase. The third is the instability towards the B/CAF phase, while the fourth seems to not be relevant.
Appendix G: Hartree and Fock couplings having opposite signs
When Landau-level mixing is strong, it could happen that the Hartree and Fock parts of a given coupling are of opposite sign. We will present sample results for the three cases (i) g z,F /g z,H < 0; g xy,F /g xy,H > 0. (ii) g z,F /g z,H > 0; g xy,F /g xy,H < 0. (iii) g z,F /g z,H < 0; g xy,F /g xy,H < 0.
We have found that if the ratio F/H of a particular coupling is negative, the magnitudes of H and F parts of that particular coupling seem to be irrelevant to the phase diagram. This is in contrast to the case when the ratio F/H is positive, where it matters a great deal whether the ratio is bigger or smaller than unity.
Consider first case (i): g z,F /g z,H < 0; g xy,F /g xy,H > 0. We set g z,F = −g z,H , and vary the ratio g xy,F /g xy,H .
FIG. 25.
Phase diagram for gz,F /gz,H = −1, and gxy,F /gxy,H = 0.75, EZ = 1 and EV = 0.0. The topology of the phase diagram changes; in particular, the CAF region no longer extends to infinity, but is enclosed by the other phases. There is no coexistence anywhere in the phase diagram.
The phase diagram for g xy,F /g xy,H < 1 is shown in Fig. 25. The topology of the phase diagram changes dramatically, but there is no coexistence anywhere. The phase diagram for g xy,F /g xy,H > 1 is shown in Fig. 26), in which there are two coexistence regions. The first one is the B/CAF phase, with BO, FM, CAF, and SVEY order, which we already encountered when g z,F /g z,H > 0. The second phase, which we call the FSVE phase, only occurs when g z,F /g z,H < 0. At E V = 0 this phase shows the coexistence of FM and either SVEX or SVEY order (they are degenerate). Fig. 27 shows the order parameters along a horizontal section of the phase diagram at g z,H = 0.8. 28 shows the changes that occur on this section when E V is turned on. The B/CAF phase changes in a familiar manner, acquiring a CDW order parameter as well as unequal SVEX and SVEY order parameters in addition to the BO and CAF order already present. The FSVE phase splits into two phases, both having some CDW order. The first has FM and unequal SVEX and SVEY order parameters, while the second is the familiar SVE+ phase.
Let us now go on to case (ii), g z,F /g z,H > 0; g xy,F /g xy,H < 0. We choose g xy,F = −1.0 * g xy,H . Fig. 29 shows the phase diagram for g z,F /g z,H < 1. The topology again changes dramatically, but there are no coexistence phases. Fig. 30 shows the phase diagram for g z,F /g z,H > 1. The familiar SVE+ phase interpolates between the FM and CDW phases. Fig. 31 shows the order parameters vs g xy,H along a section of Fig. 30 at g z,H = −1.3. One sees the firstorder transition from the BO to the SVE+ phase, and the subsequent second-order transition into the CDW phase. All other coupling constants are chosen as in Fig. 30. The system is in the BO phase at the extreme left, makes a firstorder transition into the SVE+ phase, and finally, a secondorder transition into the CDW phase at the extreme right.
Finally we turn to case (iii), g z,F /g z,H < 0; g xy,F /g xy,H < 0. The various panels of Fig. 32 show the phase diagrams for different F/H ratios. The common feature is the presence of the FSVE coexistence phase.
FIG. 1 .
1Parameters are EZ = 1.0, gz,F = gz,H = gz, gxy,F = gxy,H = gxy, EV = 0.0,this is the Kharitonov's Ultra short range limit.Here as one can see their is no coexistence phase.
FIG. 2 .
2Instability lines for the BO and CAF phases as a function of gxy,H and gz,F for EZ = 1.0, gz,H = 2.0, gxy,F = −1.0, EV = 0.0. The BO phase is unstable above the black line, while the CAF phase is unstable below the green line. The red line is where the energies of the BO and CAF phases cross. The vertical dashed line represents gxy,H = gxy,F . One can see that coexistence between BO and CAF state only occurs when 0 > gxy,H > gxy,F . The coexistence phase is shaded blue. FIG. 3. Instability lines for the CDW and FM phases as a function of gxy,F and gz,H for EZ = 1.0, EV = 0.0, gz,F = −2.7. The FM phase is unstable below the red line, while the CDW phase is unstable above the blue dotted line. The dashed horizontal line represents gz,H = gz,F . The coexistence between CDW and FM (the SVE+ phase, shaded green) occurs only for 0 > gz,H > gz,F .
FIG. 4 .
4Phase diagram for EZ = 1.0, gz,F = 1.25gz,H , gxy,F = 1.25gxy,H , EV = 0.0. The ratios gµ,F /gµ,H > 1 for both types of couplings.
FIG. 5 .
5Phase diagram for EZ = 1.0, gz,F = 0.75gz,H , gxy,F = 1.25gxy,H , EV = 0.0.The ratios of couplings ensure gxy,F /gxy,H > 1, but gz,F /gz,H < 1. The B/CAF coexistence region (shaded blue) still occurs but there is no coexistence near the CDW/FM phase boundary, which remains first-order.FIG. 6. Phase diagram for EZ = 1.0, gz,F = 1.25gz,H , gxy,F = 0.75gxy,H , EV = 0.0. Now the conditions are no longer met for the B/CAF phase to occur. However, the SVE+ phase (shaded green) does occur near the CDW/FM phase boundary, which is now split into two second-order lines. FIG. 7. Phase diagram for EZ = 1.0, gz,F = 0.75gz,H , gxy,F = 0.75gxy,H , EV = 0.0. There is no coexistence.
FIG. 8 .FIG. 9 .
89Evolution of order parameters vs EZ for gz,H = −1.2, gz,F = 1.25gz,H , gxy,H = −0.5, gxy,F = 1.25gxy,H , EV = 0.0. The ratios of the H and F couplings corresponds toFig. 4. The system is in CDW phase at EZ = 0. As EZ increases there is a second-order transition to a phase where FM and CDW order coexist. This coexistence is mediated by the presence of the SVE+ order parameter, which spontaneously breaks the U (1)s and U (1)v symmetries. For large enough EZ the system goes into FM phase.Let us first consider the case g z,F /g z,H > Evolution of order parameters vs EZ for gz,H = −0.5, gz,F = 1.25gz,H , gxy,H = −1.0, gxy,F = 1.25gxy,H , EV = 0.0.The ratios of the H and F couplings corresponds to the phase diagram ofFig. 4. The system is in BO phase at EZ = 0. As EZ increases the system undergoes a second-order transition to the intermediate B/CAF phase, where BO, CAF, and SVEY order coexist. As EZ increases further, there is another second-order transition to the pure CAF phase, and eventually a third second-order transition to the FM phase at large EZ .
FIG. 11 .
11order parameters vs EZ for gz,H = −0.2, gz,F = 0.75gz,H , gxy,H = −1.2, gxy,F = 1.25gxy,H , EV = 0.0. The ratios of the H and F couplings correspond to the phase diagram of Fig. 5. The system is in the BO phase at EZ = 0. As EZ increases there is a second-order transition to the B/CAF phase (with BO, CAF, FM, and SVEY order). For larger EZ , there is another second-order transition to the pure CAF phase. Finally, the system goes through another second-order transition into the FM phase at very large EZ . Evolution of order parameters vs EZ for gz,H = −1.5, gz,F = 1.25gz,H , gxy,H = −0.5, gxy,F = 0.75gxy,H , EV = 0.0. The ratios of the H and F couplings correspond to the phase diagram of
FIG. 12 .FIG. 13 .
1213Evolution of order parameters at EV = 0 as a function of gxy,H for gz,H = −0.45, gz,F = 1.25gz,H , gxy,F = 1.25gxy,H . The ratios of the H and F couplings correspond to the phase diagram of Fig. 4. It can be seen that the system undergoes a first-order transition from the B/CAF to the F/CDW phase. Evolution of order parameters as a function of gxy,H for gz,H = −0.45, gz,F = 1.25gz,H , gxy,F = 1.25gxy,H for vanishingly small EV = 0.01. The ratios of the H and F couplings correspond to the phase diagram of
FIG. 14 .
14For this figure the coupling ratios of gxy and gz are same as in Fig. 4 and we have considered a small valley Zeeman EV = 0.25.By comparing this figure with Fig. 4 one can see that for this choice of coupling ratios, the presence of finite EV doesn't qualitatively change the nature of the phase diagram. FIG. 15. For this figure the coupling ratios of gxy and gz are same as in Fig. 5 and we have considered a valley Zeeman field EV = 0.5.By comparing this figure with Fig. 5 one can see that for this choice of coupling ratios, the presence of finite EV reduce the coexistence phase area of B/CAF. And here CDW and CAF phases, CDW and FM phases are separated by first order lines. FIG. 16. Phase diagram at EV = 0.5 for the coupling ratios gz,F /gz,H = 1.25, gxy,F /gxy,H = 0.75. Comparing this with the phase diagram for the same ratios at EV = 0 (Fig. 6) one can see that for this choice of coupling ratios, EV has suppressed the B/CAF phase completely and shrunk the SVE+ phase.
Fig.
Fig. 15 shows the phase diagram at E V = 0.5 when the ratios are g z,F = 0.75g z,H ; g xy,F = 1.25g xy,H . As in the case of the previous figure, some of the phase boundaries move, but the topology remains the same as at E V = 0. The same is true for g z,F /g z,H = 1.25; g xy,F /g xy,H = 0.75, shown in Fig. 16 and g µ,F /g µ,H = 0.75 shown in Fig. 17.
FIG. 17 .
17Phase diagram at EV = 0.5 for the coupling ratios gµ,F /gµ,H = 0.75. There is no qualitative change as compared to the corresponding phase diagram at EV = 0(Fig. 7)
FIG. 18 .
18Phase diagram for EZ = 1, and EV = 1.4. The ratios of the Hartree and Fock couplings are gz,F = 0.75gz,H ; gxy,F = 0.75gxy,H as in
FIG. 19 .
19The coupling constants are fixed such that at EV = 0 there is no B/cAF phase. We choose EZ = 1, gz,H = 3.4, gxy,H = −2.5, gxy,F = −2.1. Varying the other two parameters, gz,F , EV , we see that there is an intermediate regime in both in which the B/CAF phase does appear.
Fig.
Fig. 19 shows a slightly different way of looking at the occurrence of the B/CAF phase as E V varies. We have fixed the Zeeman coupling at E Z = 1, and the xy interactions such that g xy,F /g xy,H < 1, implying that coexistence will not occur for E V = 0. The coupling g z,H is also fixed. We show the range of parameters in E V , g z,F where the B/CAF phase occurs. As can be seen, there is an intermediate range of E V and g z,F where the B/CAF phase appears.
FIG. 20 .
20The parameters are EZ = 1.0, gz,F = 0.8, gxy,F = −0.6. One can see from this figure also that only when (|gz,H | > |gz,F |) (for both gz,H , gz,F > 0), their can be a coexistence phase of (FM+CDW) for some range of EV . The vertical dashed line denotes gz,H = gz,F .
Fig
. 20 shows the fate of the SVE+ phase at large E V .
FIG. 22 .FIG. 23 .
2223Phase diagram for EZ = 1.0, EV = 0.0, gz,F = −1.0 * gz,H , gxy,F = 1.25 * gxy,H . The CAF now occurs in a finite region of the parameter space. There are two coexistence phases. The blue shaded region is the B/CAF phase with BO, CAF, and SVEY order. The region shaded brown shows a new type of coexistence between FM and SVEX/SVEY order. Nonzero order parameters along a horizontal cut of the phase diagram ofFig. 22at gz,H = 0.8. All other coupling constants are the same as inFig. 22. One starts in the BO phase at large negative gxy,H . There is a second-order transition into the B/CAF phase with BO, CAF, and SVEY order. Next, there is another second-order transition into the FSVE phase showing the coexistence of FM and SVEX/SVEY order, the two being degenerate. We have chosen to plot SVE-= |τxσx −τyσy| /2, which is continuous across the transition. Finally, for larger gxy,H the system goes into the FM phase.
FIG. 24 .
24Phase diagram for the ratios gz,F /gz,H = 1.25 > 0 and gxy,F /gxy,H = −1 < 0. The other couplings are EZ = 1, EV = 0. While the topology has changed, the SVE+ phase that interpolates between the CDW and FM phases occurs here as well.
sêz are unit vectors indicating the directions of the state on the valley and spin Bloch spheres, respectively. The spinors |n and |s are defined in the standard way
FIG. 26 .
26Phase diagram for gz,F /gz,H = −1, and gxy,F /gxy,H = 1.25, EZ = 1 and EV = 0.0. The blue shaded region is the B/CAF phase, which has BO, FM, CAF, and SVEY order. The brown shaded region is a new coexistence phase with FM and either of SVEX or SVEY order.
FIG. 27 .
27Nonzero order parameters vs gxy,H for gz,H = 0.8. All other coupling constants are the same as inFig. 26.The B/CAF phase shows BO, FM, and SVEY (=SVE-) order, while the FSVE phase shows FM and SVEY order.
Fig.
Fig. 28 shows the changes that occur on this section when E V is turned on. The B/CAF phase changes in a familiar manner, acquiring a CDW order parameter as well as unequal SVEX and SVEY order parameters in addition to the BO and CAF order already present. The FSVE phase splits into two phases, both having some CDW order. The first has FM and unequal SVEX and SVEY order parameters, while the second is the familiar SVE+ phase. Let us now go on to case (ii), g z,F /g z,H > 0; g xy,F /g xy,H < 0. We choose g xy,F = −1.0 * g xy,H .
FIG. 29 .
29For this phase diagram we have considered EZ = 1.0, EV = 0.0, gz,F = 0.75 * gz,H , gxy,F = −1.0 * gxy,H . One can clearly see that as here gxy,F = −1.0 * gxy,H CAF is the stable phase for gxy,H > E Z 2 . FIG. 30. Phase diagram for EZ = 1.0, EV = 0.0, gz,F = 1.25 * gz,H , gxy,F = −1.0 * gxy,H . The topology of the phase diagram is different from the earlier cases, but now the SVE+ coexistence phase appears between the FM and CDW phases. FIG. 28. Order parameters vs gxy,H for various values of EV , with EZ = 1.0, gz,F = −1.0, gz,H = 0.8, gxy,F = 1.25gxy,H . (a) For a tiny EV = 0.05 the FSVE phase splits into two phases. One of them has CDW, FM, and unequal SVEX and SVEY order parameters, while the other is the familiar SVE+ phase with CDW, FM, and SVE+ order.(b) As EV increases to 0.2, the B/CAF and FSVE phases shrink, while the SVE+ phase expands at their expense. (c) At EV = 0.5 the FSVE phase has vanished, leaving behind the B/CAF and SVE+ phases.
FIG. 31 .
31Nonzero order parameters vs gxy,H for gz,H = −1.3.
. The system finally goes into the FM phase at very large EZ .1.10
1.15
1.20
1.25
1.30
1.35
1.40
E Z
0.0
0.2
0.4
0.6
0.8
1.0
FM
CAF
BO
CDW
SVE+
SVE−
FIG. 21. Evolution of order parameters vs EZ for gz,H =
0.9, gz,F = 0.75gz,H , gxy,H = −0.8, gxy,F = 0.75gxy,H , EV =
1.4. The ratios of the H and F couplings correspond to the
phase diagram of Fig. 18. The system is in the B/CO phase
at EZ = 0. As EZ increases there is a second-order transition
to the B/CAF phase (with BO, CDW, CAF, FM, SVEX and
SVEY order). For larger EZ , there is another second-order
transition to the SVE+ phase(with FM and CDW order).
C. Hartree and Fock parts of g z/xy having opposite
signs
− 6g xy,F + 4g xy,H + 2g z,F 1 2 (g xy,F + 2g xy,H + g z,F ) − (g xy,F + 2g xy,H + g z,F ) − − 6g xy,F + 4g xy,H + 2g z,Fwhere
A =
1
4
E 2
Z (g xy,F −g xy,H )
g 2
xy,F
E 2
Z (g xy,F +g xy,H )
4g 2
xy,F
−E V 1 −
E 2
Z
4g 2
xy,F
1
2 E 2
Z (g xy,F +g xy,H )
4g 2
xy,F
1
4
E 2
Z (g xy,F −g xy,H )
g 2
xy,F
Appendix C: BO phaseThe angles β 1 and β 2 never appear in the expression for the HF energy of the BO state. Thus, β 1 and β 2 are Appendix E: CDW phase The CDW phase is described by the anglesThe angles β 1 , β 2 never appear in the expression of the ground state energy, which isThe angles β 1 and β 2 also do not appear in the Hessian matrix, which has the following formNote that one eigenvalue is always zero. This does not indicate instability but rather the fact that one of the four continuously varying angles does not appear in the Hessian. The lines of instability of the CDW phase areThe first is the instability toward the B/CO phase, while the second is the instability towards the SVE+ phase.Appendix F: SVE+ phaseThe SVE+ phase occurs near the boundary of the FM and CDW phases. In addition to the FM and CDW order parameters, this phase also has a nonzero expectation value ofOrdering the rows and columns as K ↑, K ↓, K ↑, K ↓, the projector matrix has the form ∆ SV E+ (ξ) =
= 1.0, EV = 0.0, gz,F = −0.75 * gz,H , gxy,F = −0.75 * gxy,H (b) EZ = 1.0, EV = 0.0, gz,F = −0.75 * gz,H , gxy,F = −1.25 * gxy,H (c) EZ. F Gz, H /Gz, H < 0; Gxy,F /Gxy, Phase diagrams for case (iii). = 1.0, EV = 0.0, gz,F = −1.25 * gz,H , gxy,F = −1.25 * gxy,H (d) EZ = 1.0, EV = 0.0, gz,F = −1.25 * gz,H , gxy,F = −0.75 * gxy,H . The brown shaded region denotes the FSVE phase in all the figuresFIG. 32. Phase diagrams for case (iii) gz,F /gz,H < 0; gxy,F /gxy,H < 0. (a) EZ = 1.0, EV = 0.0, gz,F = −0.75 * gz,H , gxy,F = −0.75 * gxy,H (b) EZ = 1.0, EV = 0.0, gz,F = −0.75 * gz,H , gxy,F = −1.25 * gxy,H (c) EZ = 1.0, EV = 0.0, gz,F = −1.25 * gz,H , gxy,F = −1.25 * gxy,H (d) EZ = 1.0, EV = 0.0, gz,F = −1.25 * gz,H , gxy,F = −0.75 * gxy,H . The brown shaded region denotes the FSVE phase in all the figures.
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|
[] |
[
"Uncovering Extreme Nonlinear Dynamics in Solids Through Time-Domain Field Analysis",
"Uncovering Extreme Nonlinear Dynamics in Solids Through Time-Domain Field Analysis"
] |
[
"P D Keathley \nResearch Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"S V B Jensen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark\n",
"M Yeung \nResearch Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"M R Bionta \nResearch Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"L B Madsen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark\n"
] |
[
"Research Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Department of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark",
"Research Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Research Laboratory of Electronics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Department of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark"
] |
[] |
Time-domain analysis of harmonic fields with sub-cycle resolution is now experimentally viable due to the emergence of sensitive, on-chip techniques for petahertz-scale optical-field sampling. We demonstrate how such a time-domain, field-resolved analysis uncovers the extreme nonlinear electron dynamics responsible for high-harmonic generation within solids. Time-dependent density functional theory was used to simulate harmonic generation from a solid-state band-gap system driven by near-to mid-infrared waveforms. Particular attention was paid to regimes where both intraband and interband emission mechanisms play a critical role in shaping the nonlinear response. We show that a time-domain analysis of the harmonic radiation fields identifies the interplay between intra-and interband dynamical processes underlying the nonlinear light generation. With further analysis, we show that changes to the dominant emission regime can occur after only slight changes to the peak driving intensity and central driving wavelength. Time-domain analysis of harmonic fields also reveals, for the first time, the possibility of rapid changes in the dominant emission mechanism within the temporal window of the driving pulse envelope. Finally, we examine the experimental viability of performing time-domain analysis of harmonic fields with sub-cycle resolution using realistic parameters. arXiv:2210.07899v1 [physics.optics]
|
10.1103/physrevb.107.054302
|
[
"https://export.arxiv.org/pdf/2210.07899v1.pdf"
] | 252,907,675 |
2210.07899
|
8260efe6da4a2c682b2a4b69b8d4951d503d9cc6
|
Uncovering Extreme Nonlinear Dynamics in Solids Through Time-Domain Field Analysis
(Dated: October 17, 2022)
P D Keathley
Research Laboratory of Electronics
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
S V B Jensen
Department of Physics and Astronomy
Aarhus University
DK-8000Aarhus CDenmark
M Yeung
Research Laboratory of Electronics
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
M R Bionta
Research Laboratory of Electronics
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
L B Madsen
Department of Physics and Astronomy
Aarhus University
DK-8000Aarhus CDenmark
Uncovering Extreme Nonlinear Dynamics in Solids Through Time-Domain Field Analysis
(Dated: October 17, 2022)
Time-domain analysis of harmonic fields with sub-cycle resolution is now experimentally viable due to the emergence of sensitive, on-chip techniques for petahertz-scale optical-field sampling. We demonstrate how such a time-domain, field-resolved analysis uncovers the extreme nonlinear electron dynamics responsible for high-harmonic generation within solids. Time-dependent density functional theory was used to simulate harmonic generation from a solid-state band-gap system driven by near-to mid-infrared waveforms. Particular attention was paid to regimes where both intraband and interband emission mechanisms play a critical role in shaping the nonlinear response. We show that a time-domain analysis of the harmonic radiation fields identifies the interplay between intra-and interband dynamical processes underlying the nonlinear light generation. With further analysis, we show that changes to the dominant emission regime can occur after only slight changes to the peak driving intensity and central driving wavelength. Time-domain analysis of harmonic fields also reveals, for the first time, the possibility of rapid changes in the dominant emission mechanism within the temporal window of the driving pulse envelope. Finally, we examine the experimental viability of performing time-domain analysis of harmonic fields with sub-cycle resolution using realistic parameters. arXiv:2210.07899v1 [physics.optics]
I. INTRODUCTION
High-harmonic generation (HHG) has proved a fruitful medium for studying extreme nonlinear interactions between intense pulses of light and matter. Through the study of HHG in atoms and molecules, we have developed a deep understanding of how energy is exchanged between light and electrons on the attosecond timescale [1]. As this work has grown and matured, it has ushered in the rapidly developing fields of attosecond science and technology [2].
In the past decade researchers have begun exploring HHG from solid-state systems [3][4][5]. Compared to atomic and molecular systems, solid-state systems open new avenues for fundamental exploration. Due to the added role of the solid's bandstructure, and the importance of electron-electron interactions, HHG in solids promises to be an interesting tool for probing attosecond to femtosecond dynamical interactions between light and a variety of materials. Indeed, HHG in solids has already been used to reconstruct band structure properties of solids [6][7][8][9], measure the Berry curvature [10], and to track phase transitions in strongly-correlated materials [11]. Unfortunately, along with these new avenues for exploration comes additional complexity in experimental analysis. In particular, both intraband and interband processes result in harmonics with similar non-perturbative intensity scaling laws. These similarities make it difficult, and in some cases impossible, to use spectral information alone to disentangle the physical mechanisms underlying HHG in solids. * These authors contributed equally.
Both theoretical and experimental work [6,[12][13][14][15] shows that field-sensitive information is critical to understanding HHG in solids as the harmonic fields carry unique signatures of the underlying emission mechanisms. While prior work has focused on field-sensitive information gained through phase-and polarizationresolved measurements [6,[13][14][15][16], characterization of the harmonic field waveforms in the time-domain would provide new insights into the underlying dynamics of the generation process. Motivated by advances in experimental methods for petahertz-scale optical-field sampling [17][18][19][20][21][22][23][24], here we investigate: (1) the physics that would be revealed through a time-domain analysis of the high-harmonic fields generated in solids; and (2) the viability of petahertz-scale field sampling techniques for experimentally measuring harmonic fields directly in the time-domain assuming realistic experimental parameters.
In Sec. II we examine how the interplay between the intra-and interband emission channels manifests in timedomain field signatures. Using time-dependent density functional theory (TDDFT), we show that the timedomain structure of the generated harmonic fields naturally reveals the dominant HHG emission mechanisms and their corresponding electron dynamics. We demonstrate how Fourier analysis can be used to break apart and study the temporal structure of both intraband and interband processes in further detail. Our analysis reveals that only moderate changes in peak intensity and central driving wavelength can alter the dominant emission mechanism resulting in dramatic changes to the temporal structure of the harmonic fields. This dramatic change in the temporal structure of the harmonic fields stands in stark contrast to the minimal changes observed in the HHG spectra.
Our time-domain field analysis in Sec. II also shows Normalized high harmonic spectra for each condition simulated: driving wavelength of 2 µm with a peak intensity of 4 × 10 10 W/cm 2 (bottom, blue curve); driving wavelength of 2 µm with a peak intensity of 5 × 10 10 W/cm 2 (central, orange curve); and driving wavelength of 2.3 µm with a peak intensity of 4 × 10 10 W/cm 2 (top, green curve). Each curve is normalized differently for visual clarity. For each case the same system was used as a generation medium, having a bandgap of roughly 3.3 eV.
that, unlike HHG from gasses, with HHG from solids it cannot be assumed, even for driving pulses containing tens of cycles, that the emitted fields are semi-periodic in time. In particular, we observe that under certain conditions the dominant emission mechanism can suddenly switch from intra-to interband over a sub-cycle region of time within the pulse envelope of the driving waveform. This complex emission process and resulting lack of periodic structure in the generated harmonic fields means that phase-resolved techniques requiring a certain level of periodicity are in general inadequate for the study of HHG from solids (e.g. techniques similar to reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) that track the interference phase from harmonic to harmonic [13,25]). We find that measurements having both sub-cycle time resolution and broad spectral coverage are required for a general understanding of solid-state HHG.
Our findings in Sec. II highlight the ripe opportunity for applying recent advances in petahertz-scale field sampling [17] to the characterization and study of HHG in solids. In Sec. III we show that emerging optical field sampling techniques, in particular those that leverage field-driven tunneling ionization [26,27] and nanostructures for field enhancement [24,28], place such measurements within reach. The ex-situ nature of these fieldsampling approaches makes them ideal for examining the origins of extreme nonlinearities in solids across a variety of material platforms. Beyond resolving fundamental scientific questions, experimental characterization of harmonic fields will also be vital to the eventual use of HHG from solids in applications designed to leverage the attosecond-scale temporal structure of the harmonic field waveforms in time.
II. HHG SIMULATIONS AND ANALYSIS
Our simulations of HHG in band gap materials are based on time-dependent density functional theory (TDDFT) [29]. It allows for a self-consistent ab initio investigation into the dynamics of a laser-driven manybody system. It succeeds in accounting for dynamical electron-electron interactions [30,31] and is suitable for describing experimental features [32]. Furthermore, TDDFT has been vital to study the mechanisms of HHG in solids [33] especially as the real-space description allows for introducing dopants [34], topology [35] or finitesystem behavior [36,37]. For technical details on the TDDFT simulations see Appendix A.
We investigate characteristic features of the HHG mechanism using a generic model of a band gap material with a band gap energy of roughly 3.3 eV. We selected two driving wavelengths for comparison: 2 µm (≈ 0.62 eV) and 2.3 µm (≈ 0.54 eV) and apply a pulse with a vector potential A D (t) having a 30-cycle sin 2 envelope function. Here the subscript D denotes the driving pulse. For each wavelength a range of intensities were simulated. We focus on intermediate intensity levels on the order of 10 10 W/cm 2 . We expect equivalent dynamics for a wide range of systems if modifying the laser parameters to accommodate the specific material properties.
We start our analysis by examining the harmonic spectra as shown in Fig. 1. While there are differences in the spectra, it is impossible from this spectral data alone to determine the interplay of underlying emission mechanisms. Both intra-and interband processes result in oddorder harmonic generation with non-perturbative scaling of the harmonic strength with intensity. As we will show, similar-looking spectra resulting from both mechanisms mask dramatic differences in the temporal field structure.
For our time-domain analysis, we start by examining the nonlinear system response from the 2-µm driver having a peak intensity of 4×10 10 W/cm 2 as shown in Fig. 2. The squared electric field from harmonic orders (HO) ≥ 2 are shown in green. These fields were calculated by applying a high-pass filter to remove the fundamental response oscillating at the driving frequency. It is useful for analysis to plot the square of the driving field (blue curve) and vector potential (orange curve) as a guide to the eye. As has been noted in prior work, intra-and interband responses can be distinguished by their phase relationship with the driving field and vector potential [12,14]. Specifically, the interband response is concentrated under the peaks of the squared vector potential (zeros of the squared field), and the intraband response is concentrated under the peaks of the squared driving field (zeros of the vector potential) [12]. Note that for the total nonlinear response in Fig. 2 electric field is consistent with a dominance of intraband processes in the overall high harmonic emission response.
Through Fourier analysis, it is possible to peel apart the emission behavior even further by examining the time-domain squared field response of selected harmonic regions. Such analysis is shown in Figs. 2(b) and (c). In Fig. 2(b) we look at the fields generated only from HO 3-7, and in Fig. 2(c) we look at the fields only from HO 7-11. Note that we chose to break apart the response at HO 7 as this is just above the band gap energy of the material (3.3 eV lies between HO 5 and 6). Consistent with prior theoretical and experimental analysis [12][13][14], harmonics below the bandgap have an intraband character with high-harmonic energy concentrated under the peakes of the squared driving field. This is in stark contrast to harmonics above the bandgap which have an interband character, with high-harmonic energy concentrated under the peaks of the squared vector potential (zeros of the driving field).
To gain insight into the time-domain behavior of the intraband generation process it is useful to consider the semiclassical model for an intraband electron wavepacket [38,39]. This model has been applied to understand various features of HHG in solids in the long wavelength regime [3,10,[40][41][42]. For space and time-inversion symmetric samples, an intraband wavepacket centered at position x and wavevector k is governed byk = −E D (t) andẋ = ∂ε(k)/∂k , within the electric-dipole approximation and where we use atomic units. The driving electric field is given by E D (t) = − ∂A D (t)/∂t and ε(k) denotes the dispersion, which we extract from the TDDFT calculation. We consider a single trajectory that is initiated at the Γ-point with k(t = 0) = 0 and x(t = 0) = 0. Details of the semiclassical modelling are given in Appendix B. The generated field E gen,x (t) of the semiclassical intraband electron can be expressed from the current and fulfills the following equation
E gen,x (t) ∝ dj x (t) dt = d 2 ε(k) dk 2 k(t) × E D (t),(1)
where the curvature of the dispersion generally can be expressed from the effective mass tensor m −1 eff (k) = d 2 ε(k) dk 2 . An outcome of this is two characteristics of the intraband generation mechanism, (i) the appearance of higher-order harmonics is a result of the non-parabolic curvature of the dispersion (ii) the emission of harmonics in the time-domain is proportional to the driving electric field. This is observed in Fig. 3, where the emitted field vanishes when the driving electric field vanishes, and has maxima under the driving electric field maxima.
When comparing the semiclassical response to the intraband dominated below bandgap harmonics of the TDDFT calculation in Fig. 3, one observes that a single trajectory of the semiclassical intraband model is not sufficient to explain the full dynamics of the multi-electron system. If considering the interference of multiple semiclassical trajectories, we expect the spectral features to improve. The phase-relationship with the driving electric field is, however, independent of the initial conditions for the semi-classical trajectories and will thus persist in the interference of multiple trajectories. This characteristic is clearly present in the below bandgap TDDFT dynamics, providing a guideline to decipher signatures of the intraband generation processes in convoluted emission signals.
To study the impact of a change in driving wavelength on the time-domain fields, we repeat the above analysis when keeping the peak intensity fixed at 4 × 10 10 W/cm 2 but with a change of the driving wavelength to 2.3 µm as shown in Fig. 4. Unlike for the 2-µm driver, now the emission is dominated by interband-like emission, with the greatest field concentration under the peaks of the squared vector potential. The analysis of the HO regions generally follows the corresponding response from the 2-µm driver, with intraband-like emission dominating emission from HO 3-7, and interband-like emission dominating from HO 7-11. Referring back to Fig. 1 we note that the overall response is consistent with the shift in strength of the harmonics toward higher HOs for the case of the 2.3-µm driver relative to that of the 2-µm driver. Interestingly, the time-domain study also reveals that the harmonic radiation is less concentrated under the peaks of either the squared driving field or squared vector potential as for the case of the 2-µm driver. Finally, we study the impact of a change to the peak driving intensity by keeping the driving wavelength fixed at 2 µm while increasing the peak intensity to 5 × 10 10 W/cm 2 as shown in Fig. 5. Interestingly, for this case we observe a dramatic shift in the emission response from intraband-like to interband-like within the time window of the driving pulse envelope. In Fig. 5(a) we see that for times before roughly 96 fs the field energy is concentrated mainly under the peaks of the squared driving field, while just after 96 fs the fields rapidly concentrate under the peaks of the squared vector potential with a significant reduction in the duration of each halfcycle burst.
Looking at Figs. 5(b) and (c), we find on closer inspection that this switch in dominance occurs due to the unique character of the emission response within the time window of the pulse envelope from the two harmonic regions. At earlier times, HO 3-7, exhibiting intraband-like behavior, dominate the emission response. The half-cycle emission bursts in this window are concentrated under the peaks of the squared driving field, and experience a sudden drop in intensity near 96 fs. On the other hand, the emission response for HO 7-11 is dominant beyond 96 fs. As with earlier cases, these harmonics have an interband character with the field bursts concentrated under the peaks of the squared vector potential. They are also shorter in duration than for HO 3-7.
Our findings in this section highlight the need for subcycle, field-resolved techniques in order to develop a full understanding of the precise temporal character of the high-harmonic radiation response in solids.
III. SAMPLING HHG FIELDS
In this section we examine the experimental viability of performing time-domain field-analysis of HHG from solids. There are two fundamental constraints that have to be satisfied. First, adequate temporal resolution of the field measurement process is required. The needed resolution to capture up the N th harmonic would be roughly T cyc,D /(2N ) where T cyc,D is the cycle time of the driving field given by T cyc,D = λ D /c, where λ D is the driving wavelength and c the speed of light. For example, assuming a driving wavelength of 2 µm, to capture the timedomain field information up to the 9 th harmonic would require a temporal resolution of roughly 370 as or better. Second, the technique has to have adequate sensitivity to the signal field. Given the driving field strengths for the results in Sec. II were on the order of 0.5 GV m −1 , and assuming a generation efficiency on the order of 10 −6 , we can estimate that a sensitivity to harmonic field strengths on the order of MV m −1 is needed.
Several techniques for sampling optical fields with fewto sub-femtosecond resolution have now been demonstrated [17][18][19][20][21][22][23][24]43]. One category of these techniques has shown particular promise for the purpose of sampling HHG from solids: Tunneling Ionization with a Perturbation for the Time-Domain Observation of an Electric Field (TIPTOE) [26,27]. The TIPTOE technique leverages the sub-femtosecond electron tunneling response from gas-phase and solid-state systems driven by strong optical fields [21,24,[26][27][28]. In the gas phase, TIPTOE has demonstrated the capability of sampling fields with frequencies in excess of 1 PHz [27] (i.e. subfemtosecond temporal resolution). More recently, it has been shown that the sensitivity of TIPTOE can be significantly enhanced using nanostructures [24,28]. In particular, in Ref. [24] sensitivity down to 600 kV m −1 was demonstrated in the near-infrared using gold nanoantennas. Encouraged by these demonstrations, in the following we numerically explore the feasibility of a TIPTOE measurement based on optical tunneling from a metal for the examination of the HHG fields analyzed in Sec. II.
We show a notional schematic for the sampling of HHG from a solid-state system using TIPTOE in Fig. 6(a). An external laser pulse is incident on a beam-splitter (BS) with the transmitted pulse becoming the driving field E D (t) for the HHG, and the reflected field becoming the gate field E G (t) for driving the TIPTOE measurement. The generated harmonics are filtered by the dichroic mirror (DM) in order to isolate HO such that measured HO ≥ 2. The reflected harmonic fields become the signal E sig (t) for the TIPTOE measurement. In other words, E sig (t) is the field resulting from filtering out the fundamental from E gen (t). The delay of E G (t) relative to E sig (t) is referred to as τ , which is controlled by the lower delay stage. For the TIPTOE measurement, one records the oscillations of a cross correlation current I cc (τ ) generated via optical-field-driven tunneling from a gas-phase or solid-state system as a function of the delay τ . The cross correlation current is approximated as:
I cc (τ ) ∝ ∞ −∞ dΓ dE EG(t−τ ) × E sig (t)dt,(2)
where Γ(E) is the tunneling rate as a function of field E [24,26]. Here, we model the optical-field-driven tunneling from a metal surface using a Fowler-Nordheim emission rate as described in Refs. [24,44,45]. We note, however, that other systems, such as gases, semiconductors, or molecules, could be used for generating I cc . Our choice was motivated by the ability of metals to achieve significant field enhancements for improved sensitivity to the signal field [24,28]. We start with a gate field E G (t) at the metal surface with a work function of 5.1 eV. The work function chosen is close to that of gold, and representative of most metals, with typical work functions varying between 4 to 6 eV. The gate field E G (t) was modeled as a Gaussian pulse having a central wavelength of 2 µm, full-width at half-maximum duration of 15 fs, and peak field strength of 7.5 GV m −1 . The tunneling response from the gate field Γ(E G (t)) is plotted in Fig. 6(b). The peak gate field strength was chosen to be in line with prior work examining optical tunneling from solids [24,44,46,47] and to provide adequate temporal resolution. The pulse duration was chosen to ensure a single dominant sub-cycle emission window in time. We realize that the duration of the gate field is shorter than that of the harmonic driving field provided in Sec. II. In practice, this could be accommodated via spectral filtering of a short pulse either as part of the transmission response of the beam-splitter, or as a separate element just after the beam-splitter.
We then take the signal field E sig (t) to be that of the harmonic fields as shown in Fig. 5, with a peak field strength of 1 MV m −1 . We calculate the cross-correlation current I cc (τ ) as defined in Eq. (2). The sampling process imposes its own transfer functioñ
H Det (ω) = F dΓ dE ED(t) *
which is plotted in Fig. 6(c). To extract the sampled signal field information from I cc (τ ) we then calculated it by taking E sampled (t) = F −1 F I cc /H Det (ω) . We note that in practice the electromagnetic response of the emitter structure (for example, that of a nanoantenna or nanostructured surface if they are being used for increased sensitivity) would also have to be accounted for in the calculation and analysis of I cc (τ ). However, such a broad examination of various systems and their electromagnetic responses is beyond the scope of this work. Here we focus our attention on fundamental limitations arising from the tunneling response itself.
In Figs. 6(d-f) we compare the harmonic fields from Fig. 5 to the sampled fields calculated through our solidstate TIPTOE analysis. Note that the sampled fields reconstruct the input fields with a high degree of accuracy. In particular, the sampled field output accurately tracks the sudden transition from intra-to interband-like emission near 96 fs. In Figs. 6(e) and (f) we compare the quality of the sampled data in greater detail using Gabor transformations of the TDDFT and sampled field data. The tunneling response clearly has sufficient temporal resolution to accurately track the harmonic fields with no loss of information up to the 9 th harmonic. However, beyond the 9 th harmonic, |H Det (ω)| drops off suddenly as shown in Fig. 6(c), preventing accurate sampling of higher-frequency fields under the conditions simulated.
However, the bandwidth might be extended by various means. For example, by using a metal having a higher work function to increase the nonlinearity of the tunneling response, or by using a few-cycle gate pulse with a higher central frequency.
IV. SUMMARY AND CONCLUSION
In this work we have demonstrated how analysis of time-domain fields from HHG clearly reveals the relative contributions of intra-and interband emission mechanisms. We used TDDFT models that account for harmonic fields generated from both mechanisms within a driven linear atomic chain of atoms. We exploited the fact that the fields of intraband harmonics concentrate under the peaks of the squared driving field, while interband harmonics concentrate under the peaks of the squared vector potential to study the interplay between these two emission mechanisms as both a function of HO, as well as in time during the evolution of a single driving pulse.
As with prior work [12,13], we found that harmonics above the bandgap were dominated by interband processes while those below the bandgap were dominated by intraband processes but influenced by the coupling of both intra-and interband processes. Using Fourier analysis we were able to isolate and study the temporal response of harmonics from both regimes. A semiclassical analysis of the intraband generation process provided an emission that was temporally confined to the shape of the driving electric field. This characteristic feature of intraband harmonics was compared to the many-electron TDDFT harmonic emission signal, and found useful as a guideline to decipher harmonics originating from intraor interband mechanisms.
We also made observations that were uniquely enabled through the analysis of the harmonic fields in time. In particular, we observe that complex temporal field structures can arise where intra-and interband dominance can evolve from one temporal region to another within the envelope of the driving pulse. Furthermore, this evolution was found to occur over the duration of just a single cycle of the driving field. Given this observation, we conclude that time-averaged, phase-resolved techniques that assume the harmonics are well-described as a periodic pulse train having a smooth envelope (e.g. RABBITT and related techniques) are inadequate for a general analysis of the dynamics of HHG in solids. To properly characterize and analyze rapid shifts in emission dynamics like those observed in Fig. 5 would necessitate measurements having both sub-cycle time resolution and broad spectral coverage.
Finally, we explored the feasibility of emerging opticalfield sampling methods for the experimental time-domain analysis of high harmonic fields generated in solids. Given the need for temporal resolution of hundreds of attoseconds and ∼MV m −1 field sensitivity, we found the TIPTOE method [26,27] using a metallic tunneling medium [24,28] to be a compelling choice. Our calculations show that under realistic experimental conditions, TIPTOE from metals offers sufficient temporal resolution and spectral coverage to enable time-domain harmonic field analysis.
Our findings strongly support the pursuit of timedomain field measurements for the analysis of HHG in solids. Methods for sub-cycle field analysis are rapidly advancing [17]. We are confident that these technologies will become instrumental for uncovering fundamental aspects of ultrafast strong-field processes in condensed matter directly in the time domain.
VI. DATA AND CODE AVAILABILITY
The simulation data and code used for analysis and plot generation for this work can be found at https: //github.com/qnngroup/manu-uncovering-extremenonlinear-dynamics-in-solids-through-time-doma in-field-analysis.git.
Appendix A: TDDFT simulations
We represent the electrons by auxiliary noninteracting time-dependent Kohn-Sham (KS) orbitals ϕ σ,i (x, t) with spin σ = {↑, ↓}, which are obtained by imaginary time propagation. When driven by an electromagnetic field, described by the vector potential A D (t), the electron dynamics can be captured along the laser polarization direction with a one-dimensional model. Thus, in atomic units, the electron-nuclear interaction is described with a static softened, = 2.25, Coulomb potential v ion (x) = −
ϕ σ,i (x, t) = −∂ 2 x /2 − iA D (t) ∂ x +ṽ KS [n σ ] (x, t) ϕ σ,i (x, t)
using the Crank-Nicolson method with a predictorcorrector step and an absorbing boundary potential [48,49]. − [6n σ (x, t)/ π] 1/3 . These include the dynamic electron-electron interactions through the density n (x, t) = σ=↑,↓ n σ (x, t) and spin density n σ (x, t) = Nσ−1 i=0 |ϕ σ,i (x, t)| 2 . We consider a chargeand spin-neutral system such that the number of electrons with a given spin is N ↑,↓ = ZN/2. The spatial grid contains 21250 grid points of size 0.1. Macroscopic propagation effects are not accounted for, as these are suppressed for thin targets [50,51], which can be produced for experiments [41]. Similarly, we disregard nondipole effects [42] as these are negligible if describing the generated field as E gen,x (t) ∝ dj x (t)/dt , with j x (t) being the current measured along the polarization axis of the driving field. A temporal grid with 95000 grid points and a step of 0.1 ensured convergence.
tions.
We use the symmetric dispersion in the vicinity of the Γ-point of the previously obtained TDDFT bandstructure, and expand as a Fourier series ε(k) = [1 + n c n cos(nka)] /4a 2 , with coefficients c n = −37.4496δ n,1 +2.9303δ n,2 −3.6618δ n,3 +0.9685δ n,4 − 1.1009δ n,5 +0.5821δ n,6 −0.4201δ n,7 +0.3697δ n, 8 . We propagate a single-electron wavepacket trajectory of the semiclassical equations, initiated at the Γ-point, k(t = 0) = 0 at x(t = 0) = 0, as is commonly done in the literature [7,10,15]. Explicitly for such material parameters Eq. (1) can be further written as The first term, arising from the curvature of the dispersion, gives rise to the harmonic contributions in the generated electric field, and is dependent on the initial conditions of the semiclassical wavepacket. The second term is independent of initial conditions, and confines the emitted field to the temporal shape of the driving electric field.
FIG. 1. Normalized high harmonic spectra for each condition simulated: driving wavelength of 2 µm with a peak intensity of 4 × 10 10 W/cm 2 (bottom, blue curve); driving wavelength of 2 µm with a peak intensity of 5 × 10 10 W/cm 2 (central, orange curve); and driving wavelength of 2.3 µm with a peak intensity of 4 × 10 10 W/cm 2 (top, green curve). Each curve is normalized differently for visual clarity. For each case the same system was used as a generation medium, having a bandgap of roughly 3.3 eV.
FIG. 2 .
2(a), the concentration of harmonic energy aligns with the peaks of the squared Time-domain fields generated by various harmonic contributions for a driving wavelength of 2 µm and peak intensity of 4 × 10 10 W/cm 2 . (a) All HO ≥ 2, (b) HO from 3-7, and (c) HO from 7-11.
semiclassical and TDDFT model. All driving conditions the same as forFig. 2. (a) Time-domain fields for all generated HO ≥ 2 of the semiclassical model (green), and HO 3-7 of the TDDFT model. Note that just as for the lower HO of the TDDFT model, the semiclassical model predicts HH fields that are concentrated under peaks of the square of the driving electric field. (b) A comparison of the spectra of the two models. The semiclassical model is dominated by the lower-order harmonics and drops off rapidly with increasing HO. On the otherhand, due to the presence of interband processes and more rich accounting of electron-electron interactions, an extended plateau forms in the TDDFT response. The TDDFT harmonics have rich structuring and a broader bandwidth in comparison with the harmonics predicted by the simple semiclassical response.
FIG. 4 .
4Time-domain fields generated by various harmonic contributions for a driving wavelength of 2.3 µm and peak intensity 4 × 10 10 W/cm 2 . (a) All HO ≥ 2, (b) HO from 3-7, and (c) HO from 7-11.
FIG. 5 .
5Time-domain fields generated by various harmonic contributions for a driving wavelength of 2 µm and peak intensity of 5 × 10 10 W/cm 2 . (a) All HO ≥ 2. Note the sudden transition from intraband-to interband-type radiation at t ≈ 95 fs. (b) HO from 3-7, and (c) HO from 7-11.
FIG. 6 .
6Simulating harmonic field sampling using solid-state TIPTOE. (a) Notional schematic of field-sampling experiment using TIPTOE. Abbreviations: BS -beam-splitter; CM -curved mirror; DM -drilled mirror; HCM -holey curved mirror. (b) Calculation of Γ(EG(t)) representing the instantaneous electron tunneling response from the metal surface. (c) Magnitude of the sampling transfer function |HDet(ω)|. (d-f) Comparison of direct TDDFT field output and the result of our simulation of the sampling response. (d) TDDFT output |Esig(t)| 2 (shaded green) and sampled field response |E sampled (t)| 2 (solid red) as a function of time. (e-f) Gabor transform spectrograms of the TDDFT fields (e) and sampled fields (f) showing results up to the 9 th harmonic.
is based in part upon work supported by the Air Force Office of Scientific Research under award number FA9550-18-1-0436. M. Y. Acknowledges support from the National Science Foundation Graduate Research Fellowship Program, Grant No. 1745302. L. B. M. and S. V. B. J. acknowledges support from the Danish Council for Independent Research (GrantNo.9040-00001B). S. V. B. J. further acknowledges support from the Danish Ministry of Higher Education and Science.
[(x − x i ) 2 + ] −1/2 formed by a string of N = 100 ions of nuclear charge Z = 4 placed at x i = [i − (N − 1) /2] a and separated by lattice constant a = 5.3. The KS-orbitals are propagated through the time-dependent KS equation i∂ t
The time-dependent KS potential v KS [{n σ }] (x, t) = v ion (x)+v H [n] (x, t)+v xc [{n σ }] (x, t) contains the Hartree potential v H [n] (x, t) = dx n (x , t) [(x − x ) 2 + ] −1/2 ,and the local spin-density exchange-correlation potential v xc [{n σ }] (x, t)
Appendix B: Semiclassical simulationsThe intraband generation mechanism is clarified by a numerical simulation of the semiclassical equa-
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|
[] |
[
"Semantic Segmentation with Active Semi-Supervised Representation Learning",
"Semantic Segmentation with Active Semi-Supervised Representation Learning"
] |
[
"Aneesh Rangnekar [email protected] \nRochester Institute of Technology Rochester\nNYUSA\n",
"Christopher Kanan [email protected] \nUniversity of Rochester Rochester\nNYUSA\n",
"Matthew Hoffman \nRochester Institute of Technology Rochester\nNYUSA\n"
] |
[
"Rochester Institute of Technology Rochester\nNYUSA",
"University of Rochester Rochester\nNYUSA",
"Rochester Institute of Technology Rochester\nNYUSA"
] |
[] |
Obtaining human per-pixel labels for semantic segmentation is incredibly laborious, often making labeled dataset construction prohibitively expensive. Here, we endeavor to overcome this problem with a novel algorithm that combines semi-supervised and active learning, resulting in the ability to train an effective semantic segmentation algorithm with significantly lesser labeled data. To do this, we extend the prior state-of-the-art S4AL algorithm by replacing its mean teacher approach for semi-supervised learning with a self-training approach that improves learning with noisy labels. We further boost the neural network's ability to query useful data by adding a contrastive learning head, which leads to better understanding of the objects in the scene, and hence, better queries for active learning. We evaluate our method on CamVid and CityScapes datasets, the de-facto standards for active learning for semantic segmentation. We achieve more than 95% of the network's performance on CamVid and CityScapes datasets, utilizing only 12.1% and 15.1% of the labeled data, respectively. We also benchmark our method across existing stand-alone semi-supervised learning methods on the CityScapes dataset and achieve superior performance without any bells or whistles.Getting labels for supervised learning problems is challenging, especially for semantic segmentation where these labels are needed on a per-pixel level. The most widely used methods for reducing the need for labels are semi-supervised learning (SSL) and pool-based active learning (AL). Semi-supervised learning proposes strategies to use the unlabeled dataset alongside labeled samples, typically by maintaining an exponential moving average of the network to predict pseudo labels[18,24,37], and pool-based active learning queries the most informative samples within the unlabeled data pool, in terms of pixels, regions, or entire images, to add to the labeling pool via a predetermined scoring mechanism[4,13,29,31]. Recently, both EquAL [13] and S4AL [29] have pioneered combining active and semisupervised learning for semantic segmentation, resulting in significant gains compared to using either in isolation. However, these methods are complicated -the former relies on enforcing consistency in the predictions without accounting for their correctness, and the latter
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10.48550/arxiv.2210.08403
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[
"https://export.arxiv.org/pdf/2210.08403v1.pdf"
] | 252,918,457 |
2210.08403
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d8cca7187aa4fb12330106a5e794e13b263c534c
|
Semantic Segmentation with Active Semi-Supervised Representation Learning
Aneesh Rangnekar [email protected]
Rochester Institute of Technology Rochester
NYUSA
Christopher Kanan [email protected]
University of Rochester Rochester
NYUSA
Matthew Hoffman
Rochester Institute of Technology Rochester
NYUSA
Semantic Segmentation with Active Semi-Supervised Representation Learning
RANGNEKAR, KANAN, HOFFMAN: S4AL+ 1
Obtaining human per-pixel labels for semantic segmentation is incredibly laborious, often making labeled dataset construction prohibitively expensive. Here, we endeavor to overcome this problem with a novel algorithm that combines semi-supervised and active learning, resulting in the ability to train an effective semantic segmentation algorithm with significantly lesser labeled data. To do this, we extend the prior state-of-the-art S4AL algorithm by replacing its mean teacher approach for semi-supervised learning with a self-training approach that improves learning with noisy labels. We further boost the neural network's ability to query useful data by adding a contrastive learning head, which leads to better understanding of the objects in the scene, and hence, better queries for active learning. We evaluate our method on CamVid and CityScapes datasets, the de-facto standards for active learning for semantic segmentation. We achieve more than 95% of the network's performance on CamVid and CityScapes datasets, utilizing only 12.1% and 15.1% of the labeled data, respectively. We also benchmark our method across existing stand-alone semi-supervised learning methods on the CityScapes dataset and achieve superior performance without any bells or whistles.Getting labels for supervised learning problems is challenging, especially for semantic segmentation where these labels are needed on a per-pixel level. The most widely used methods for reducing the need for labels are semi-supervised learning (SSL) and pool-based active learning (AL). Semi-supervised learning proposes strategies to use the unlabeled dataset alongside labeled samples, typically by maintaining an exponential moving average of the network to predict pseudo labels[18,24,37], and pool-based active learning queries the most informative samples within the unlabeled data pool, in terms of pixels, regions, or entire images, to add to the labeling pool via a predetermined scoring mechanism[4,13,29,31]. Recently, both EquAL [13] and S4AL [29] have pioneered combining active and semisupervised learning for semantic segmentation, resulting in significant gains compared to using either in isolation. However, these methods are complicated -the former relies on enforcing consistency in the predictions without accounting for their correctness, and the latter
Introduction
relies on complex data augmentation schemes for convergence. Here we propose S4AL+, which greatly simplifies the S4AL method by replacing its mean-teacher framework with self-training, thereby eliminating the use of complex data augmentations [38,40].
Most methods for active learning rely on cross-entropy loss for training their networks [13,29,39]. While cross-entropy is a widely used loss function in semantic segmentation, it operates on a individual pixels of the semantic map and fails to take into consideration possible cues of similarity and differences in areas throughout the entire image or set of images to strengthen its learning further (Fig. 1). The computer vision community has developed several algorithms to address this problem, notably Conditional Random Fields (CRF) [5], Affinity Nets [23], Region Mutual Information Loss (RMI) [43] and Contrastive Learning [22,42]. We hypothesize that boosting cross-entropy performance with any of these methods can improve the quality of pseudo labels, thereby improving the quality of queried labeling instances.
In this work, we show how active learning for semantic segmentation can be improved with a straightforward technique: at each active learning cycle, we produce pseudo labels for the dataset with the self-training framework [38] and leverage contrastive representation learning to improve the boundaries between different classes [22]. Our combination of self-training with contrastive representation learning in S4AL+ enables querying superior samples for active learning, which results in more efficient learning. While S4AL+ was designed for active learning, we also demonstrate state-of-the-art performance on semisupervised learning.
Background
Pool-based Active Learning for Semantic Segmentation is a technique for ranking unlabeled data points on their importance based on machine learning methods including, but not limited to, consistency [13,15,21,32], diversity [10,33] and feature level learning [4,29,31,39]. These frameworks can be further classified based on their approach for querying the data (image [10,15,33,39], region [4,13,21,29,32] or pixel [31]). Notably, EquAL [13], Minimax [10], and S4AL [29] proposed including the unlabeled dataset as part of the training protocol to achieve better image-level (Minimax) or region-level (EquAL, S4AL) results. Unlike these approaches, we show that adding a simple contrastive embed-ding head and self-training to boost the networks' quality of pseudo labels is sufficient to attain state-of-the-art results.
Semi-Supervised Learning for Semantic Segmentation commonly learns representations using a teacher-student framework [12,14,18,25,26,38,46], and recently, combining teacher-student with contrastive embeddings [1,22,37,45]. Most teacher-student frameworks use the mean teacher framework as their foundation [35], wherein a slowly updated version (teacher) of the continuously updated model (student) predicts reliable pseudo labels on the unlabeled data points for joint training. The exceptions to this line of approach are ST++ [40] and USRN [14] which use self-training based learning [38]. These algorithms complement our work, which differs in two aspects of learning. Unlike ST++, we use pixellevel (local) metrics instead of image-level (global) metrics for determining the quality of pseudo labels, and unlike USRN, we use contrastive representation learning to improve the quality of pseudo labels instead of clustering to improve the data distribution.
Supervised Contrastive Learning stems from adapting the one-to-many contrastive loss to the many-to-many pipeline for mapping similarities and differences between data points in the representation space [20]. Multiple frameworks aim to boost the pixel-wise predictions in semantic segmentation by incorporating contrastive learning with memory banks (prototypes) for supervised learning [36,42,44] and semi-supervised learning [1,37,45]. Our work employs a much lighter technique, ReCo [22], which continuously updates the contrastive embeddings per data iteration and achieves a comparative performance to its memory banks counterparts.
Our Active Learning Framework & Algorithm: S4AL+
Active learning methods work with an initial labeled pool (D L , x l ) and a relatively larger unlabeled pool (D U , x u ), intending to query instances from D U for labeling. This task is performed at every active learning cycle (AL T ), whose stopping criteria (AL C ) is either preset or determined as the number of cycles required to achieve comparable performance to supervised learning (D L , D U −→ D L ).
We build S4AL+ on the foundations of S4AL [29], self-training [38] and contrastive representation learning [22]. Following S4AL, we query the instances x u at every AL T at region level using the average entropy of the network prediction's in the region. At the end of every active learning cycle, we query for labeled data from D U by ranking the regions according to network's entropy and we account for pseudo labels by considering every pixel whose prediction confidence is above a fixed threshold. In this manner, the network has richer labels within the unlabeled pool at the beginning of the next active learning cycle and does not need to iteratively predict pseudo labels every data iteration [29,35].
To ensure high quality within the queried regions, we investigate allowing the network to 'peep' into the data within the unlabeled pool by using semi-supervised learning. We achieve this with two simple adjustments: 1) We train the network with the self-training framework [35,38], and 2) We add a contrastive embedding head to strengthen class-wise representations [22] (Fig. 2). We briefly discuss the motivations and functionality of both our modifications, before discussing our findings.
DeepLabv3+
Predictions Representations Figure 2: Schematic showing how our modified DeepLabv3+ framework employs contrastive representation learning to improve the performance of pseudo labels per learning cycle, which improves the performance of our active learning framework. Note how road keys group together (red) and ensure distance from sidewalk (orange), and similarly, bus keys (yellow) separate from train (green). The confidence in the prediction map highlights the keys for smart selection during contrastive learning (black arrow).
Self-training
Motivation. Mean-teacher based approaches for semi-supervised learning, wherein an exponentially moving average of the student network is used to predict pseudo labels on D U , requires complex data augmentations (variants of CutMix [41] or ClassMix [26]) and regularizations to function successfully on semantic segmentation. Multiple research studies in the areas of semi-supervised learning and active learning ( [18,29,37]) use modified data augmentation pipelines to ensure robust pseudo labels. On the contrary, self-training uses a relatively simple data augmentation scheme, while managing to achieve comparable performance.
The self-training pipeline for semi-supervised semantic segmentation can be summarized in the following steps: where N S , N T , wa, sa, CE indicate student network, teacher network, weak data augmentation, strong data augmentation, and cross-entropy loss respectively. Self-training also relies on injecting noise into the network for augmentations, which we achieve with Dropout for MobileNetV2 and Stochastic Depth for the ResNet variants in our framework [16,19,30,34]. This pipeline can be retrofitted for active learning with two minor adjustments. First, we modify Step 5 to repeat itself every AL T , until we reach the stopping criteria cycle AL C .
1. Train the network N T on D L wa with CE,
We also query x u within D U for labeling during every iteration of Step 2. As a result, we obtain the relatively easier set of labels in D U via pseudo labeling and the harder set of labels in D U via manual annotations at each active learning cycle (AL T ). This results in a larger portion of information available for the network during the next training phase of the active learning cycle (AL T ).
Contrastive Representation Learning
Motivation. We face two challenges when directly applying the framework for semantic segmentation: 1) for classification, the entire feature representation for the network corresponds to a single image category, and 2) cross-entropy loss works on a per-pixel basis and ignores the possibility of learning from other pixels in the image (and the batch) to improve its understanding. Hence, we explore the field of supervised contrastive learning and augment our network with a contrastive embedding head for learning meaningful representations (2). Specifically, we adapt the ReCo loss [22] into our training pipeline due to its simplicity and lower memory overhead. We modify our network for semantic segmentation by adding another decoder head that accounts for learning representations and is trained via the ReCo loss (Fig. 2). Assuming that r indicates the representations obtained in parallel to the network's class-wise predictions, and C is the total number of classes present in the dataset during the mini-batch under training, Eqn. 1 summarizes the fundamentals of the ReCo loss [22] as follows:
L ReCo = ∑ c∈C ∑ r q ∼R c q − log exp(r q · r c,+ k /τ) exp(r q · r c,+ k /τ) + ∑ r − k ∼R c k exp(r q · r − k /τ) ,(1)
where • R c q represents the positive set containing all representations whose ground truth labels belong to class c,
• r c,+ k represents the positive anchor, which is the mean of r q per class c, • r − k are all representations within R c k , which is the negative key set containing all representations whose ground truth label is not class c, and • τ is the scalar temperature control parameter.
Eqn. 1 accounts for all positive and negative keys present within a mini-batch whose size can grow exponentially based on the number of images and their resolutions. ReCO alleviates this problem by constructing a pair-wise similarity graph among the mean representations of every class (r c,+ k , c ∈ C) and actively sampling for a meaningful set of negative keys r − k per R c q , whose quantity is a tunable hyper-parameter (RC K ). This ensures that semantically different classes, for example, Bus and Vegetation, are rarely sampled as a pair for learning, while semantically similar classes like Bus and Train are seen more often.
Similarly, r q are chosen based on the threshold RC δ s for every corresponding prediction confidence from the network and a tunable hyper-parameter (RC Q ) which determines the quantity, similar to RC K . RC δ s ensures that positive keys belong to areas within the prediction map which can benefit from representation learning. In this manner, ReCo ensures that only the most informative sets (both positive and negative) are trained while maintaining a minimal computation overhead. We replace the CE loss function in Steps 1 and 3 with a combination of CE and ReCo loss to further boost the network's learning potential.
Summary: S4AL+ vs. Other Methods
Our approach differs from S4AL, and other approaches in the semi-supervised and active learning literature in three ways: 1) We do not use the mean teacher pipeline to predict pseudo labels at every mini-batch iteration, in this manner, we save on the training time and also potential bias from the labels imbalance being carried forward every step, 2) We do not rely on heavy augmentations like CutMix [41] or ClassMix [26], and 3) We use contrastive learning with self-training to utilize the unlabeled pool to its maximum potential (Fig. 1).
Experiments and Results
Before presenting our results, we first introduce the metrics and datasets used in experiments.
Datasets, metrics and training protocol
Datasets. We conduct experiments on the CamVid, and CityScapes datasets [3,8]. CamVid has a default resolution of 720 × 960, with 11 classes of interest, and CityScapes has a default resolution of 1024 × 2048, with 11 classes of interest. For active learning, we downsample CamVid and CityScapes to 360 × 480 and 688 × 688, respectively, for training and evaluation, following [29,39]. For semi-supervised learning, we crop images within the CityScapes dataset at 768 × 768 for training, and maintain the original resolution for evaluation, following [18,37].
Metrics. We evaluate our algorithm's performance for active learning by measuring the proportion of additional labeled data required to reach more than 95% performance in terms of mean IoU (mIoU) on fully supervised learning. We evaluate our algorithm's performance for semi-supervised learning by measuring the mIoU achieved using a specific % of fully supervised data as the labeled data.
Network Architectures. We use MobileNetV2 [30] with a modified stride of 16 as the backbone for all experiments on active learning (following [29,39]), except in comparison to EquAL [13], wherein we use ResNet-50 with a modified stride of 8. We also use ResNet-50 [16] and ResNet-101-DeepStem [17], with modified stride of 8, for experiments on semi-supervised learning as per the protocol in the comparative state-of-the-art. We use the DeepLabv3+ framework for semantic segmentation throughout all our experiments [6].
Network Optimization. We begin all our experiments with an initial learning rate of 1×10 −2 and use the "poly" learning schedule to gradually decrease the learning rate, similar to [6,29,39]. We use SGD for optimization and a weight decay of 0.0001 in all our experiments. We train the networks for 100 epochs on the CamVid and CityScapes datasets for active learning, and 240 epochs on the CityScapes dataset for semi-supervised learning. For all data augmentation schemes, we use random application of image resizing and horizontal flipping as the weak augmentations set (wa), and Gaussian blur, Color Jitter, and CutOut [9] as the strong augmentations set (sa). We do not create our own splits, but instead use the data splits provided in individual benchmarks to minimize disparity in our findings. Tables 1 and S1 contain the details of our training procedures on active learning and semi-supervised learning respectively. Figure 3: We demonstrate that our framework, S4AL+, works the best for active learning on the CamVid (a) and CityScapes (b) datasets. Both graphs show the mIoU improvements relative to the amount of labeled data utilized in comparison with the previous state-of-the-art frameworks, DEAL [39] and S4AL [29], and the supervised learning performance.
Results: Active Learning
We conduct experiments for the full active learning system with the CamVid and CityScapes datasets with MobileNetV2 encoder and DeepLabv3+ framework. In both cases, we begin with 10% of the data as our labeled dataset and gradually query for additional labeled data per active learning cycle ( [29,39]). We query four regions per image of 30 × 30 and 43 × 43 for each active learning cycle on CamVid and CityScapes, respectively, following [29]. We also assign pseudo labels to the pixels that meet our prediction confidence threshold of 0.7 in every image in the unlabeled data pool. Fig. 3 shows our results on the CamVid and CityScapes dataset. Our approach achieves over 95% of the full dataset performance with only 12.1% of the labeled pixel data for the CamVid dataset, and 15% labeled pixel data on the CityScapes dataset. We observe a significantly larger boost in the performance during the final active learning cycles, as we allow the network to train for a longer time, motivated by [2,29]. Fig. 4 shows the difference in outputs from S4AL and S4AL+, highlighting the advantage of using contrastive learning to learn better object features. We refer the reader to Tables S3 and S4 in the supplemental for a discussion on the class-wise performance on both datasets.
We further compare our system to EquAL [13], another region-based selection method, that uses a ResNet-50 encoder with DeepLab-v3+ [6,16]. Starting with 8% labeled data and a budget of 12% labeled data on the CamVid dataset, our approach achieved an mIOU of 66.4 on CamVid, compared to 63.4 from EquAL and 65.3 from S4AL. When starting with 3.5% data on the CityScapes dataset, our approach achieves a mIoU of 67.5 with only 10% labeled data, compared to EquAL's 67.4 with 12% data and S4AL's 66.7 with 10% data, thus demonstrating superior performance of our method on multiple training processes.
Image
Ground Truth S4AL S4AL+ Figure 4: Comparison of our framework to the previous state-of-the-art S4AL. The areas of interest are highlighted and refer to regions wherein our approach supersedes S4AL in terms of correct pixel prediction.
Results: Semi-supervised Semantic Segmentation
We compare our approach without the active learning pipeline to semi-supervised learning methods. Specifically, we do not query for any additional data from the unlabeled pool of images and only use ground truth for those images that fall in the training set. We train the network on the labeled training data, assign the pseudo labels based on the prediction confidence, and retrain the network jointly with labeled and (pseudo-labeled) unlabeled data twice, before reporting our results.
For semi-supervised Learning, we vary the percentage (%) of labeled images within training data. Fig. 5(a) shows the performance of our framework in terms of previous stateof-the-art approaches. We compare S4AL+ with Mean Teacher [35], Cross Consistency Training (CCT) [27], Cross Pseudo Supervision (CPS) [7], Adaptive Equalized Learning (AEL) [18] and Unreliable Pseudo Labels (U 2 PL) [37] which use the ResNet-101-DeepStem encoder. We observe a slight drop in the performance on the 1/16 labeled data scenario, but perform at-par or better than other approaches in all other scenarios. Our method is also superior in terms of efficiency as 1) it does not use multiple sets of encoders and decoders (like CCT and CPS), 2) does not complex memory bank mechanisms to actively sample for long tail distribution classes (MT, AEL and U 2 PL), and 3) does not actively obtain pseudo labels at every data iteration (MT, AEL and U 2 PL). We refer the readers to Table S2 in the supplemental for in-depth results on the CityScapes dataset. Table 1: Summary of all Active learning experiments on CamVid and CityScapes datasets. LD and ULD indicate the percentage of labeled data and unlabeled data respectively. We observe that using 1:2 labeled to unlabeled image ratio, and training for longer epochs, on the final active learning cycle, result in superior gains with respect to mIoU on both datasets.
Discussion
Our results enable us to answer two questions:
1. Does self-training help the case of active learning? Yes. As a stand-alone adjustment, self-training boosts the initial mIoU by 1.8 points on the CamVid dataset and 1.2 points on the CityScapes dataset when using the MobileNetV2 encoder.
2. Does representation learning help the case of active learning? Yes again. We observe a boost of 2 points in the initial mIoU for the CamVid dataset and 0.8 points on the CityScapes dataset with the MobileNetV2 encoder. We also verify this for semisupervised learning in Fig. 5(b) and Table S2, wherein we use variants of a stronger ResNet encoder.
Conclusion
We propose a solution for improving active learning for semantic segmentation motivated by the desire to eliminate dependency on data augmentation schemes that involve randomness to ensure robust pseudo labels. We achieve this by formulating active learning as a stepwise semi-supervised learning problem, using self-training compared to the popular teacherstudent-based framework. We enrich this framework with a representation learning head that ensures the network can maximize its learning potential. While both our components seem reasonably straightforward, we emphasize that our goal was not to create something entirely new but to develop a solution from the existing literature. S4AL+ is a strong, but elegant, method for active and semi-supervised learning, with potential for improvement (Fig. 6), as it achieves state-of-the-art results in both domains. Figure 5: (a) Comparison of our framework in terms of mIoU to the previous state-of-theart approaches for semi-supervised learning on CityScapes dataset. (b) Comparison of improvements gained by using contrastive learning (ReCo [22]) for semi-supervised learning on CityScapes dataset.
Image
Ground Truth S4AL S4AL+ S4AL+ (SSL_R101D) Figure 6: Knowledge distillation? S4AL+ still makes incorrect predictions under cases of extra-similar object appearances. However, we show that this is a limitation of the encoder, MobileNetV2, by comparing its outputs post active learning to the stand-alone output of the same approach trained under the semi-supervised learning framework (with a similar amount of initial labeled data) and the significantly more powerful ResNet-101-DeepStem encoder.
Supplemental Material
We will release all code towards reproducing the results in this paper post publication. We provide a full summary of our experiments on semi-supervised learning in Table S1. Table S2 reports the results on CityScapes for semi-supervised learning as a comparative analysis to previous state-of-the-art and class-wise performance on the IoU metric. Tables S3 and S4 report the class-wise IoU performance on CamVid and CityScapes datasets for Active learning respectively. We train all our models on 4 Nvidia V100 GPUs with 16GB, and use Pytorch [28] with Pytorch Lightning [11] for all our experiments. We also use 16-bit precision as needed, especially for training ResNet101-DeepStem encoder based DeepLabv3+ models. Table S2: Semi-supervised learning results on the CityScapes dataset in terms of labeled images (the first column indicates the ratio of data usage, and the second column indicates the number of labeled samples). All numbers are reported on mIoU metric using a ResNet-101-DeepStem as the encoder with DeepLab-v3+, except those marked with *, which indicates a ResNet-50 encoder to ensure fairness [40]. SupOnly stands for supervised training on the portion of labeled data, and SupOnly + Rep indicates a representation learning head is added to the decoder along with the prediction head.
Figure 1 :
1An example from the CityScapes dataset for Active learning: (a) Shows the image, (b) The ground truth labels, (c) The predictions from a network trained only with crossentropy loss, and (d) The predictions from a network trained with contrastive representation learning. The red box highlights the improved predictions in for the train and the nearby vegetation over standard training with cross-entropy loss.
2 .
2Obtain pseudo labels for D U from N T with a determined threshold, 3. Jointly train D L wa and D U sa on N S with CE, 4. N S −→ N T , 5. Repeat 2 to 4 till convergence,
Table S3 :
S3Active learning results on the CamVid dataset compared to the existing state of art, highlighting the total amount of data required. The results are reported on the IoU metric, and the numbers within the brackets () indicate the total labeled data usage.Sky
Building
Column
Pole
Road
Side
walk
Tree
Sign
Fence Car
Pedestrian Bicyclist mIoU
S4AL [29] (13.8%)
91.51 82.30
15.54
91.52 72.05 73.59 40.11 31.17 78.99 48.65
49.55
61.36
S4AL+ (12.1%)
91.08 81.99
16.62
93.88 79.21 75.65 37.39 34.27 80.31 51.20
47.56
62.65
Supervised (100.0%) 91.48 81.36
20.84
93.90 79.30 75.17 40.58 33.76 79.97 49.83
52.86
63.55
Table S4 :
S4Active learning results on the CityScapes dataset compared to the existing state of art, highlighting the total amount of data required. The results are reported on the IoU metric, and the numbers within the brackets () indicate the total labeled data usage.Road
Side
walk
Building Wall
Fence Pole
Traffic
Light
Traffic
Sign
Vegetation Terrain
DEAL [39] (40%)
95.89 71.69
87.09
45.61 44.94 38.29 36.51
55.47
87.53
56.90
S4AL [29] (16.3%) 96.53 80.16
86.63
48.15 46.41 35.10 43.78
58.07
88.91
61.50
S4AL+ (15.1%)
97.38 79.07
87.87
43.69 46.47 32.24 41.94
57.35
89.14
57.89
Supervised (100%) 97.58 80.55
88.43
51.22 47.61 35.19 42.19
56.79
89.41
60.22
Sky
Pedestrian Rider
Car
Truck Bus
Train
Motorcycle Bicycle
mIoU
DEAL [39] (40%)
91.78 64.25
39.77
88.11 56.87 64.46 50.39
38.92
56.59
61.64
S4AL [29] (16.3%) 92.08 65.14
39.75
90.52 64.15 65.31 41.53
46.18
58.97
63.62
S4AL+ (15.1%)
92.47 67.23
43.44
90.34 49.47 66.37 54.72
47.34
64.01
63.60
Supervised (100%) 92.69 65.12
37.32
90.67 66.24 71.84 63.84
42.35
61.84
65.30
AcknowledgementsThis work was supported by the Dynamic Data Driven Applications Systems Program, Air Force Office of Scientific Research under Grant FA9550-19-1-0021. We gratefully acknowledge the support of NVIDIA Corporation with the donations of the Titan X and Titan Xp Pascal GPUs used for this research, and DataCrunch.io for the usage of Nvidia V100 GPUs.Table S1: Summary of all Semi-supervised learning experiments on the CityScapes datasets. LD indicates the total number of labeled data images used following previous state-of-the-art protocols. We use a 1:1.5 labeled to unlabeled image ratio as that is the maximum we are able to fit on the GPU, and use contrastive representation learning for all experiments. We report the results on the mIoU metric.
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[] |
[
"FINITE GROUPS ACTING ON SEVERI-BRAUER SURFACES",
"FINITE GROUPS ACTING ON SEVERI-BRAUER SURFACES"
] |
[
"Constantin Shramov "
] |
[] |
[] |
We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero.
|
10.1007/s40879-020-00448-3
|
[
"https://arxiv.org/pdf/2006.14671v1.pdf"
] | 220,128,060 |
2006.14671
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bb11dd91aadfb85a5c9e7a4b1164e1585a580254
|
FINITE GROUPS ACTING ON SEVERI-BRAUER SURFACES
25 Jun 2020
Constantin Shramov
FINITE GROUPS ACTING ON SEVERI-BRAUER SURFACES
25 Jun 2020
We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero.
Introduction
A Severi-Brauer surface over a field K is a surface that becomes isomorphic to P 2 over the algebraic closure of K. Concerning birational automorphisms of non-trivial Severi-Brauer surfaces (i.e. those that are not isomorphic to P 2 over the base field) the following is known.
Theorem 1.1 ([Sh20a, Theorem 1.2(ii)]). Let S be a non-trivial Severi-Brauer surface over a field K of characteristic zero. Then every finite group acting by birational automorphisms of S is either abelian, or contains a normal abelian subgroup of index 3.
The following result was proved in [Sh20b]. We denote by µ n the cyclic group of order n.
Theorem 1.2. Let p ≡ 1 (mod 3) be a prime number. Consider the non-trivial homomorphism µ 3 → Aut(µ p ), and let G p ∼ = µ p ⋊ µ 3 be the corresponding semidirect product. Then there exists a number field K and a non-trivial Severi-Brauer surface S over K such that the group Aut(S) contains the group G p .
Due to the work of I. Dolgachev and V. Iskovskikh [DI09a], a classification of finite groups acting by birational selfmaps of P 2 over an algebraically closed field of characteristic zero is available. The purpose of this paper is to give a classification of finite groups acting by birational selfmaps on non-trivial Severi-Brauer surfaces over fields of characteristic zero. We will show that possible finite subgroups of automorphism groups of non-trivial Severi-Brauer surfaces are cyclic groups of certain orders, and semidirect product of cyclic groups with µ 3 of certain kind; possible finite subgroups of birational automorphism groups of non-trivial Severi-Brauer surfaces are finite subgroups of automorphism groups together with the group µ 3 3 .
1
To formulate a precise assertion, we need to introduce additional notation. Given a positive integer n, denote by µ * n the multiplicative group of residues modulo n that are coprime to n. Let G be a semidirect product of µ n and µ 3 corresponding to a homomorphism χ : µ 3 → µ * n ∼ = Aut(µ n ).
If n = p r i i , where p i are pairwise different primes congruent to 1 modulo 3, we say that G is balanced if the compositions of χ with all natural projections µ * n → µ * p r i i are injective. An example of such a semidirect product is the group µ 3 ∼ = µ 1 ⋊ µ 3 itself. We point out that a balanced semidirect product of a given order may be not unique. The reader is referred to §2 for details. Theorem 1.3. Let n be a positive integer, and let G be a finite group. The following assertions hold.
(i) Let n be a positive integer. There exists a field K of characteristic zero and a non-trivial Severi-Brauer surface over K such that the group Bir(S) of birational automorphisms of S contains an element of order n if and only if n = 3 r p r i i , where p i are primes congruent to 1 modulo 3, and r 1 (in other words, n is not divisible by 9 and not divisible by primes congruent to 2 modulo 3). In this case the group Aut(S) contains an element of order n as well.
(ii) Let G be a finite group. There exists a field K of characteristic zero and a nontrivial Severi-Brauer surface over K such that the group Aut(S) contains a subgroup isomorphic to G if and only if there is a positive integer n divisible only by primes congruent to 1 modulo 3 such that G is isomorphic either to µ n , or to µ 3n , or to a balanced semidirect product µ n ⋊µ 3 , or to the direct product µ 3 ×(µ n ⋊µ 3 ), where the semidirect product is balanced. (iii) Let G be a finite group. There exists a field K of characteristic zero and a nontrivial Severi-Brauer surface over K such that the group Bir(S) contains a subgroup isomorphic to G if and only if either G is one of the groups listed in assertion (ii), or G ∼ = µ 3 3 . One can reformulate Theorem 1.3 as a description of the sets of all finite orders of elements and all finite subgroups of the groups Aut(S) and Bir(S), where S varies in the set S of all non-trivial Severi-Brauer surfaces over all fields of characteristic zero. Namely, denote AO = {n | n ∈ Z 1 , and there exists S ∈ S such that Aut(S) contains an element of order n} , BO = {n | n ∈ Z 1 , and there exists S ∈ S such that Bir(S) contains an element of order n} , AG = {G | |G| < ∞, and there exists S ∈ S such that Aut(S) contains a subgroup isomorphic to G} , BG = {G | |G| < ∞, and there exists S ∈ S such that Bir(S) contains a subgroup isomorphic to G} . Also, denote by N the set of all positive integers of the form n = p r i i , where p i are primes congruent to 1 modulo 3, and denote B = {G | G ∼ = µ n ⋊ µ 3 is a balanced semidirect product for some n ∈ N } .
2
In these notation, Theorem 1.3 claims that
AO = BO = N ∪ {3n | n ∈ N } , AG = {µ n | n ∈ N } ∪ {µ 3n | n ∈ N } ∪ B ∪ {µ 3 × G | G ∈ B} ,
BG = AG ∪ µ 3 3 . As we will see in Remark 4.2, every Severi-Brauer surface over a field of characteristic zero has an automorphism of order 3. However, over some fields birational automorphism groups of non-trivial Severi-Brauer surfaces may be rather poor in finite subgroups. For instance, in the case of the field Q of rational numbers, Theorem 1.3 implies the following simpler classification.
Corollary 1.4. Let S be a non-trivial Severi-Brauer surface over Q, and let G be a finite subgroup of Bir(S). Then G is a subgroup of µ 3 3 . In particular, G is abelian. Using the terminology of [Pop14, Definition 1], we see from Corollary 1.4 that the Jordan constant of the birational automorphism group of every non-trivial Severi-Brauer surface over Q equals 1, while for a non-trivial Severi-Brauer surface over an arbitrary field of characteristic zero it does not exceed 3 by Theorem 1.1. Note that the Jordan constant of the birational automorphism group of the projective plane over Q equals 120 by [Yas17, Theorem 1.11]. I do not know if the groups µ 2 3 or µ 3 3 can actually be embedded into birational automorphism groups of some non-trivial Severi-Brauer surfaces over Q.
It would be interesting to obtain a complete classification of finite groups acting on each non-trivial Severi-Brauer surface depending on the arithmetic properties of the base field and the correspondning central simple algebra, similarly to what was done for conics in [GA13], cf. [Bea10]. Also, it would be interesting to find out if there exists an (infinite) extension K ′ of Q and a non-trivial Severi-Brauer surface S ′ over K ′ such that the group Aut(S ′ ) contains all the groups listed in Theorem 1.3(ii), and Bir(S ′ ) contains all the groups listed in Theorem 1.3(iii).
The plan of the paper is as follows. In §2 we collect some auxiliary assertions concerning semidirect products of cyclic groups. In §3 we gather auxiliary facts concerning Severi-Brauer surfaces and Galois theory. In §4 we construct examples of finite groups acting on non-trivial Severi-Brauer surfaces. In §5 we describe the possible finite orders of automorphisms of non-trivial Severi-Brauer surfaces. In §6 we classify finite subgroups of automorphism groups of non-trivial Severi-Brauer surfaces and prove Theorem 1.3. In §7 we prove Corollary 1.4.
Notation. Given a field K, we denote byK its algebraic closure. For a variety X defined over K, we denote by XK its extension of scalars toK.
Acknowledgements. I am grateful to S. Gorchinskiy, L. Rybnikov, A. Trepalin, and V. Vologodsky for useful discussions. Special thanks go to D. Osipov who spotted a gap in a preliminary version of the paper and suggested several improvements of the exposition. I was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project "5-100", and by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".
Semidirect products
In this section we collect some elementary assertions concerning semidirect products of certain finite cyclic groups.
Let n be a positive integer. Write
n = k i=1 p r i i ,
where p i are pairwise different prime numbers. There is a canonical isomorphism
µ n ∼ = k i=1 µ n (p i ), where µ n (p i ) ∼ = µ p r i i is the p i -Sylow subgroup of µ n .
For the multiplicative group µ * n of residues modulo n that are coprime to n, one has a canonical isomorphism
µ * n ∼ = k i=1 µ n (p i ) * , where µ n (p i ) * ∼ = µ * p r i i . If p i > 2, we have µ n (p i ) * ∼ = µ p r i −1 i (p i −1) , see for instance [Lan65, § II.2].
Recall that the group Aut(µ n ) is canonically identified with µ * n . Thus, for every homomorphism χ : µ 3 → µ * n we can construct a semidirect product G ∼ = µ n ⋊ µ 3 . Vice versa, every semidirect product corresponds to some homomorphism χ.
Definition 2.1. Suppose that n is divisible only by primes congruent to 1 modulo 3, and let χ : µ 3 → µ * n be a homomorphism. We say that χ is balanced if its composition with each of the projections µ * n → µ n (p i ) * is an embedding. We say that a semidirect product G of µ n and µ 3 corresponding to the homomorphism χ is balanced if χ is balanced.
Example 2.2. Let n = 1. Then the set of prime divisors of n is empty, and thus Definition 2.1 does not impose any condition on a semidirect product. Thus, the semidirect product µ 3 ∼ = µ 1 ⋊ µ 3 is balanced.
Example 2.3. Let p be a prime number congruent to 1 modulo 3. Then the group G p described in Theorem 1.2 is a balanced semidirect product of µ p and µ 3 . Moreover, this group is the unique balanced semidirect product of µ p and µ 3 .
Example 2.4. Let p 1 and p 2 be distinct prime numbers congruent to 1 modulo 3, and let G p 1 ∼ = µ p 1 ⋊ µ 3 be the balanced semidirect product. Then the group G p 1 × µ p 2 is isomorphic to a semidirect product µ p 1 p 2 ⋊ µ 3 which is not balanced. Indeed, the image of the corresponding homomorphism
χ : µ 3 → µ * p 1 p 2 ∼ = µ p 1 −1 × µ p 2 −1
is contained in the subgroup µ p 1 p 2 (p 1 ) * ∼ = µ p 1 −1 of µ * p 1 p 2 . We point out that for a given positive integer n a balanced semidirect product of order 3n may be not unique.
Example 2.5. Let p 1 and p 2 be distinct prime numbers congruent to 1 modulo 3, and let n = p 1 p 2 . Then µ * n ∼ = µ p 1 −1 × µ p 2 −1 , and each of the (cyclic) groups µ n (p i ) * ∼ = µ p i −1 contains a unique element δ i of order 3. Set d 1 = δ 1 δ 2 and d 2 = δ 1 δ −1 2 . Let σ be a generator of µ 3 , and let χ 1 , χ 2 : µ 3 → µ * n be the homomorphisms that send σ to d 1 and d 2 , respectively. Construct the groups G 1 and G 2 as semidirect products of µ n and µ 3 corresponding to the homomorphisms χ 1 and χ 2 , respectively. We claim that G 1 is not isomorphic to G 2 . Indeed, while it is easy to construct an automorphism ζ of µ * n such that χ 2 = ζ • χ 1 , there does not exist an inner automorphism ζ of µ * n like this (that is, the automorphism ζ cannot be obtained as a conjugation with an element of µ * n ), because the group µ * n is abelian. Therefore, the groups G 1 and G 2 are not isomorphic to each other by [BE99,Theorem 5.1].
The next result shows that all non-balanced semidirect products have a structure similar to what we see in Example 2.4.
Lemma 2.6. Let n be a positive integer divisible only by primes congruent to 1 modulo 3, let χ : µ 3 → µ * n be a homomorphism, and let G be a semidirect product of µ n and µ 3 corresponding to χ. Then χ is balanced if and only if the only element of µ n that commutes with the subgroup µ 3 is trivial. Moreover, if χ is not balanced, then there exists an isomorphism
(2.7) G ∼ = (µ n 1 ⋊ µ 3 ) × µ n 2 ,
where n 1 n 2 = n and n 2 > 1.
Proof. Suppose that χ is not balanced. Then there exists a prime divisor p of n such that the composition of χ with the projection µ * n → µ n (p) * is trivial. Hence, the group µ 3 commutes with the p-Sylow subgroup µ n (p) ∼ = p r of µ n . Therefore, we get a required isomorphism G ∼ = (µ n 1 ⋊ µ 3 ) × µ n 2 , where n 1 = n p r and n 2 = p r . Conversely, suppose that χ is balanced. Denote by d ∈ µ * n the image of a generator of µ 3 with respect to χ, so that d 3 ≡ 1 (mod n). Assume that there exists a non-trivial element t ∈ µ n that commutes with µ 3 . Considering µ n as the group of residues modulo n, we can write td ≡ t (mod n). Since µ n is isomorphic to a direct product of its Sylow subgroups, we conclude that there is a prime divisor p of n such that the image t p of t in the p-Sylow subgroup µ n (p) ∼ = µ p r of µ n is non-trivial. We have
(2.8) t p d ≡ t p (mod p r ).
Also, we know that d ≡ 1 (mod p r ) because the homomorphism χ is balanced, and d 3 ≡ 1 (mod p r ). We see from (2.8) that d ≡ 1 (mod p). Let s be the maximal integer such that d ≡ 1 (mod p s ); then 1 s < r. Write
d ≡ cp s + 1 (mod p s+1 ),
where c ≡ 0 (mod p). One has
1 ≡ d 3 ≡ 3cp s + 1 (mod p s+1 ).
Since the numbers 3 and c are not divisible by p, this gives a contradiction.
Remark 2.9. In the notation of Lemma 2.6, one can choose n 1 and n 2 so that the semidirect product µ n 1 ⋊ µ 3 in (2.7) is balanced.
We conclude this section by two simple general observations concerning semidirect products.
Lemma 2.10. Let n and m be positive integers, and let d be an integer such that d m ≡ 1 (mod n). Let H be a group generated by an element x of order n and an element y of order m subject to relation
(2.11) xy = yx d .
Suppose that the cyclic subgroups generated by x and y have trivial intersection.
Then H ∼ = µ n ⋊ µ m , and the semidirect product structure corresponds to the homomorphism µ m → µ * n sending a generator of µ m to d ∈ µ * n . Proof. We see from relation (2.11) that every element of H can be written in the form y r x s for some positive integers r and s. Therefore, H is a quotient of the semidirect prod-uctH ∼ = µ n ⋊ µ m corresponding to the homomorphism µ m → µ * n that sends a generator of µ m to d ∈ µ * n . On the other hand, if for some r 1 , r 2 , s 1 , and s 2 one has y r 1 x s 1 = y r 2 x s 2 , then y r 1 −r 2 = x s 2 −s 1 . Since the cyclic subgroups generated by x and y have trivial intersection, this implies that r 1 ≡ r 2 (mod m) and s 1 ≡ s 2 (mod n). Hence |H| = mn = |H|, which means that H ∼ =H.
Lemma 2.12. Let G be a finite group that does not contain elements of order 9, and let H be a normal subgroup of index 3 in G.
Then G ∼ = H ⋊ µ 3 .
Proof. Choose a preimage g of a generator of the group G/H ∼ = µ 3 with respect to the projection ψ : G → G/H. The order of g is divisible by 3, but is not divisible by 9. Thus it equals 3m for some positive integer m not divisible by 3. Hence the element g ′ = g m has order 3, and ψ(g ′ ) generates G/H. This means that the subgroup F ∼ = µ 3 in G generated by g ′ has trivial intersection with H, and thus G is a semidirect product of H and F .
We remark that one cannot drop the assumption about the absence of elements of order 9 in Lemma 2.12. Indeed, the group µ 9 contains a normal subgroup µ 3 of index 3, but it is not isomorphic to a (semi)direct product of two copies of µ 3 . Note also that under a stronger assumption that H does not contain elements of order 3 the assertion of Lemma 2.12 becomes a particular case of a much more general Schur-Zassenhaus theorem, see e.g. [Isa08, Theorem 3.5].
Central simple algebras
In this section we gather auxiliary facts concerning Severi-Brauer surfaces, and also some assertions from Galois theory.
We refer the reader to [Art82] and [Kol16] for the basic theory of Severi-Brauer surfaces and higher-dimensional Severi-Brauer varieties. The foundational fact of this theory is that Severi-Brauer varieties of dimension m over a field K are in one-to-one correspondence with central simple algebras of dimension (m + 1) 2 over K, with P m corresponding to the algebra of (m + 1) × (m + 1)-matrices. A Severi-Brauer variety over K is trivial if and only if it has a K-point.
Given an algebra A, we denote by A * the multiplicative group of its invertible elements. The following characterization of the automorphism group of a Severi-Brauer variety is well-known, see Theorem E on page 266 of [Châ44], or [ShV18, Lemma 4.1].
Lemma 3.1. Let X be a Severi-Brauer variety over a field K corresponding to a central simple algebra A. Then Aut(X) ∼ = A * (K)/K * .
Let L/K be a Galois extension with Galois group isomorphic to µ 3 . Choose an element a ∈ K * and a generator σ of Gal (L/K). Then one can associate with σ and a a cyclic algebra A, which is a central simple algebra over K, see [GS06,§2.5] or [GSh18, Exercise 3.1.6]. Explicitly, A is generated over K by L and an element α subject to relations α 3 = a and λα = ασ(λ), λ ∈ L.
Lemma 3.2 (see e.g. [GSh18, Exercise 3.1.6(i)]). Suppose that in the above notation the element a is not contained in the image of the Galois norm N L/K . Then A is not isomorphic to a matrix algebra.
It appears that the above construction is responsible for all possible central simple algebras of dimension 9 over an arbitrary field K (with a trivial exception of the situation when K has no cyclic extensions of degree 3, in which case there are no central simple algebras of dimension 9 over K apart from the matrix algebra anyway). Namely, the following result can be found in [Wed21] or [GS06, Chapter 7, Exercise 9].
Lemma 3.3. Let A be a central simple algebra of dimension 9 over a field K. Suppose that A is not isomorphic to the matrix algebra. Then A is a cyclic algebra constructed from some cyclic Galois extension L/K and some element a ∈ K * . Lemma 3.4. Let A be a central simple algebra of dimension q 2 over a field K, where q is a prime number. Then A is either the matrix algebra, or a division algebra.
We will also need the following facts from Galois theory.
Theorem 3.5 ([CF67, § III.1]). Let n be a positive integer, and let ξ be a primitive n-th root of unity. Then Q(ξ)/Q is a Galois extension, and Gal (Q(ξ)/Q) ∼ = µ * n . Lemma 3.6. Let ξ 1 and ξ 2 be primitive roots of unity of degrees n 1 and n 2 , respectively. Suppose that n 1 and n 2 are coprime, and set M = Q(ξ 1 ). Then M(ξ 2 )/M is a Galois extension, and
Gal (M(ξ 2 )/M) ∼ = µ * n 2 . Proof. Note that Q(ξ 1 ) ∩ Q(ξ 2 ) = Q, see for instance [Lan65, § VIII.3]
. Also, we know from Theorem 3.5 that Q(ξ 2 )/Q is a Galois extension, and Lemma 3.7. Let L/K be a finite Galois extension of number fields. Then the Galois norm map N L/K : L * → K * is not surjective.
Gal (Q(ξ 2 )/Q) ∼ = µ * n 2 .
Construction of subgroups
In this section we construct examples of finite groups acting on non-trivial Severi-Brauer surfaces. It is easy to produce a non-trivial Severi-Brauer surface with an action of the group µ 2 3 . Example 4.1 (see [Sh20a,Example 4.7]). Let K be a field of characteristic different from 3 that contains a non-trivial cubic root of unity ω. Let a and b be elements of K such that b is not a cube in K, and a is not contained in the image of the Galois norm for the field extension L/K, where
L = K 3 √ b .
If K is a number field, then an element b like this always exists for obvious reasons, and an element a exists by Lemma 3.7. Consider the algebra A over K generated by variables u and v subject to relations
u 3 = a, v 3 = b, uv = ωvu.
Then A is a central simple algebra, and moreover a division algebra; actually, one can construct it as a cyclic algebra over K associated with (a generator of the Galois group of) the cyclic extension L/K and the element a. Thus A corresponds to a non-trivial Severi-Brauer surface S. One can see that u and v generate a non-abelian subgroup of order 27 and exponent 3 in A * (sometimes called the Heisenberg group). Their images in A * /K * generate a group isomorphic to µ 2 3 , which is embedded as a subgroup into Aut(S) by Lemma 3.1.
Remark 4.2. Let S be a Severi-Brauer surface over a field K. Then S has an automorphism of order 3. Indeed, if S ∼ = P 2 , then this is obvious. Otherwise it follows from Lemma 3.3 that S corresponds to some cyclic algebra. In the notation of §3, the algebra A contains the element α ∈ K such that α 3 ∈ K. Thus α defines an automorphism of S of order 3 by Lemma 3.1.
The proof of the next result is similar to that of Theorem 1.2, see [Sh20b].
Lemma 4.3. Let n > 1 be an integer divisible only by primes congruent to 1 modulo 3. Let G ∼ = µ n ⋊ µ 3 be a balanced semidirect product. Then there exists a number field K and a non-trivial Severi-Brauer surface S over K such that the group Aut(S) contains G.
Proof. Let ξ denote a primitive root of unity of degree n, and let L = Q(ξ). Then L/Q is a Galois extension with the Galois group isomorphic to µ * n by Theorem 3.5. Let χ : µ 3 → µ * n be the homomorphism giving rise to the semidirect product structure on G. Consider χ as a homomorphism χ : µ 3 → Gal (L/Q) , and let K ⊂ L be the subfield of invariants of the subgroup χ(µ 3 ) ⊂ Gal (L/Q). Then L/K is a Galois extension with the Galois group Gal (L/K) isomorphic to µ 3 . Let σ be a generator of Gal (L/K). Since χ is balanced, we conclude from Lemma 2.6 that for all 1 m < n one has σ(ξ m ) = ξ m ; thus ξ m ∈ K. Explicitly, σ sends ξ ∈ L to ξ d , where d ∈ µ * n is the image of a generator of µ 3 under the embedding χ : µ 3 → µ * n . In particular, one has d 3 ≡ 1 (mod n).
By Lemma 3.7 there exists an element a ∈ K such that a is not contained in the image of the Galois norm of the field extension L/K. Consider the cyclic algebra A over K associated with σ and a. We know that A is a central simple algebra. However, A is not a matrix algebra by Lemma 3.2. Since the dimension of A over K equals 9, we conclude that A is a division algebra by Lemma 3.4. The algebra A contains the root of unity ξ ∈ L and an element α such that α 3 = a and relation (4.4) ξα = αξ d holds.
Denote byα andξ the images in the quotient group A * /K * of the elements α and ξ, respectively, and denote byĜ the subgroup of A * /K * generated byα andξ. Since α 3 ∈ K and α ∈ K, the order ofα in A * /K * equals 3 as well. Similarly, one has ξ n = 1 ∈ K, and ξ m ∈ K for all 1 m < n. Thus the order ofξ in A * /K * equals n. Since n is coprime to 3, the cyclic groups generated byξ andα have trivial intersection. Furthermore, relation (4.4) impliesξα =αξ d . HenceĜ is the semidirect product of µ n and µ 3 corresponding to χ by Lemma 2.6, so thatĜ ∼ = G.
We see that the elementsξ andα generate a subgroup isomorphic to G in A * /K * . On the other hand, for a Severi-Brauer surface S corresponding to A one has Aut(S) ∼ = A * /K * by Lemma 3.1.
Remark 4.5. If in the notation of the proof of Lemma 4.3 we choose χ : µ 3 → Gal (L/Q) to be a non-balanced homomorphism, and set K to be the field of invariants of the group χ(µ 3 ), then for some 1 m < n the element ξ m is invariant under the action of the Galois group Gal (L/K). This means that ξ m is an element of K. Therefore, while the subgroup of A * generated by ξ and α has a quotient isomorphic to the non-balanced semidirect product µ n ⋊µ 3 corresponding to χ, its image in Aut(S) ∼ = A * /K is isomorphic to a (balanced) semidirect product µ n ′ ⋊ µ 3 for some n ′ < n, cf. Remark 2.9.
The following result is proved by a blend of constructions used in Example 4.1 and Lemma 4.3.
Lemma 4.6. Let n 1 be an integer divisible only by primes congruent to 1 modulo 3. Let G ′ ∼ = µ n ⋊ µ 3 be a balanced semidirect product, and let G ∼ = µ 3 × G ′ . Then there exists a number field K and a non-trivial Severi-Brauer surface S over K such that the group Aut(S) contains G.
Proof. Let ω and ξ denote primitive roots of unity of degrees 3 and n, respectively; let τ be a cubic root of 2. Set M = Q(ω) and L = M(ξ, τ ). We observe that
M(τ ) ∩ M(ξ) = M.χ ′ : µ 3 → µ * n
giving rise to the semidirect product structure on G ′ . Let K ⊂ L be the subfield of invariants of the subgroup χ(µ 3 ) ⊂ Gal (L/M). Then L/K is a Galois extension such that Gal (L/K) ∼ = µ 3 . A generator σ of Gal (L/K) sends ξ ∈ L to ξ d , where d is some integer such that d 3 ≡ 1 (mod n), and sends τ ∈ L to ωτ . Since the homomorphism χ ′ is balanced, we conclude from Lemma 2.6 that for all 1 m < n one has σ(ξ m ) = ξ m ; thus ξ m ∈ K.
By Lemma 3.7 there exists an element a ∈ K such that a is not contained in the image of the Galois norm of the field extension L/K. Consider the cyclic algebra A over K associated with σ and a. We know that A is a central simple algebra, and A is not a matrix algebra by Lemma 3.2. Thus A is a division algebra by Lemma 3.4. The algebra A contains the n-th root of unity ξ and an element α such that α 3 = a and relation (4.4). holds. Furthermore, A contains τ , and (4.7) τ α = ωατ.
Denote byα,ξ, andτ the images in the quotient group A * /K * of the elements α, ξ, and τ , respectively. Denote byĜ ′ the subgroup of A * /K * generated byα andξ, and byĜ the subgroup generated byα,ξ, andτ . The order ofα in A * /K * equals 3. The order ofξ in A * /K * equals n, and the cyclic groups generated byξ andα have trivial intersection. Using relation (4.4), we see thatĜ ′ is the semidirect product of µ n and µ 3 corresponding to the homomorphism χ ′ , and soĜ ′ ∼ = G ′ (cf. the proof of Lemma 4.3).
Since ω ∈ K, we see from relation (4.7) that the groupĜ ′′ generated byα andτ is isomorphic to µ 2 3 (cf. Example 4.1). In particular,τ has order 3 and commutes withα. Since τ and ξ are elements of the field L ⊂ A, they commute with each other, and thusτ commutes withξ. Henceτ commutes with the groupĜ ′ generated byα andξ. Since n is coprime to 3, the groupĜ ′′ has trivial intersection with the cyclic group generated byξ. This implies that the cyclic group µ 3 generated byτ has trivial intersection with the groupĜ ′ . Thus we conclude that
G ∼ = µ 3 ×Ĝ ′ ∼ = µ 3 × G ′ ∼ = G.
Therefore, the elementsξ,α, andτ generate a subgroup isomorphic to G in A * /K * . Thus the automorphism group of the Severi-Brauer surface corresponding to A contains G by Lemma 3.1.
Lemmas 4.3 and 4.6 imply the following.
Corollary 4.8. Let n = 3 r p r i i be a positive integer, where p i are primes congruent to 1 modulo 3, and r 1. Then there exists a number field K and a non-trivial Severi-Brauer surface S over K such that the group Aut(S) contains an element of order n.
Automorphisms of prime order
In this section we describe the possible finite orders of automorphisms of non-trivial Severi-Brauer surfaces.
Corollary 4.8 shows that for every positive integer n not divisible by 9 and not divisible by primes congruent to 2 modulo 3 there exists a non-trivial Severi-Brauer surface over a field of characteristic zero with an automorphism of order n. We complete this picture by showing that these are all possible finite orders of automorphisms of non-trivial Severi-Brauer surfaces. Let us start with a general observation.
Lemma 5.1. Let S be a non-trivial Severi-Brauer surface over a field of characteristic zero. Let g be a non-trivial automorphism of finite order of S. Then g has exactly three fixed points on SK, and these points are transitively permuted by the Galois group Gal K /K . In particular, g cannot have a unique isolated fixed point on SK.
Proof. Since g has finite order, it either has exactly three fixed points on SK, or it acts on SK with a unique isolated fixed point. Suppose that the latter is the case. Since the action of Gal K /K on S commutes with the action of g, the isolated fixed point must be Gal K /K -invariant, and thus defined over K. This is impossible because the Severi-Brauer surface S is non-trivial. Therefore, g has exactly three fixed points on SK; since S has no points over K, these three points have to be transitively permuted by Gal K /K . Proof. Suppose that the Severi-Brauer surface S is non-trivial, and consider the action of g on SK ∼ = P 2 K . Letg be a preimage of g under the natural projection π : SL 3 K → Aut P 2
K ∼ = PGL 3 K .
The order ofg equals either p or 3p. Since p is not divisible by 3, we can multiplyg by an appropriate scalar matrix so that the order ofg equals p. Moreover, the elementg of order p such that π(g) = g is unique in SL 3 K , and hence it is defined over the field K. Let ξ 1 , ξ 2 , and ξ 3 be the eigen-values ofg; these are p-th roots of unity. According to Lemma 5.1, the element g has exactly three fixed points on P 2 K , and these points are transitively permuted by the group Gal K /K . In other words, the roots of unity ξ 1 , ξ 2 , and ξ 3 are pairwise different and form a Gal K /K -orbit. Hence the image Γ of Gal K /K in the automorphism group Aut(µ p ) ∼ = µ p−1 of the multiplicative group µ p of p-th roots of unity has an orbit of order 3. This means that the order of Γ is divisible by 3. However, the number p − 1 is not divisible by 3 by assumption, which gives a contradiction.
The argument from the proof of Lemma 5.2 cannot be applied to an automorphism of a Severi-Brauer surface whose order is divisible by 3, since in this case we cannot always find a preimage of such an automorphism in SL 3 K defined over the base field K. However, we can study such elements using a slightly modified argument.
Lemma 5.3. Let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains an element g of order 9. Then S ∼ = P 2 .
Proof. Suppose that the Severi-Brauer surface S is non-trivial. According to Lemma 5.1, the element g has exactly three fixed points P 1 , P 2 , and P 3 on P 2 K , and these points are transitively permuted by the group Gal K /K . Let T i ∼ =K 2 be the Zariski tangent space to P 2 K at the point P i . Then g acts faithfully on T i . Let ξ ′ i and ξ ′′ i be the eigen-values of this action. Choose some preimageg of g under the projection π : SL 3 K → PGL 3 K ;
we emphasize thatg may not be defined over K. Let ξ 1 , ξ 2 , and ξ 3 be the eigen-values ofg corresponding to the fixed points P 1 , P 2 , and P 3 of g, respectively. Relabelling the numbers ξ ′ i and ξ ′′ i if necessary, we may assume that
ξ ′ 1 = ξ 2 ξ 1 , ξ ′′ 1 = ξ 3 ξ 1 , ξ ′ 2 = ξ 3 ξ 2 , ξ ′′ 2 = ξ 1 ξ 2 , ξ ′ 3 = ξ 1 ξ 3 , ξ ′′ 3 = ξ 2 ξ 3 .
Note that ξ ′ i and ξ ′′ i are 9-th roots of unity, while ξ i may not be 9-th roots of unity but are always 27-th roots of unity, becauseg 9 is a scalar matrix in SL 3 K .
If for some i = j we have ξ 3 i = ξ 3 j , then the non-trivial automorphism g 3 of S has infinitely many fixed points on SK, and thus S ∼ = P 2 by Lemma 5.1. This gives some restrictions on the values of ξ ′ i and ξ ′′ i . For instance, let ω be a non-trivial cubic root of unity, let υ be one of the numbers 1, ω, or ω 2 . If ξ ′ 1 = υξ ′ 2 , then ξ 2 ξ 1 = υ ξ 1 ξ 2 , so that υξ 2 1 = ξ 2 2 . Thus ξ 6 1 = ξ 6 2 , and hence ξ 3 1 = ξ 3 2 because both ξ 1 and ξ 2 are 27-th roots of unity. Similarly, if ξ ′ 1 = υξ ′′ 3 , then ξ 2 ξ 1 = υ ξ 2 ξ 3 , and thus υξ 1 = ξ 3 and ξ 3 1 = ξ 3 3 . In each of these cases we see that S ∼ = P 2 , which contradicts our assumption.
Since the points P 1 , P 2 , and P 3 are transitively permuted by the Galois group Gal K /K , the action of Gal K /K transitively permutes the non-ordered pairs
(5.4) {ξ ′ 1 , ξ ′′ 1 }, {ξ ′ 2 , ξ ′′ 2 }, {ξ ′ 3 , ξ ′′ 3 }.
(Note that at the same time the collection of eigen-values ξ 1 , ξ 2 , ξ 3 may be not preserved by the Galois group, sinceg is not necessarily defined over the field K.) Thus there exists an element γ ∈ Gal K /K such that γ(ξ ′ 1 ) and γ 2 (ξ ′ 1 ) are contained in the second and the third pairs in (5.4), respectively, and γ 3 (ξ ′ 1 ) = ξ ′ 1 . Such an element defines an automorphism of order 3 of the cyclic subgroup ofK * generated by ξ ′ 1 . Since ξ ′ 1 is a 9-th root of unity, one has γ(ξ ′ 1 ) = υ −1 ξ ′ 1 and γ 2 (ξ ′ 1 ) = υ −2 ξ ′ 1 , where υ is one of the numbers 1, ω, or ω 2 .
The above computation shows that γ(ξ ′ 1 ) = ξ ′′ 2 and γ 2 (ξ ′ 1 ) = ξ ′′ 3 , so that γ(ξ ′ 1 ) = ξ ′ 2 and γ 2 (ξ ′ 1 ) = ξ ′ 3 . In other words, we have
ξ ′ 1 = υξ ′ 2 = υ 2 ξ ′ 3 . Thus ξ 2 ξ 1 = υ ξ 3 ξ 2 = υ 2 ξ 1 ξ 3 . 12
Hence ξ 2 2 ξ 1 = υξ 3 = ξ 2 1 ξ 2 , so that ξ 3 1 = ξ 3 2 . This again means that S ∼ = P 2 , and gives a contradiction. Let us summarize the results of the last two sections.
Corollary 5.5. Let n be a positive integer. Then there exists a field K of characteristic zero, a non-trivial Severi-Brauer surface S over K, and an element of order n in the group Aut(S), if and only if n = 3 r p r i i , where p i are primes congruent to 1 modulo 3, and r 1.
Proof. The "if" part of the assertion is given by Corollary 4.8. The "only if" part follows from Lemmas 5.2 and 5.3.
Subgroups
In this section we classify finite subgroups of automorphism groups of non-trivial Severi-Brauer surfaces and prove Theorem 1.3.
Lemma 6.1. LetK be an algebraically closed field of characteristic zero, and letǦ ∼ = µ 2 p , where p is a prime number, be a finite subgroup of GL 3 K . Suppose thatǦ does not contain non-trivial scalar matrices. Then there exists an element ofǦ whose fixed point locus on P 2 K contains a unique isolated point. Proof. Let ξ be a non-trivial p-th root of unity, and letG be the group generated byǦ and the scalar matrix with diagonal entries equal to ξ. Obviously, one has |G| = p 3 , and the image ofG in PGL 3 K coincides with that ofǦ. Since the total number of elements of order p in GL 3 K equals p 3 − 1, all of them must be contained inG. In particular, this applies to the matrix
A = ξ 0 0 0 1 0 0 0 1 .
The fixed locus of the image of A in PGL 3 K is a union of a line and an isolated fixed point in P 2 K . Corollary 6.2. Let p = 3 be a prime number, and let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains a subgroup G ∼ = µ 2 p . Then S ∼ = P 2 . In other words, for p 3 every finite abelian p-group acting on a non-trivial Severi-Brauer surface over a field of characteristic zero is cyclic.
Proof. Consider the action of G on SK ∼ = P 2 K . Consider the natural projection π : SL 3 K → Aut P 2 K ∼ = PGL 3 K . SetG = π −1 (G), so thatG is a group of order 3p 2 . LetǦ be the p-Sylow subgroup ofG. Then π gives an isomorphism betweenǦ and G. Moreover, since p = 3, the groupǦ does not contain non-trivial scalar matrices. By Lemma 6.1 there exists an element g ∈ G such that the set of fixed points of g on P 2 K contains a unique isolated point. Thus the required assertion follows from Lemma 5.1. Corollary 6.3. Let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains a subgroup G ∼ = µ 2 3 . Suppose that G has at least one fixed point on SK ∼ = P 2 K . Then S ∼ = P 2 . Proof. Consider the natural projection π : SL 3 K → PGL 3 K . SetG = π −1 (G), so thatG is a group of order 27. Since G has a fixed point on P 2 K , the natural three-dimensional representation V ofG is reducible. Recall that the dimension of every irreducible representation ofG either equals 1, or is divisible by 3. Hence V splits into a sum of three one-dimensional representations ofG, which means that the groupG is abelian. Choose the elementsg 1 andg 2 inG whose images in PGL 3 K generate the group G. Theng 1 andg 2 generate a subgroupǦ ∼ = µ 2 3 inG such thatǦ does not contain non-trivial scalar matrices. By Lemma 6.1 there exists an element g ∈ G such that the set of fixed points of g on P 2
K contains a unique isolated point. Thus the required assertion follows from Lemma 5.1.
Corollary 6.4 (cf. [Sh20a, Lemma 4.1(iii),(iv), Lemma 4.6]). Let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains a finite 3-group G. Suppose that G is not a subgroup of µ 2 3 . Then S ∼ = P 2 . Proof. Suppose that the Severi-Brauer surface S is non-trivial. Since G is a 3-group of order at least 27, it contains a subgroup G ′ of order exactly 27. Furthermore, the group G, and thus also G ′ , does not contain elements of order greater than 3 by Lemma 5.3. Therefore, if G ′ is abelian, then G ′ ∼ = µ 3 3 ; if G ′ is non-abelian, then it is the Heisenberg group of order 27 and exponent 3. In both cases G ′ contains a subgroup Z ∼ = µ 3 that commutes with every element of G ′ .
We claim that G ′ contains a subgroup µ 2 3 acting on SK ∼ = P 2 K with a fixed point. Indeed, let g 1 be a generator of Z, let g 2 be an element of G ′ not contained in Z, and let g 3 be an element of G ′ not contained in the subgroup µ 2 3 generated by g 1 and g 2 . According to Lemma 5.1, the element g 1 has exactly three fixed points P 1 , P 2 , and P 3 on P 2 K . Since g 2 commutes with g 1 , it preserves the set {P 1 , P 2 , P 3 }. Therefore, g 2 either fixes each of the points P i , or permutes them transitively. In the former case the group µ 2 3 generated by g 1 and g 2 acts on P 2 K with fixed points. So, we may assume that g 2 defines a cyclic permutation of the points P 1 , P 2 , and P 3 . Similarly, the element g 3 preserves the set {P 1 , P 2 , P 3 }, and we may assume that it defines a cyclic permutation of P 1 , P 2 , and P 3 . This means that either the element g 2 g 3 or the element g 2 g −1 3 preserves the points P 1 , P 2 , and P 3 . Together with g 1 this element generates a group µ 2 3 acting on P 2 K with a fixed point.
Now Corollary 6.3 shows that S ∼ = P 2 .
While the group µ 2 3 can act on a non-trivial Severi-Brauer surface over a field of characteristic zero, it appears that such an action imposes strong restrictions on other finite groups acting on this Severi-Brauer surface.
Lemma 6.5. Let p > 3 be a prime number, and let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains a subgroup G ∼ = µ 2 3 ×µ p . Then S ∼ = P 2 .
Proof. Suppose that the Severi-Brauer surface S is non-trivial. The group µ 2 3 ⊂ G contains four subgroups isomorphic to µ 3 . According to Lemma 5.1, each of these subgroups has exactly three fixed points on SK. Furthermore, none of the points fixed by two different subgroups µ 3 can coincide with each other, since otherwise S ∼ = P 2 by Corollary 6.3. On the other hand, since the group µ p ⊂ G commutes with µ 2 3 , it maps the fixed points of every element g of µ 2 3 again to points fixed by g. Therefore, µ p has four distinct invariant sets of three points on SK ∼ = P 2 K . Since p > 3, each of the 12 points in the union of these sets is µ p -invariant. This is impossible by Lemma 5.1.
The next lemma shows that non-trivial non-balanced semidirect products do not appear as subgroups of automorphism groups of Severi-Brauer surfaces over fields of characteristic zero.
Lemma 6.6 (cf. Remark 4.5). Let n be a positive integer divisible only by primes congruent to 1 modulo 3, and let G be a semidirect product of µ n and µ 3 . LetK be an algebraically closed field of characteristic zero. Suppose that G ⊂ PGL 3 K . Then G is either isomorphic to µ n × µ 3 , or is a balanced semidirect product.
Proof. Assume that G is a non-balanced semidirect product. By Lemma 2.6, for some positive integers n 1 and n 2 such that n 1 n 2 = n and n 2 > 1 there is an isomorphism
G ∼ = G 1 × µ n 2 ,
where G 1 ∼ = µ n 1 ⋊ µ 3 . Let us also assume that G is not isomorphic to µ n × µ 3 . Then G, and thus also G 1 , is not abelian; in particular, we have n 1 > 1.
Consider the projection π : SL 3 K → PGL 3 K , and setG 1 = π −1 (G 1 ). The groupG 1 is not abelian, and its order 3n 1 is not divisible by 2. Thus the natural three-dimensional representation ofG 1 is irreducible. Let g be a generator of the subgroup µ n 2 ⊂ G, and choose a preimageg of g with respect to π. Since the order n 2 of g is coprime to 3, we can multiplyg by an appropriate scalar matrix and assume that the order ofg equals n 2 as well. We claim thatg commutes withG 1 . Indeed, let γ be an element ofG 1 . Since π(γ) ∈ G 1 commutes with π(g) = g, the commutator κ =gγg −1 γ −1 is a scalar matrix in SL 3 K . On the other hand,g n 2 is the identity matrix, so it commutes with γ. Writing g n 2 γ = κ n 2 γg n 2 , one concludes that κ n 2 is the identity matrix, and thus κ is the identity matrix itself.
We see thatg commutes withG 1 . Therefore, by Schur's lemmag acts in the threedimensional representation by a scalar matrix. This is impossible, because the image g ofg in PGL 3 K is non-trivial. The obtained contradiction shows that G ∼ = µ n × µ 3 .
Remark 6.7. For an alternative proof of Lemma 6.6, consider two elements g 1 , g 2 ∈ PGL 3 K such that the orders of g i are finite, at least one of the orders is greater than 3, and the fixed point locus Fix(g i ) of each g i consists of three distinct points. One can notice that g 1 and g 2 commute with each other if and only if Fix(g 1 ) = Fix(g 2 ). Applying this criterion to the generators g 2 and g of the subgroups µ n 2 and µ 3 of G appearing in (2.7) (and assuming that Fix(g 2 ) and Fix(g) consist of three points), and then to g and a generator of the subgroup µ n of G, we conclude that G is isomorphic to a direct product of µ n and µ 3 . The case when the fixed locus of some of the elements g 2 or g consists of a line and an isolated point is also easy to analyze.
Let us summarize the above results.
Corollary 6.8. There exists a field K of characteristic zero and a non-trivial Severi-Brauer surface over K such that the group Aut(S) contains a finite subgroup isomorphic to G if and only if there is a positive integer n divisible only by primes congruent to 1 modulo 3 such that G is isomorphic either to µ n , or to µ 3n , or to a balanced semidirect product µ n ⋊ µ 3 , or to the direct product µ 3 × (µ n ⋊ µ 3 ), where the semidirect product is balanced.
Proof. The "if" part of the assertion is given by Corollary 4.8 and Lemmas 4.3 and 4.6. Let us prove the "only if" part. Let S be a non-trivial Severi-Brauer surface over a field of characteristic zero, and let G be a finite subgroup of Aut(S).
To start with, assume that G is abelian. Then G is isomorphic to the product of its p-Sylow subgroups G(p) for all p dividing |G|. By Corollary 6.2, for p = 3 the group G(p) is cyclic. By Corollary 6.4, the group G(3) is isomorphic to a subgroup of µ 2 3 . Moreover, if G(3) ∼ = µ 2 3 , then all the groups G(p) for p = 3 are trivial by Lemma 6.5. We conclude that G is isomorphic either to µ 2 3 , or to µ n , or to µ 3n , where n 1 is the product of |G(p)| for all prime divisors of |G| different from 3. In the former case, we can consider G ∼ = µ 2 3 as a product of µ 3 and a balanced semidirect product µ 1 ⋊ µ 3 ∼ = µ 3 . In the latter two cases, all prime divisors p of n are congruent to 1 modulo 3 by Lemma 5.2. This proves the required assertion in the case when G is abelian. Now assume that G is not abelian. By Theorem 1.1, the group G contains a normal abelian subgroup H of index 3. Since G does not contain elements of order 9, one has G ∼ = H ⋊ µ 3 by Lemma 2.12. According to the above argument, H is isomorphic either to µ 2 3 , or to µ n , or to µ 3n , where n is divisible only by primes congruent to 1 modulo 3. In the first case |G| = 3|H| = 27, which is impossible by Corollary 6.4. In the second case we have G ∼ = µ n ⋊ µ 3 , and G is not isomorphic to µ n × µ 3 because it is non-abelian. Thus G is a balanced semidirect product by Lemma 6.6. In the third case (6.9)
G ∼ = µ 3n ⋊ µ 3 ∼ = (µ 3 × µ n ) ⋊ µ 3 ∼ = µ 3 × (µ n ⋊ µ 3 ),
because there are no non-trivial homomorphisms from µ 3 to µ * 3 ∼ = µ 2 . Furthermore, the semidirect product µ n ⋊µ 3 on the right hand side of (6.9) is not a direct product, because the group G is non-abelian. Hence this semidirect product is balanced by Lemma 6.6. This completes the proof in the case when G is non-abelian.
The following assertion is implied by [ Proposition 6.10. Let S be a non-trivial Severi-Brauer surface over a field of characteristic zero, and let G be a finite subgroup of Bir(S). Then either G is isomorphic to a subgroup of Aut(S), or G is a subgroup of µ 3 3 . Furthermore, there exists a non-trivial Severi-Brauer surface over a field of characteristic zero whose birational automorphism group contains a subgroup isomorphic to µ 3 3 . Now we are ready to prove our main result.
Proof of Theorem 1.3. Assertion (i) is given by Corollary 5.5 and Proposition 6.10. Assertion (ii) is given by Corollary 6.8. Assertion (iii) follows from Proposition 6.10.
Remark 6.11. There is a number of results on finite subgroups of multiplicative groups of division algebras, starting from the classical papers [Her53] and [Ami55]. They can hardly be used to conclude anything about automorphism groups of Severi-Brauer varieties, since a finite group in the quotient A * /K * is not necessarily an image of a finite subgroup of A * . However, every finite subgroup in A * projects to a finite subgroup of A * /K * . It would be interesting to find out which results on finite subgroups of multiplicative groups of division algebras can be recovered from geometric properties of Severi-Brauer varieties.
Field of rational numbers
In this section we discuss finite groups acting by birational automorphisms of non-trivial Severi-Brauer surfaces over Q and prove Corollary 1.4.
It is well known that the group PGL 3 (Q) does not contain elements of prime order p 5, see e.g. [DI09b,§1]. The following assertion is an analog of this result for automorphism groups of non-trivial Severi-Brauer surfaces.
Lemma 7.1. Let S be a Severi-Brauer surface over the field Q such that the group Aut(S) contains an element g of prime order p 5. Then S ∼ = P 2 .
We give two slightly different proofs of Lemma 7.1. The first of them uses the same approach as the proofs of Lemmas 5.2 and 5.3.
First proof of Lemma 7.1. Suppose that the Severi-Brauer surface S is non-trivial, and consider the action of g on SQ ∼ = P 2 Q . Letg be a preimage of g under the natural projection π : SL 3 Q → PGL 3 Q .
Since p is not divisible by 3, we can multiplyg by an appropriate scalar matrix so that the order ofg equals p. Moreover, the elementg of order p such that π(g) = g is unique in SL 3 Q , and hence it is defined over the field Q.
Let ξ 1 , ξ 2 , and ξ 3 be the eigen-values ofg; these are p-th roots of unity. According to Lemma 5.1, the element g has exactly three fixed points on P 2 Q , and these points are transitively permuted by the group Gal Q /Q . Thus the roots of unity ξ 1 , ξ 2 , and ξ 3 are pairwise different and form a Gal Q /Q -orbit. However, the group Gal Q /Q acts transitively on the p − 1 primitive roots of unity of degree p by Theorem 3.5. Since p − 1 > 3, this gives a contradiction.
The second proof uses the properties of central simple algebras.
Second proof of Lemma 7.1. Let A be the central simple algebra of dimension 9 over Q corresponding to the surface S. By Lemma 3.1 there exists an element x ∈ A * such that x ∈ Q and x p = b for some b ∈ Q.
Assume that A is a division algebra. Then x generates a subfield L Q inside A. The field L is contained in some maximal subfield of A. On the other hand, any maximal subfield of A has degree 3 over Q, see [Bou58, § VIII.10.3]. Hence L also has degree 3 over Q (and is maximal itself). Therefore, there exists an irreducible polynomial F (t) of degree 3 over Q such that x is a root of F (t). Thus F (t) divides the polynomial t p − b. Note that if b is not a p-th power of a rational number, then the polynomial t p − b is irreducible over Q, see for instance [Lan65,Theorem VIII.9.16]. Since p > 3 by assumption, one cannot have F (t) = t p − b, and hence the polynomial t p − b must be reducible. Thus we see that (7.2) b = c p for some c ∈ Q, and the roots of F (t) have the form x = ξ 1 c, ξ 2 c, and ξ 3 c, where ξ 1 , ξ 2 , and ξ 3 are pairwise different p-th roots of unity. We see that ξ 1 , ξ 2 , and ξ 3 form a Gal Q /Q -orbit. This is impossible by Theorem 3.5. Alternatively, one can conclude from (7.2) that the polynomial F (ct) divides the p-th cyclotomic polynomial Φ p (t), and use the fact that the latter is irreducible over Q.
The obtained contradiction shows that A is not a division algebra. By Lemma 3.4, this means that A is the matrix algebra, so that S ∼ = P 2 .
For an alternative proof (and a more general statement) of Lemma 7.1 we refer the reader to [Ser07,Theorem 6].
Finally, we prove Corollary 1.4.
Proof of Corollary 1.4. By Proposition 6.10, we may assume that the group G is contained in Aut(S). Then the order of G is divisible only by primes not exceeding 3 by Lemma 7.1.
On the other hand, we know from Theorem 1.3(i) (or from Lemma 5.2) that the order of G is odd. Therefore, in this case G is a 3-group, and thus it is a subgroup of µ 2 3 by Corollary 6.4.
The next fact is immediately implied by the theorem of Wedderburn about the structure of central simple algebras, see [GS06, Theorem 2.1.3].
Thus the assertion follows from [Lan65, Theorem VIII.1.4]. The next result is well known to experts; it follows for instance from [Ste89, Theorem 1(b)].
Indeed, this intersection contains M and is contained in M(τ ). The degree of the extension M(τ )/M equals 3, so we only need to check that M(τ ) is not contained in M(ξ). On the other hand, if M(τ ) ⊂ M(ξ), then Q(τ ) is a subfield of M(ξ) as well. SinceM(ξ) = Q(ω, ξ) = Q(ωξ),we know from Theorem 3.5 that the Galois group Gal (M(ξ)/Q) is cyclic. Hence every subfield of M(ξ) is normal over Q, which is not the case for Q(τ ).Note that M(ξ)/M is a Galois extension with Galois group isomorphic to µ * n by Lemma 3.6, and M(τ )/M is a Galois extension with Galois group isomorphic to µ 3 .Since the extension L/M can be represented as the composite of the extensions M(τ )/M and M(ξ)/M, we conclude that Gal (L/M) ∼ = Gal (M(τ )/M) × Gal (M(ξ)/M) ∼ = µ 3 × µ * n , see for instance [Lan65, Theorem VIII.1.5]. Let χ : µ 3 → Gal (L/M) be the homomorphism such that its composition with the projection Gal (L/M) → Gal (M(τ )/M) is an isomorphism µ 3 ∼ −→ µ 3 , and the composition with the projection Gal (L/M) → Gal (M(ξ)/M) is the homomorphism
Lemma 5.2 (cf.[Sh20a, Lemma 4.2]). Let p ≡ 2 (mod 3) be a prime number. Let S be a Severi-Brauer surface over a field K of characteristic zero such that the group Aut(S) contains an element g of order p. Then S ∼ = P 2 .
Sh20a, Proposition 3.7], [Sh20a, Corollary 4.5], and [Sh20c, Theorem 1.2].
Finite subgroups of division rings. S Amitsur, Trans. Amer. Math. Soc. 80S. Amitsur. Finite subgroups of division rings. Trans. Amer. Math. Soc. 80 (1955), 361-386.
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In Vector bundles and complex geometry. A Beauville, Contemp. Math. 522Amer. Math. SocFinite subgroups of PGL 2 (K)A. Beauville. Finite subgroups of PGL 2 (K). In Vector bundles and complex geometry, pp. 23-29. Contemp. Math. 522, Amer. Math. Soc., Providence, RI, 2010.
Construction of finite groups. H Besche, B Eick, J. Symbolic Comput. 274H. Besche, B. Eick. Construction of finite groups. J. Symbolic Comput. 27 (1999), no. 4, 387-404.
Première partie: Les structures fondamentales de l'analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples. N Bourbaki, Actualités Sci. Ind. 231261HermannÉléments de mathématiqueN. Bourbaki.Éléments de mathématique. 23. Première partie: Les structures fondamentales de l'analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples. Actualités Sci. Ind., no. 1261, Hermann, Paris, 1958.
Variations sur un thème de H. Poincaré. F Châtelet, Ann. Sci.École Norm. Sup. 3F. Châtelet. Variations sur un thème de H. Poincaré. Ann. Sci.École Norm. Sup. (3), 61 (1944), 249-300.
Algebraic number theory. J. W. S. Cassels and A. FröhlichAcademic PressLondonAlgebraic number theory. Edited by J. W. S. Cassels and A. Fröhlich. Academic Press, Lon- don;
Finite subgroups of the plane Cremona group. In Algebra, arithmetic, and geometry: in honor of Yu. I Dolgachev, V Iskovskikh, Birkhäuser Boston. IProgr. Math.I. Dolgachev, V. Iskovskikh. Finite subgroups of the plane Cremona group. In Algebra, arith- metic, and geometry: in honor of Yu. I. Manin. Vol. I, pp. 443-548, Progr. Math. 269, Birkhäuser Boston, Boston, MA, 2009.
On elements of prime order in the plane Cremona group over a perfect field. I Dolgachev, V Iskovskikh, Int. Math. Res. Not. IMRN. 18I. Dolgachev, V. Iskovskikh. On elements of prime order in the plane Cremona group over a perfect field. Int. Math. Res. Not. IMRN 2009, no. 18, 3467-3485.
Finite group actions on curves of genus zero. M Garcia-Armas, J. Algebra. 394M. Garcia-Armas. Finite group actions on curves of genus zero. J. Algebra, 394 (2013), 173-181.
Ph, T Gille, Szamuely, Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. CambridgeCambridge University Press101Ph. Gille, T. Szamuely. Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics, 101. Cambridge University Press, Cambridge, 2006.
Unramified Brauer group and its applications. Translations of Mathematical Monographs, 246. S Gorchinskiy, C Shramov, American Mathematical SocietyProvidence, RIS. Gorchinskiy, C. Shramov. Unramified Brauer group and its applications. Translations of Mathematical Monographs, 246. American Mathematical Society, Providence, RI, 2018.
Finite multiplicative subgroups in division rings. I Herstein, Pacific J. Math. 3I. Herstein. Finite multiplicative subgroups in division rings. Pacific J. Math. 3 (1953), 121- 126.
M Isaacs, Finite group theory. Graduate Studies in Mathematics, 92. Providence, RIAMSM. Isaacs. Finite group theory. Graduate Studies in Mathematics, 92. AMS, Providence, RI, 2008.
Severi-Brauer varieties; a geometric treatment. J Kollár, arXiv:1606.04368J. Kollár. Severi-Brauer varieties; a geometric treatment. arXiv:1606.04368 (2016)
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Jordan groups and automorphism groups of algebraic varieties. V Popov, Automorphisms in birational and affine geometry. ChamSpringer79V. Popov. Jordan groups and automorphism groups of algebraic varieties. in Automorphisms in birational and affine geometry, pp. 185-213, Springer Proc. Math. Stat., 79, Springer, Cham, 2014.
Bounds for the orders of the finite subgroups of G(k). J.-P Serre, EPFL PressLausanneGroup representation theoryJ.-P. Serre. Bounds for the orders of the finite subgroups of G(k). Group representation theory, 405-450, EPFL Press, Lausanne, 2007.
Birational automorphisms of Severi-Brauer surfaces. C Shramov, Sb. Math. 2113C. Shramov. Birational automorphisms of Severi-Brauer surfaces. Sb. Math., 211 (2020), no. 3, 466-480.
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The Jordan constant for Cremona group of rank 2. E Yasinsky, Bull. Korean Math. Soc. 545E. Yasinsky. The Jordan constant for Cremona group of rank 2. Bull. Korean Math. Soc. 54 (2017), no. 5, 1859-1871.
Laboratory of Algebraic Geometry, 6 Usacheva str. 8119991Moscow; MoscowSteklov Mathematical Institute of Russian Academy of Sciences ; Russia National Research University Higher School of EconomicsRussia E-mail address: [email protected] Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow, 119991, Russia National Research University Higher School of Economics, Laboratory of Algebraic Geometry, 6 Usacheva str., Moscow, 119048, Russia E-mail address: [email protected]
|
[] |
[
"XHIP-II: Clusters and associations",
"XHIP-II: Clusters and associations"
] |
[
"Charles Francis ",
"Erik Anderson \n800 Morton St97520AshlandORUSA\n",
"\nElphinstone Rd\nTN34 2EGHastingsUK\n"
] |
[
"800 Morton St97520AshlandORUSA",
"Elphinstone Rd\nTN34 2EGHastingsUK"
] |
[] |
Context. In the absence of complete kinematic data it has not previously been possible to furnish accurate lists of member stars for all moving groups. There has been an unresolved dispute concerning the apparent inconsistency of the Hipparcos parallax distance to the Pleiades. Aims. To find improved candidate lists for clusters and associations represented among Hipparcos stars, to establish distances, and to cast light on the Pleiades distance anomaly. Methods. We use a six dimensional fitting procedure to identify candidates, and plot CMDs for 20 of the nearest groups. We calculate the mean parallax distance for all groups. Results. We identify lists of candidates and calculated parallax distances for 42 clusters and 45 associations represented within the Hipparcos catalogue. We find agreement between parallax distance and photometric distances for the most important clusters. For single stars in the Pleiades we find mean parallax distance 125.6 ± 4.2 pc and photometric distance 132 ± 3 pc calibrated to nearby groups of similar in age and composition. This gives no reason to doubt either the Hipparcos database or stellar evolutionary theory. PACS: 98.35.Df 98.35.Pr
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10.1134/s1063773712110023
|
[
"https://export.arxiv.org/pdf/1203.4945v1.pdf"
] | 119,285,733 |
1203.4945
|
59b73056ae35b5f09a7a13ee81787bf3ce05ff04
|
XHIP-II: Clusters and associations
Charles Francis
Erik Anderson
800 Morton St97520AshlandORUSA
Elphinstone Rd
TN34 2EGHastingsUK
XHIP-II: Clusters and associations
1Galaxy: open clusters and associations; Galaxy: solar neighbourhood
Context. In the absence of complete kinematic data it has not previously been possible to furnish accurate lists of member stars for all moving groups. There has been an unresolved dispute concerning the apparent inconsistency of the Hipparcos parallax distance to the Pleiades. Aims. To find improved candidate lists for clusters and associations represented among Hipparcos stars, to establish distances, and to cast light on the Pleiades distance anomaly. Methods. We use a six dimensional fitting procedure to identify candidates, and plot CMDs for 20 of the nearest groups. We calculate the mean parallax distance for all groups. Results. We identify lists of candidates and calculated parallax distances for 42 clusters and 45 associations represented within the Hipparcos catalogue. We find agreement between parallax distance and photometric distances for the most important clusters. For single stars in the Pleiades we find mean parallax distance 125.6 ± 4.2 pc and photometric distance 132 ± 3 pc calibrated to nearby groups of similar in age and composition. This gives no reason to doubt either the Hipparcos database or stellar evolutionary theory. PACS: 98.35.Df 98.35.Pr
Introduction
Complete and accurate memberships of kinematic groups are important for the study of Galactic dynamics, for tests of stellar evolutionary theory, and to estimate cluster distances, required to establish the cosmological distance scale. However, the absence of complete kinematic information and inevitable uncertainties in stellar parallaxes create difficulties in ascertaining the validity of lists of candidate stars for individual clusters. Furthermore, it is now established that the local velocity distribution contains six clearly defined stellar streams (e.g., Famaey et al., 2005;Anderson, 2009, 2012 and references cited therein). In consequence, accidental alignments between proper motions and positions are far more probable than would be the case for a well mixed distribution.
We define moving groups as stars sharing a common motion and localized in a region of space. They are distinguished from streams, which are all-sky motions and which are parts of the spiral structure of the Galaxy (Francis & Anderson, 2009. Moving groups will be termed 'clusters' if they are gravitationally bound, and 'associations' otherwise. Associations typically consist of young stars originating in the same process, resulting from the collisions between outward bound gas clouds (corresponding to the Hyades stream) and clouds following the spiral arm. In this paper a separation parameter will be defined, and used to determine whether a group is likely to be a cluster or an association.
Because of the likelihood of chance alignments it is important to calculate group memberships on the basis of complete kinematic data, and with the most accurate and extensive information available. The Extended Hipparcos Compilation ("XHIP", is based on Hipparcos the New Reduction of the Raw Data ("HIP2", van Leeuwen, 2007) gives radial velocities for 46 392 Hipparcos stars, together with metallicities for 18 549 stars and multiplicity information from the Catalog of Components of Double & Multiple Stars (Dommanget & Nys, 2002) and The Wash-ington Visual Double Star Catalog, version 2010-11-21 (Mason et al., 2001-2010. It is therefore appropriate that XHIP recalculates group memberships for all clusters and associations containing a reasonable number of Hipparcos stars with known radial velocities.
We tested a number of groups using membership lists taken from the literature, and all groups listed in the New catalog of optically visible open clusters and candidates (V3.0) (Dias et al., 2002(Dias et al., -2010 or in Karchenko et al. (2005a&b) with distances less than 800 pc in either catalogue, at which point the groups have become too distant to contain many Hipparcos stars, and parallax distances have become meaningless. We used a six dimensional (position × velocity) fitting procedure which results in improved accuracy for group memberships as indicated by a larger numbers of candidates and/or smaller dispersion than found in lists obtained from the literature. We found evidence in Hipparcos stars for 87 clusters and associations, but we note that it does not necessarily follow from lack of evidence that a group does not exist, since it may contain less bright stars. 42 groups show sufficient separation from the surrounding star field to be classed as probable clusters.
The distance to the Pleiades has been the subject of an unresolved dispute. Van Leeuwen (2009) has suggested that the difference between the Hipparcos and other estimates is due to a failure of stellar evolutionary theory, while others have questioned the Hipparcos astrometry. Van Leeuwen calculated the Pleiades parallax distance using a weighted mean. We observe that weighting can lead a systematic error. After recalculating without weighting we find a distance within 2σ error bounds of most other estimates and conclude that this gives no reason question either stellar evolutionary theory or the Hipparcos astrometry. We also resolve anomalous results concerning the distance of the Diamond cluster and Blanco 1, and, because of the importance of cluster distances to the cosmological distance scale, we recalculate all parallax distances without weighting.
Section 2. reviews the algorithm by which we determine cluster memberships in XHIP. In section 3. we describe the method used to determine cluster distance. Section 4. describes the colour-magni-tude diagrams (CMDs) for a number of significant groups. Section 5. contains remarks on particular groups. Section 6. comments on the Pleiades distance anomaly, finding that the most likely cause for any remaining anomaly is random error. Our conclusions are summarised in section 7..
Group membership
The traditional method for ascertaining membership of the Hyades has been the convergent point method (e.g. Perryman et al., 1998). The convergent point method does not make use of parallax distance, which has to be added as a separate constraint. With the accuracy of parallaxes now available for stars at the distance of the Hyades it is no longer the best method (Van Leeuwen, 2009) for the Hyades. For groups at greater distance, with lower proper motions and (importantly) lower radial velocities, errors dominate and the convergent point method is unreliable. Van Leeuwen's approach was to first find groups in the celestial sphere (RA and DE coordinates) and then to constrain the other variables. In consequence, group membership is determined by a cylindroid in position × velocity space, rather than the more natural choice of an ellipsoid described below, and because the initial selection depends on only two variables it is difficult to accurately determine cluster size so that his lists typically omit candidates which we find at the edges of a cluster. We also have a database with many more, and more accurate, radial velocities than previous studies, and it makes sense to use this additional information in finding improved candidate lists.
To identify the membership of a moving group it is necessary to match both the position and the velocity of each star to within a region of a six-dimensional space, where distance is determined from Hipparcos parallax measurements. We have rejected the notion that membership can be decided from other factors, such as stellar composition, because indicators of composition, such as [Fe/ H] are notoriously uncertain (as is seen, for example, in XHIP, in which measurements of [Fe/H] from many sources were compared and combined).
The list of group members is determined iteratively, by testing each star in XHIP under the condition:
(2.1) for a fixed value of n, where x runs over the dynamical variables, Plx, RA, DE, RV, pmRA, pmDE, except for groups occupying a large region of the sky, when x runs over the Cartesian variables X, Y, Z, U, V, W, where X is directed towards the Galactic centre, Y is the direction of rotation, Z is directed to the Galactic north pole, and U, V, W are the respective velocity components. x and σ x are the mean and standard deviation for the group satisfying eq. (2.1) in the previous iteration. It is necessary to use Cartesian variables when the angular size of the cluster substantially affects radial velocity and proper motions, but it is better to use radial variables where possible because they correspond more closely to the actual measurements and because radial distance errors are much greater than angular errors, and hence the error ellipsoid is aligned with the radial direction. Cartesian variables were used for the Hyades cluster, the Ursa Major association, and the AB Doradus, β Pictoris and Tucana/Horologium moving groups.
At each iteration, all stars in XHIP with known RV that satisfy eq. (2.1) are included in the new group list. Then the means and variances of the variables x for the group are recalculated and the iteration is repeated. The convergence criterion is simply that the candidate list remains unchanged in an iteration.
We initialized the iteration using n = 3.5 and a list of group stars taken from the literature, when available, or group coordinates given in Dias et al. or Karchenko et al., together with typical group dimensions. The Hyades was initialised from the list given by Perryman et al. (1998). The β Pictoris, AB Doradus, Tucana/ Horologium, TW Hydrae associations and the η Chamaeleontis cluster were initialised from Zuckerman & Song (2004). Upper Centaurus Lupus, Lower Centaurus Crux, Upper Scorpius, Vel OB2, Collinder 121, Per OB2, Cas-Tau, Lac OB1, Cep OB2, and Cep OB6 were initialised from de Zeeuw et al. (1999). The cluster was considered "found" when the group became stable under iteration of eq. (2.1). For associations, the found groups are not necessarily those intended by those who originally named them. For example, ASCC 18 and ASCC 20 converge to the same list of stars. For a number of associations, convergence was found for larger groups than those originally intended. Our lists contain overlaps. For example, the Orion Molecular Cloud Complex contains a number of groups including the Orion Nebula (M42), the Horsehead Nebula, Collinder 70, NGC 1980, NGC 1981, and the Running Man Nebula (NGC 1977 which have similar motions and which can be regarded as parts of the larger group. In the case of an overlap more than one comma separated group is listed in column GrpName of XHIP.
If an initial list containing at least four Hipparcos stars with known radial velocities was not found from the data in any of our sources we rejected the cluster. Otherwise we found a revised list by iterating with eq. (2.1). Usually the size of the candidate list increased or stabilised at each iteration, but for some groups the number of candidates reduced before stabilising. If the number of candidates continued to increase without stabilising, or if the dispersion of any of the dynamical variables was unusually large for a group we used n = 3, which generally resulted in a more compact list of candidates. We then tried n = 3.5 again. If a stable group was found under the iteration we increased n by 0.5 and repeated the iteration to establish the largest half integral value of n for which a stable group under the iteration could be found.
For most well-known clusters our 6-dimensional fitting procedure leads to a larger number of candidates and/or smaller dispersion than is found in lists obtained from the literature. For groups using spherical coordinates, we extended the list of candidate stars to include stars without known radial velocities by reducing the right-hand side of eq. (2.1) so as to leave the standard deviation of each dynamical variable similar to its value for the core group of stars with complete dynamical information. This method does not apply to groups found using Cartesian coordinates.
Eq. (2.1) can be justified both dynamically and statistically. The dynamical argument has intuitive appeal, but the statistical argument may be preferred because it incorporates both measurement errors and physical dispersion, does not depend on approximation, and applies to unbound groups. Dynamically, for a gravitationally bound group, the further a star is from the centre of the group the nearer its velocity must be to that of the centre of gravity of the cluster. This can be described using a linear approximation for the gravitational potential energy at distance r from the centre of the cluster, such that r 2 + kv 2 ≈ constant, where v is velocity relative to the centre of the cluster, and the constant k is determined by the ranges of values of r and v taken for each star over time. Since r and v are bounded for cluster stars, we may impose eq. (2.1) as a criterion for cluster membership. Statistically eq. (2.1) defines the
x x - ( ) 2 σ x 2 ------------------ x ∑ n 2 <
interior of an ellipsoid in position × velocity space, centred at the mean for the cluster, and with axes proportional to the standard deviation for each variable. Thus our procedure generalises to 6D the familiar principle of discarding outliers beyond a given number of standard deviations from the mean.
Group identifications can only be given with certainty for stars with known radial velocities, and accurate parallaxes. In practice, many stars still do not have known radial velocities and, even using HIP2, parallax errors lead to distance errors much greater than the size of the group. Consequently, there remains some uncertainty in group memberships. Associations are not gravitationally bound, and do not have the ellipsoidal shape suggested by eq. (2.1). Memberships of associations are necessarily less accurate than those of clusters, but eq. (2.1) still has merit, by providing an objective criterion by which the existence of a group and a list of candidate stars can be established.
n is a measure of the concentration of a group, and of its separation from the surrounding star field, and (because it is rigorously defined) may be preferred to the Trumpler classifications I -IV. Since they are gravitationally bound, clusters are more compact than associations. As a result the largest value of n giving convergence can be expected to be greater for a cluster than for an association. In practice, we found that a maximum value of n = 3.5 is typical for an association. For clusters n ≥ 5 is usual. The lower value of n for associations shows poor dynamical separation from surrounding stars, indicating that the majority of associations are just randomly dense regions arising in much greater processes in which stars are formed. This was also shown by the fact that a number of associations found in the Pleiades stream have overlapping memberships. We rejected groups with n < 3.5 or with fewer than four Hipparcos stars with known radial velocities.
Seven groups have separation n = 4.5: Feigelson 1, α Perseus, Cep OB6, ASCC 122, the Christmas Tree cluster, ASCC 115, and ASCC 113. These are all young groups, and can be classed as OB associations. However, both α Perseus and the Christmas Tree cluster are generally regarded as clusters, and have 1σ radii less than 10 pc. It seems probable that a separation of n = 4.5 is indicative of a gravitationally bound group, with less good separation from the surrounding star field than one would expect of a mature cluster, due to the similar motions of other stars in the complex in which the group formed. We have therefore classed 42 groups with n ≥ 4.5 as clusters, and 45 groups with n ≤ 4 as associations.
Parallax distance
Van Leeuwen (2009) calculated group parallax distances using weighted mean parallax. Weighting by inverse squared error is generally a sound method to combine different measurements and reduce uncertainty, but it requires that there is no correllation between the values and weights. We confirmed that there is no correlation between parallax and parallax error for the entire population (there is no reason why there should be), but the method used by Hipparcos compares parallax to the surrounding star field. Although van Leeuwen has rejected this as a cause of systematic error in stellar parallaxes, he did not exclude the possibility of an effect on parallax errors. We tested the correlation between parallax and parallax error for forty clusters (omitting two with overlaps between groups) using a two tailed Student's t-test. We found that eight contain correlations significant at 95% or more, and three significant at 99% or more. For the Diamond cluster (one of those for which van Leeuwen reported a distance anomaly) the correlation between parallax and parallax error is -0.6 and is significant at 99.95%. In the case of the Pleiades the correlation is -0.2 and is significant at 86%. Both positive and negative correlations are found. A detailed analysis of the reasons for these correlations would be complex and have little value, but the existence of such correlations means that an unweighted mean must be used when calculating cluster distance. For more distant stars, van Leeuwen derived cluster parameters from a combined analysis of the abscissa residuals for the member stars. This method uses intermediate astrometric data for single cluster members. Because of the importance of avoiding weighting, in all cases we have calculated distance from the mean parallax. This is valid even for distant clusters where some stated parallaxes are negative. It results in larger stated errors than a method using a weighted mean, but we believe this is correct, so as to avoid unknown systematic errors.
Clusters and associations within 300 pc
Cluster N N vL Sep R XHIP R vL R D R K 1σ
Clusters and associations beyond 300 pc
Cluster N N vL Sep R XHIP R vL R D R K 1σ
In practice, in many cases the distance estimate is little changed by the use of weighting in the calculation of mean parallax, but the result of using weighting is notably low for the Pleiades, Omicron Velorum (IC 2391) and α Perseus (table 1), and for the Diamond cluster (NGC 2516), IC 4665 (table 2). In three cases, NGC 2232, Trumpler 10 and NGC 2547, for which van Leeuwen found greater distances, the difference is largely accounted for by changes in the candidate lists.
The full list of clusters and properties is given in groups.dat, which is part of XHIP, available as CDS Catalog V/137B. groups.dat is described in table 3. Distances given in groups.dat are based on the full candidate list, including stars for which radial velocities are unknown. In practice we found small differences, well within error bounds, when these stars were excluded. For (nearby) groups whose membership is calculated using Cartesian variables Dist is mean distance, otherwise Dist is found from mean parallax. In table 1 and table 2, R XHIP is Dist, given in groups.dat.
Because of difficulties in the solution from the raw data, parallaxes are inevitably less accurate for multiple star systems than for single stars. In most cases restricting to single stars makes little difference to results, but it is worth remarking that for the group of three clusters for which anomalously low parallax distances were found by van Leeuwen, the Pleiades, Blanco 1 and the Diamond cluster (NGC 2516), restricting to single stars caused an increase in the distance estimate, in the case of Blanco 1 by as much as 5%.
Colour-magnitude diagrams
We plotted the CMD for three nearby moving groups, AB Doradus, β Pictoris, and Tucana/Horologium, together with the Ursa Major association, the Hyades, the Coma star cluster, Praesepe, the Pleiades, the Southern Pleiades and Omicon Velorum (figure 1). The motions of AB Doradus, β Pictoris, and Tucana/Horologium, ensure that they consist of young stars. We plotted only single stars, since they have more accurate parallaxes and more accurate luminosities, leading to a tighter distribution. For the Pleiades we used an applied reddening E(B -V) = 0.034 mag for RA > 056.818968º, and E(B -V) = 0.057 mag otherwise (Taylor, 2008). For other clus-ters E(B -V) was taken from Karchenko et al. We used extinction equal to 3.1 times reddening (e.g. Gibson & Nordsieck, 2008).
The tight form of the CMD in figure 1 is a measure of the accuracy of Hipparcos parallaxes for near stars and clusters. In particular there is good agreement between the positions of β Pictoris and Tucana/Horologium and AB Doradus. The small difference between the position of these groups and the Hyades is accounted for by the greater age and metallicity of the Hyades. The Pleiades does appear slightly below and to the left of the other groups. This could in part be caused by an underestimate of the effect of extinction, but is most probably the result of random errors (section 6.).
Label Units Explanation
Name ---Group name AltName ---Alternative group name Count ---Number of Hiipparcos candidate stars n ---Separation parameter (27) It is possible to read the relative ages of particular groups from the positions of the main sequence cutoff and the main sequence for early types. Thus the Hyades appears to be a little older than Praesepe, not coeval as is sometimes assumed (e.g. Dobbie et al., 2006). Praesepe and the Hyades are both members of the Hyades stream, formerly known as the Hyades supercluster, and have similar motions, but it is known that streams contain stars with a wide range of ages (e.g., Chereul et al., 1998Chereul et al., , 1999. Praesepe does appear coeval with the Coma cluster, but is clearly not related as they have very different motions. Figure 1 confirms that these clusters are older than the Pleiades, AB Doradus, β Pictoris, and Tucana/Horologium.
We plotted the CMD for single stars in the Hyades & Pleiades clusters, NGC 2451A, α Perseus, Blanco 1, Collinder 135, the Ptolemy cluster, NGC 7092, NGC 2232, Collinder 140, the diamond cluster and the Scorpio-Centaurus association (figure 2).
With the exception of the Hyades and Pleiades, which are included in both CMDs for comparison, most of these groups are younger than those in figure 1, as may be seen from the difference in the curve followed by stars on the main sequence. In accordance with the predictions of evolutionary theory, early types are below and to the left of the corresponding position in figure 1, while for colours in the range ~0.4 < B -V <~0.8 mag we find pre-main sequence stars lying above the main sequence. Greater scatter is also seen in figure 2 due to the spread of distances of stars in the Scorpius-Centaurus association.
Notable groups
Ursa Major association
The Ursa Major association (Collinder 285) has a core containing 15 Hipparcos stars with complete kinematic information and Figure 1: CMD for single stars in the Pleiades cluster, the Hyades cluster, the β Pictoris, AB Doradus and Tucana/Horologium moving groups, the Ursa Major association, the Coma star cluster, Praesepe, the Southern Pleiades and Omicron Velorum. Cluster distance is used for the Pleiades, Coma, Praesepe, Southern Pleiades and Omicron Velorum, and parallax distance for individual stars is used for the other groups. Reddening has been applied. n = 6. Two stars, HIP 65327 and HIP 59496 are added to the nucleus given by King & Villarreal (2003) and by Soderblom & Mayor (1993). The value of n is exceptionally high for an association, showing low velocity dispersion and good separation from surrounding stars. This supports the notion that the Ursa Major association is the remnant of an open cluster. There are no further candidates in Hipparcos for which radial velocities are not available. No convergence is found from extended lists of Ursa Major candidates (e.g. Soderblom & Mayor, 1993;King & Villarreal, 2003). We believe that such extended lists contain stars with similar velocities taken from the Sirius stream, and should not be regarded as belonging to the Ursa Major association.
Hyades cluster
The Hyades (Melotte 25) are sufficiently close that inaccuracies due to change in angular position must be excluded when calculating group membership. Errors in distance are modest and membership is best calculated using the dynamical variables X, Y, Z, U, V, and W. We restricted to stars with better than 20% parallax errors. We identified 190 Hipparcos stars in the Hyades cluster with known radial velocities, and with cluster separation n = 5.5. From the list of 197 Hyades candidates given by Perryman et al. (1998), eight do not have known radial velocity, two more have parallax errors greater than 20%, and 14 are rejected as outside our dynamical cluster definition, leaving 173 in our list. Six dimensional fitting resulted in more compact distributions in position and velocity space. We may expect our estimate of the distance to the Hyades to be slightly low because the constraint that parallax errors are less than 20%, which is necessary to calculate corrected motions in UVW space, introduces a selection bias toward nearer stars. As only two stars were rejected by this criterion, the bias is small. In fact our distance to the centroid of the cluster is 45.5 ± 0.8 pc, a little less than the mean distance 46.1 ± 0.6 pc (as one expects), slightly below other estimates but well within a 1σ error.
Pleiades stream
Velocities found for Lower Centaurus Crux, Upper Centaurus Lupus, Tucana/Horologium, and β Pictoris are very similar, and close to the modal velocity of the Pleiades stream, which consists (in the main) of very young stars. Many other stars have similar velocities. Similarly, AB Doradus has a similar velocity to the Pleiades cluster and numerous associations with similar motions take part in the Orion Molecular Cloud Complex. This supports the model of spiral structure described by Francis & Anderson (2009 in which star forming regions are formed from collisions between outward bound gas clouds and inward bound gas following the spiral arm. These collision processes take place continuously and on a large scale. Associations are best regarded as denser regions of a much greater process, such that associations formed near to each other in space and time are likely to have similar velocities and composition. De Zeeuw et al. (1999) found membership lists for a number of OB associations, without benefit of known radial velocities, using a variant on the convergent point method. The convergent point method works well for the Hyades, which is at relatively low distance, has relatively high proper motion, and, importantly, has high radial velocity. However, this method is unreliable for associations at greater distances such that errors are large compared to proper motion, and such that the convergent point is sufficiently far from the cluster that non-linear affects due to the curvature of the celestial sphere come into play.
OB associations
The Scorpius-Centaurus association contains three sub-associations, Lower Centaurus Crux, Upper Centaurus Lupus, and Upper Scorpius. The list for Upper Scorpius from de Zeeuw et al. contains 120 candidates, over a region with 1σ radius 13.7 pc., and with radial velocity dispersion 8.9 km s -1 , high for stars which are supposed to be kinematically related. Our fitting procedure restricted the 1σ radius to 8.7 pc and radial velocity dispersion to 4.7 km s -1 . Dispersion in proper motion was also around half of that of the original list. We found 82 candidates in a region with less than 4% of the original 6-volume. Our final list for Upper Centaurus Lupus contains 169 members, compared to 221 in the list from de Zeeuw et al., in a region of less than 7% of the 6-volume. Our list for Lower Centaurus Crux contains 187 members, compared to 180 in the list from de Zeeuw et al., in a region less than 18% of the size.
After initialising Vel OB2 using coordinates from Karchenko et al., we found 14 candidates, including 9 with known radial velocities, with separation n = 4 and distance 389 ± 85 pc, centered at RA =122.2º, DE = -48.9º, and with radial velocity 14.8 ± 1.9 km s -1 . After initialising from the list of 93 candidates for Vel OB2 given by de Zeeuw et al. we found 73 candidates, in a region of 5% of the size. Only 39 of our candidates are in common with those of de Zeeuw et al.'s list. Only 28 of our candidates have known radial velocities. We found a distance of 510 ± 18 pc, at RA = 122.3º, DE = -46.7º, radial velocity 22.2 ± 1.0 km s -1 . Only three candi-dates are in common with our list for the nearer group, and we conclude that these are separate associations. We have called the nearer group Vel OB2A and the further group Vel OB2B.
De Zeeuw et al. list 41 candidates for Per OB2. These are widely dispersed in position and velocity space (radial velocity dispersion is 20 km s -1 ). Our test converged to essentially the same group of candidates as for IC 348, and we have not listed Per OB2 separately in the data file. De Zeeuw et al. list 96 candidates for Lac OB1, also with a wide dispersion in position and velocity. Our list contains 68 members in a region with less than 6% of the 6-volume and a distance of 466 ± 17 pc. 31 of these have known radial velocity and 47 are contained in de Zeeuw's list. This group has a substantial overlap with ASCC 122. De Zeeuw et al. list 71 candidates for Cep OB2. Our list contains 13 members in a region with less than 0.4% of the 6-volume, and at a distance 882 ± 85 pc. 12 have known radial velocity and 7 are given in de Zeeuw's list. There is a substantial overlap between this group and NGC 7160. De Zeeuw et al. list 20 candidates for Cep OB6. Our list contains 11 members in a region with 18% of the 6-volume, and at a distance 275 ± 6 pc and with separation n = 4.5. 9 have known radial velocity and 10 are given in de Zeeuw's list. Baumgardt (1998) reported the existence of Collinder 132, Roslund 5, and NGC 2451B as uncertain. We found 8 Hipparcos candidates for Collinder 132 with separation n = 5.5, whereas Baumgardt gave 12 candidates, none of which appear in our list. We found 18 Hipparcos stars in Roslund 5 with separation n = 3.5, compared to 5 Hipparcos stars given by Baumgardt, of which four are on our list. We identified 19 candidates for NGC 2451B with separation is n = 3.5. We thus confirm that these groups exist, and that NGC 2451B and Roslund 5 have characteristics of an association while Collinder 132 is a cluster.
Confirmed groups
Unidentifiable groups
In general it is not the case that one can exclude the existence of a moving group because no evidence for that group is found in Hipparcos. Exceptions arise for stellar associations in the Pleiades stream, because these are groups of young stars which ought to contain early types and should be represented in Hipparcos, and for certain lists given in the literature which are sufficiently dispersed in velocity and position space that they cannot be considered as moving groups.
Because the majority of stars participate in spiral structure, and therefore belong to streams, chance alignments are far more probable than would be the case for a well-mixed distribution, which has generally been assumed in the absence of a convincing explanation of spiral structure. It transpires that stars given as candidates for the "HIP 98321" group by Madsen, Dravins, and Lindegren (2002) are spread over a wide region of space and have a wide velocity dispersion. We concluded that they are not representative of a cluster or an association, but are merely stars in the Pleiades stream.
Similarly, the "HR 1614" group (Eggen, 1978) is not a moving group as such, but is a collection of stars in the Hercules stream. Similarities in stellar composition in HR 1614 reflect the known similarities in composition of the Hercules stream, which has been shown to consist predominately of stars predating the formation of spiral structure, about 9 Gyrs ago (Francis & Anderson, 2009).
We found no evidence for the TW Hydrae association or the η Chamaeleontis cluster which contain few Hipparcos stars with known radial velocities in the Pleiades stream. If these were true near associations they would be expected to contain early type stars from recent star formation and should be better represented in Hipparcos.
Of six stars listed by van Leeuwen (2009 & personal correspondence) as candidates for NGC 6633, five have known radial velocities, but these have widely discrepant values, with standard deviation, 17.3 km s -1 . It is impossible to say that these candidates are representative of a group.
According to Karchenko et al. and Dias et al., Collinder 121 is at a distance of 1100 pc -beyond the range of our tests. De Zeeuw et al. list 103 stars for Collinder 121. These have a mean parallax distance of 568 pc, but are widely dispersed in both velocity and position. We did not find any signs that they contain a moving group. The list given by de Zeeuw et al for Cas-Tau contains 83 stars. These are spread over a huge region of space. It is not possible to identify any core group or common motion by initialising a search from this list.
The Pleiades distance anomaly
The distance to the Pleiades has been calculated by many methods (e.g., Percival, Salaris & Groenewegen, 2005;Zwahlen et al., 2004;Turner, 1979;Soderblom et al., 2005;Xiaopei Pan, Shao & Kulkarni, 2004). The majority of determinations have found results in the range 132-135 pc (see van Leeuwen, 2009, table 3), but the Hipparcos satellite caused consternation by finding a distance of only 118 parsecs by measuring the parallax of stars in the cluster, a method which might be expected to yield the most direct and accurate result. In 2009, Van Leeuwen gave the revised distance, 120.2 ± 1.9 pc, using HIP2.
For the extended sample of 70 stars the mean parallax distance is 124.1 ± 2.7 pc. The major part of the increase results because we calculated the distance from the mean parallax without weighting, while Van Leeuwen found mean parallax weighted by inverse error squared. Because parallax distances to multiple star systems are inherently less accurate, and may contain an unquantified systematic error, a probably better estimate is obtained by considering only single stars. From 45 single stars we obtain a distance of 125.6 ± 4.2 pc. This is typically within 2σ of the total error of the value found by other methods.
Van Leeuwen (2009) suggested that unknown factors in stellar evolution may result in an unmodelled difference between the positions of the Pleiades and Hyades on the CMD, and that using nearby stars of similar age and composition to the Pleiades might improve the calculation of the Pleiades distance from luminosity. In fact there is a small difference between the position of the Hyades on the CMD and the position of other groups. This is expected since the Hyades has greater metalicity and age. AB Doradus, β Pictoris, and Tucana/Horologium (and in particular AB Doradus) are thought to be similar in age and composition to the Pleiades. Indeed, AB Doradus has very close motion, and is thought to be coeval and to have an origin in space very close to the Pleiades (Ortega et al. 2007). Based on these three near groups we find photometric distance, 132 ± 3 pc, slightly less than estimates based on the Hyades. We conclude that the Hipparcos parallax distance for the Pleiades is low, but consistent within measurement errors.
Van Leeuwen grouped the Diamond cluster (NGC 2516) together with the Pleiades and Blanco 1 as having anomalously low parallax distances. For the Diamond cluster, Van Leeuwen used 11 stars, of which we have rejected three, while our list contains 21 stars, 19 with radial velocities. For our complete list of 21 stars, we found distance 388 ± 22 pc, and for singles stars we found distance 395 ± 24 pc. These figures are in good agreement with photometric distances to the Diamond cluster given by van Leeuwen. The difference in membership accounts for much of the anomaly reported by Van Leeuwen; we calculate 360 ± 14 pc, compared to 342.5 pc calculated by van Leeuwen for the same stars.
We found 16 candidates for Blanco 1, 15 with radial velocities, with separation n = 6. Van Leeuwen found 13 candidates, of which we reject one. For our list of 16 stars we found a distance of 205 ± 15 pc. For 13 single stars we found 214 ± 14 pc, compared to 207 ± 12 pc given by van Leeuwen. Hipparcos stars in Blanco 1 are too early to obtain a reliable photometric distance, but Stauffer et al. (2010) provide data for a larger sample of less bright candidate single stars, from which we find photometric distance 235 ± 10 pc. This discrepancy between photometric and parallax distances is not excessive.
Comparison between the calculated parallax distances from XHIP and distances given by Dias et al. and Karchenko et al. shows a small number of other discrepancies, which may be due to differences in candidate lists, but no systematic difference between photometric and parallax distance.
Conclusion
By making use of up to date radial velocity information to fit 3D velocity as well as position, we have found improved lists of candidates in the Hipparcos data for 87 clusters and associations. We have defined a parameter n, for separation from surrounding field stars, and used it to divide these into 42 likely clusters and 45 associations. We have calculated size and parallax distance and size for these moving groups.
We observed that the calculation of the distance to the Pleiades from Hipparcos data by van Leeuwen contains a subtle systematic error, which biases the calculation towards a low result, and that parallax distances and luminosities of binaries and multiple star systems are inherently less accurate than those of single stars. For our sample of 45 single Pleiades stars we found a mean parallax distance 125.6 ± 4.2 pc. This result is less than 2σ below most estimates by other methods, and 1.2σ less than the photometric distance, 132 ± 3 pc, calibrated to AB Doradus, β Pictoris, and Tucana/ Horologium, which are thought to be similar in age and composition. The Pleiades parallax distance is thus no longer a cause for concern.
Data XHIP can be retrieved from the Centre de Données astronomiques de Strasbourg (CDS Catalog V/137B). The list of found groups and their properties is contained in the file groups.dat.
Figure 2 :
2CMD for single stars in the Hyades & Pleiades clusters, NGC 2451A, α Perseus, Blanco 1, Collinder 135, the Ptolemy cluster, NGC 7092, NGC 2232, Collinder 140, the diamond cluster and the Scorpio-Centaurus association. Parallax distance for individual stars is used for the Hyades and cluster distance is used for the other groups. Scorpio-Centaurus uses distances for Lower Centaurus Crux, Upper Centaurus Lupus and Upper Scorpius. Reddening has been applied. There is greater scatter than in figure 1 because of the spread of distances in the Scorpius-Centaurus association, and because of pre-main sequence stars lying above the main sequence, as well as very young stars lying below it.
Table 1: The number of members, N, and cluster distance, R XHIP , are given for the full candidate lists, including multiple stars and stars with unknown radial velocity. N vL and R vL are as given by van Leeuwen (2009). R D and R K are distances given by Dias et al. and Karchenko et al. Sep is the separation parameter, n, which describes the concentration of the group and its kinematic separation from field stars.rad
1σ rad
n
pc
pc
pc
pc
pc
deg
Ursa Major
15
8
6
25.2 ± 0.3
25.3
25
3.2
7.3
β Pictoris
43
3.5
27.8 ± 2.1
24.6
118
AB Doradus
28
3.5
29.3 ± 2.4
22.1
131
Hyades
190
150
5.5
46.1 ± 0.6
46.4 ± 0.5
45
8.1
10.1
Tucana / Horologium
73
3.5
50.1 ± 1.2
48.1
73.8
Coma
43
27
6
86.3 ± 1.3
86.7 ± 0.9
96
87
7.2
4.8
Feigelson 1
7
4.5
110.8 ± 7.0
114
3.9
2.0
Lower Centaurus Crux
187
3.5
122.4 ± 1.4
19.1
9.0
Pleiades
70
53
6
124.1 ± 2.7
120.2 ± 1.9
133
130
5.8
2.7
Upper Centaurus Lupus
169
3.5
143.6 ± 2.0
23.3
9.3
Upper Scorpius
82
3.5
144.0 ± 2.7
8.7
3.5
Southern Pleiades (IC 2602) 29
15
6
148.8 ± 3.2
148.6 ± 2.0
161
160
3.4
1.3
Omicron Velorum (IC 2391)
20
11
5.5
165.6 ± 8.0
144.9 ± 2.7
175
176
2.7
0.9
Platais 2
7
4
178 ± 22
201
201
8.2
2.6
Platais 9
7
8
178.9 ± 7.5
174
200
6.9
2.2
Praesepe
40
24
6
181.8 ± 8.5
181.5 ± 6.0
187
187
7.3
2.3
NGC 2451A
20
14
6.5
186.0 ± 4.8
183.5 ± 3.7
189
188
6.0
1.8
α Perseus
80
50
4.5
186.6 ± 4.4
172.4 ± 2.7
185
190
9.6
2.9
Blanco 1
16
13
6
205 ± 15
207 ± 12
269
269
3.6
1.0
Collinder 135
14
5
247.3 ± 5.6
316
319
8.4
2.0
NGC 1647
6
6
248 ± 38
540
539
4.4
1.0
Stephenson 1
15
4
265 ± 16
390
373
4.5
1.0
Ptolemy (NGC 6475)
22
20
5.5
266 ± 11
270
301
300
3.3
0.7
IC 348
28
3.5
274 ± 16
385
394
9.9
2.1
Cep OB6
11
4.5
274.8 ± 5.7
14.9
3.1
Platais 4
30
3.5
280 ± 18
276
276
16.2
3.3
NGC 7092
14
7
6
292 ± 15
303
326
311
4.7
0.9
Stock 2
12
4
294 ± 28
303
380
6.2
1.2
Collinder 359
10
4
299 ± 27
249
640
8.7
1.7
Table 2 :
2Columns as in table 1.
Table 3 :
3Description of groups.dat, contained in CDS Catalog V/ 137B.
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New Praesepe white dwarfs and the initial mass-final mass relation. D Dobbie, MNRAS. 369D. Dobbie et al., 2006, New Praesepe white dwarfs and the initial mass-final mass relation. MNRAS, 369, 383-389.
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. N V Kharchenko, A E Piskunov, S Roeser, E Schilbach, R.-D , N. V. Kharchenko, A.E. Piskunov, S. Roeser, E. Schilbach, R.-D.
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The Washington Visual Double Star Catalog. B D Mason, G L Wycoff, W I Hartkopf, G G Douglass, C E Worley, CDS Catalog: B/wds. 122B.D. Mason, G.L. Wycoff, W.I. Hartkopf, G.G. Douglass, C.E. Wor- ley, 2001-2010, The Washington Visual Double Star Catalog, Astron. J. 122, 3466. CDS Catalog: B/wds.
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[
"Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars",
"Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars",
"Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars",
"Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars"
] |
[
"C Lemon \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"T Anguita \nDepartamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700\n\nLas Condes\nSantiagoChile\n\nMillennium Institute of Astrophysics\nMonseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile\n",
"M Auger \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n",
"F Courbin \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"A Galan \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"R Mcmahon \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n",
"F Neira \nDepartamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700\n\nLas Condes\nSantiagoChile\n\nMillennium Institute of Astrophysics\nMonseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile\n",
"M Oguri \nCenter for Frontier Science\nChiba University\n1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan\n\nDepartment of Physics\nGraduate School of Science\nChiba University\n1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan\n",
"P Schechter \nMIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA\n\nMIT Department of Physics\n02139CambridgeMAUSA\n",
"A Shajib \nDepartment of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA\n\nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n",
"† ",
"T Treu \nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n",
"C Lemon \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"T Anguita \nDepartamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700\n\nLas Condes\nSantiagoChile\n\nMillennium Institute of Astrophysics\nMonseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile\n",
"M Auger \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n",
"F Courbin \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"A Galan \nInstitute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland\n",
"R Mcmahon \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n",
"F Neira \nDepartamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700\n\nLas Condes\nSantiagoChile\n\nMillennium Institute of Astrophysics\nMonseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile\n",
"M Oguri \nCenter for Frontier Science\nChiba University\n1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan\n\nDepartment of Physics\nGraduate School of Science\nChiba University\n1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan\n",
"P Schechter \nMIT Kavli Institute for Astrophysics and Space Research\n02139CambridgeMAUSA\n\nMIT Department of Physics\n02139CambridgeMAUSA\n",
"A Shajib \nDepartment of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA\n\nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n",
"† ",
"T Treu \nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n"
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"Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Institute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland",
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"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA",
"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA",
"Institute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland",
"Departamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700",
"Las Condes\nSantiagoChile",
"Millennium Institute of Astrophysics\nMonseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile",
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Institute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland",
"Institute of Physics\nLaboratoire d'Astrophysique\nEcole Polytechnique Fédérale de Lausanne (EPFL)\nObservatoire de SauvernyCH-1290VersoixSwitzerland",
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Departamento de Ciencias Fisicas\nUniversidad Andres Bello\nFernandez Concha 700",
"Las Condes\nSantiagoChile",
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"Center for Frontier Science\nChiba University\n1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan",
"Department of Physics\nGraduate School of Science\nChiba University\n1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan",
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"Department of Astronomy and Astrophysics\nUniversity of Chicago\n60637ChicagoILUSA",
"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA",
"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA"
] |
[
"MNRAS",
"MNRAS"
] |
We report the spectroscopic follow-up of 175 lensed quasar candidates selected using Gaia Data Release 2 observations followingLemon et al. (2019). Systems include 86 confirmed lensed quasars and a further 17 likely lensed quasars based on imaging and/or similar spectra. We also confirm 11 projected quasar pairs and 11 physical quasar pairs, while 25 systems are left as unclassified quasar pairs -pairs of quasars at the same redshift, which could be either distinct quasars or potential lensed quasars. Especially interesting objects include 8 quadruply imaged quasars of which two have BAL sources, an apparent triple, and a doubly lensed LoBaL quasar. The source redshifts and image separations of these new lenses range between 0.65-3.59 and 0.78-6.23 respectively. We compare the known population of lensed quasars to an updated mock catalogue at image separations between 1 and 4 arcseconds, showing a very good match at < 1.5. At > 1.5, only 47% of the predicted number are known, with 56% of these missing lenses at image separations below 1.5 . The missing higher-redshift, small-separation systems will have fainter lensing galaxies, and are partially explained by the unclassified quasar pairs and likely lenses presented in this work, which require deeper imaging. Of the 11 new reported projected quasar pairs, 5 have impact parameters below 10 kpc, almost tripling the number of such systems, which can probe the innermost regions of quasar host galaxies through absorption studies. We also report four new lensed galaxies discovered through our searches, with source redshifts ranging from 0.62 to 2.79.
|
10.1093/mnras/stac3721
|
[
"https://export.arxiv.org/pdf/2206.07714v1.pdf"
] | 249,712,144 |
2206.07714
|
4b1189bf4b14724cde0b200dd45add3a0e8224e1
|
Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars
2022
C Lemon
Institute of Physics
Laboratoire d'Astrophysique
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Observatoire de SauvernyCH-1290VersoixSwitzerland
T Anguita
Departamento de Ciencias Fisicas
Universidad Andres Bello
Fernandez Concha 700
Las Condes
SantiagoChile
Millennium Institute of Astrophysics
Monseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile
M Auger
Institute of Astronomy
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
Kavli Institute for Cosmology
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
F Courbin
Institute of Physics
Laboratoire d'Astrophysique
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Observatoire de SauvernyCH-1290VersoixSwitzerland
A Galan
Institute of Physics
Laboratoire d'Astrophysique
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Observatoire de SauvernyCH-1290VersoixSwitzerland
R Mcmahon
Institute of Astronomy
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
Kavli Institute for Cosmology
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
F Neira
Departamento de Ciencias Fisicas
Universidad Andres Bello
Fernandez Concha 700
Las Condes
SantiagoChile
Millennium Institute of Astrophysics
Monseñor Nuncio Sotero Sanz 100, Oficina 1047500011Providencia, SantiagoChile
M Oguri
Center for Frontier Science
Chiba University
1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan
Department of Physics
Graduate School of Science
Chiba University
1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan
P Schechter
MIT Kavli Institute for Astrophysics and Space Research
02139CambridgeMAUSA
MIT Department of Physics
02139CambridgeMAUSA
A Shajib
Department of Astronomy and Astrophysics
University of Chicago
60637ChicagoILUSA
Department of Physics and Astronomy
University of California
90095Los AngelesCAUSA
†
T Treu
Department of Physics and Astronomy
University of California
90095Los AngelesCAUSA
Gravitationally lensed quasars in Gaia -IV. 150 new lenses, quasar pairs, and projected quasars
MNRAS
0002022Accepted XXX. Received YYY; in original form ZZZPreprint 17 June 2022 Compiled using MNRAS L A T E X style file v3.0gravitational lensing: strong -quasars: general -methods: observational
We report the spectroscopic follow-up of 175 lensed quasar candidates selected using Gaia Data Release 2 observations followingLemon et al. (2019). Systems include 86 confirmed lensed quasars and a further 17 likely lensed quasars based on imaging and/or similar spectra. We also confirm 11 projected quasar pairs and 11 physical quasar pairs, while 25 systems are left as unclassified quasar pairs -pairs of quasars at the same redshift, which could be either distinct quasars or potential lensed quasars. Especially interesting objects include 8 quadruply imaged quasars of which two have BAL sources, an apparent triple, and a doubly lensed LoBaL quasar. The source redshifts and image separations of these new lenses range between 0.65-3.59 and 0.78-6.23 respectively. We compare the known population of lensed quasars to an updated mock catalogue at image separations between 1 and 4 arcseconds, showing a very good match at < 1.5. At > 1.5, only 47% of the predicted number are known, with 56% of these missing lenses at image separations below 1.5 . The missing higher-redshift, small-separation systems will have fainter lensing galaxies, and are partially explained by the unclassified quasar pairs and likely lenses presented in this work, which require deeper imaging. Of the 11 new reported projected quasar pairs, 5 have impact parameters below 10 kpc, almost tripling the number of such systems, which can probe the innermost regions of quasar host galaxies through absorption studies. We also report four new lensed galaxies discovered through our searches, with source redshifts ranging from 0.62 to 2.79.
INTRODUCTION
Gravitationally lensed quasars offer several unique probes of astrophysics and cosmology. Their variable sources allow measurements of the time delays between images, which can be used to constrain the Hubble constant through time-delay cosmography (Shajib et al. 2020;Harvey 2020). The compact nature of the source allows a probe of the mass at compact scales in the lensing galaxy through ★ E-mail: [email protected] † NHFP Einstein Fellow microlensing; the brightness variations caused by stars as they pass in front of each quasar image can be used to constrain the Initial Mass Function (Jiménez-Vicente & Mediavilla 2019; Schechter et al. 2014) and place limits on the abundance of primordial black holes (Mediavilla et al. 2017). Simultaneously, microlensing can constrain the innermost structure of quasars at high-redshift (Hutsemékers & Sluse 2021; Paic et al. 2022). The particular brightness of quasar sources provides an excellent tool for studying the geometry and kinematics of intervening matter (Rubin et al. 2018;Okoshi et al. 2021;Lemon et al. 2022). This brightness also means that lensed quasars are often source-selected (Myers et al. 2003;Oguri et al. 2006), in contrast with lensed galaxy samples which are selected through their bright lensing galaxies. This offers a chance to create a well-defined statistical sample down to small separations, which can be used to probe cosmology and galaxy evolution parameters (Chae 2003;Oguri et al. 2012). These studies are often limited by the small number of lenses that suit their particular purpose, for example requiring four well-separated images with time delays above 10 days for time-delay cosmography. Recent optical to near-infrared surveys now provide an efficient way to select high-confidence quasar candidates across the whole sky (e.g. Stern et al. 2012), and have the resolution and depth to identify whether such systems are lensed, i.e. whether there are multiple images and a lensing galaxy present (e.g. Chan et al. 2022). However, lensed quasars are particularly rare, and outnumbered by convincing contaminant systems, mainly composed of compact star-forming galaxies and chance quasar/star projections (Treu et al. 2018). The all-sky space-based catalogues from Gaia offer a particularly effective way of removing these contaminants. In the previous papers of this series, we have shown that combining the Gaia catalogue detections and parameters with WISE and ground-based optical imaging can efficiently remove the common contaminants of lensed quasar searches (Lemon et al. 2017(Lemon et al. , 2018(Lemon et al. , 2019. In this paper, we continue our spectroscopic follow-up campaign of lensed quasar candidates selected with the techniques developed in these papers.
In Section 2, we describe the selection techniques and datasets used for candidate selection. Follow-up spectroscopy and imaging are described in Section 3. Section 4 details the characteristics of individual systems, and a general discussion on the samples and sub-samples are discussed in Section 5. We conclude in Section 6.
CANDIDATE SELECTION
Candidates were selected following Lemon et al. (2019), and represent the follow-up of different Right Ascension and Declination ranges, as well as untargeted systems in previously covered sky, due to lack of spectroscopic follow-up time. Two searches begin from a catalogue of quasars and quasar candidates. For these searches we used the Milliquas catalogue (Flesch 2015), which consists of confirmed quasars and highly likely quasar candidates based on X-ray, radio, and optical and infrared colour selection. We supplement this catalogue with objects using colours from the Wide-field Infrared Survey Explorer (WISE). We take objects with W1-W2 > 0.4 (in Vega), which is less strict than most quasar selections in order to include systems with possibly large contamination from the lensing galaxy, or sources with non-standard quasar colours. Our final search begins from a morphological galaxy catalogue. Below, we briefly describe the selection techniques for lensed quasars within these quasar and galaxy samples, making use of Gaia data release 2 (GDR2, Gaia Collaboration et al. 2018). Further details on selection numbers and completeness of recovering existing samples can be found in Lemon et al. (2019).
Multiple Gaia detections around quasar catalogues
This search is refined from that originally performed in Lemon et al. (2018), namely searching for multiple Gaia detections around red ALLWISE detections (Mainzer et al. 2011). The search consists of visually inspecting ground-based colour images of systems satisfying varying W1-W2 colour cuts, local stellar densities and thresholds on the significance of proper motion for the different numbers of Gaia detections associated to the system. The Milliquas catalogue is also used as a starting catalogue for this selection.
Modelling unWISE pixels
The main contaminants of the search detailed above are quasar+star projections, since they will have multiple Gaia detections, often lacking the necessary proper motion precision to remove the star as a moving object, and having quasar-like colours in the infrared, radio, or X-ray due to blending issues in WISE (PSF FWHM of ∼5 ). Ground-based imaging can often rule out such pairs through optical colour differences or a lack of a lensing galaxy, however contaminants still remain and the number of objects is too large to visually inspect. Lemon et al. (2019) introduced a forced model fitting of WISE data based on Gaia positions to extract the infrared colours of nearby point sources and thus classify the multiple components within such systems. We used the unWISE W1 and W2 data (Lang et al. 2016), an unblurred coaddition of all WISE data to-date. Following local Gaia density cuts, proper motion significance cuts, and finally cuts in the G-W1 vs. W1-W2 colour space, we obtained a classification procedure to retain nearly 99 per cent of known lensed quasars, while removing ∼80 per cent of previously followed-up contaminants.
Gaia detections offset from LRGs
Since both search techniques presented above rely heavily on source selection, we also investigate using galaxies as a starting catalogue. We use spectroscopic galaxies from SDSS DR12, and keep all systems with one Gaia detection within 1 and 3.5 , or two Gaia detections within 4.5 (and separated by less than 4.5 ). The need for the single detection to be further than 1 is based on known lensed quasars satisfying such a constraint, and reduces contamination. We also use morphological galaxies from Pan-STARRS, requiring at least one Gaia detection between 1 and 3 away. Proper motion and astrometric excess noise cuts are performed on all matching Gaia detections to further remove contamination from stars and star-forming galaxies.
OBSERVATIONS
Spectroscopy
William Herschel Telescope: ISIS
55 systems were observed with the Intermediate-dispersion Spectrograph and Imaging System (ISIS) on the WHT on the nights of 11-12 Feb. 2019. The R158R and R300B gratings were used for the red and blue arms respectively, with the standard 5300Å dichroic and GG495 second order cut filter in the red arm. This provided dispersions of 1.81Å pixel −1 and 0.86Å pixel −1 for the red and blue arms.
Nordic Optical Telescope: ALFOSC
43 systems were observed with grism #4 and the Alhambra Faint Object Spectrograph and Camera (ALFOSC) on the NOT on 1-2 October 2019, providing a dispersion of 3.3Å pixel −1 . All spectra were reliably deblended, except J2017+6204, for which high-resolution imaging confirms it as a lensed quasar.
New Technology Telescope: EFOSC2
79 systems were observed with grism #13 of the ESO Faint Object Spectrograph and Camera version 2 (EFOSC2) over four runs (NTT1, 0104.A-0194(A): 22-24 Oct. 2019; NTT2, 0104.A-0194(B): 18-20 Jan. 2020; NTT3, 106.218K.001: 19-21 Oct. 2020;and NTT4, 106.218K.002: 15-17 Jan. 2021), providing a dispersion of 2.77Å pixel −1 .
General reduction procedure
Since several spectrographs were used for long-slit spectroscopic follow-up of our candidates, we give a brief description of the common steps. The extraction follows that outlined in Lemon et al. (2022), and fits multiple Moffat components, the parameters for which are determined from the binned data themselves. In particular, the following steps are taken:
• After bias-subtracting each image, cosmic rays are found and masked as all pixels above a certain threshold in the absolute Laplacian of the data. For the brightest objects this value is increased depending upon visual inspection of the cosmic ray mask.
• The sky background is subtracted by determining the median value within the pixels either side of the trace of the object (between 20 and 80 pixels away). We later use this same background to correct the wavelength calibration to bright sky lines. The region is also used to determine the sky background noise at each spectral pixel (i.e., wavelength).
• A Poisson noise map is created using the detector gain, and added in quadrature with the sky noise to provide a pixel noise map.
• A wavelength solution is determined from fitting the lines of an arc exposure (HeNe or CuAr) taken the afternoon before observations. An absolute shift is calculated for each exposure by measuring the positions of the Na 5892Å, [O ]-5577Å and 6300Å sky lines. This is often a significant shift ( 1 pixel) and shows variation with wavelength, and so absolute wavelength measurements should not be trusted to more than the approximate pixel dispersion value of each instrument/grism combination.
Imaging
Deep imaging is key to identifying the lensing galaxy or counterimage in lensed quasar candidates, and thus we use several imaging datasets throughout this work. Principally we use grizY imaging from Pan-STARRS (Chambers et al. 2016), and the grz imaging from the DESI Legacy Imaging Surveys (Dey et al. 2019). When higher-resolution imaging is required or a system is investigated for other catalogue detections, we check the Canadian Astronomy Data Centre archive, Vizier catalogues within 5 arcseconds (Ochsenbein et al. 2000), and nearby NED detections and associated references. For some quadruply-imaged systems HST multi-band imaging and models are presented in Schmidt et al. (2022).
General imaging analysis
When analysing individual systems, we often want to fit the pixels with a combination of analytic profiles. Most commonly, we want to identify if a lensing galaxy is present between two bright quasar PSFs. For each image available, we use a nearby star in the field and fit this as a Moffat profile to determine the PSF of the system (Moffat 1969). We then fit a square 10×10 cutout around the system, using the relevant number of point sources. This process is performed simultaneously in all bands (or possibly a single band), and if a galaxy profile is required we use a Sérsic profile (Sérsic 1963). The free parameters are the position(s) of the point sources, the position(s) of the galaxies, and their associated effective radii, Sérsic indices, and ellipiticity parameters. To find the best-fit parameters we use the Markov chain Monte Carlo (MCMC) sampler Python package (Foreman-Mackey et al. 2013).
Results
The resulting classifications from analysing both the available imaging and spectroscopy of each system are listed in Table 1, with the selection method, maximum image separation, and Gaia magnitudes and proper motion significances.
We broadly classify the systems into the following categories: definite lensed quasars, likely lensed quasars (based on the subjective opinion and experience of the authors), unclassified quasar pairs (systems in which two quasars are spectroscopically resolved and are at the same redshift however no convincing lensing galaxy is seen but also for which there is no clear evidence for them being physically distinct quasars), binary quasars (Section 5.2), projected quasars (Section 5.3), and contaminant systems (definitely not lenses). We note that our use of unclassified quasar pairs is akin to Nearly Identical Quasars (NIQs) which has been used in recent literature (e.g. Anguita et al. 2018), however the change is simply to include those systems which have low signal-to-noise data and could readily be classified as a likely lensed quasar or binary quasar given deeper spectra.
Extracted imaging and spectra of systems are respectively shown in Figures 1 and 3 for confirmed lensed, 4 and 5 for likely lensed quasars, and 6 and 7 for unclassified quasar pairs. We also show the spectra for binary quasars and projected quasars in Figures 8 and 9 respectively.
NOTES ON INDIVIDUAL SYSTEMS
In this section we comment on any interesting aspects of the individual systems, including their photometry, spectra, and mass modelling. We divide the systems into sections based on their final classification.
Confirmed Lensed Quasars
J0030-3358
A jump in the continuum of the fainter image of this double is likely associated to the 4000Å break of the lensing galaxy, as associated Ca H, K and G-band absorption features are also seen, placing the lens at z=0.715. This is supported by the Legacy Survey photometric redshift estimate of 0.84±0.06 (Zhou et al. 2021), with a small discrepancy likely due to blending with the quasar light. There is absorption on the blue side of the C line seen in both images, however this could also be attributed to Milky Way absorption since it is coincident with z=0 Ca H and K absorption.
J0116+4052
The spectra show two quasars at z=1.85 with very similar continuum and broad emission line profiles, suggesting the system is a likely lens. Only shallow Pan-STARRS imaging exists for this system and shows no significant residuals upon subtraction of two PSFs. Follow-up OSIRIS imaging reveals a third source closest to the Western image, which we take as a detection of the lensing galaxy ( Figure 2).
J0152-2448
This double system at = 1.69 was also independently discovered by Spiniello et al. (2019).
J0200-1509
The EFOSC2 data shows a blended quasar spectrum at z=1.185. Given the two obvious Gaia detections and red residual flux upon subtraction of two PSFs in Legacy Survey imaging, which is well-fit by simultaneously including a Sersic which falls between the two PSFs, we designate this system as a lens.
J0232-2429
This system was originally selected by Lemon et al. (2018), who classified it as a likely quasar+star projection, despite poor seeing during observations and the small separation of 1.44 . However, we obtained further spectroscopy to confirm this classification, and resolve two quasars at = 1.58 with similar emission line profiles and continuum. There is residual red flux in the PSF-subtracted Legacy Survey image, consistent with a lensing galaxy. We therefore upgrade the classification to a secure lens. The system is in the outskirts of a cluster (∼ 1 from the centre) which has galaxies with photometric redshifts of ∼0.85. The quasar is radio loud with J0116+4052 J2017+6204 J2132+3635 Figure 2. OSIRIS imaging of three systems. White scale bars represent 1 arcsecond. North is up, East is left. Galaxies can be seen between the quasar images for J0116+4052 ( = 1.85) and J2017+6204 ( = 1.73), however nothing is seen for J2132+3635 ( = 1.215), therefore it is likely a physical binary.
detections in NVSS (4.7±0.7 mJy at 1.4GHz) and VLASS (2.7±0.4 mJy at 2-4GHz).
J0247+7706
The ALFOSC-NOT spectra clearly resolve two quasar traces at z=2.73 and many narrow absorption lines common to both components. Faint hints of an extended galaxy are seen in the PSF subtracted Pan-STARRS imaging. When adding a Sersic component to the fit, flux is allocated in all bands to this component, and its best-fit position is between the two PSFs closer to the fainter image. Given this and the very similar spectra, we classify this system as a lens, however deeper imaging should be taken to verify the detection of the lensing galaxy and characterise its properties.
J0247-6349
The seeing for this spectrum was particularly poor, leading to low signal-to-noise and a blended spectrum. The 2D spectral modelling allocates emission line flux to both components, however cannot be taken as reliable. Nevertheless, the emission lines are clearly from a z=2.295 quasar, and the Legacy Survey images show two point sources either side of a red galaxy. Therefore we classify this system as a lens. This is supported by the independent spectra and discovery of this system by Spiniello et al. (2019).
J0310-5545
This double was selected as a single Gaia detection near a possible LRG, however the Gaia detection is not associated to one of the quasar images but to a nearby star. The possibility of two further point sources around the nearby galaxy was noticed serendipitously during visual inspection. Deep EFOSC2-NTT spectra aligned to capture the galaxy and two faint sources clearly show three traces, two of which contain emission lines of a quasar at z=2.31, and the middle trace being an LRG at z=0.298, therefore confirming this system as a lensed quasar.
J0325-2232
The spectra show emission lines of a z=1.35 quasar in both components, however the continua and relative emission line amplitudes are clearly discrepant, with stronger broad emission lines in the redder image. Legacy Survey imaging shows a galaxy between the two components, which is well fit by a Sersic positioned between the PSFs and slightly nearer the South-Western image, B. In the -band of both Pan-STARRS and Legacy Survey imaging, A is brighter than B, however B is brighter in both Gaia (by 0.21 mag in the broad G filter) and in the ground-based red filters, even when including the galaxy in the fit. Absorption is present around 4785Å and 4890Å in only the redder image. This system is likely a lens with contamination and reddening by the lensing galaxy coupled with microlensing. Deeper spectra will be useful for understanding the nature of the discrepant spectra.
J0329-0208
This double lensed quasar lies 3.5 from the centre of the cluster MACS J0329-0211. The HST coverage of the cluster narrowly misses this lens system, however Chandra ACIS-I imaging does cover the system, revealing an extended X-ray source with a measured flux of 4.11×10 −14 erg cm −2 s −1 in the 0.3-8 keV range (Wang et al. 2016). The lensing galaxy may be a member of this z=0.45 galaxy cluster, however there are no obvious absorption lines in our spectra around this redshift.
J0339-6121
This double shows signs of a LoBAL quasar source at ≈1.89, with the troughs blueshifted by around 0.028c, typical of many BAL quasars (Rankine et al. 2020). While the absorption around 5762Å could be attributed to Fe -UV48 triplet, it is more likely to be due to the CrII-2062Å triplet since we do not see strong evidence for the defining FeLoBAL FeII UV1 and UV2 lines. The flux ratio of the images across wavelength is approximately constant (2.4:1) but shows clear differences around several absorption lines, suggesting microlensing is present in the system, a particularly useful tool for constraining the quasar source geometry (Hutsemékers et al. 2010;Sluse et al. 2015).
J0350-4611
The NTT-EFOSC2 spectra of this double are low signal-to-noise but each component shows emission lines of a z=1.505 quasar. The C profile is not apparent, and there is a significant drop in flux at the expected wavelength and blueward of C in both components. The lens is seen in the Legacy Survey PSF-subtracted residuals. Deeper spectra should be taken to understand the nature of this system, as it is possibly a useful lensed BAL quasar.
J0405-3730
The 2D spectra of this double show the resolved traces of the images of a quasar at z=3.585, but also three narrow emission lines located between the two quasar traces, but much closer to the brighter trace. These lines are consistent with O , H and S at z=0.2295, and a narrow absorption line in the brighter trace can also be attributed to Na at this redshift. The imaging of this system shows that at least two galaxies are responsible for the lensing, one between the two images but offset and highly elliptical and another offset. We postulate that the lens is a merger between two galaxies, triggering star formation explaining the aforementioned narrow lines. Highresolution imaging will be needed to better understand this system.
J0416+7428
This double has a very low redshift lens (z=0.098), which appears to be part of a group with two nearby galaxies apparent in the Pan- . Spectra of the confirmed lenses in the observed frame. The most prominent quasar emission lines are marked with vertical lines, and details on certain spectra can be found in Section 4. In some cases one or several spectra have been offset for visualisation, and the offset is marked on the y-axis. STARRS imaging. In the 2D spectra, narrow emission lines are seen both 4.6 arcseconds from the main lensing galaxy in the direction of the more distant image, and also 13 arcseconds in the other direction coincident with diffuse extended continuum emission for the latter. These lines are consistent with H and N emission at the lens galaxy redshift, therefore we attribute the former to a star-forming region in the outskirts of the main lensing galaxy, and the latter to star formation in the outskirts of the galaxy group member to the West. The lensing galaxy has an i-band brightness of 16.22. The low redshift of the lensing galaxy is rare amongst lensed quasars, with the only known lensed quasar with a redshift lower than this being the Einstein cross (Huchra et al. 1985). Another double in our sample, J1526-1400, also has a low-redshift, bright lensing galaxy, so we investigate the expected number of lenses as a function of lensing galaxy brightness after describing J1526-1400 in Section 4.1.42.
J0457-7820
This system was selected through the Gaia multiplet WISE search, with three Gaia detections ( =18.81, 19.07, and 19.60) and W1-W2=0.95. Archival NOAO DeCAM imaging shows three bright blue point sources around a galaxy. The follow-up NTT-EFOSC2 spectra, positioned to capture light from all three images simultaneously, are low signal-to-noise but clearly show three traces of a z=3.145 quasar. Three -band and three -band exposures, each of 120s were taken with the Magellan Instant Camera, MagIC. A second galaxy is apparent to the North-West of the system. All six exposures were fit simultaneously as two galaxies and three point spread functions, with the PSF being fit as a grid of pixels, and constrained directly from the data (following Ostrovski et al. 2018). Figure 10 shows the colour-image stack (where the green channel is a median of the and bands), and residuals. No counterimage is detected. We repeat the image subtraction but artificially include a point source of increasing brightness until it is obvious in the residuals. This provides an upper limit on any counterimage brightness being 1% as bright as image B, i.e. a -band magnitude of ∼24.0. One possibility is that this is a rare three image system caused by the straddling of two galaxies. Shin & Evans (2008) provide a comprehensive catalogue of image configurations caused by lensing of two separated galaxies, modelled as Singular Isothermal Spheres. Several configurations allow for three image regions within the resulting caustics, where the third image is not necessarily strongly demagnified. Given that all images lie between the two galaxies, the most likely configuration is 3B-1, following the labelling of Shin & Evans (2008). This naked catastrophe occurs when the two astroid caustics merge in a beak-to-beak calamity (see, e.g., Kassiola et al. 1992;Orban de Xivry & Marshall 2009), but the pseudo-caustics do not overlap, creating a naked merged cusp region. Introducing external shear and mass ellipticity can create naked cusps which are not necessarily merged, again creating source plane regions corresponding to three bright images.
To test which is the relevant scenario for J0457-7820, we perform mass models based on the astrometry from fitting the MagIC imaging. We use only the positions to constrain the mass model, since microlensing, extinction, and variability over the time delay can cause optical image fluxes to vary significantly from their macromodel values. This gives us only 6 free parameters. A single SIE model fixed to the main lensing galaxy, predicts a fourth image, only 2.5% the brightness of B. Such an image could be consistent with our 1% detection threshold coupled with reddening and microlensing demagnifying such an image, however, the model is unphysical with an axis ratio of 0.2, and the image positions are not well recovered. Including shear in the model recovers the image positions exactly, but also predicts a fourth image, about 1% as bright as B. The lensing galaxy mass is also less elliptical (0.14) but a large shear of 0.35 is required, in the direction of the second galaxy.
We note that this model has more free parameters than constraints, however some parameters are still well-constrained. This model can be forced to have three images with very little change in the mass model parameters and well-fit images ( 2 ∼ 0.4). Given the shear direction, we investigate two component mass models. SIS+SIS or SIS+SIE models do not recover the three image positions. An SIE+SIE model fits the images and does not predict a fourth image, however the galaxies are predicted to have large ellipticities and lie in the same direction (∼25 degrees North of West). Adding shear to this model reduces the ellipticities of the mass components and still recovers only three images, however, the lens parameters are not as well constrained. We show the caustics of this best-fit model in Figure 10. In this instance the cusps do not merge, but since the model is still under-constrained further constraints from the lensed source host galaxy (and limits on a possible fourth image) will be needed to determine more reliable caustic configurations.
J0607-2152
This quad was confirmed during the 2019 WHT observing run, in which the system was targetted at two position angles: through the brightest image and each of the other two Gaia detections. The traces are resolved in the red arm, however in the blue arm of one observation, the images are not resolved, hence we show only the red arm extraction for the fainter image in Figure 3. The extracted fluxes for the brightest image between the two exposures are in good agreement in the red arm, however in the blue arm they are in disagreement likely due to slit losses and blending. Figure 11 shows the presence of two further objects between and to the East of the 3 Gaia detections. Including a galaxy and a further PSF fits the data to the noise, so we expect that these are the lensing galaxy and fourth quasar image. Using these 5 positions, we are not able to fit an SIE+shear model to the data, suggesting a more complex lensing model is required, or the fourth image is much closer to A. This system was also independently confirmed by Stern et al. (2021). They suggest that the fourth image is between the Northern components, however this is not supported by the aforementioned residuals.
J0608+4229
This system has three Gaia detections with =18.22, 19.86, 20.12. It was confirmed during the Feb 2019 WHT observing run, in which the system was targeted at two position angles: both through the brightest Gaia detection and each of the other two detections. The forced fitting extracts very similar spectra for each component. The multiple extractions of the brightest image are in good agreement, with only a slight mismatch in the relative amplitudes in the red arm, likely due to the flux-matching procedure of the blue and red arm spectra. Modelling the system as three PSFs reveals residual flux just south of the brightest image, and modelling the systems as four PSFs fit the Pan-STARRS data to the noise, as shown in Figure 12. We note, however, that no lensing galaxy is seen upon PSF subtraction, but this is often expected to be the case for higher redshift sources (here, z=2.34). An SIE model fit to the four image positions shows A is observed brighter than the model, and D is observed fainter. Since D is a saddle point in this model, microlensing could be the cause of its relative demagnification (Schechter & Wambsganss 2002). This system was also independently confirmed by Stern et al. (2021).
J0628-7448
Archival DECam data of this system reveal significant residuals after subtracting two PSFs, and flux is seen in the residual 2D NTT-EFOSC2 spectra towards redder wavelengths. Given also the similarity of the spectra, we classify this system as a lens. The system is associated with an XMM Slew Survey catalogue detection with a flux of 1.6±0.8×10 −12 erg cm −2 s −1 in the 0.2-12keV band.
J0635+6452
This is a large flux-ratio, 3.06 -separation double. The Pan-STARRS gri data appear consistent with only a PSF+galaxy, however a Gaia detection associated with the galaxy suggested the presence of a counterimage, which our WHT spectrum confirms.
The fainter image has dropped from =20.57 to =20.86 between GDR1 and GDR2.
J0643+2725
This double has a bright lensing galaxy at z=0.185. In the Pan-STARRS imaging, the quasar image closest to the lensing galaxy is the brightest (and is likely resolved since the Gaia astrometric excess noise -a catalogue parameter correlating with extendedness -is 2.57, a small value compared to most galaxies).
J0659+1629
This lens is a short axis quad, with a large Einstein radius, ∼2.3 . We obtained spectra of the two brightest images, confirming the source to be a quasar at z=3.09. The best fit SIE+shear model has a magnification of 25, and a modest shear of 0.06. The model predicts image D to be 60% fainter than observed. Since this system has a large time delay between image D, we expect that variability over the time delay can explain this discrepancy. This system was
J0734+1915
The 2D spectrum shows the traces of two quasars at z=0.745 however the signal to noise of the fainter quasar is low, making comparison difficult. Both Legacy Survey and Pan-STARRS imaging reveal significant residuals in the PSF-subtracted image, which are well fit when including a galaxy which falls betweeen the two PSFs but closer to the fainter one. We classify this system as a lens, however we suggest prioritising high-resolution imaging or deeper spectroscopy of this system to verify the lensing hypothesis.
J0803+3908
This system was originally selected with the unWISE modelling technique, given it has two Gaia detections separated by 0.91 . Good-seeing CFHT -band data reveal a fold configuration quad. Figure 13 shows these data, with a 4 PSF fit and subtraction, showing faint extended residuals possibly due to a lensing galaxy. Very good seeing conditions allowed spatially resolved spectra between the bright pair and the Northern image, confirming the source to be at z=2.97. Schindler et al. (2018) confirmed this source as a bright quasar at z=2.975 in their North Galactic Cap sample of the Extremely Luminous Quasar Survey (ELQS-N), however they did not comment on any possibility of it being gravitationally lensed. Such catalogues present a good opportunity for finding compact lensed quasars thanks to the magnification bias of lensing.
J0818-2613
This system was confirmed during the 2019 WHT observations. It was discovered as four Gaia detections associated with a red ALLWISE detection. Spectra were taken at two position angles, each through the bright pair, and clearly show that the source is a z=2.155 BAL quasar with multiple velocity outflows (e.g., Korista et al. 1993). The system is in a high stellar density environment, since it is at a galactic latitude of +5.4 • , an area which has been purposefully omitted from previous searches due to high contamination rates. The strong constraints from requiring four Gaia detections around photometric quasars allows us to now probe such high-density regions. The Gaia magnitudes are 17. 51, 17.58, 19.73, and 19.94, and the system has 3 ROSAT counts within 25 . The system was targeted for spectroscopic follow-up despite the high stellar density and odd configuration, since the SEDs from optical to infrared wavelengths for all images are similar, and a faint lensing galaxy is seen upon subtraction of the PSFs. The image configuration is peculiar for a quad. When fitting an SIE+shear model to the 4 image positions and relative fluxes, the lensing galaxy position is recovered to within 0.5 . A model using the galaxy position provides a total magnification of 58, a galaxy mass ellipticity of 0.61, and position angle 34 degrees East of North, with a strong shear of 0.38 at a position angle 126 degrees East of North. This model seems highly unphysical with such a large shear and orthogonal mass and shear axis directions, and is perhaps indicative of the lensing potential not being simple. Indeed, the large Einstein radius of 2.96 is suggestive of a galaxy group or cluster acting as the lens. The lens was also independently confirmed by Stern et al. (2021).
J0826+7002
The system was originally targeted in a GDR1 search (Lemon et al. 2018), as it was selected as a Gaia singlet near a morphological Pan-STARRS galaxy, however the WHT spectra were not deep enough to confirm the fainter object as a quasar. It was targeted again after it was recovered by the same search technique in GDR2, and better seeing conditions and lower airmass observations reveal both Gaia detections to be quasars at z=1.62. Both images are detected in GDR1 and GDR2, but the separation is 5.82 , hence the system was not discovered by the unWISE modelling or WISE-Gaia doublet selection. Pixel modelling shows three lensing galaxies lying closer to the brighter image. There is a third Gaia detection in Gaia EDR3 on the Western galaxy of these three, perhaps hinting at a third image of the quasar blended with the galaxy. High-resolution imaging is needed for this system.
J0833+2612
This lens was selected as a single Gaia detection offset from a Pan-STARRS morphological galaxy (Section 2.3). The slit was positioned at 167.4 degrees East of North, and clearly resolves two traces of a BAL quasar at z≈3.26. The DECALS residual imaging shows a likely faint counterimage, confirming this system as a quad. CFHT rand z-band imaging also show a counterimage, and our PSF-subtracted modelling of these data is shown in Figure 14.
There is a coincident signal in VLASS at the location of the system, suggesting the source could be radio-loud.
J0907+6224
This double has both SDSS and BOSS spectra. It has a source redshift of z=1.86 and a separation of 2.48 . The BOSS spectrum shows two narrow absorption features in the C line. The different continua between the BOSS and SDSS spectra is likely attributed to the different fibre apertures (2 and 3 respectively), and contamination from the lensing galaxy. The SDSS pipeline catalogues this as two objects, with an i-band magnitude of 18.93 for the Northern component, allowing a colour selection from SQLS to select this as a potential lens. The likely reason this was not selected was due to a colour difference in gr between the two components; 0.54 and 1.01 for the quasar and companion respectively (Oguri et al. 2006).
J0911-0948
This double lens is associated with an XMM serendipitous source with a flux of 8.6±1.3×10 −14 erg cm −2 s −1 in the 0.2-12keV band (Webb et al. 2020).
J0918-0220
This double was confirmed during the 2019 WHT observations, revealing a source redshift of 0.803, and a lens redshift of 0.460. This system was also independently selected as a lens candidate by Sonnenfeld et al. (2020) and confirmed in Jaelani et al. (2021) who report consistent redshifts.
J0921+3020
This 2.93 -separation double has a source redshift of z=3.33, with one bright image ( =18.66 with SDSS and BOSS spectra), and one faint image ( =20.86) blended with a galaxy. The lens redshift of this double is seen in the difference spectra between the two components, showing clear absorption lines at z=0.428. There are two galaxy clusters surrounding this lens, with spectroscopic redshifts from SDSS spectra of z=0.334, z=0.427. Therefore, the lens belongs to the latter cluster. The system lies in the Smithsonian Hectospec Lensing Survey (SHELS) four degree field, and both objects have SHELS spectra of z=3.33 quasars (Geller et al., 2014). The system has radio and X-ray counterparts in FIRST (4.33±0.14 mJy at 1.
J0936-1211
The NTT-EFOSC2 spectrum shows two clear traces, one of a z=2.00 quasar with a very blue continuum, and one red trace with absorption features of a z=0.260 galaxy. There are hints of broad emission features including C and the blue continuum in this latter trace. Indeed, the two Gaia detections coupled with the alignment of the background quasar within 2.1 of the foreground galaxy strongly suggests the presence of a counterimage. Two PSFs and a galaxy are required for clean residuals of the Pan-STARRS imaging, with the second PSF in the location of an expected counterimage, namely 0.9 from the galaxy opposite the bright image. We therefore classify J0936-1211 as a lensed quasar, however, deeper imaging is needed to definitively confirm this.
J1003+0651
This double has three GDR2 detections, centred on a SDSS spectroscopic galaxy at z=0.225. A feature around 4300Å is indicative of a quasar, and subtraction of the SDSS model galaxy template reveals quasar emission lines at z=2.56. Our WHT data resolve two quasar traces at the same redshift, either side of the lensing galaxy, thus confirming this system as al lensed quasar.
J1008+0046
This system has a spectroscopic redshift of z=1.511 from the 2df quasar redshift survey (Croom et al. 2004). Our follow-up WHT resolve two quasars at this redshift and HSC imaging clearly reveals a lensing galaxy.
J1037+0018
The NTT-EFOSC2 spectra show two quasars at z=2.462, in agreement with a 2dF spectrum at z=2.464. This system has HSC DR3 data, which clearly shows a lensing galaxy upon subtraction of the two PSFs.
J1041+1710
The NTT-EFOSC2 spectra resolve two quasars at z=2.00, with two broad absorption line features in both the C and Si lines. A lensing galaxy is seen in the PSF-subtracted Legacy Survey imaging.
J1233-3542
This double has two bright Gaia detections ( =18.54, 19.56) separated by 2.08 . The WHT spectrum reveals particularly narrow quasar emission lines at z=2.28. Archival DECam giz imaging shows clear residuals upon subtraction of two PSFs, thus we classify this system as a lens.
J1247-3253
This double has two bright components (G=18.27, 19.09) separated by 1.85 . NTT-EFOSC2 spectroscopy reveals very similarly shaped spectra of quasars at z=1.66. Shallow archival griz DECam imaging shows residual flux in the z-band, which we take as a detection of the lensing galaxy.
J1255+0737
WHT-ISIS long-slit spectra were taken with the slit position angle at 5.6 degrees East of North. The spectra reveal two quasars at z=2.145.
In Gaia EDR3 there is a third detection associated to a galaxy 2 arcseconds to the East, which could be due to a another image of the source blended with the galaxy, as in the case of SDSSJ1405+0959 (Rusu et al. 2014). High-resolution imaging is needed to see if this is an interesting case of a lens with two lens planes and three images.
J1303+1816
A G=12 star lies 15 from this lens, offering a good opportunity for natural guide star adaptive optics observations of the system. The galaxy lies in a group, with one nearby galaxy having an SDSS spectroscopic redshift of z=0.410, and a further one with z=0.473. Clear signs of a z=0.46 lensing galaxy are seen in the spectrum of the fainter image.
J1307+0642
This system was first selected as a lens candidate by Lemon et al. (2019), however, the spectra were too shallow to detect quasar emission lines. The new 1200s WHT-ISIS exposure shows clearly two traces of a z=2.03 quasar, with the fainter image blended with a z=0.23 galaxy. A SDSS spectrum confirms our redshift of the lensing galaxy. Archival Chandra data -originally from observations of 3C281, which is situated less than 3 arcminutes from this systemresolve two X-ray point sources coincident with two optically blue point sources either side of a bright galaxy. The fluxes of the two components are 3.7±1.9 10e-14 and 0.9±0.4 10e-14 erg cm −2 s −1 in the broad 0.5-7keV band.
J1326+3020
The WHT spectrum reveals two traces, one of a quasar at z=1.852, and another of a galaxy at z=0.339. The Legacy Survey imaging shows a blue point source in the model-subtracted residual image, suggesting this as a faint counterimage. Indeed there are hints of the Ly line in the blue arm raw pixels of the WHT-ISIS data, thus we classify this as a lens. The lens is a member of a cluster, whose members were targeted for spectroscopy by Holden et al. (1999). They obtained a spectrum for the lensing galaxy of this system, reporting a broad emission line around 5437Å, which they attributed to MgII, giving a tentative redshift of z=0.94, however our spectra confirm this is in fact the C ] line in agreement with an archival eBOSS spectrum centred on the galaxy.
J1526-1400
This double has the lowest redshift source yet discovered for lensed quasars, z=0.648, and a particularly bright, low-redshift lensing galaxy at z=0.096. Similar to J1003+0651, the lensing galaxy is detected by Gaia. Given the brightness of this lens ( =16.02), and that of J0416+7428, we can consider how many bright lensing galaxies are expected based on the OM10 mocks. With the two doubles presented here, and Q2237+030 (Huchra et al. 1985), we have 3 lenses with multiple Gaia detections, and lensing galaxies brighter than i=16.5. This seems to be consistent with the 4-7 expected across the whole sky based on OM10 but it is perhaps surprising given that our selection techniques are based mostly on source colours. Given these small-number statistics, investigating spectroscopic samples of bright galaxies up to z=0.1 is needed to reveal whether there is an overabundance of low-redshift lensing galaxies compared to the mocks.
J1550+0221
This double has a BOSS spectrum, and was selected as a lens candidate independently by Spiniello et al. (2018) using pixel analysis of Kilo-Degree Survey imaging.
J1945-2857
This double has two Gaia detections separated by 2.69 , with clearly resolved spectra of a z=2.56 quasar in the NTT-EFOSC2 data. The 2D PSF-subtracted spectrum shows residuals around the Ly line below both images, suggesting either a PSF not well-fit by the Moffat used (however no such residuals are seen anywhere else in the subtraction), or Ly is spatially resolved, suggesting a bright lensed host galaxy or even secondary quasar source. High-resolution imaging is needed for this system.
J2017+6204
Two NOT-ALFOSC 900s spectra were taken at position angles of 124.2 and 50.7 degrees East of North, however both showed a blended trace of a z=1.72 BAL quasar. Adaptive optics imaging with OSIRIS on Keck 2 resolves the four images and lensing galaxy, and is shown in Figue 2. The system was also independently observed by Stern et al. (2021) who obtained unresolved spectroscopy, corroborating our reported redshift.
J2110-3755
The NTT-EFOSC2 spectrum is low signal-to-noise, however, two traces are clearly visible, with at least one associated to a z=1.50 quasar. Given the extended residuals in the Legacy Survey -band image when fit with just two PSFs and the extended MgII emission across both traces, we classify this system as a lens.
J2205+1019
While the extracted 1D spectra only show quasar emission lines in one trace, the residual 2D spectrum clearly shows Lyman-flux opposite the confirmed quasar, thus confirming the presence of a counterimage. The lens is particularly low redshift at z=0.108.
J2205-3727
This quad was selected as a Gaia double around a red WISE detection and visually inspected using archival DECam data, with subsequent spectroscopic confirmation with NTT-EFOSC2. The slit was positioned at 86.4 degrees East of North to pass through the brightest image of the cusp, and the far counterimage. Gaia EDR3 only catalogues one image, compared to two in GDR2.
J2213-5926
We identify a likely lens redshift of z=0.545, however deeper spectra are required to confirm this.
J2316+0610
This system was originally selected by Lemon et al. (2019) who suggested a possible source redshift of z=1.96, however their spectra were too noisy to conclusively confirm the system as a lens. Our deeper NOT data clearly resolve two quasars at z=1.955, thus confirming this system as a lens. The source is a BAL quasar with multiple components, confirmed by a recent eBOSS spectrum. We measure a lens redshift of z=0.378, consistent with the redshift of a nearby galaxy with a BOSS spectrum.
Likely Lensed Quasars
These systems are all classified as likely lensed quasars. Most are lacking the detection of a lensing galaxy, but have particularly similar resolved spectra, and thus should be prioritised for follow-up high-resolution imaging. For some systems the spectra are too noisy to confirm the presence of a counterimage, however the imaging suggest such a counterimage to be present, and thus require deeper spectroscopy.
J0138+4841
This particularly bright, small separation pair ( =16.42, 17.45, and Δ =0.69 ) shows a clear broad emission line at 6078Å. The 2D spectrum modelling extracts very similar spectra for the two components with a flux ratio smoothly increasing from 1.8 at 4000Å to 2.4 at 9000Å, the latter of which is in good agreement with the Gaia flux ratio (2.6). The brighter component shows narrow absorption at the wavelength of C ]. Modelling the Pan-STARRS data reveals extended flux, which can be reasonably well-fit around the 2PSFs when including a Sersic profile which falls between the two sources, suggesting it is a lensing galaxy. Given the small-separation and the only available imaging data being low-resolution, we suggest obtaining additional high-resolution imaging before designating this system as a sure lens. This system is also associated to both a ROSAT X-ray source and a radio source in VLASS (1.65±0.35mJy at 2-4GHz).
J0149-6532
The imaging for this system clearly shows two blue PSFs either side of a luminous red galaxy (LRG). The spectrum of the brighter blue source is low signal-to-noise and shows only hints of a quasar spectrum at z=0.944.
J0221+0555
The resolved ALFOSC-NOT spectra show very similar z=1.52 quasar profiles, with both components having strong absorption bluewards of the C ] line and similar narrow C profiles. Since no residual lens galaxy is seen in the imaging.
J0326-3122
This system was already reported by Schechter et al. (2017) as a lensless twin at z=1.34, with narrow absorption at z=0.504. We obtained EFOSC-NTT spectra of the system across a larger wavelength range, revealing the MgII emission line of the components. Our results qualitatively agree with those of Schechter et al. (2017). A galaxy is seen about 1 North of the system in the PSF subtracted Legacy Survey images, possibly responsable for the z=0.504 absorption in image A.
J0326-4950
The Legacy Survey imaging residuals for this system show an extended galaxy coincident with one of the Gaia detections, as well as possible faint blue point source. The spectra are low signal-to-noise but the galaxy is determined to be at z=0.229 through emission lines. The other blue point source is likely a z=1.72 quasar, however this is not definite. The system could still be acting as a lens, so deeper imaging and spectroscopy are required. The system is associated with a ROSAT X-ray source.
J0332-6608
This system consists of two bright point sources ( =18.30, 19.46) separated by 0.94 . NTT-EFOSC2 spectra reveal similar emission lines of a z=1.97 quasar, with a constant flux ratio of ∼2.5 except for a differing depth of an absorption line at 4860Å. Only Legacy Survey and -band data is available, and no obvious lens is seen upon subtracting the two PSFs.
J0346+2154
This 0.99 -separation system was targetted for NOT-ALFOSC spectroscopy, which confirmed the presence of two quasar spectra at = 2.365. The system was also independently selected by Krone-Martins et al. (2019) as a lens candidate. They obtained resolved spectroscopy and suggest this is most likely a lensed quasar given the similar narrow absorption asssociated with the blue wing of C . We also note this absorption line, and striking similarity between the two spectra, with the only slight differences being in the height of the C and C ] lines, which is likely explained by microlensing. We model CFHT MegaCam stacked -band data as 2 PSFs, however the PSF model is not good enough to clearly tell if there are residuals from a possible lensing galaxy.
J0916-2848
While the continua of the two components are quite different, the emission lines have similar profiles. If the system is a lens, the different continua can be explained by the presence of a lensing galaxy and/or microlensing, as has been seen in other systems, for example GRALJ2343+0435 (
J1008-2911
The EFOSC2 extracted spectra show two quasars at z=2.49 with very similar line profiles and continua, suggesting this is a likely lens. However, no residuals are seen in either the Pan-STARRS PSF-subtracted image or archival DECam gri imaging.
J1019-1322
The NTT-EFOSC2 spectra reveal two quasars at z=2.325 with similar continua and line profiles, however no galaxy is seen in the Pan-STARRS imaging residuals.
J1033-8249
NTT-EFOSC2 spectra of this 2.33 -separation double reveals two quasars with similar profiles at z=1.69. The spectra appear to have a red continuum, however slit losses due to atmospheric refraction and high airmass observations cannot be entirely ruled out despite our correction model. Shallow archival NOAO data exist for this system, however no obvious lens light is seen upon subtracting two point sources.
J1036-8544
NTT-EFOSC2 spectra of this 1.22 -separation double reveals two quasars with similar continua and MgII lines at z=1.09. Shallow archival NOAO data clearly show the two point sources detected by Gaia. Subtracting two PSFs from the available r and i band data reveal some faint residuals around the system, likely due to a lensing galaxy. We currently classify this system as a likely lens, requiring deeper imaging data.
J1103-1005
This 0.72 -separation system has two bright Gaia detections (G=17.79, 18.01), and NTT-EFOSC2 long-slit spectroscopy shows spatially resolved traces of a z=1.29 quasar. The emission line profiles are very similar, with a difference in the slope of the continua in the blue. The flux ratio varies smoothly from 1.9 to 1 between 4000 and 8000Å respectively. This is a very likely lens with the spectrum differences attributed to reddening by a lensing galaxy and/or microlensing. There are hints of an extended source in the Pan-STARRS i-band PSF-subtracted image, however we suggest higher-resolution imaging is required to confirm this system as a lens.
J1322+7852
This system was selected as a Gaia double with a red WISE source. In Pan-STARRS and Legacy Survey imaging, there is clearly one blue point source next to a galaxy, and WHT-ISIS spectroscopy shows two traces. The galaxy trace shows both an old stellar population with an obvious 4000Å break and common LRG absorption lines, but also narrow star formation emission lines. The trace of the blue PSF is almost featureless and low signal-to-noise, however binning the spectrum reveals broad emission lines of a z=2.165 quasar. No spectral features of this quasar appear in the trace of the galaxy as a possible counterimage, however such features would be expected to be too faint if a counterimage exists. The Legacy Survey residuals show a very faint possible counterimage. Deeper spectra and/or imaging are needed to confirm this as a counterimage.
J1350+3155
An archival SDSS spectrum of this system shows a galaxy at z=0.211. Our spectrum corroborates this redshift for the galaxy, and also shows that the nearby blue PSF is a quasar at z=1.895, however no quasar emission lines are seen in the combined 1500s extraction of the galaxy. We expect that the quasar is strongly lensed given the small separation between the Gaia detections of 1.44 , and the high velocity dispersion of the galaxy (238 kms −1 ). We estimate the Einstein radius of this lens given this dispersion, the source and lens redshifts, and assuming a singular isothermal sphere mass profile, obtaining ≈1.35 . Generally, images of such a profile are separated by twice the Einstein radius, i.e. 2.7 . The likeliest solution is that the counterimage is highly demagnified, either through microlensing, or extinction/reddening by the lensing galaxy, or some combination of both. High-resolution imaging for this system should be prioritised to search for a counterimage.
J2244-0550
NTT-EFOSC2 spectra show both components to be quasars at z=2.84 with almost identical profiles. However, no lens residuals are seen in the Pan-STARRS or Legacy Survey PSF-subtracted images.
J2255+8009
The NOT-ALFOSC spectroscopy resolves two quasars with very similar spectra and a pronounced drop in flux below 4650Å, either due to intrinsic quasar emission or line-of-sight absorption. Two deep absorption features are seen at 4607Å and 4650Å consistent with absorption from Ca H and K at z=0.172, however, no other clear absorption features at this redshift are seen in the spectra. The emission lines are likely associated with a z≈2.8 quasar, implying that Ly is strongly absorbed (and perhaps these two absorption lines are due to proximate damped Ly systems). The Pan-STARRS data are well fit by two PSFs.
Unclassified Quasar Pairs
The following systems have resolved spectroscopy showing two quasar components at consistent redshifts, however they lack the spectral signal-to-noise and/or lens galaxy detection to be classified as a (likely) lensed quasar.
J0124-6334
The Legacy Survey image of this system shows two point sources of different colours, and subtracting 2 PSFs shows clear extended residuals between the PSFs. When including a Sersic profile, the data are well fit, with the position of the Sersic falling between the two PSFs closer to the fainter image, consistent with that expected of a lensing galaxy in a typical double configuration. The best-fit Sersic index is ∼1, typical of spiral galaxies. The spectra are low signal-to-noise and overlap due to poor seeing and small separation, however binning the spectra shows large differences between them, with different continua, broad emission line profiles, and relative intensities (e.g. O is present in the fainter image). Thus the lensing scenario is unlikely, but not necessarily ruled out due to differential extrinsic effects like strong absorption and/or microlensing. If these are indeed distinct quasars, the projected separation of 0.87 is 6.52 kpc at z =1.30.
J0125-1012
The two components of this system have similar spectra of a z=1.22 quasar, but no definite lensing galaxy is detected in the available imaging.
J0127-1441
The NTT-EFOSC2 spectra reveal the two components to be similar quasars at z=1.754, however there is only a faint red source ( < 22) South of the system seen in the PSF subtracted image. Given the wider separation of this system, 3.00 , and assuming it is a lens, we can estimate the faintest possible lensing galaxy brightness. We use the Oguri & Marshall (2010) catalogue and simply search this catalogue for mock lenses with similar image separations (within 0.5 ) and source redshifts (within 0.25), and consider the lens magnitudes and redshifts of the remaining systems as the possible range of lensing galaxy parameters for J0127-1441. 279 systems are recovered, with median and 1 uncertainties in redshift and -band magnitude of 0.44 ± 0.18 and 19.08 ± 1.01 respectively. The lens properties of the system with the faintest lensing galaxy are z=0.974 and =20.6. It is thus unlikely that the faint source is a lensing galaxy.
J0130+0725
The spectra show two quasars at z=1.54 with different emission line profiles, as there is prominent absorption in the C and MgII lines of the fainter quasar. This is likely explained by the system being a projected pair with some small velocity separation, and line-ofsight absorption by the host galaxy of the closer (brighter) quasar. A higher resolution archival eBOSS spectrum indeed resolves both the MgII 2800Å and C 1549Å absorption doublets (Blanton et al. 2017). No residuals are seen in the PSF-subtracted images. Intrinsic absorption along different sightlines to the same source quasar cannot necessarily be ruled out.
J0133+0816
The spectra reveal two quasars at z=1.27, with small differences in the widths of the C ] and MgII emission lines. Modelling deep CFHT riz Megacam stacked data as two PSFs reveal no obvious lensing galaxy.
J0146-6510
The two components have very similar NTT-EFOSC2 spectra of a quasar at z=1.255, however no lens is seen in the Legacy Survey -band residuals after fitting 2 PSFs. The system is associated with a radio source in GLEAM (0.555±0.011 Jy at 170-231MHz, Hurley-Walker et al. 2017), and SUMSS (214±9 mJy at 843MHz, Mauch et al. 2003), and an X-ray source (Slew Survey XMMSL2 Catalogue, 2.3±1.1×10e-15 Wm 2 , Saxton et al. 2008).
J0311+0550
The NOT spectrum of this system reveals the two components to have similar spectra of a = 0.777 quasar, however modelling the Pan-STARRS data as two point sources does not show any significant residuals.
J0527-2431
The NTT-EFOSC2 spectrum shows two resolved traces with emission lines of a z=1.437 quasar, but with differences in the line profiles and relative intensities. Deeper spectra and/or imaging are required to definitively classify the two components as distinct quasars, separated by 1.46 , i.e. 12.5 kpc.
J0642+5337
The NOT-ALFOSC resolved spectrum shows two quasars at z=1.86, with several narrow absorption lines in the brighter component. This is likely a distinct quasar pair due to the lack of lens in the PSFsubtracted Legacy Survey image, the different C profile, and the sky separation of 3.17 .
J0723+4739
This pair shows no residuals in PSF-subtracted Legacy Survey imaging, and the peaks of the MgII profiles are offset in velocity. Given possible extrinsic absorption in a lensing configuration causing this difference, we categorise the system as an unclassified quasar pair, but likely a binary quasar. If the components are indeed distinct quasars, their projected separation would be 10.9 kpc (1.58 ). Deeper spectra and imaging are required. This system was targeted by Lemon et al. (2018) but their results were inconclusive due to low signal-to-noise.
J0728+2607
The NOT-ALFOSC spectrum shows two quasars at z=1.025, with large differences in their emission line profiles and continua, thus making it a likely physical quasar pair. Their projected separation is 17.6 kpc at z=1.025 (2.15 ).
J0816+2339
Despite the WHT spectrum being low signal-to-noise, the two components are clearly quasars at z=1.22. This is likely a distinct quasar pair given the slight differences in the MgII profile.
J0833-0721
The NTT-EFOSC2 spectrum shows two similar traces of a z=0.828 quasar, however there is a small offset in the O -4960 and 5008Å lines, which appears to be real in the 2D spectrum. This would immediately rule out the lensing hypothesis, however the spectra are particularly similar, and small wavelength variations in the spatial direction could explain the apparent offset. No lens is seen in the Pan-STARRS imaging residuals. Further spectra are required to confirm a velocity difference in the O lines. If the quasars are indeed distinct, the projected separation would be 9.39 kpc.
J0909-0749
The NTT-EFOSC2 spectra resolve this 0.81 -separated pair, showing clear z=1.075 quasar emission lines in each component, however the continua are significantly different. Modelling the Pan-STARRS data as 2 point sources shows no significant residuals in the redder bands. The system is associated with a VLASS detection with a flux of 2.6±0.3mJy.
J0941-2443
The NTT-EFOSC2 spectrum shows two traces, each with an emission line around 5400Å. This is most likely MgII, as then other common emission lines would fall outside the wavelength range of our spectrum. C ] would then lie at the very bluest end of our spectrum, and indeed an increase in flux may hint at the red wing of this emission line. The MgII profiles are different, and there is no apparent lensing galaxy in the PSF-subtracted Legacy Survey or Pan-STARRS imaging. This is likely a distinct quasar pair, with a projected separation of 15.8 kpc (2.25 ).
J1045+3433
WHT-ISIS spectroscopy reveals two components with similar spectra of a z=1.205 quasar. No lens seen in Legacy Survey residuals but emission lines and continua are similar enough to warrant deeper follow-up imaging.
J1102+3421
The WHT-ISIS spectrum reveals two quasars at z=1.405, however strong absorption is seen in the C line of only the fainter component. Coupled with no lens residuals in the PSF-subtracted Pan-STARRS residuals, this system is likely a physical quasar pair, with the fainter component at slightly higher redshift than the bright component, however differential line-of-sight absorption in a lensing scenario cannot necessarily be ruled out.
J1202+0703
This 1.21 -separated pair shows two spatially resolved traces of a z=2.187 quasar in the NTT-EFOSC2 spectrum, with each trace having similar continua and profiles, except for absorption in the C line of the fainter trace. No lens is seen in the residuals.
J1428+0500
This system was previously followed up by Lemon et al. (2019) who concluded the two components had spectra of quasars at z=1.38. Our new WHT-ISIS spectra now reveal the C profile and place both quasars at z=1.375, with differing MgII profiles due to absorption in the fainter component. The Legacy Survey data are well fit by two PSFs. If the quasars are distinct, the projected separation of 2.23 translates to 17.0 kpc at z=1.375.
J1449-2025
EFOSC2-NTT spectroscopy shows that both components of this 1.17 -separation pair are quasars at z=0.594 with similar spectra. A two PSF fit to the Pan-STARRS imaging shows significant residuals around the system, as expected if the system is either a lens (residuals due to a lensing galaxy) or simply a distinct quasar pair (residuals due to the quasar hosts given the low redshift). Adding a single Sersic fits these residuals and this component falls between the two PSFs. If this system is a distinct pair, the projected separation would be 7.9 kpc.
J1821+6005
The NOT-ALFOSC spectrum shows two traces of quasar at z=2.052 with similar continua, but a slight difference in the shape of the C line. Lemon et al. (2018) also presented WHT-ISIS follow-up of this system, concluding the system to be a distinct quasar pair. No lens is seen in the PSF-subtracted Pan-STARRS, Legacy Survey, or CFHT -band imaging.
J1832+5349
The two components of this system are separated by 3.03 , and NOT-ALFOSC spectra reveal two quasars at z=1.16, with similar MgII emission line profiles and continua. However, no lensing galaxy is seen in the Legacy Survey residuals after PSF subtraction. In Pan-STARRS there is a hint of residual flux between the two quasars in the Y-band. Given the separation and relatively low redshift, we expect this is a distinct quasar pair.
J2008+0438
This system lies in a high stellar density environment, at a galactic latitude of -14 • . One trace of the NOT-ALFOSC spectra is clearly a quasar at z=1.70, however the other does not have similarly obvious broad emission lines. The latter trace has a broad emission line at the same wavelength as the MgII line of the first trace, however the telluric absorption band obscures the red wing. There is a hint of a line at the same wavelength as C ] of the first trace suggesting this source is also a quasar at z=1.70, but no C emission is seen.
J2057+0217
The NOT-ALFOSC spectra show two components with similar continua but different emission line profiles suggesting a distinct quasar pair. If true, the sky separation of 1.16 would correspond to a physical projected separation of 8.9 kpc at z=1.52. The WHT-ISIS observations of this system by Lemon et al. (2018) do not reveal a C emission line as we see in these newer spectra, which is likely due to unaccounted slit losses.
J2341-1557
NTT-EFOSC2 spectra of this small-separation system (1.02 ) reveal two quasars at z=1.535 with different continua and relative emission line strengths. It is likely a distinct quasar pair -and thus would have a projected separation of 8.8 kpc.
Physical Quasar Pairs
J0041-5350
Resolved spectra show that each component of this system is a quasar at z=0.55, however the lensing scenario is ruled out since the O profiles are very different. The extended asymetric flux seen in the ground-based imaging is likely a sign of an ongoing merger between the two quasar host galaxies. The sky separation of 1.06 translates to 6.2 kpc, making it one of the smallest separation confirmed quasar binaries above z=0.5 (Chen et al. 2022).
J0315-3522
Legacy Survey imaging shows two point sources and a galaxy offset to one side of the fainter component. NTT-EFOSC2 spectra reveal two quasars at z=0.457, with residuals around O -5008Å suggesting spatially resolved emission. Given the difference between the spectra for both the continua and emission line profiles, and lack of lensing galaxy candidate in the imaging, we designate this system as a distinct quasar pair. The separation is 1.30 at z=0.457 -a projected separation of 6.9 kpc.
J1116-2122
Given the 3.29 separation and relatively low redshift sources (z=0.709), we categorise this as a distinct quasar pair. The projected separation is 23.9 kpc.
J2121+1713
The spectra reveal two quasars at z=0.63, however, the emission lines and continua are markedly different. Given also the low redshift, large separation of 2.89 , and lack of lensing galaxy in the residuals, we classify this system as a distinct quasar pair. The projected physical separation is 17.9 kpc.
J2132+3635
The NOT-ALFOSC spectrum shows two quasars at z=1.215 with slight differences in the MgII profile. Follow-up OSIRIS imaging reveals no lensing galaxy (Figure 2), thus we designate this as a distinct quasar pair. The sky separation of 1.33 translates to a physical projected separation of 9.9 kpc.
J2256+2223
Despite similar profiles of a z=0.75 quasar, the O narrow emission lines are slightly offset in wavelength. This cannot be a calibration drift in the spatial direction since the sky emission lines stay within the same pixel centroid across the width of the system and over several tens of arcseconds. Therefore, this is a distinct quasar pair with sky separation of 1.63 , translating to 10.7 kpc in projected separation at z=0.75.
J2322+0916
ALFOSC-NOT spectroscopy shows that both components of this pair are quasars at z=1.20, in agreement with the SDSS spectrum of the fainter component. The emission line shapes and continua are different, and no residuals are seen in the PSF-subtracted Legacy Survey imaging. Therefore, we classify this system as a distinct quasar pair, with projected separation of 37.9 kpc (4.51 ).
J2355-4553
The NTT-EFOSC2 spectrum reveals two quasars at z=2.85, however one has strong broad absorption lines. We suggest this must imply that the two quasars are distinct, and thus have a projected separation of 16.7 kpc given their 2.20 sky separation.
Projected Quasar Pairs
J0027+0438
The spectra show a bright quasar at z=1.972, and an AGN (broad H emission) with extended host galaxy at z=0.1935. The projected separation of the system is 1.92 . While no signs of a counterimage are seen in the imaging, the lensing scenario is still plausible as the quasar host galaxy requires only a modest Einstein radius, ∼1 , and would explain the particularly large apparent brightness of the quasar through an associated magnification. High resolution imaging or deeper spectroscopy will be required to rule out the lensing hypothesis fully.
J1820+3747
This projected quasar pair system is associated with an X-ray source in ROSAT. The quasar redshifts are 0.614 and 0.754, and Legacy Survey imaging and photometric redshifts suggests the system is coincident with a cluster at the background quasar source redshift.
J2302-4154
One trace shows C ] and MgII emission lines of a z=1.22 quasar, while the other only shows one emission line at 5540Å, and no other clear emission lines. This is most likely MgII, since at this redshift no other common quasar emission lines are expected to fall within the wavelength range of our spectra. This would put the second component at a lower redshift of 0.98, with the background quasar passing at 8.7 kpc in projection.
Other
J0336-3244
One trace of this system is a = 0.545 quasar, however the trace of the second object is noisy, with hints of z=0 absorption lines, i.e. it is possibly a star. Deeper spectra are needed to verify this.
J0819+0457
The signal-to-noise of the fainter trace is too low to determine any robust features. It is likely that this is a distinct quasar pair or quasar-star projection, however deeper spectra are needed.
J0938+0629
The spectrum shows resolved narrow line emission at z=0.363, however only the brighter component has broad emission lines. We therefore classify this as a quasar+galaxy pair.
J1140+2303
The resolved spectra show a z=2.405 quasar and a likely star given H and H absorption at z=0. However, there are also similarities between the two spectra, in particular around the wavelength of Lyman alpha, which is unlikely to be due to extraction problems given the wide separation of 3.63 . We suggest this is a quasar-star projection, however, deeper spectroscopy and/or imaging is needed to definitively conclude this.
J1442-0857
The extracted spectra reveal galaxies at z=0.164, the redder one of which has strong narrow line emission (OII, H , NII, SII). The redder component also appears to be coincident with a bright radio detection in VLASS.
J1752+0826
This system appears as two blue point sources either side of a galaxy, however follow-up WHT-ISIS spectra reveal likely z=0 absorption lines in both traces, alongside narrow line emission at the redshift of the galaxy, z=0.3195. Such similarities to double quasar configurations are expected at high stellar densities. This system lies at a galactic latitude of b=15 • .
Lensed galaxies
As a by-product of looking for lensed quasars, lensed galaxies may be discovered semi-serendipitously since they share unique morphological similarities with lensed quasars, namely multiple blue detections around luminous red galaxies. We present four new lensed galaxies that were found in our searches, two with single Gaia detections and two with two blue SDSS detections around a red galaxy.
J0343-2828
This lens was selected as a potential lensed quasar candidate through the single Gaia detection offset method using DR2 ( 2 = 21.29), and looked promising given the compact PSFs. However, spectroscopic follow-up with the NTT in October 2019 confirmed the source as a z=1.655 galaxy, with a lens at z=0.385. The coordinates are: R.A., Dec. = (55.7978,. HST imaging (HST Proposal 15652, P.I. :Treu) shows that indeed the images are not consistent with point sources, and further show two lensing galaxies with a fifth image between them (Schmidt et al. 2022).
J0500-5534
This lens was selected as a potential lensed quasar through the single Gaia detection offset method in Gaia DR2 ( DR2 = 20.28) and a marginally red colour of W1-W2=0.53. The coordinates are: R.A., Dec. = (75.1017, -55.5796). The DES imaging is well-fit by four point sources and a Sersic galaxy profile, with the positions forming a short-axis cusp configuration, which is indeed well-fit by an SIE+shear lens model. Follow-up spectroscopy with FIRE on the 6.5-meter Baade Magellan telescope shows clearly resolved narrow emission lines of a z=0.620 galaxy. Figure 15 shows the 1D spectrum, with the following emission lines marked: H , [S ], and He . In EDR3, there is an additional detection for the counterimage and is the only known lensed galaxy with two Gaia detections to our knowledge ( DR2 = 20.84, 20.98).
J0920+4521
This system was discovered through a search for multiple blue catalogued detections in SDSS within a few arcseconds of a red photometric galaxy. Legacy Survey imaging shows a blue arc East of a group of four galaxies all with similar colours (see Figure 15). Two of these galaxies have existing SDSS spectra, one an LRG at z=0.546, and the other with narrow emission lines at z=0.956. However, closer inspection of the latter spectrum shows also an LRG spectrum consistent with z=0.546, suggesting the emission lines are from a background source, possibly a counterimage of the observed arc. Follow-up spectroscopy with ISIS on the WHT were taken on 31 March 2017 at two position angles -along the arc, and through the brightest part of the arc and the galaxy with the spectrum with narrow emission lines. The spectra are shown in Figure 15. The arc shows clear absorption lines of a z=2.633 galaxy, and the galaxy spectrum is qualitatively similar to the SDSS one, showing the narrow OII emission of a z=0.956 source. One possibility is that the main lensing galaxy is indeed at z=0.956 and any signs of a z=0.546 LRG are due to contamination or blending with a nearby galaxy. However, this is unlikely given that the ground-based imaging shows clearly similar colours between the multiple galaxies. We suggest the most likely situation is that there are two main lensing galaxies at z≈0.55 with two sources at z=0.956 and z=2.633. While the lensing potential may be complex, if well-constrained, the system can be a very useful tool for cosmography (e.g. Collett & Auger 2014).
J1059+4251
This system was discovered through a search in SDSS imaging for multiple photometric quasar candidates around photometric red galaxies. Follow-up spectroscopy with ISIS on the WHT on 31 March 2017 targeting the bright arc reveals a z=2.793 galaxy. Figure 15 shows the 1D spectrum and the DECaLS grz colour image of the system. HST imaging reveals three lensing galaxies and a complex lensing configuration. This system is examined in detail in Citro et al. (2021).
DISCUSSION
Lenses
Sample characteristics
We can compare several features of the lenses we have discovered with those previously known in order to elucidate possible biases in our selection methods and/or those of previous discovery methods. For the following analysis, we compile a list of 224 lensed quasars from the literature, mainly from the database of Lemon et al. (2019), but including those from more recent publications (e.g., Desira et al. 2022). Since our goal is to find lenses with optically bright images detected by Gaia, we remove lenses with no Gaia detections, or unmeasured source redshifts. From the known sample, this eliminates 38 lenses, mainly composed of CLASS radio-loud sources, lenses with faint optical images (More et al. 2016;Chan et al. 2022), particularly reddened or high-redshift quasar sources (e.g., Fan et al. 2019), and two systems which have Gaia detections but unmeasured or insecure source redshifts: B2108+213 from McKean et al. (2005) and J2218-3322 from Chen et al. (2022). It also removes one object from the new sample, J0310-5545, which was serendipitously discovered because of a Gaia detection from a nearby star. This leaves 85 lenses (8 quads, 1 triple, 76 doubles) in our new sample, and 186 (2 five-image lenses, 41 quads, 3 triples, 140 doubles) in the previously known sample. We can immediately compare the multiplicity ratio of the known sample against those of this paper: the ratio of lenses with more than two images to those with just two is 46 to 140 (1 to 3.0), and 9 to 76 (1 to 8.4) respectively. These very different ratios are likely due to the ease of finding quads (generally easily recognisable even without the presence of a bright lensing galaxy) compared to doubles coupled with the bias of following up quad candidates and reporting them in earlier papers or by others (e.g., Schechter et al. 2017;Lucey et al. 2018). It cannot be taken as a bias in our selection since our selection criteria recover all optically bright quads.
In Figure 16, we show the source redshift and maximum image separation of these two samples, marking the 16th, 50th, and 84th percentiles, which highlights two marginal differences: (i) the sources of the lensed quasars from this work have a higher median redshift than the known lens sample (1.94 and 1.86 respectively), likely due to redshift constraints of previous searches relying on UV-excess for pre-selection (Oguri et al. 2006); and (ii) the image separation distribution is marginally more peaked than that of the existing sample (medians of 1.83 and 1.77, and standard deviations of 0.92 and 1.07 respectively). While not significantly different populations, these differences can be explained by the cutoff at larger separations in our original selection criteria, and also the cutoff at lower separations from our candidates which are likely lenses but do not show lensing galaxies in the shallow existing imaging. The counterparts in the existing sample have already received the necessary deeper follow-up imaging. Including these unclassified quasar pairs as lenses would remove this difference at smaller separations, but exacerbate the redshift offset between the samples. See Section 5.1.3 for a comparison of these samples to mock catalogues, to assess the relevant completeness of known lensed quasars.
Comparing Gaia DR2 to Gaia EDR3
We investigate the differences in detection of lensed quasar images between Gaia DR2 and early Gaia DR3 (EDR3), not only to elucidate the completeness and biases in our current and previous selections, but also as a potential probe of discovering new lenses through catalogue parameter comparison. We combine the known and new samples of lensed quasars from Section 5.1.1, resulting in 274 lenses (now including those without known source redshifts). We inspect cutouts of the lenses with the detections from both data releases overlaid and note the following:
• 241 (88%) have equal numbers of Gaia detections between data releases; however in 3 cases detections have been replaced: HS0810+2554 (Reimers et al. 2002), the DR2 detections coincide with one of the bright pair and the Northern image, while EDR3 replaces the Northern image with the South-Eastern image -interestingly, the first data release catalogues the same images as EDR3; PSJ0840+3550 (Lemon et al. 2018), the counterimage was catalogued in DR2 but in EDR3 it has been removed and the galaxy is catalogued; and SDSSJ1251+2935 (Kayo et al. 2007), the merging pair was catalogued in DR2, but now has just the Southern image of this pair and the Western image catalogued in EDR3.
• 25 (9%) have gained one detection. For 19 of these, the detection is associated with the lensing galaxy. For 5 (3 quads, 2 doubles), the detection is associated with another image. And for 1, PMNJ0134-0931, it did not previously have a detection in DR2.
• 7 have lost a detection. 5 (1 quad, 4 doubles) have lost an image detection, 1 double (J1003+0651) has lost a galaxy detection, and 1 quad (SDSSJ1640+1932) has lost both of its image detections and gained a detection of the lensing galaxy. Figure 17 shows an assortment of lens systems with the DR2 and EDR3 detections overlaid. Interestingly, for H1413+117, the Gaia position for image C is offset from the true position, i.e. from HST imaging, and has shifted closer to this true position in EDR3, however, it is still inconsistent with the position given the EDR3 positional uncertainty of ∼6 milliarcseconds. Conversely, the positions of the two EDR3 detections in HE1113-0641 seem to be further from the true positions than in DR2.
To investigate further these positional offsets as a possible method for finding compact quads and other lenses, we match images between DR2 and EDR3. We record the -band magnitude difference and positional offset for all lensed quasar images clearly belonging to the same component (i.e. galaxy or specific image). For one image of 2M1310-1714 there is no G magnitude in EDR3, so this single image is omitted from our investigation. The distribution of these offsets in position and photometry are shown in Figure 18. The largest positional offsets are those of images in small separation quads, namely H1413+117, HE1113-0641, WFI2026-4536, and J2218-3322 (see Figure 17). In the absence of the raw Gaia data, these offsets suggest a promising avenue for discovering more compact quads through large astrometric offsets between Gaia data releases. However, to properly assess the efficacy of such a method would require understanding the same metric distribution in common contaminants, which is beyond the scope of this paper. There are also several systems with large photometric differences between data releases. These are all systems with bright, low-redshift, lensing galaxies. In the case of the lensing galaxy of J1003+0651, a large increase in flux (positive DR2 − DR3 ) is seen, and can be attributed to the counterimage being removed in the most recent data release (see Figure 17). Conversely, several galaxies have a large reduction in flux (e.g., J1526-1400 and J2205+1019), despite having the same detections in both releases. This is likely explained by a combination of deblending problems with a nearby counterimage and the known increase in variability for elongated galaxies due to average scanning direction differences (Riello et al. 2021).
Comparison to mock catalogues
We can also compare the current sample of lenses to those expected from a mock catalogue to understand the limitations of our selection. For this we use an updated version of the mock catalogue from Oguri & Marshall (2010). The updates include adopting a galaxy velocity function for all types of galaxies from Bernardi et al. (2010) Figure 17. Overlay of DR2 and EDR3 catalogue detections on the best available imaging data for various interesting cases. Blue plus symbols are EDR3, while red crosses are DR2. The white scalebar represents 0.5 arcseconds. The filter has been chosen to match the peak of the Gaia G bandpass when possible. most complete; systems with single Gaia detections can have much fainter images that prevent spectroscopic confirmation. To compare the mock catalogue to the true sky, we limit the area of sky to the typical areas searched for lenses, namely those not too highly contaminated by stars. We define this as any sky area with local Gaia detection densities lower than 20,000 detections per square degree (as measured by the number of detections within a 100 arcsecond radius) and within the Pan-STARRS footprint, i.e. declinations above -30 degrees. This leaves 53.9% of the full-sky. We also include limits on image separation of 1 to 4 arcseconds, and source redshifts below 4. These criteria keep 121 known lensed quasars.
To provide mock quasars in the catalogue with Gaia magnitudes, we follow Lemon et al. (2019) by using SDSS quasars with both Gaia detections and measured i-band magnitudes, and draw a G-i colour for each mock source from the known quasars within 0.025 in redshift. The number of mocks that pass, normalised to the sky area used for the real lens sample, is ∼214. The image separation and source redshift distributions are shown in Figure 19. The predicted numbers are in good agreement with the known lenses for larger image separations and redshifts below 1.5. This is naturally explained by prioritising lens candidates with clear signs of a possible lensing galaxy, which is favoured by lower redshift sources and larger image separations.
The Figure 19. Distribution of image separations against source redshifts for a statistical sample of known lensed quasars (red) as described in the text. Circles are doubles, while squares are quads. Also overlaid in the histograms are the predicted numbers satisfying the same criteria from an updated version of the Oguri & Marshall (2010) mock catalogue; blue is an overprediction relative to observed numbers, and red is an under-prediction.
the number of observed lenses by 44%, suggesting a discrepancy within their imposed galaxy velocity dispersion function, quasar luminosity function, or shear distribution.
Physical quasar pairs
We classify 11 systems as distinct quasar pairs at very similar redshifts. Following Hennawi et al. (2010), we take our operational definition of a binary quasar to have measured redshifts within 2000 kms −1 . This accounts for possible offsets of emission lines from the systemic redshift due to the dynamics of the broad line region, inflows/outflows, and absorption, while also allowing for peculiar velocities of the two quasars while still remaining in the same halo. The best-fit redshifts are found by minimising the 2 of fitting a quasar template using P QSOF within a range of redshifts (Guo et al. 2018). Uncertainties are not quoted since they depend strongly on the prior for potential velocity offsets of certain lines. A summary of key parameters is given in Table 2. Of particular interest are the smallest-separation distinct quasar pairs, which probe the latest stage in the merger process and are the least well-characterised population. Only recently have binary quasars below sub-10kpc been discovered above redshift 1 (Inada et al. 2008;Anguita et al. 2018;Tang et al. 2021), due to typical ground-based observing conditions and blending at such small angular separations (typically under 1 arcsecond). Lemon et al. (2022) argued that the discovery of the first lensed binary quasar (with a sub-10kpc source) likely suggests a larger population of unlensed small-separation binaries. Several convincing candidate systems are already being found using Gaia and HST follow-up imaging (Shen et al. 2021;Chen et al. 2022).
We therefore expect to find such systems as a byproduct of a lensed quasar search. However, systems with residual flux between the putative lensed images are preferentially observed for spectroscopy, which explains our bias towards low-redshift distinct quasar pairs relative to the lensed quasar sources (median redshifts of 0.71 and 1.94 respectively).
Projected quasars
We report 11 new projected quasar pairs, with angular separations ranging from 0.98 -4.64 , and physical impact parameters from 6.2 -34.3 kpc. We summarise the key parameters of these pairs in Table 3. Such systems are valuable for understanding quasar feedback by probing the host galaxy of the foreground quasar through absorption of the background quasar continuum (see the Quasars probing quasars paper series, e.g. Hennawi et al. 2006). Studies are typically limited to probing regions above 25 kpc from the foreground quasars (e.g. Johnson et al. 2015), since samples are built only from well-separated quasars (more likely to both have been targeted for spectroscopy). Of particular interest are the systems with small transverse proper distances (i.e. under 20 kpc) since they are both intrinsically rare (P( ) ∼ 2 ), and are harder to discover due to observational constraints (e.g. systems become blended in ground-based imaging and are therefore not selected for spectroscopic follow-up as quasars). There are discrepancies between simulations and observations regarding Ly absorption statistics below 100kpc in quasar host galaxies (Prochaska et al. 2013;Sorini et al. 2020), suggesting observational constraints on the absorption properties at small projected separations will be key to constraining more advanced feedback models. The QSO pair Spectral Database of Findlay et al. (2018) lists only 4 projected quasar pairs with angular separations under 3 arcseconds and velocity differences above 2000 kms −1 (after removing known lensed quasars and spurious cross-matches), compared to ∼3000 such pairs with projections under 30 arcseconds. Sub-10-kpc-separation projected pairs from the literature have come from lens searches: DESJ0215-4728 ( =6.3 kpc, fg =0.467, 1.07 ) from Anguita et al. (2018), J0240-0208 ( =7.6 kpc, fg =1.058, Shalyapin et al. (2018) and Anguita et al. (2018), and J2251+0016 ( =10.0 kpc, fg =0.41, 1.86 ) from Tang et al. (2021). The projected quasars presented in this work almost triple the number of systems probing the foreground galaxies within 10 kpc (from 3 to 8 systems).
The number of such systems expected at each separation can be predicted through well-constrained quasar luminosity functions, along with the predicted luminosity and redshift distributions. We investigate this using the quasar luminosity function from Ross et al. (2013). We recover both the redshift distribution and absolute numbers of the spectroscopically confirmed quasar sample in a 500 square degree patch of SDSS. We estimate a sky density of quasars (0 < < 4) with -band magnitude above 20.7 of ≈30.0 per square degree. In an area of 20,000 square degrees, this predicts 110 pairs within 5 arcseconds, and 17 pairs within 2 arcseconds. Several other studies report sub-2 projected pairs (Inada et al. 2008;More et al. 2016;Schechter et al. 2017). Including the previously mentioned systems, the total is ∼10, below the predicted number which was a conservative value given some components of the known systems have > 20.7. Designing searches to find these missing projected pairs should be considered in future lensed quasar and quasar pair searches.
Finally, we note that some of these systems could still be acting as strong lenses when the foreground galaxy is massive enough, and any counterimage is blended with the foreground host and is undetected in both the available low signal-to-noise imaging and spectra. High-resolution follow-up imaging would provide strong constraints on any such strong lensing, and, in the case of no detection, can still be used to obtain constraints on the mass profile .
Contaminants
It is prudent to reflect on the main contaminants from our search, principally to aid future discovery methods. Given the utility and rarity of small-separation projected and physical quasar pairs, we include them in our true positive sample. Of the 175 observed systems, 25 are confirmed as contaminant systems: 17 quasar+star/nonquasar, 5 star-forming galaxies, and 3 star pairs. Of the stellar pairs, two have red WISE colours similar to those of quasars, which are rare but a well-known contaminant in infrared-colour selection of quasars, and one (J1752+0826) was selected as a Gaia detection near a galaxy and whose Pan-STARRS image shows a very convincing possible lensed quasar. Of the 5 star-forming galaxies, one was a potential quad candidate selected through the single offset method, and the rest were Gaia multiplets around red WISE quasar candidates. Finally, of the 17 quasar+star pairs, 15 were selected as Gaia multiplets around WISE quasars, and 2 as Gaia singlets near galaxies. We expect that future improvements in the Gaia catalogue -through more precise catalogue measurements -to continue to help remove such contaminants.
CONCLUSION
We have presented spectroscopic follow-up of 175 lensed quasar candidates, confirming 103 new lensed quasars, 17 of which simply lack photometric detection of the lensing galaxy, however, whose spectra are either very similar or whose imaging shows a faint counterimage. We also report 25 further unclassified quasar pairs which likely contain a mixture of lenses and distinct quasar pairs at the same redshift, however, further data is required in each case. We present 11 projected quasar pairs, and 11 binary pairs, which substantially increases the population of known systems at angular separations of a few arcseconds and below.
We used a mock catalogue of lensed quasars with an updated quasar luminosity function, galaxy velocity dispersion function, and treatment of shear to compare the current population of known lensed quasars with multiple Gaia detections. We show that the majority of lensed quasars with sources below redshift 1.5 are known, however, almost half of lenses with sources above z=1.5 are yet to be discovered, which mostly have image separations below 1.5 . The likely and possible lensed quasars discovered in this work represent already a subset of these missing lenses.
With the goal of creating a magnitude-limited sample of lensed quasars, future Gaia data releases should help through increasing completeness in detections at small separations, and reliable classification of quasars through increased precision of proper motions and parallaxes, as well as better variability constraints due to a longer baseline. To test for the completeness and biases of each lens search, other searches making use of spectroscopic and cadenced photometric datasets will be crucial. Such searches in LSST, making use of detecting nearby or extended variability, should not only be an efficient method for selecting lensed quasars but also for selecting small-separation physical quasar pairs and projected quasars (Kochanek et al. 2006;Chao et al. 2020). As targets become fainter and more numerous -with up to 400 lensed quasars with measured time delays in LSST (Liao et al. 2015) -a complete survey for spectroscopic redshifts must be carefully designed to convert these delays into a powerful probe of measuring the Hubble constant, with special consideration to minimising follow-up of common contaminants.
Figure 1 .
1Colour images of the confirmed lensed quasars. HSC gri, Legacy Survey grz, and Pan-STARRS gri are used for the majority of the cutouts (with preference in the given order). Other imaging is shown as explained in the relevant system subsection of Section 4. White scale bars are 1 . North is up, East is left.
Figure 3
3Figure 3. Spectra of the confirmed lenses in the observed frame. The most prominent quasar emission lines are marked with vertical lines, and details on certain spectra can be found in Section 4. In some cases one or several spectra have been offset for visualisation, and the offset is marked on the y-axis.
Figure 3 Figure 3 Figure 4 .
334-Colour images of the probable lensed quasars, requiring deeper imaging or spectra. Scalebars are 1 arcsecond.
Figure 5 .Figure 6 .
56Spectra of very likely lensed quasars. Details of the spectra are explained inFigure 3. Colour images of the unclassified quasar pairs.
Figure 7 .
7Spectra of unclassified quasar pairs. Details of the spectra are explained inFigure 3.
Figure 8 .Figure 9 .
89Spectra of binary quasars, with strong evidence for being distinct, and thus not graviationally lensed. Details of the spectra are explained inFigure 3. Spectra of projected quasar pairs. Details of the spectra are explained inFigure 3.
Figure 10 .
10(a) MagIC g and i-band imaging of J0457-7820; (b) with a 3 PSF model subtracted, (c) with galaxies and PSFs subtracted, (d) fiducial mass model caustics with source position overlaid (see text for details).
Figure 11 .
11Analysis of Pan-STARRS imaging of J0607-2152. Top row is for gri, while the bottom is just r-band. (a) original data, (b) with a 3 PSF model subtracted, (c) with 4 PSFs subtracted from a 4 PSF + galaxy model, (d) residuals of the 4 PSF + galaxy model.
Figure 12 .Figure 13 .
1213Analysis of Pan-STARRS imaging of J0608+4229. Top row is for gri, while the bottom is just r-band. (a) original data, (b) with a 3 PSF model subtracted, (c) with a 4 PSF model subtracted. (a) CFHT r-band image of J0803+3908 with a 1 scalebar, (b) model with PSF positions overlaid, (c) residual image possibly showing extended emission from the lensing galaxy. also independently selected by Delchambre et al. (2019) as a lens candidate, and independently confirmed by Stern et al. (2021).
Figure 14 .
14(a) CFHT rz colour image of J0833+2612 with a 1 scalebar, (b) data with 3 PSFs subtracted showing the galaxy and bluer counterimage, (c) residuals after subtracted a 4 PSF + Sersic model.
4GHz), SZA (2.9m Jy at 5GHz, 1.4 mJy at 31GHz,Muchovej et al. 2010), VLASS (3.5±0.5 mJy at 2-4GHz, clearly resolved into two components,Gordon et al. 2021), and XMM-Newton (3.0±0.8×10 −14 erg cm −2 s −1 in the 0.2-12keV band,Webb et al. 2020).
Krone-Martins et al. 2019). A prominent absorption line is seen in only the fainter component at 4463Å. No significant residuals are detected upon PSF subtraction in the Pan-STARRS data.
Figure 15 .Figure 16 .
1516Spectra and grz images of serendipitously discovered lensed galaxies. Fluxes of the spectra are normalised to have a peak value of 1, and the x-axis is wavelength in Angstroms. Cutouts are from DECaLS and are 10 arcseconds on the side. Image separation against source redshift for previously known lensed quasars (blue) and those presented in this work (red). Circles are doubles, squares are lenses with three or more images. Only those lenses falling within the image separation and source redshift limits of the plot are included in the histograms. Lines are overlaid for the 16th, 50th, and 84th percentiles of each distribution.
Figure 18 .
18-band magnitude difference against positional offset between DR2 and EDR3 for all Gaia detections in lensed quasars. Several outliers have been labelled and are discussed further in the text. The pixel scale of the Gaia detectors in the scanning direction (59 mas) is plotted as a dashed vertical line.
mock lens catalogue of Yue et al.
Table 1 .
1Summary of observed targets and results based on spectroscopy and imaging. Selections are abbreviated as follows: W=WISE catalogue, M=Milliquas catalogue with multiple Gaia detections (D, T, Q for double, triple, and quartet); GP=Gaia singlet offset from a galaxy. Separations are the largest of the Gaia separations if more than two Gaia DR2 detections are associated to the system, or based on pixel modelling. PMSIG and runs are described in the text.Name
R.A.
Dec.
Selection
Gaia G
Sep. ( )
PMSIG
Run
Classification
J0021+1927
5.4936
19.4646
WD, MD
20.77, 19.96
2.89
1.05, 2.74
NOT
projected QSOs, z=1.045, 1.09
J0027+0438
6.9580
4.6443
WD, MD
17.65, 20.18
1.92
0.56, -
NOT
projected QSOs, z=0.1935, 1.972
J0029-0414
7.3861
-4.2472
WD, MD
19.27, 20.43
1.11
1.13, -
NTT3
z=0.518 QSO + star
J0030-3358
7.6740
-33.9767
GP
20.89
2.03
-
NTT1
lens, z=1.58, lens =0.715
J0032-4523
8.1130
-45.3884 WD, MD
20.57, 20.21
2.33
0.12, 0.56
NTT3
projected QSOs, z=1.667, 1.74
J0041-5350
10.4496 -53.8460 WD, MD
20.79, 20.85
1.06
1.29, -
NTT1
QSO pair, z=0.55
J0045-3937
11.3665 -39.6262
WD
19.70, 18.98
1.14
1.39, 0.60
NTT2
lens, z=1.85
J0116+4052 19.1635
40.8811
WD, MD
19.84, 18.96
1.28
-, 1.07
NOT
lens, z=1.85
J0124-6334
21.0089 -63.5788 WD, MD
20.50, 19.60
0.87
2.63, 5.19
NTT2
NIQ, z=1.30
J0125-1012
21.3174 -10.2082 WD, MD
19.26, 20.63
1.12
1.86, -
NTT2
NIQ, z=1.22
J0127-1441
21.7854 -14.6886 WD, MD
20.14, 20.53
3.01
0.14, 1.32
NTT3
NIQ, z=1.754
J0130+0725 22.5859
7.4212
WD, MD
18.98, 19.65
2.06
1.33, -
NOT
NIQ, z=1.54
J0133+0816 23.2695
8.2788
WD, MD
20.24, 20.57
1.55
-, -
NOT
NIQ, z=1.27
J0138+4841 24.5914
48.6963
WD
17.45, 16.42
0.69
-, 1.00
NOT
NIQ, z=1.172
J0146-6510
26.6467 -65.1799
WD
17.48, 18.71
1.00
3.49, 0.76
NTT1
NIQ, z=1.255
J0149-6532
27.2906 -65.5404
GP
20.58
2.74
-
NTT1
lens(?), z=0.944?, z=0.395
J0152-2448
28.0797 -24.8105 WD, MD
20.12, 18.66
1.91
3.81, 2.05
NOT
lens, z=1.69
J0156-2751
29.1039 -27.8562 WD, MD
20.40, 20.46
1.50
0.68, 1.81
NTT1
lens, z=2.97
J0200-1509
30.0872 -15.1609 WD, MD
20.43, 19.72
0.78
-, -
NTT1
lens, z=1.185
J0209-3841
32.3047 -38.6961 WD, MD
20.60, 20.21
1.23
1.06, 1.35
NTT1
lens, z=2.87
J0221+0555 35.4474
5.9192
WD, MD
19.96, 19.39
0.86
1.08, 1.19
NOT
NIQ, z=1.52
J0232-2429
38.0655 -24.4942 WD, MD
18.00, 19.44
1.44
2.01, 2.97
NTT1
lens, z=1.578
J0246-0131
41.6793
-1.5305
WD, MD
19.05, 20.10
1.09
4.06, -
NTT2
z=0.503 QSO + star
J0247+7706 41.8664
77.1014
WD
19.49, 20.20
1.94
1.30, 1.10
NOT
lens, z=2.73
J0247-2646
41.8764 -26.7729
WD
20.49, 20.78
1.22
-, -
NTT1
lens, z=1.47
J0247-0800
41.9561
-8.0150
GP
20.55, 20.70
1.68
-, -
NTT2
lens, z=3.28, lens =0.198
J0247-6349
41.9780 -63.8232 WD, MD
19.78, 20.94
1.49
0.29, -
NTT1
lens, z=2.295
J0249+2606 42.4595
26.1141
WD, MD
18.81, 20.15
3.21
2.18, 1.99
NOT
projected QSOs, z=1.514, 2.68
J0310-5545
47.7029 -55.7534
GP
3.57
NTT3
lens, z=2.31, lens =0.298
J0311+0550 47.7708
5.8367
WD, MD
18.36, 20.01
1.36
0.49, 1.21
NOT
NIQ, z=0.777
J0315-3522
48.9099 -35.3827 WD, MD
20.22, 17.92
1.30
1.06, 2.18
NTT2
QSO pair, z=0.457
J0325-2232
51.4511 -22.5409 WD, MD
19.74, 19.52
1.01
0.28, 1.29
NTT1
lens, z=1.35
J0326-3122
51.5284 -31.3816 WD, MD
20.68, 19.47
1.43
2.05, 2.16
NTT3
NIQ, z=1.345
J0326-4950
51.7287 -49.8337 WD, MD
19.68, 20.36
1.63
1.25, 0.64
NTT2
z=0.229 GAL + z=1.72 QSO
J0329-0208
52.4280
-2.1379
WD
19.99, 19.80
1.32
0.69, 0.25
NTT3
lens, z=1.352
J0332-6608
53.0841 -66.1400 WD, MD
18.30, 19.46
0.94
7.98, 1.90
NTT2
NIQ, z=1.97
J0336-3244
54.0816 -32.7410
WD
20.78, 18.13
1.29
-, 5.21
NTT2
z=0.565 QSO + other
J0339-6121
54.7870 -61.3625
WD
18.62, 19.52
1.26
0.30, 0.83
NTT1
lens, z=2.89
J0346+2154 56.5458
21.9124
WD, MD
18.98, 18.76
0.99
-, 4.55
NOT
NIQ, z=2.365
J0346-6414
56.5579 -64.2417
GP
19.41
1.91
-
NTT1
lens, z=2.96
J0347-2154
56.7690 -21.9095
GP
20.01, 19.46
1.87
-, 2.38
WHT/NTT1 lens, z=0.81, lens =0.187
J0350-4611
57.7146 -46.1858 WD, MD
20.66, 19.68
1.30
1.52, 2.86
NTT1
lens, z=1.505
J0355-5624
58.8040 -56.4147 WD, MD
20.93, 20.03
1.29
-, 2.14
NTT1
lens, z=1.86
J0401-2514
60.4504 -25.2439 WD, MD
19.03, 20.22
1.45
1.67, 1.60
WHT
lens, z=1.32
J0405-3730
61.2721 -37.5128
WD
20.25, 20.57
1.70
0.58, 0.35
NTT2
lens, z=3.585, lens =0.2295
J0408+6333 62.2485
63.5522
WD
14.77, 16.89
0.99
3.19, 2.13
NOT
stars
J0416-5606
64.1783 -56.1073 WD, MD
20.08, 20.81
1.04
1.66, -
NTT1
lens, z=1.45
J0416+7428 64.1972
74.4827
WD
19.74, 19.02
2.64
1.96, 3.57
WHT
lens, z=0.900 lens =0.097
J0429-2246
67.4161 -22.7676
WD
17.42, 19.98
1.80
1.08, -
NTT2
galaxies, z=0.206
J0436+7851 69.1080
78.8530
WD, MD
19.00, 18.66
4.64
1.44, 0.82
NOT
projected QSOs, z=0.765, 1.975
J0440-0905
70.0482
-9.0911
WD, MD
19.92, 19.19
1.69
2.38, 2.15
WHT
lens, z=2.105
J0457-7820
74.3483 -78.3466
WT
19.60, 18.81,
19.07
4.89
1.12, 1.14, 1.50
NTT1
lens, z=3.145
J0504-2446
76.1616 -24.7752
GP
20.50, 19.22
2.05
1.48, 5.44
WHT
lens, z=0.771
J0526-3933
81.5470 -39.5630
WD
19.99, 20.69
1.41
1.88, -
NTT1
lens, z=2.41
J0527-2431
81.7608 -24.5328
WD
19.53, 20.57
1.46
2.60, -
NTT2
NIQ, z=1.437
J0544+4350 86.1448
43.8386
WD
19.64, 20.25
2.04
0.40, 1.09
WHT
lens, z=3.11
J0553+0910 88.2723
9.1812
WD, MD
19.71, 19.73
1.11
2.47, 2.63
NOT
projected QSOs, z=1.08, 1.85
J0601-2220
90.2664 -22.3464 WD, MD
20.65, 20.43
0.98
0.44, 2.12
NTT2
projected QSOs, z=0.809, 0.836
J0607-2152
91.7954 -21.8716
WT
20.86, 19.32,
20.90
1.57
-, 2.05, -
WHT
quad, z=1.305
J0608+4229 92.1725
42.4935
WT, MT
20.13, 18.23,
19.86
1.27
-, 3.94, -
WHT
quad, z=2.346
J0628-7448
97.2299 -74.8010 WD, MD
17.89, 20.50
2.61
2.27, 1.96
NTT2
lens, z=2.692
combined with the redshift evolution predicted by the Illustris cosmological hydrodynamical simulation (Torrey et al. 2015) as described in detail in Oguri (2018), adopting cosmological parameters from the latest Planck satellite mission result (Planck Collaboration et al. 2020), and adopting an improved redshift-dependent model of the external shear.We decide to compare only systems with multiple Gaia detections, since this is where we expect the current selection to beHS0810+2554
PSJ0840+3550
RXJ0911+0551
J1003+0651
HE1113−0641 SDSSJ1251+2935
H1413+117
WFI2026−4536
J2205−3727
Table 2 .
2Summary of confirmed binary quasars. † The brighter component of J2355-4553 is a BAL quasar, and thus is compatible with two redshifts even when masking the absorption features. We quote the more likely lower redshift (2.832) and relevant velocity difference, but a redshift of 2.8506 (Δ = 60kms −1 ) is also possible.Name
Δv (kms −1 ) Δ ( )
(kpc)
J0041-5350
0.5489, 0.5504
290
1.06
6.8
J0315-3522
0.4572, 0.4573
20
1.3
7.6
J0707+4109
0.5114, 0.5149
690
2.96
18.3
J0805+3550
1.6604, 1.6761
1760
1.13
9.6
J0939-0109
0.2240, 0.2247
170
1.9
6.8
J1116-2122
0.7094, 0.7106
210
3.29
23.6
J2121+1713
0.6298, 0.6307
170
2.89
19.8
J2132+3635
1.2135, 1.2148
180
1.33
11
J2256+2223
0.7506, 0.7534
480
1.63
12
J2322+0916
1.2064, 1.2066
30
4.51
37.4
J2355-4553 †
2.8320, 2.8498
1390
2.20
17.2
Table 3 .
3Summary of confirmed projected quasar pairs.Name
Δ ( )
(kpc)
J0021+1927 1.045
1.09
2.89
23.4
J0027+0438 0.194 1.972
1.92
6.2
J0032-4523
1.667
1.74
2.34
19.8
J0249+2606 1.514
2.68
3.21
27.2
J0436+7851 0.765 1.975
4.64
34.3
J0553+0910
1.08
1.85
1.11
9.0
J0601-2220
0.809 0.836
0.98
7.4
J1820+3747 0.614 0.754
3.10
20.9
J2302-4154
0.98
1.22
1.22
9.7
J2314+0323 0.315 0.606
1.87
8.6
J2318+0250
1.83
1.945
3.23
27.2
0.94 ) from
© 2022 The Authors arXiv:2206.07714v1 [astro-ph.GA] 15 Jun 2022 2 C.Lemon et al.
MNRAS 000, 1-30(2022)
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-30(2022)
ACKNOWLEDGEMENTSWe would like to thank Amanda S. Bosh and Stephen E. Levine for obtaining MagIC observations of lensed quasars. This work is supported by the Swiss National Science Foundation (SNSF)BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program "The Emergence of Cosmological Structures" Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 11433005).The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NE-OWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration.The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.
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[
"Existence of finite test-sets for k-power-freeness of uniform morphisms Existence of finite test-sets for k-power-freeness of uniform morphisms",
"Existence of finite test-sets for k-power-freeness of uniform morphisms Existence of finite test-sets for k-power-freeness of uniform morphisms"
] |
[
"G Richomme [email protected] ",
"F Wlazinski [email protected] \nLaRIA\nUniversité de Picardie Jules Verne\nUniversité de Picardie Jules Verne\n33 Rue Saint Leu80039Amiens cedex 01France\n",
"G Richomme ",
"F Wlazinski ",
"\nLaRIA\nLaboratoire de Recherche en Informatique d'Amiens\nUniversité de Picardie Jules Verne -CNRS FRE\n2733 33, rue Saint Leu80039Amiens cedex 01France\n",
"\nLaRIA RESEARCH REPORT : LRR\n\n"
] |
[
"LaRIA\nUniversité de Picardie Jules Verne\nUniversité de Picardie Jules Verne\n33 Rue Saint Leu80039Amiens cedex 01France",
"LaRIA\nLaboratoire de Recherche en Informatique d'Amiens\nUniversité de Picardie Jules Verne -CNRS FRE\n2733 33, rue Saint Leu80039Amiens cedex 01France",
"LaRIA RESEARCH REPORT : LRR\n"
] |
[] |
A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. We provide such an algorithm when k ≥ 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite test-sets for k-power-freeness.
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10.1016/j.dam.2007.04.024
|
[
"https://arxiv.org/pdf/cs/0512051v1.pdf"
] | 1,189,746 |
cs/0512051
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493a34cb16505b5a844b681387b15da9b2b0fef8
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Existence of finite test-sets for k-power-freeness of uniform morphisms Existence of finite test-sets for k-power-freeness of uniform morphisms
13 Dec 2005 2005-10 (December 2005) a , 1st February 2008
G Richomme [email protected]
F Wlazinski [email protected]
LaRIA
Université de Picardie Jules Verne
Université de Picardie Jules Verne
33 Rue Saint Leu80039Amiens cedex 01France
G Richomme
F Wlazinski
LaRIA
Laboratoire de Recherche en Informatique d'Amiens
Université de Picardie Jules Verne -CNRS FRE
2733 33, rue Saint Leu80039Amiens cedex 01France
LaRIA RESEARCH REPORT : LRR
Existence of finite test-sets for k-power-freeness of uniform morphisms Existence of finite test-sets for k-power-freeness of uniform morphisms
13 Dec 2005 2005-10 (December 2005) a , 1st February 2008Formal LanguagesCombinatorics on wordsk-power-free wordsmorphismstest-sets
A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. We provide such an algorithm when k ≥ 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite test-sets for k-power-freeness.
Introduction
Repetitions in words is a recurrent subject of study in Combinatorics on Words. The reader can consult for instance [7,13,14,15] for surveys of results and applications. The interest for such regularities dates back to the works of A. Thue [23,24] (see also [3,4]) who, one century ago, provided examples of some repetition-free words, more precisely some square-free and overlap-free words. The construction of some of these words is simple: they are generated as fixed points of free monoid morphisms. An example is the fixed point (denoted Θ ω (a)) of the morphism Θ defined by Θ(a) = abc, Θ(b) = ac and Θ(c) = b:
Θ ω (a) = abcacbabcbacabcacbacabcb . . . This word is k-power-free [9,24] for any integer k ≥ 2, that is, it does not contain any word on the form u k with u non-empty. May be strangely, for any k ≥ 2, the morphism Θ is not itself k-powerfree: it does not map all k-power-free words on k-power-free words (Θ(ab k−1 a) = ab(ca) k bc). So where'as any k-power-free morphisms generates a k-power-free word, the converse does not hold.
F. Mignosi and P. Séébold [16] have proved that it is decidable whether a morphism generates a k-power-free word: more precisely they proved that, given a word w and a morphism f , it is decidable whether the language {f n (w) | n ≥ 0} is k-power-free. However, given an integer k ≥ 3, to decide if a morphism is k-power-free is still an open problem even if some partial results have been achieved especially for morphisms acting on binary alphabets and for 3-power-free morphisms on ternary alphabets [2,11,12,21,25]. We note that the case k = 2 was solved by M. Crochemore [8]. We also observe that properties of k-power-free morphisms are badly known (see for instance [20]) despite of some efforts in the eighties [11,12] when relations between morphisms and variable-length codes (in the sense of [5]) were studied.
A related problem is the study of overlap-free morphisms: an overlap-free word is a word that does not have any factor of the form auaua with a a letter and u a word; an overlap-free morphism is a morphism preserving overlap-freeness. The study of overlap-free binary morphisms provides ideas of simple tests that can be extended to other classes of morphisms like k-power-free morphisms. For instance, the monoid of overlap-free binary endomorphisms is finitely generated. Unfortunately this is no longer true for both larger alphabets and k-power-free morphisms [8,18,21]. Another simple idea is to test overlap-freeness using a finite set of overlap-free words, called test-set for overlapfreeness [6,19]. Recently [22] we have shown that, in the general case, a finite test-set exists for overlap-freeness of morphisms defined on an alphabet A if and only if A is a binary alphabet. But if we consider only uniform morphisms (the images of the letters have all the same length), such test-sets always exist. Note that the study of uniform overlap-free morphisms is natural since all overlap-free binary endomorphisms are uniform. Another reason to study uniform morphisms is provided by Cobham's theorem stating that a word is automatic if and only if it is the image under a 1-uniform morphism of a fixed point of a uniform morphism (see for instance [1]). Finally let us mention that uniform morphisms are sometimes easier to use to give examples of infinite words with particular properties, as done for instance in [17] where a finite test-set is provided for morphisms mapping α + -power-free words onto β + -power-free words when α and β are two rational numbers with 1 ≤ α ≤ β ≤ 2.
We started the study of test-sets for k-power-freeness of morphisms in [21] where we obtained a result similar to the case of overlap-freeness: for k ≥ 3, a finite test-set exists for k-power-freeness of morphisms defined on an alphabet A if and only if A is a binary alphabet. The purpose of this paper is to complete this work showing that, as for overlap-freeness, there always exist test-sets for k-power-freeness of uniform morphisms (see Theorem 3.1). Up we know, the existence of such test-sets for uniform morphisms was previously stated only for morphisms defined on two-letter [10,11,25] or three-letter alphabets [12].
Despite of the similarities between overlap-freeness and k-power-freeness, we would like to stress many differences between the two studies. Firstly, we mention that the maximal lengthes of words involved in the test-sets are different since of course in one case they depend on the parameter k and not just on the size of the alphabet. More important is the fact that we introduce a new way to tackle the decidability of repetition-freeness.
We will only consider test-sets for k-power-freeness when k ≥ 3. Indeed it is well-known that a uniform morphism is 2-power-free (that is square-free) if and only if the images of 2-power-free words of length 3 are 2-power-free: in our terminology this means that the set of 2-power-free words of length 3 is a test-set for 2-power-freeness of uniform morphisms. The test-sets we obtain are not so simple and depend on both the value of k and the cardinality of A.
We present our test-sets, main tools for the proof and the proof itself in Section 3, Section 4 and Section 5 respectively.
with the empty word ε as neutral element and A as set of generators. Given a non-empty word u = a 1 . . . a n with a i ∈ A, the length of u denoted by |u| is the integer n that is the number of letters of u. By convention, we have |ε| = 0.
A word u is a factor of a word v if there exist two (possibly empty) words p and s such that v = pus. We also say that v contains the word u (as a factor). If p = ε, u is a prefix of v. If s = ε, u is a suffix of v. A word u is a factor (resp. a prefix, a suffix ) of a set of words X, if u is a factor (resp. a prefix, a suffix) of a word in X.
Let w be a word and let i, j be two integers such that 0 ≤ i − 1 ≤ j ≤ |w|. We denote by w[i..j] the factor u of w such that there exist two words p and s with w = pus, |p| = i − 1, |pu| = j. Note that, when j = i − 1, we have w[i..j] = ε. When i = j, we also denote by w[i] the factor w[i..i] which is the i th letter of w.
Given two words w and u, we denote by |w| u the number of different words p such that pu is a prefix of w. For instance, if w = abaababa, we have |w| a = 5, |w| aba = 3.
Powers of a word are defined inductively by u 0 = ε, and for any integer n ≥ 1, u n = uu n−1 : such a word is called a n-power when n ≥ 2 and u = ε. A word is k-power-free (k ≥ 2) if it does not contain any k-power as factor. A set of k-power-free words is said k-power-free.
Let us recall two well-known results of combinatorics on words:
Proposition 2.1 [13] Let A be an alphabet and u, v, w three words over A. If vu = uw and v = ε then there exist two words r and s over A and an integer n such that u = r(sr) n , v = rs and w = sr.
Lemma 2.2 [11,12] If a non-empty word v is an internal factor of vv (that is, if there exist two non-empty words x and y such that vv = xvy) then there exist a non-empty word t and two integers
i, j ≥ 1 such that x = t i , y = t j and v = t i+j .
Let A, B be two alphabets. A morphism f from A * to B * is a mapping from A * to B * such that for all words u, v over A, f (uv) = f (u)f (v). When B does not have any importance, we will say that f is a morphism on A or that f is defined on A. A morphism on A is entirely known by the images of the letters of A. When B = A, f is called an endomorphism (on A). Given an integer L, f is L-uniform if for each letter a in A we have |f (a)| = L. A morphism f is uniform if it is L-uniform for some integer L ≥ 0. Given a set X of words over A, and given a morphism f on A, we denote by f (X) the set {f (w) | w ∈ X}.
A morphism f on A is k-power-free if and only if f (w) is k-power-free for all k-power-free words w over A. For instance, the empty morphism ǫ (∀a ∈ A, ǫ(a) = ε) is k-power-free.
Main result
Let us recall that in all the rest of this paper A is an alphabet containing at least two letters and k ≥ 3 is an integer.
Our main result (Theorem 3.1) is the existence of test-sets for k-power-freeness of uniform morphisms whatever is A and k: A test-set for k-power-freeness of uniform morphisms on A is a set T ⊆ A * such that, for any uniform morphism f on A, f is k-power-free if and only if f (T ) is k-power-free.
This existence is provided by the set
T A,k = U k,A ∪ (kPF(A) ∩ V k,A )
where U k,A , kPF(A) and V k,A are defined as follows:
• U k,A is the set of k-power-free words over A of length at most k + 1,
• kPF(A) is the set of all k-power-free words over A, and
• V k,A is the set of words over A that can be written a 0 w 1 a 1 w 2 . . . a k−1 w k a k where a 0 , a 1 , . . . , a k are letters of A and w 1 , w 2 , . . . , w k are words over A verifying ||w i | − |w j || ≤ 1 and |w i | a ≤ 1; ∀1 ≤ i, j ≤ k and ∀a ∈ A.
In the previous definition, the inequality |w i | a ≤ 1 means that any letter of A appears at most once in w i . In particular, it follows that max{|w|
| w ∈ T A,k } ≤ b k,A where b k,A = k × Card(A) + k + 1.
Theorem 3.1 T A,k is a test-set for k-power-freeness of uniform morphisms on A.
An immediate consequence is the following corollary that gives a simple bound for the length of the words whose images we have to check to verify the k-power-freeness of a morphism:
Tools
In this section we recall or introduce some useful tools. May be the reader will read them when needed in the proof of Theorem 3.1, but we would like to present the novelties of our approach (from Section 4.2).
ps-morphisms
A morphism f is a ps-morphism (Keränen [11] called it ps-code) if f (a) = ps, and f (b) = ps ′ , f (c) = p ′ s with a, b, c ∈ A (possibly c = b), p, s, s ′ , p ′ in B * then necessarily b = a or c = a. Any any ps-morphism is injective. A basic result about these morphisms is: [11,12] If all the k-power-free words of length at most k + 1 have a k-power-free image by a morphism f , then f is a ps-morphism.
Lemma 4.1
Decomposition of k-powers
One situation that we will quickly meet in the proof of Theorem 3.1 is: f is a L-uniform ps-morphism (L ≥ 0), w is a k-power-free word such that f (w) contains a k-power u k and |w| ≥ k + 1. In this case, Lemma 4.2 below will enable us to decompose u k using factors of f (w) (see also Figure 1).
We observe that (possibly by replacing w by one of its factors) we can consider that u k is directly covered by f (w). This means that u k is not a factor of the image of a proper factor of w. More precisely, if p 0 and s k are the words such that f (w) = p 0 u k s k then |p 0 | < L and |s k | < L. The present situation verifies: Lemma 4.2 Let f be a uniform morphism and let k ≥ 3 be an integer. A k-power u k (u = ε) is directly covered by the image of a word w of length at least k + 1 if and only if there exist words (p i ) i=0,...,k , (s i ) i=0,...,k , (w i ) i=1,...,k and letters (a i ) i=0,...,k such that:
(1) w = a 0 w 1 a 1 . . . a k−1 w k a k , (2) f (a i ) = p i s i (0 ≤ i ≤ k), (3) s 0 = ε,(4)p i = ε (1 ≤ i ≤ k), (5) u = s i−1 f (w i )p i (1 ≤ i ≤ k). = / ε u k = / ε = / ε = / ε = / ε u u u p s 1 1 p s 2 2 s −1 k p s k k p k −1 0 k −1 0 w 1 w k k w 2 2 s p = f ( ) w 0 0 f ( ) a ( ) ( ) a a ( ) f f f f ( ) ( ) a ( ) f ( ) f a f Figure 1: (p i , s i , x i , w i ) i=0,...,k -decomposition of u k in f (w)
Proof of Lemma 4.2. By definition f is uniform: Let L be the integer such that |f (b)| = L for each letter b. Clearly if words (p i ) i=0,...,k , (s i ) i=0,...,k , (w i ) i=1,...,k and letters (a i ) i=0,...,k verify Conditions (1) to (5), then |w| ≥ k + 1 and u k is directly covered by f (w).
Assume now that u k is covered by f (w) with |w| ≥ k + 1. Let p 0 and s k be the words such that f (w) = p 0 u k s k . For each integer ℓ between 0 and k, let i ℓ be the least non-zero integer such that pu ℓ is a prefix of f (w[1.
.i ℓ ]). Since u k is covered by f (w), i 0 = 1, i k = |w| and i 0 ≤ i 1 ≤ i 2 ≤ . . . ≤ i k .
If i ℓ = i ℓ+1 for (at least) one integer ℓ between 0 and k − 1, then |u| ≤ |f (a ℓ )| = L. For any integer m between 0 and k − 1, since f is L-uniform and |u| ≤ L = |f (a m )|,
i m + 1 ≥ i m+1 (i m = i m+1 or i m + 1 = i m+1 ). Hence |w| = i k ≤ i ℓ+1 + (k − ℓ − 1) = i ℓ + (k − ℓ − 1) ≤ (i 0 + ℓ) + (k − ℓ − 1) = k: a contradiction. So i 0 < i 1 < i 2 < . . . < i k .
We define for each integer ℓ between 1 and k the words
w ℓ = w[i ℓ−1 + 1 . . . i ℓ − 1] and p ℓ such that f (w[1..i ℓ+1 − 1])p ℓ = p 0 u ℓ . Moreover let a ℓ = w[i ℓ ] for 0 ≤ ℓ ≤ k.
By construction for 0 ≤ ℓ ≤ k − 1, the word p ℓ is a non-empty prefix of f (a ℓ ) and so we can consider the word s ℓ such that f (a ℓ ) = p ℓ s ℓ . Up to now by construction, we have Conditions (1), (2), (4) and (5). Since u k is covered by f (w), Condition (3) is also verified.
k + 1, if (p i , s i , a i , w i ) i=0,.
..,k is a (4k + 4)-uple such that w 0 = ε and the other 4k + 3 words verify Conditions (1) to (5) of Lemma 4.2, we will say that u k has a
(p i , s i , a i , w i ) i=0,...,k -decomposition in f (w), or that (p i , s i , a i , w i ) i=0,...,k is a decomposition of u k in f (w).
Non-synchronized decompositions of k-powers
Between all decompositions that a k-power can have in the image of a word by a L-uniform morphism f , Lemma 4.5 will allow us to eliminate the following possibility:
Definition 4.4 Let (p i , s i , a i , w i ) i=0,.
..,k be as in Definition 4.3. When |s i | = |s i+1 | for an integer i between 1 and k − 2, the decomposition is said synchronized (with respect to images of factor of w), or shortly that the k-power u k is synchronized in f (w).
Let us make several remarks about this definition.
First it is immediate that a decomposition (p i , s i , a i , w i ) i=0,...,k of a k-power is synchronized if and only if for all integers i, j with 1 ≤ i < j ≤ k − 1, we have |s i | = |s j |. Since f is uniform, and since f (a ℓ ) = p ℓ s ℓ (for all ℓ, 1 ≤ ℓ ≤ k − 1), it is also equivalent that |p i | = |p j | for all 1 ≤ i < j ≤ k − 1, or that |p i | = |p i+1 | for all 1 ≤ i ≤ k − 2.
One aspect may appear strange: why do not we allow i = 0 in the definition of a synchronized decomposition? This is due to the dissymmetry brought by Conditions (3) and (4) in the definition of a decomposition. Assume that (
p i , s i , a i , w i ) i=0,...,k is a synchronized decomposition of a k-power u k in f (w) with f L-uniform. Since |s 1 | = |s 2 |, we have |p 1 | = |p 2 |. Moreover u = s 0 f (w 1 )p 1 = s 1 f (w 2 )p 2 . Thus p 1 = p 2 and s 0 f (w 1 ) = s 1 f (w 2 ). When s 1 = ε, since also p 1 = ε, we have 0 < |s 1 | < L. In this case s 0 = s 1 . But when s 1 = ε, since s 0 = ε, we have s 0 = s 1 , p 0 = ε and s 0 = f (a 0 ) = s 1 .
Of course we do not consider i = k − 1 in the definition of a synchronized decomposition simply because s k is not a factor of u k .
Lemma 4.5 Let f be a uniform ps-morphism defined on an alphabet A, and let k ≥ 3 be an integer.
Any k-power directly covered by the image by f of a k-power-free word of length at least k + 1 is not synchronized.
Proof of Lemma 4.5.
By definition f is uniform: Let L be the integer such that |f (b)| = L for each letter b. Assume there exists a k-power u k that has a synchronized decomposition (p i , s i , a i , w i ) i=0,...,k in f (w), where w is a k-power-free word. By hypothesis s i = s j and p i = p j for all 0 < i < j < k. We denote s = s 1 and p = p 1 . From u = sf (w 2 )p = sf (w k )p k , we deduce that |p| = |p k | mod L. Since 0 < |p|, |p k | ≤ L, we get p k = p. Hence u = s 0 f (w 1 )p and u = sf (w i )p for all 2 ≤ i ≤ k. We have seen before the lemma's statement that s 0 = s when s = ε and s 0 = f (a 0 ) when s = ε.
Assume first s = ε and s 0 = f (a 0 ). Since f is injective, we get a 0 w 1 a 1 = w i a i for all 2 ≤ i ≤ k. Thus w = (a 0 w 1 a 1 ) k . This contradicts the fact that w is k-power-free.
So s = ε and s 0 = s. Since f is injective, w i = w 1 for all 1 ≤ i ≤ k and a i = a 1 for all
1 ≤ i ≤ k − 1. Hence w = a 0 (w 1 a 1 ) k−1 w 1 a k . Since w is k-power-free, a 0 = a 1 and a k = a 1 . Let a = a 1 , b = a k , c = a 0 , p ′ = p 0 and s ′ = s k : f (c) = p ′ s, f (b) = ps ′ . From f (a) = ps, we deduce that f is not a ps-morphism.
We end this section with some examples of non-synchronized k-powers.
Example 4.6 f (a) = baaba, f (b) = bcdab, f (c) = cdabc, f (d) = dbaab : f (abcd) = baab(abcd) 3 baab.
The decomposition of (abcd) 3 The decomposition of (12345123452) 3 in f (154216322) is given by a 0 = 1, a 1 = 4, a 2 = 6, a 3 = 2,
in f (abcd) is given by a 0 = a, a 1 = b, a 2 = c, a 3 = d, w 1 = w 2 = w 3 = ε, p 0 = baab = s 3 , s 0 = a, p 1 = bcd, s 1 = ab, p 2 = cd, s 2 = abc, p 3 = d. b f(a) f(b) d c b a a a b b a b c d f(d) f(c) a b c d b af (1) (5) (4) (2) (1) (6) (3) (2) (2) f f f f f f fw 1 = 5, w 2 = 21, w 3 = 32, p 0 = ε, p 1 = 452, p 2 = 52, p 3 = 2, s 0 = f (a 0 ), s 1 = 1, s 2 = 12, s 3 = 345.
Reduction of a k-power
In this section, we introduce the key technic of the proof of Theorem 3.1. It consists in the possibility to reduce the length of k-powers in order to consider only k-powers covered by the image of a word in V k,A .
Proposition 4.8 Let f be an injective uniform morphism on A.
If there exists a k-power-free word W of length at least k + 1 such that U k is directly covered by f (W ) then there exists a word w of length at least k + 1 such that w ∈ V k,A , |w| ≤ |W | and f (w) covers a k-power u k . Moreover the k-powers u k and U k are both synchronized or both non-synchronized.
This proposition is a direct corollary of Lemma 4.9 (to be used inductively) whose idea is illustrated by Figure 4.
We denote by Reduced(U k , W ) the set of pairs (u k , w) that can be obtained in conclusion of Proposition 4.8. Lemma 4.9 (Reduction lemma) Let f be an injective uniform morphism on A and let w be a word over A. We assume that there exists a non-empty word u such that the k-power u k has a
(p i , s i , a i , w i ) i=0,..,k -decomposition in f (w)
. We also assume that there exist an integer 1 ≤ ℓ ≤ k and a letter a in A such that w ℓ = x ℓ y ℓ z ℓ and both x ℓ and y ℓ end with a. Then: Proof of lemma 4.9.
1. For all integers i such that 1 ≤ i ≤ k, there exist three words x i , y i , z i such that w i = x i y i z i , |s ℓ−1 f (x ℓ )| − |f (a)| < |s i−1 f (x i )| ≤ |s ℓ−1 f (x ℓ )| and |y i | = |y ℓ |. 2. Let u ′ = s ℓ−1 f (x ℓ z ℓ )p ℓ and w ′ = a 0 k i=1 (x i z i a i ). The k-power (u ′ ) k has a (p i , s i , a i , x i z i ) i=0,..,k - decomposition in f (w ′ ). 3. |w ′ | < |w|.f (x 1 ) k −2 k −1 −1 k k −1 k −1 k k v v f (x ) f (x k ) k −1 k v' v' −1 ) l ) l f (x ) ( f a) ( f a) = s q −1 f ( q q x y z q ) p q ) l f ( ) y ) l ) l f ( ) z 2 1 1 2 v v v 1 2 f (x ) 2 v' v' v' i f (x i ) i i i
1. By definition f is uniform: Let L be the integer such that |f (b)| = L for each letter b.
Let i be an integer such that
1 ≤ i ≤ k. We have u = s i−1 f (w i )p i = s ℓ−1 f (x ℓ y ℓ z ℓ )p ℓ .
Let us observe that:
|s i−1 | ≤ |s ℓ−1 f (x ℓ )| ≤ |s i−1 f (w i )| Indeed, since |f (x ℓ )| = 0 (x ℓ ends with a), we have |s i−1 | ≤ |f (a i−1 )| = L = |f (x ℓ )| ≤ |s ℓ−1 f (x ℓ )|. Moreover |s ℓ−1 f (x ℓ )| ≤ |s ℓ−1 f (x ℓ )|+|f (y ℓ )|−|f (a)| ≤ |s ℓ−1 f (w ℓ )|−|f (a)| ≤ |s ℓ−1 f (w ℓ )|+|p ℓ |−|p i | = |s i−1 f (w i )|.
Thus we can define x i as the greatest prefix (maybe empty) of w i such that s i−1 f (x i ) is a prefix of s ℓ−1 f (x ℓ ). Since f is uniform, we have:
|s ℓ−1 f (x ℓ )| − |f (a)| < |s i−1 f (x i )| ≤ |s ℓ−1 f (x ℓ )| It follows that |s i−1 f (x i )| ≤ |s ℓ−1 f (x ℓ )| < |s ℓ−1 f (x ℓ y ℓ )|. Let y i be the greatest word such that x i y i is a prefix of w i and s i−1 f (x i y i ) is a prefix of s ℓ−1 f (x ℓ y ℓ ). Let z i be the word such that w i = x i y i z i Let v ′ i be the word such that s i−1 f (x i y i )v ′ i = s ℓ−1 f (x ℓ y ℓ ). We have v ′ i f (z ℓ )p ℓ = f (z i )p i . Assume |v ′ i | ≥ L. The definition of y i implies that z i = ε. The equality |v ′ i f (z ℓ )p ℓ | = |p i | with |p ℓ | = 0 is incompatible with |p i | ≤ L. Thus |v ′ i | < L. It follows: |s ℓ−1 f (x ℓ y ℓ )| − |f (a)| < |s i−1 f (x i y i )| ≤ |s ℓ−1 f (x ℓ y ℓ )|
From this double inequality and the previous one concerning |s i−1 f (x i )|, we deduce that |f (y ℓ )|− |f (a)| < |f (y i )| < |f (y ℓ )| + |f (a)|.
Since f is uniform, it follows that |f (y ℓ )| = |f (y i )| and |y i | = |y ℓ | (see Figure 4).
For all integers 1
≤ i ≤ k, let v i be the word such that s i−1 f (x i )v i = s ℓ−1 f (x ℓ ). By definition of x i , we have 0 ≤ |v i | < |f (a)|. Moreover f (y i z i )p i = v i f (y ℓ z ℓ )p ℓ . Since |y i | = |y ℓ |, we get |s ℓ−1 f (x ℓ )| = |s ℓ−1 f (x ℓ y ℓ )| − |f (y ℓ )| = |s i−1 f (x i y i )v ′ i | − |f (y i )| = |s i−1 f (x i )v ′ i |. It follows that |v i | = |v ′ i |.
Since x ℓ and y ℓ both end with a and since |v i | = |v ′ i | < |f (a)|, it follows that v i and v ′ i are both suffixes of f (a) and so
v i = v ′ i . Let w ′ = a 0 k i=1 (x i z i a i ). For all integers i such that 1 ≤ i ≤ k, we have u ′ = s ℓ−1 f (x ℓ z ℓ )p ℓ = s i−1 f (x i )v i f (z ℓ )p ℓ = s i−1 f (x i z i )p i . Thus, f (w ′ ) = p 0 s 0 k i=1 f (x i z i )p i s i = p 0 k i=1 s i−1 f (x i z i )p i s k = p 0 u ′k s k .
3. Since y ℓ = ε (y ℓ ends with a), we have |w ′ | < |w|.
Let us give an example of reduction: . This morphism is not 3-power-free (it is not a ps-morphism). We observe (see Figure 5) that f (17185429a2163bc322) contains the cube (12345178123462345123452) 3 . This 3-power can be reduced on two ways. First, using the fact that f (1) appears twice in the first occurrence of u, we can obtain the cube (123462345123452) 3 in the image of f (1854a216c322) as shown by Figure 6. Second, using the fact that f (3) appears twice in the first occurrence of u, we can obtain the cube (12345123452) 3 in the image of f (154216322) as shown by Figure 7. (1)
(7) (1) (8) (5)(4)
(2) (9) ( a )
(1)
(3) b) ( c ) ((6)
(2)
f f f f f f f f f f f f f f f f f(a) (2) (1) (6) c ) ((8)(5)(4)f (1) (5)(4)
(2) (1)
(3)
(2) We observe in Example 4.10 that the two possible reductions verify the first Reduction Rule, and the different words obtained are both in V k,A . The one chosen will be the first reduction according to the second rule.
f f f f f f f
We end with two remarks (using notations from Lemma 4.9) that will be useful in the end of Theorem 3.1. The first remark is a direct consequence of the first part of Lemma 4.9. To understand the second remark, we observe that since |u| = |s ℓ−1 f (x ℓ y ℓ z ℓ )p ℓ | = |s j−1 f (x j y j z j )p j | for all integers j such that 1 ≤ j ≤ k, we also have |f (z j )p j | − |f (a)| < |f (z ℓ )p ℓ | ≤ |f (z j )p j |.
Remark 4.11
1. If there exists an integer q such that x q = ε then x ℓ = a and |s ℓ−1 | < |s q−1 |.
2.
If there exists an integer q such that z q = ε then z ℓ = ε and |p ℓ | ≤ |p q |. More precisely given two words W and U with W k-power-free, |W | ≥ k + 1 and U k directly covered by f (W ), we will construct (using this proposition) some words w and u such that (u k , w) belongs to Reduced(U k , W ), |w| ≤ |W | and w ∈ V k,A . Moreover the decomposition of u k in f (w) will be non-synchronized. Since the word w belongs to V k,A and since f (T A,k ) is k-power-free, we can see that w is not k-power-free, and so there exists a non-empty word v such that v k is a factor of w. We will be able to prove that this situation will be possible only if k = 3 and |v| = 1.
More precisions on the reduction
But when k = 3, the following example shows that there can exists words w and u such that u k has a non-synchronized decomposition in f (w): so we will need to be more precise in our use of the reductions. Thus this 5-uniform morphism f is a ps-morphism for which there exists a non-synchronized k-power. We let the reader verify that f is a 3-power-free morphism and so f (T 3,{1,2,3,4,5,6,7,8,9} ) is 3-power-free.
We now explain how we tackle the situation k = 3 and |v| = 1. As we have just seen by Example 8, there can exist words w ∈ V k,A and u such that u k has a non-synchronized decomposition in f (w). We will show that, under all current hypotheses, w and u cannot be obtain by successive reductions from the words W and U define in the previous section. For this purpose, we will be more precise on the way the reductions are made to obtain a couple (u k , w) in Reduced(U k , W ). Actually one can observe that if a word does not belong to V k,A , there can exist many different ways to reduce it using Lemma 4.9. We will apply the two following additional rules (with the notations of Lemma 4.9): 1. |x ℓ | a = 1 and |y ℓ | a = 1 2. if there exist an integer 1 ≤ ℓ ′ ≤ k and a letter a ′ in A such that w ℓ ′ = x ℓ ′ y ℓ ′ z ℓ ′ and both x ℓ ′ and y ℓ ′ end with a ′ and such that (ℓ, a) = (ℓ ′ , a ′ ), then
|s ℓ−1 f (x ℓ )| < |s ℓ ′ −1 f (x ℓ ′ )|.
These rules mean that we always made the leftmost reduction possible. The determinism introduced by these rules will be a key element of the proof.
Proof of Theorem 3.1
In this section, we prove Theorem 3.1 which means: given any L-uniform morphism f on A (with L ≥ 0 an integer), f is k-power-free if and only f (T A,k ) is k-power-free.
Let f is a uniform morphism from A * to B * where B is an alphabet not necessarily equals to A, and let L be the integer such that |f (b)| = L for each letter b. The "only if" part of the theorem follows immediately from the definition and the "if" part is also immediate when L = 0. Thus from now on L ≥ 1. We assume that f (T A,k ) is k-power-free and we show (by contradiction) that f is k-power-free.
Since U k,A ⊆ T A,k , by Lemma 4.1, we have Fact 1 f is a ps-morphism.
Let us recall that this implies that f is injective
Assume by contradiction that f is not k-power-free. We first make a crucial choice.
Choice 1 : let W be a k-power-free word of smallest length such that f (W ) directly covers a k-power.
Let U be a word such that U k is directly covered by f (W ). Since U k,A ⊆ T A,k , |W | ≥ k + 1. Let (p i , s i , a i , W i ) i=0,...,k be a decomposition of U k in f (W ). By Lemma 4.5, this decomposition is not synchronized, that is, s i = s j and p i = p j for all integers i, j with 0 < i < j < k.
Applying iteratively the Reduction Lemma 4.9 with the deterministic rules chosen in Section 4.5, we construct some words w and u such that (u k , w) belongs to Reduced(U k , W ), |w| ≤ |W | and w ∈ V k,A . We know that the decomposition of u k in f (w) is (p i , s i , a i , w i ) i=0,...,k for some words (w i ) i=0,...,k .
Let us observe that since the decomposition (p i , s i , a i , W i ) i=0,...,k is not synchronized, it follows the definition that (p i , s i , a i , w i ) i=0,...,k is also not synchronized.
Since w ∈ V k,A and f (kPF(A) ∩ V k,A ) is k-power-free, we deduce that:
Fact 2 w is not k-power-free.
Choice 2 : let v k be a smallest k-power factor of w (v = ε). This fact is a consequence of the following proposition which is a corollary of the well-known Fine and Wilf's theorem (see [13,14] for instance).
We denote v 1 , v 2 words such that w = v 1 v k v 2 .
Proposition 5.1 [11] Let x and y be two words. If a power of x and a power of y have a common factor of length at least equal to |x| + |y| − gcd(|x|, |y|) then there exist two words t 1 and t 2 such that x is a power of t 1 t 2 and y is a power of t 2 t 1 with t 1 t 2 and t 2 t 1 primitive words. Furthermore, if |x| > |y| then x is not primitive.
Proof of Fact 3. Assume the opposite. By Proposition 5.1, there exist two words t 1 , t 2 and two integers n 1 , n 2 such that f (v) = (t 1 t 2 ) n 1 and u = (t 2 t 1 ) n 2 . Since u = ε and v = ε, we have t 1 t 2 = ε, n 1 ≥ 1 and n 2 ≥ 1. If n 1 ≥ 2, f (v ⌈k/2⌉ ) = (t 1 t 2 ) n 1 ⌈k/2⌉ contains the k-power (t 1 t 2 ) k . Since k ≥ 3, ⌈k/2⌉ < k, and so |v ⌈k/2⌉ | < |v k | ≤ |w| ≤ |W |. By choice of v, v ⌈k/2⌉ is k-power-free: this contradicts Choice 1 on W . So n 1 = 1. We get |u| = |f (v) n 2 | = n 2 |f (v)| and so |u| = 0 mod L. For all integers j between 1 and k, |us j | = |s j−1 f (w j )p j s j | = |s j−1 f (w j a j )|, and so |s j | = |s j−1 | mod L. But for j ≥ 1, p j = ε, so that |s j | < L. It follows that |s j | = |s j−1 | for all j ≥ 2. This contradicts the fact that the decomposition of u k is not synchronized.
Fact 4 |v| = 1 and k = 3
The proof of this fact is made of three steps.
Step 4.1 If |f (v)| ≥ |u| then |v| = 1 and k = 3. Proof. Since v = ε, we can write v = xv ′ = v ′′ y for two letters x, y and two words v ′ , v ′′ . Since
f (w) = f (v 1 )f (v) k f (v 2 ) = pu k s with |p| < L and |s| < L, the word C = f (v ′ v k−2 v ′′ ) is a common factor of f (v k ) and u k . We have |C| = |f (v) k | − |f (xy)| = kL|v| − 2L. When |v| ≥ 2 or when k ≥ 4, |C| ≥ 2L|v| = 2|f (v)| ≥ |f (v)| + |u|.
So by Fact 3, we cannot have |v| ≥ 2 or k ≥ 4, that is (since v = ε), we must have |v| = 1 and k = 3.
Step 4.2 If |f (v)| < |u| then |v| = 1.
Proof. Let us assume by contradiction that |v| ≥ 2. There exist two letters x, y and a word v ′ such that
v = xv ′ y. Since |f (v 1 )f (v) k f (v 2 )| = |f (w)| = |pu k s| ≥ |u k | > |f (v k )|, we have |v 1 v 2 | ≥ 1. If v 1 = ε, we get x = a 0 and f (x) = p 0 s 0 . Since |f (xv ′ yv k−1 )| = |f (v) k | < |u k | ≤ |pu k |, the word C = s 0 f (v ′ yv k−1 )
is a prefix of u k and so a common factor of f (v) k and u k . Let us recall that w ∈ V k,A . This means in particular that |w 1 | y ≤ 1 and so, since w = xw 1 a 1
k i=2 w i a i starts with xv ′ yxv ′ y, w 1 is a prefix of v ′ yxv ′ . Consequently |u| ≤ |s 0 f (v ′ yxv ′ y)|. It follows that |C| = |s 0 f (v ′ yxv ′ y)| + (k − 2)|f (v)| ≥ |u| + |f (v)|. This contradicts Fact 3. So v 1 = ε.
Similarly we can prove that v 2 = ε and so v k = (xv ′ y) k is a factor of w 1 k i=2 a i w i . Thus f (v) k is a common factor of f (v) k and u k . Since w ∈ V k,A , we have |w i | x ≤ 1 and |w i | y ≤ 1 for all 1 ≤ i ≤ k. This implies that |xv ′ yxv ′ y| ≥ |w i | + 2 for all 1 ≤ i ≤ k and thus |f (v) 2 | = |f (xv ′ yxv ′ y)| ≥ |u|. Consequently |f (v 3 )| ≥ |f (v)| + |u|, and once again we have a contradiction with Fact 3.
Step 4.3 If |f (v)| < |u| then k = 3. Proof. By the previous step, we know that |v| = 1.
So v = x for a letter x. Since |f (v 1 )f (x) k f (v 2 )| = |f (w)| = |pu k s| ≥ k|u| > k|f (x)|, we have |v 1 v 2 | ≥ 1. If v 1 = ε, x k is a prefix of w, x = a 0 and f (x) = p 0 s 0 . Since |s 0 f (x) k−1 | ≤ |f (x) k | < |u k | and since s 0 f ( k i=1 w i a i ) = u k s k , the word s 0 f (x) k−1
is a prefix of u k and so a common factor of f (v) k and u k . Since w ∈ V k,A , |w 1 | x ≤ 1. This implies w 1 = ε or w 1 = x and so |u| = |s 0 f (w 1 )
p 1 | ≤ |s 0 f (xx)|. If k ≥ 4, |s 0 f (x) k−1 | ≥ |s 0 f (xx)| + |f (x)| ≥ |f (v)| + |u|: this contradicts Fact 3. So k = 3.
When v 2 = ε, symmetrically we can prove k = 3. Now we consider the case where v 1 = ε and v 2 = ε. The word f (x) k is a common factor of f (v) k and u k . Let us recall that w = a 0 k i=1 w i a i , and |w i | x ≤ 1 for each integer i with 1 ≤ i ≤ k. Since here x k is a factor of w 1
k−1 i=2 a i w i+1 , there must exist an integer i, 1 ≤ i ≤ k such that w i = x. Thus |u| + |f (x)| = |s i−1 f (w i )p i | + |f (x)| ≤ |f (x 4 )|. This contradicts Fact 3 when k ≥ 4. So k = 3.
We now make a break in the proof of the theorem to explain the situation. Up to now, we have proved this theorem when k ≥ 4 showing that, when f (T A,k ) is k-power-free, there cannot exist words like w and u such that w ∈ V k,A and u k has a non-synchronized decomposition in f (w). Example 4.12 shows that this is possible when k = 3. Consequently when dealing with Case k = 3 (and |v| = 1), we have to consider the sequence of reductions of the couple (U k , W ) into the couple (u k , w). This will occur only in Cases 3 and 7 below. Actually Example 4.12 belongs to Case 3.
then, since f (x) = p 1 s 1 = v 3 v ′ 3 , v 3 is a prefix of p 1 and, since f is injective, yy is a factor of w 2 : this contradicts the fact that w ∈ V k,A .
So |p 1 | < |v 3 | and consequently
|s 1 | > |v ′ 3 | (since f (x) = p 1 s 1 = v 3 v ′ 3 )
. We observe that s 0 = s 1 v ′ 3 and s 1 is a suffix of s 0 (remember f (x) = p 0 s 0 ), that is s 0 = s 1 v ′ 3 = v 4 s 1 for a word v 4 . Lemma 2.1 implies the existence of words α, β and of an integer r such that s 1 = (αβ) r α, v ′ 3 = βα and s 0 = (αβ) r+1 α (and v 4 = αβ). Note that r ≥ 1 and αβ = ε since |s 1 | > |v ′ 3 | = 0. Thus the words f (xy) contains the factor s 0 v ′ 3 = (αβ) r+2 α which contains the 3-power (αβ) 3 . Since xy ∈ T A,k , this contradicts the 3-power-freeness of f (T A,k ).
Case 1.b: |s 0 | < |s 1 |
We have f (x) = p 0 s 0 = p 1 s 1 and u starts with both s 0 and s 1 . Let v 3 and v 4 be the words such that v 3 s 0 = s 1 = s 0 v 4 . Lemma 2.1 implies the existence of words α, β and of an integer r such that v 3 = αβ, s 0 = (αβ) r α, v 4 = βα. Since u starts both with s 0 f (x) and with s 1 , the word v 4 is a prefix of f (x). Thus f (xx) contains the factor s 1 v 4 = (αβ) r+2 α. Since xx ∈ T A,k and f (T A,k ) is 3-power-free, we have r = 0, that is s 1 = s 0 βs 0 .
If |f (x)| ≥ |βs 0 | + |s 1 |, then f (x) = v 4 ts 1 = βs 0 ts 0 βs 0 for a word t and p 1 = βs 0 t: u = s 0 f (x)p 1 = s 0 βs 0 ts 0 βs 0 βs 0 t. Since the word uu starts with s 1 f (y) = s 0 βs 0 f (y) and since |f (y)| = |f (x)|, we have f (y) = ts 0 βs 0 βs 0 . It follows that f (yx) contains the 3-power (s 0 β) 3 : this contradicts the 3-power-freeness of f (T A,k ).
So |f (x)| < |βs 0 | + |s 1 |. Since u starts with s 0 f (x) and with s 1 = s 0 βs 0 , there exists a word v 5 such that f (x) = (βs 0 )v 5 . Let us recall that f (x) = p 1 s 0 (βs 0 ). Lemma 2.1 implies the existence of some words γ, δ and of an integer s such that p 1 s 0 = γδ, βs 0 = (γδ) s γ and v 5 = δγ. The word uu starts both with s 0 f (x) = s 0 βs 0 v 5 and with s 1 f (y) = s 0 βs 0 f (y). Consequently v 5 is a prefix of f (y) and f (xy) starts with (γδ) s+2 γ. Since xy ∈ T A,k and f (T A,k ) is 3-power-free, s = 0: γ = βs 0 , p 1 s 0 = βs 0 δ, v 5 = δβs 0 and f (x) = βs 0 δβs 0 . Let us recall that |p 1 | + |s 1 | = |f (x)| < |βs 0 | + |s 1 |, that is |p 1 | < |βs 0 |. Consequently from p 1 s 0 = βs 0 δ, we deduce that δ is a suffix of s 0 . Thus f (xxy) contains the factor s 0 f (x)v 5 that ends with (δβs 0 ) 3 . Since x = y, xxy ∈ T A,k . We have a contradiction with the 3-power-freeness of f (T A,k ). This ends the proof of impossibility of Case 1.
As already said, Case 9 is symmetric to Case 1. Moreover Case 5 can be treated as previously. Hence Proof. Since Cases 3 and 7 are symmetric, from now on we only consider Case 3. Let us consider the sequence of words obtained by successive reductions of W leading to w. More precisely, let (ν i , σ i ) 1≤i≤m be the couple of words such that (ν 1 , σ 1 ) = (U, W ), (ν m , σ m ) = (u, w) and for each i, 1 ≤ i < m, (ν k i+1 , σ i+1 ) is the word in Reduced(ν k i , σ i ) obtained by applying Lemma 4.9 with the additional Reduction Rules chosen in Section 4.5. By the reduction process, we know that each one of the k-powers ν k j (1 ≤ j ≤ m) has a (p i , s i , a i , w i,j ) 0≤i≤3 -decomposition in f (σ i ) for some words (w i,j ) 1≤i≤3 .
By hypotheses of Case 3, σ m = w contains the 3-power x centered in a 1 . We mean more precisely that w 1 = w m,1 ends with x (and so is not the empty word), a 1 = a m,1 = x and w 2 = w m,2 starts with x (and is also not the empty word). On other part, σ 1 = W is 3-power-free. Thus there exists an integer q with 1 ≤ q < m such that σ q does not contains xxx centered in a 1 where'as σ j contains xxx centered in a 1 for all j such that q + 1 ≤ j ≤ m. To simplify temporarily the notation, we set W 1 = σ q , W 2 = σ q+1 , U 1 = ν q and U 2 = ν q+1 .
By the reduction process, there exist words (x i , y i , z i ) 1≤i≤3 (set also x 0 = y 0 = z 0 = ε) such that U 3 1 has a (p i , s i , a i , x i y i z i ) i=0,...,3 -decomposition in f (W 1 ) and U 3 2 has a (p i , s i , a i , x i z i ) i=0,...,3decomposition in f (W 2 ). Since W 2 is obtained from W 1 by the Reduction Lemma 4.9 there exist an integer 1 ≤ ℓ ≤ 3 and a letter a in A such that both x ℓ and y ℓ end with a and |s ℓ−1 f (x ℓ )| − |f (a)| < |s i−1 f (x i )| ≤ |s ℓ−1 f (x ℓ )| and |y i | = |y ℓ |. By the Reduction Rule 1, |x ℓ | a = |y ℓ | a = 1.
Finally let us stress that by definition of W 1 and W 2 , we assume that x 1 z 1 ends with x, x 2 z 2 starts with x and that either x 1 y 1 z 1 does not end with x or x 2 y 2 z 2 does not start with x. We end in two steps showing first that x 1 y 1 z 1 must end with x, and second that x 2 y 2 z 2 must start with x: This contradicts the previous sentence.
Step 1: x 1 y 1 z 1 must end with x Assume by contradiction that x 1 y 1 z 1 does not end with x. Since x 1 z 1 ends with x, we have z 1 = ε and y 1 ends with b = x (since x ℓ and y ℓ ends with the same letter, it also means that l = 1). By Remark 4.11 (2), z ℓ = ε and |p ℓ | ≤ |p 1 |. Thus U 1 = s 0 f (x 1 y 1 )p 1 = s ℓ−1 f (x ℓ y ℓ )p ℓ with |y 1 | = |y ℓ |, x 1 (= x 1 z 1 ) ends with x and both x ℓ and y ℓ end with a. Let c be the first letter of y 1 (see figure 11). Figure 11:
f( ) a a' a" f( ) a a' a" ) l ) l ) l ) l f(b) U f f x 0 s ( (y p f(c s −1 p f( ) y 1 ) 1 ) f(x f(x ) ) 1 ) 1
Let a ′′ be the suffix of f (a) such that p 1 = a ′′ p ℓ and let a ′ be the prefix of f (a) such that f (a) = a ′ a ′′ . Since f (b)p 1 and f (a)p ℓ are both suffixes of U 1 , we get that f (b) ends with a ′ . Since
|f (y 1 )p 1 | = |f (y ℓ )a ′′ p ℓ |, we get that |s 0 f (x 1 )a ′′ | = |s 0 f (x 1 y 1 )p 1 | + |a ′′ | − |f (y 1 )p 1 | = |s ℓ−1 f (x ℓ )|.
So f (x) ends with a ′ and f (c) starts with a ′′ . Since p 1 and so a ′′ are prefixes of f (x), by a length criterion, it follows that f (x) = a ′′ a ′ .
If c = x, bx 2 c is 3-power-free and f (bx 2 c) contains the 3-power (a ′ a ′′ ) 3 : this contradicts the 3-power-freeness of f (T A,k ).
Thus c = x. If |x ℓ | ≥ 2, let e be the letter such that x ℓ ends with ea. Since y 1 contains b and c with b = x = c, we have |y ℓ | = |y 1 | ≥ 2. Let d be the first letter of y ℓ . We have d = a and e = a since |x ℓ | a = |y ℓ | a = 1. Since f (y 1 )p 1 = a ′′ f (y ℓ )p ℓ and since f (y 1 ) starts with f (x) = a ′′ a ′ , we get that f (d) starts with a ′ . Since s 0 f (x 1 )a ′′ = s ℓ−1 f (x ℓ ) and since f (x 1 ) ends with f (x), we get that f (e) ends with a ′′ . It follows that f (ea 2 d) contains (a ′′ a ′ ) 3 although ea 2 d is a 3-power free word: this contradicts the 3-power-freeness of f (T A,k ).
Thus c = x and |x ℓ | = 1. Consequently x ℓ = a. Since |s ℓ−1 f (x ℓ )| − |f (a)| < |s 0 f (x 1 )| ≤ |s ℓ−1 f (x ℓ )| and since x 1 ends with x, we have x 1 = x. Thus U 2 = s 0 f (x)p 1 . Since |u| ≥ |s 0 f (x)p 1 |, we deduce that U 2 = u and W 2 = w. It follows that u = s 0 f (x)p 1 .
Let us recall that moreover f (x) = p 1 s 1 and s 1 f (x) is a prefix of u. If |s 0 | < |s 1 | then f (x) is an internal factor of f (xx) and (by Lemma 2.2) f (x 2 ) contains a 3-power: this contradicts the 3-power-freeness of f (T A,k ). Thus |s 0 | ≥ |s 1 |. Let s ′′ 0 be the suffix of s 0 and p ′ 2 be the word such that s ′′ 0 f (x) = f (x)p ′ 2 and s 1 f (x)p ′ 2 is a prefix of u. By Lemma 2.1, there exist two words α and β such that s ′′ 0 = αβ( = ε), p ′ 2 = βα and f (x) = (αβ) r α for an integer r. We have |s ′′ 0 | = |s 0 |−|s 1 | ≤ |f (a 0 )|. If |s ′′ 0 | = |f (a 0 )|, then s 0 = f (a 0 ) and s 1 = ε: this contradicts the fact that u k is not synchronized in f (w). Thus |s ′′ 0 | < |f (a 0 )| = |f (x)|. Consequently r ≥ 1. Let γ be the letter such that xγ is a prefix of w 2 a 2 : p ′ 2 is a prefix of f (γ). By Fact 3, no powers respectively of f (x) and of u 3 have a common factor of length greater than |f (x)| + |u|. Hence a 0 = x. But then a 0 xγ is 3-power-free where'as f (a 0 xγ) contains (αβ) r+2 : this contradicts the 3-power-freeness of f (T A,k ), and so x 1 y 1 z 1 must end with x.
Step 2: x 2 y 2 z 2 must start with x
From what precedes, we know now that it remains to consider the case where x 2 y 2 z 2 does not start with x. We will show that this assumption leads to a final contradiction.
Since x 2 z 2 starts with x, we have x 2 = ε and y 2 starts with b = x. By Remark 4.11(1), x ℓ = a and |s ℓ−1 | < |s 1 | (and so l = 2). Thus U 1 = s 1 f (y 2 z 2 )p 2 = s ℓ−1 f (ay ℓ z ℓ )p ℓ with |y 2 | = |y ℓ |, z 2 = x 2 z 2 starts with x and y ℓ ends with a. Let c be the last letter of y 2 (see figure 12). Figure 12:
) l f( ) a a" a' f( ) b f( ) a a" a' ) l f(x ) f(c ) f(x ) ) l f( ) y ) l ) l U f s ( ) p p 1 2 f(z ) f( 2 ) y z 2 1 s −1
Let a ′ be the prefix of f (a) such that s 1 = s ℓ−1 a ′ and let a ′′ be the suffix of f (a) such that f (a) = a ′ a ′′ . Since s ℓ−1 f (a) and s 1 f (b) are both prefixes of U 1 , the word f (b) starts with a ′′ .
Since
|s 1 f (y 2 )| = |s ℓ−1 a ′ f (y ℓ )| = |s ℓ−1 f (ay ℓ )| − |a ′′ | = |U 1 | − |f (z ℓ )p ℓ a ′′ |, we have |f (z 2 )p 2 | = |U 1 | − |s 1 f (y 2 )| = |a ′′ f (z ℓ )p ℓ |.
Since f (z 2 )p 2 and a ′′ f (z ℓ p ℓ ) are both suffixes of U 1 , it follows that f (z 2 )p 2 = a ′′ f (z ℓ )p ℓ and we get that f (x) starts with a ′′ and f (c) ends with a ′ . Since a ′ is a suffix of s 1 and so of f (x), by a length criterion, we get f (x) = a ′′ a ′ .
If c = x, cx 2 b is 3-power-free and f (cx 2 b) contains (a ′ a ′′ ) 3 : this contradicts the 3-power-freeness of f (T A,k ).
Thus c = x and y 2 contains two different letters b and x. We get |y ℓ | = |y 2 | ≥ 2. Let d be the letter such that y ℓ ends with da. Since |y ℓ | a = 1, we have d = a. Since f (x)a ′′ and f (da) are both suffixes of f (y ℓ ), the word f (d) ends with a ′′ . Since x 2 z 2 = z 2 starts with x, |z 2 | x = 0. Let z ′ 2 and z ′′ 2 be the words such that z 2 = z ′ 2 xz ′′ 2 with |z ′′ 2 | x = 0. Let z ′ ℓ be the word and e be the letter such that z ′ ℓ e is the prefix of z ℓ a ℓ verifying |s ℓ−1 f (ay ℓ z ′ ℓ )| < |s 1 f (y 2 z ′ 2 x)| ≤ |s ℓ−1 f (ay ℓ z ′ ℓ e)|. Let us recall that s 1 = s ℓ−1 a ′ and so |s 1 | = |s ℓ−1 a ′ |. Moreover s 1 f (y 2 z ′ 2 x) and s ℓ−1 f (ay ℓ z ′ ℓ ) are both prefixes of U 1 , and |s 1 a ′′ | = |s l−1 | + |f (x)| = |s l−1 | mod L. Thus s 1 f (y 2 z ′ 2 x) = s 1 f (y 2 z ′ 2 )a ′′ a ′ = s ℓ−1 f (ay ℓ z ′ ℓ )a ′ . It follows that f (e) starts with a ′ . If e = a, da 2 e is 3-power-free and f (da 2 e) contains (a ′′ a ′ ) 3 : this contradicts the 3-power-freeness of f (T A,k ).
Thus e = a. Assume |z ′ ℓ e| ≤ |z ℓ |. Let us recall that the reductions are assumed to be made under two rules. The second Reduction Rule implies that, having made a reduction with, in Lemma 4.9, an integer ℓ and a letter a, then if |z ℓ | a = 0, the next |z ℓ | a reductions are made with the same integer l and the same letter a. Thus here the words σ q+2 , . . . , σ q+1+|z ′ ℓ a|a exist and are obtained using, in Lemma 4.9, the same integer ℓ and the same letter a than the ones used to reduce σ q = W 1 into σ q+1 = W 2 . Moreover ν q+1+|z ′ ℓ a|a = s 1 f (z ′′ 2 )p 2 . Since |ν q+1+|z ′ ℓ a|a | ≥ |ν m | ≥ |s 1 f (x)p 2 |, we have z ′′ 2 = ε. But since |z ′′ 2 | x = 0, we have a contradiction with the fact that w contains xxx centered in a 1 .
Thus |z ′ ℓ e| > |z ℓ |, that is, z ℓ = z ′ ℓ , e = a ℓ (= a). It follows that z ′′ 2 = ε. Since σ j contains xxx centered in a 1 for all q + 1 ≤ j ≤ m, we must have w 2 = x and u = s 1 f (x)p 2 . Let us recall that moreover f (x) = p 1 s 1 and f (x)p 1 is a suffix of u. If |p 1 | > |p 2 | then f (x) is an internal factor of f (xx) and (by Lemma 2.2) f (x 2 ) contains a 3-power: this contradicts the 3-power-freeness of f (T A,k ). Since the decomposition is not synchronized, we have |p 1 | = |p 2 |. Thus |p 1 | < |p 2 |. Let p ′′ 2 be the prefix of p 2 and s ′ 0 be the word such that p 2 = p ′′ 2 p 1 , f (x)p ′′ 2 = s ′ 0 f (x) and s ′ 0 f (x)p 1 is a suffix of u. By Lemma 2.1, there exist two words α and β such that s ′ 0 = αβ( = ε), p ′′ 2 = βα and f (x) = (αβ) r α for an integer r. Since |s ′ 0 | = |p 2 | − |p 1 | < |f (x)| (remember |p 1 | = 0), we have r ≥ 1. Let γ be the letter such that γx is a suffix of a 0 w 1 : s ′ 0 is a suffix of f (γ). By Fact 3, no power respectively of f (x) and of u 3 have a common factor of length greater than |f (x)| + |u|. Hence a 2 = x. But then γxa 2 is 3-power-free where'as f (γxa 2 ) contains (αβ) r+2 : this contradicts the 3-power-freeness of f (T A,k ). This is a final contradiction proving that Case 3 is not possible. So consequently Theorem 3.1 holds.
Conclusion
Theorem 3.1 and Corollary 3.2 lead to some natural questions: is T A,k the smaller test-set? Is the bound b k,A = k × Card(A) + k + 1 optimal? The answer to these questions are negative at least in most of the previously known cases. As already mentioned in the introduction, M. Leconte [12] has previously got a test-set when Card(A) = 3. He proved [12] : a uniform morphism f defined on a three-letter alphabet is k-power-free (k ≥ 3) if and only if the images of all k-power-free words of length at most 3k + 5 are k-power-free. We observe that in case k = 3, we obtain a better bound than M. Leconte. But in all other cases, the bound of M. Leconte is better than our. Another result shows the non-optimality of our bound b k,A . When Card(A) = 2 (and k ≥ 3), V. Keränen proved: a uniform and primitive morphism defined on a two-letter alphabet is k-power-free if and only if the images of length at most 4 are k-power-free. This bound in this result does not depend on the value of k and is far better than our general bound b k,{a,b} ≥ b 3,{a,b} = 13.
To end, let us mention further works. In this paper, we propose a new technic to tackle the decidability of k-power-freeness of uniform morphisms. We are now looking to extension of this technic to the decidability of k-power-freeness of arbitrary morphisms.
Corollary 3.2 A uniform morphism onA is k-power-free for an integer k ≥ 3 if and only if the images by f of all k-power-free words of length at most k × Card(A) + k + 1 are k-power-free.
Definition 4. 3
3When a k-power u k is directly covered by the image (by a uniform morphism f ) of a word of length at least
Figure 2 :
2Example
Figure 4 :
4To explainFigure 4, let us say that the grey parts are deleted and that the two occurrences of f (a) allow to merge the left and right non-grey parts in order to have the new k-power (u ′ ) k directly covered by the image of the new word w ′ .
Example 4. 10
10Let us consider the morphism defined by f (1) = 1234; f (2) = 2345, f (3) = 3451, f (4) = 4521, f (5) = 5123, f (6) = 5212, f (7) = 5178, f (8) = 6234, f (9) = 1781, f (a) = 2346, f (b) = 7812, f (c) = 3462
Figure 5 :
5Example
Figure 6 Figure 7 :
67second possible reduction
enable us to prove Theorem 3.1 when k ≥ 4.
Example 4 .
412 Let f be the morphism from {1, 2, 3, 4, 5, 6, 7, 8, 9} * to {a, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, b} * defined by f (1) = a0123, f (2) = 40125, f (3) = 67892, f (4) = 34012, f (5) = 56789, f (6) = 23401, f (7) = 25678, f (8) = 92340, f (9) = 1234b. We have (see Figure 8): f (1234445666789) = a(012340125678923401234) 3 b
Figure 8 :
8An example of non-synchronized 3-power Reduction rules:
Fact 3
3No powers respectively of f (v) and of u have a common factor of length greater than or equals to |f (v)| + |u| − gcd(|f (v)|, |u|).
Fact 7
7Cases 5 and 9 are not possible We end the proof of Theorem 3.1 with the proof of the final following case: Fact 8 Cases 3 and 7 are not possible.
x x x . Of course some cases are symmetric: Cases 1 and 9 (and Case 5 is very close), Cases 2 and 8, Cases 3 and 7, Cases 4 and 6. In what follows we prove that all cases are impossible since they contradict previous facts or hypotheses. Firstly: Indeed in this cases, we can see that u 3 and f (x)3have a common factor of length |u| + |f (x)| (for instance in Case 2, the common factor is s 0 p 1 s 1 x = us 1 f (x)): this contradicts Fact 3.Fact 6 Case 1 is not possible.Proof. In this case, we have f (x) = p 0 s 0 = p 1 s 1 (seeFigure 10).Let y be the first letter of w 2 a 2 (that is, w 2 = ε and y is the first letter of w 2 , or, w 2 = ε andIn both cases f (x) is an internal factor of f (xx): By Lemma 2.2, f (x) is not primitive. This implies that f (xx) contains a 3-power, a contradiction with the 3-power-freeness of f (T A,k ).So y = x. We now consider two subcases.
Automatic sequences. J.-P Allouche, J Shallit, Cambridge University PressJ.-P. Allouche and J. Shallit. Automatic sequences. Cambridge University Press, 2003.
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Axel Thue's work on repetition in words. J Berstel, 92.70LITP, septembre 1992. 4th Conf. on Formal Power Series and Algebraic Combinatorics. MontréalTechnical ReportJ. Berstel. Axel Thue's work on repetition in words. Technical Report 92.70, LITP, septembre 1992. 4th Conf. on Formal Power Series and Algebraic Combinatorics, Montréal 1992.
Axel Thue's papers on repetition in words: a translation. J Berstel, Laboratoire de Combinatoire et d'Informatique Mathématique. Montréal20Université du QuébecJ. Berstel. Axel Thue's papers on repetition in words: a translation. Technical Report 20, Lab- oratoire de Combinatoire et d'Informatique Mathématique, Université du Québec, Montréal, 1995.
Theory of codes. J Berstel, D Perrin, Academic PressSee also J. Berstel's web pageJ. Berstel and D. Perrin. Theory of codes. Academic Press, 1985. See also J. Berstel's web page.
A characterization of overlap-free morphisms. J Berstel, P Séébold, Discrete Applied Mathematics. 46J. Berstel and P. Séébold. A characterization of overlap-free morphisms. Discrete Applied Mathematics, 46:275-281, 1993.
Handbook of Formal Languages. C Choffrut, J Karhumäki, Springer1C. Choffrut and J. Karhumäki. Handbook of Formal Languages, volume 1, chapter Combina- torics of words. Springer, 1997.
Sharp characterizations of squarefree morphisms. M Crochemore, Theoretical Computer Science. 18M. Crochemore. Sharp characterizations of squarefree morphisms. Theoretical Computer Science, 18:221-226, 1982.
On irreducible alanguages and nonrationnal numbers. S , Bull. Math. de la Soc. Sci. Math. de la R.S. de Roumanie. 2169S. Istrail. On irreducible alanguages and nonrationnal numbers. Bull. Math. de la Soc. Sci. Math. de la R.S. de Roumanie, 21 (69)(3-4):301-308, 1977.
On k-repetition freeness of length uniform morphisms over a binary alphabet. V Keränen, Discrete Applied Mathematics. 9V. Keränen. On k-repetition freeness of length uniform morphisms over a binary alphabet. Discrete Applied Mathematics, 9:301-305, 1984.
On the k-freeness of morphisms on free monoids. V Keränen, Annales Academiae Scientarium Fennicae. 61Series AV. Keränen. On the k-freeness of morphisms on free monoids. Annales Academiae Scientarium Fennicae 61, Series A, 1986.
Codes sans répétition. M Leconte, LITP Université Paris 6PhD thesisM. Leconte. Codes sans répétition. PhD thesis, LITP Université Paris 6, october 1985.
M Lothaire, Reprinted in 1997 by Cambridge University Press in the Cambridge Mathematical Library. Cambridge, UKAddison-Wesley17Combinatorics on wordsM. Lothaire. Combinatorics on words, volume 17 of Encyclopedia of Mathematics. Addison- Wesley, 1983. Reprinted in 1997 by Cambridge University Press in the Cambridge Mathemat- ical Library, Cambridge, UK, 1997.
Algebraic Combinatorics on words. M Lothaire, of Encyclopedia of Mathematics. Cambridge, UKCambridge University Press90M. Lothaire. Algebraic Combinatorics on words, volume 90 of Encyclopedia of Mathematics. Cambridge University Press, Cambridge, UK, 2002.
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If a D0L language is k-power free then it is circular. F Mignosi, Séébold , ICALP 93. 700F. Mignosi and Séébold. If a D0L language is k-power free then it is circular. In ICALP 93, volume 700 of Lect. Notes in Comp. Sci., pages 507-518, 1993.
Graph coloring and combinatorics on words. P Ochem, Université de Bordeaux IPhD thesisP. Ochem. Graph coloring and combinatorics on words. PhD thesis, Université de Bordeaux I, Nov. 2005.
Some non finitely generated monoids of repetition-free endomorphisms. Information Processing Letters. G Richomme, 85G. Richomme. Some non finitely generated monoids of repetition-free endomorphisms. Infor- mation Processing Letters, 85:61-66, 2003.
Characterization of test-sets for overlap-free morphisms. G Richomme, P Séébold, Discrete Applied Mathematics. 98G. Richomme and P. Séébold. Characterization of test-sets for overlap-free morphisms. Discrete Applied Mathematics, 98:151-157, 1999.
Conjectures and results on morphisms generating k-power-free words. G Richomme, P Séébold, Int. J. of Foundations of Computer. 152ScienceG. Richomme and P. Séébold. Conjectures and results on morphisms generating k-power-free words. Int. J. of Foundations of Computer. Science, 15(2):307-316, 2004.
Some results on k-power-free morphisms. G Richomme, F Wlazinski, Theoretical Computer Science. 273G. Richomme and F. Wlazinski. Some results on k-power-free morphisms. Theoretical Com- puter Science, 273:119-142, 2002.
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Uber unendliche zeichenreihen. A Thue, Kristiania Videnskapsselskapets Skrifter Klasse I. Mat.-naturv. 7A. Thue. Uber unendliche zeichenreihen. Kristiania Videnskapsselskapets Skrifter Klasse I. Mat.-naturv, 7:1-22, 1906.
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A test-set for k-power-free binary morphisms. F Wlazinski, RAIRO Theoretical Informatics and Applications. 35F. Wlazinski. A test-set for k-power-free binary morphisms. RAIRO Theoretical Informatics and Applications, 35:437-452, 2001.
|
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[
"Evolution of the Exoplanet Size Distribution: Forming Large Super-Earths Over Billions of Years",
"Evolution of the Exoplanet Size Distribution: Forming Large Super-Earths Over Billions of Years"
] |
[
"Trevor J David [email protected] \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n\nDepartment of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA\n",
"Gabriella Contardo \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n",
"Angeli Sandoval \nDepartment of Physics and Astronomy, Hunter College\nCity University of New York\n10065New YorkNYUSA\n",
"Ruth Angus \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n\nDepartment of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA\n\nDepartment of Astronomy\nColumbia University\n550 West 120th StreetNew YorkNYUSA\n",
"Yuxi ",
"Lucy Lu \nDepartment of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA\n\nDepartment of Astronomy\nColumbia University\n550 West 120th StreetNew YorkNYUSA\n",
"Megan Bedell \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n",
"Jason L Curtis \nDepartment of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA\n",
"Daniel Foreman-Mackey \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n",
"Benjamin J Fulton \nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nIPAC-NASA Exoplanet Science Institute\n91125PasadenaCAUSA\n",
"Samuel K Grunblatt \nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n\nDepartment of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA\n",
"Erik A Petigura \nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n"
] |
[
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"Department of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA",
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"Department of Physics and Astronomy, Hunter College\nCity University of New York\n10065New YorkNYUSA",
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"Department of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA",
"Department of Astronomy\nColumbia University\n550 West 120th StreetNew YorkNYUSA",
"Department of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA",
"Department of Astronomy\nColumbia University\n550 West 120th StreetNew YorkNYUSA",
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"Department of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA",
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"California Institute of Technology\n91125PasadenaCAUSA",
"IPAC-NASA Exoplanet Science Institute\n91125PasadenaCAUSA",
"Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA",
"Department of Astrophysics\nAmerican Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA",
"Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA"
] |
[] |
The radius valley, a bifurcation in the size distribution of small, close-in exoplanets, is hypothesized to be a signature of planetary atmospheric loss. Such an evolutionary phenomenon should depend on the age of the starplanet system. In this work, we study the temporal evolution of the radius valley using two independent determinations of host star ages among the California-Kepler Survey (CKS) sample. We find evidence for a wide and nearly empty void of planets in the period-radius diagram at the youngest system ages (2-3 Gyr) represented in the CKS sample. We show that the orbital period dependence of the radius valley among the younger CKS planets is consistent with that found among those planets with asteroseismically determined host star radii. Relative to previous studies of preferentially older planets, the radius valley determined among the younger planetary sample is shifted to smaller radii. This result is compatible with an atmospheric loss timescale on the order of gigayears for progenitors of the largest observed super-Earths. In support of this interpretation, we show that the planet sizes that appear to be unrepresented at ages 2-3 Gyr are likely to correspond to planets with rocky compositions. Our results suggest that the size distribution of close-in exoplanets and the precise location of the radius valley evolve over gigayears.
|
10.3847/1538-3881/abf439
| null | 235,211,728 |
2011.09894
|
3184031bd2b1aab93868ab7beb24549ac1c7d4a7
|
Evolution of the Exoplanet Size Distribution: Forming Large Super-Earths Over Billions of Years
Trevor J David [email protected]
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Department of Astrophysics
American Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA
Gabriella Contardo
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Angeli Sandoval
Department of Physics and Astronomy, Hunter College
City University of New York
10065New YorkNYUSA
Ruth Angus
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Department of Astrophysics
American Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA
Department of Astronomy
Columbia University
550 West 120th StreetNew YorkNYUSA
Yuxi
Lucy Lu
Department of Astrophysics
American Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA
Department of Astronomy
Columbia University
550 West 120th StreetNew YorkNYUSA
Megan Bedell
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Jason L Curtis
Department of Astrophysics
American Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA
Daniel Foreman-Mackey
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Benjamin J Fulton
California Institute of Technology
91125PasadenaCAUSA
IPAC-NASA Exoplanet Science Institute
91125PasadenaCAUSA
Samuel K Grunblatt
Center for Computational Astrophysics
Flatiron Institute
10010New YorkNYUSA
Department of Astrophysics
American Museum of Natural History, Central Park West at 79th Street10024New YorkNYUSA
Erik A Petigura
Department of Physics and Astronomy
University of California
90095Los AngelesCAUSA
Evolution of the Exoplanet Size Distribution: Forming Large Super-Earths Over Billions of Years
10.3847/1538-3881/abf439Received 2020 November 19; revised 2021 March 18; accepted 2021 March 23; published 2021 May 14Unified Astronomy Thesaurus concepts: Exoplanets (498)Exoplanet evolution (491)Exoplanet astronomy (486)Super Earths (1655)Mini Neptunes (1063) Supporting material: machine-readable table
The radius valley, a bifurcation in the size distribution of small, close-in exoplanets, is hypothesized to be a signature of planetary atmospheric loss. Such an evolutionary phenomenon should depend on the age of the starplanet system. In this work, we study the temporal evolution of the radius valley using two independent determinations of host star ages among the California-Kepler Survey (CKS) sample. We find evidence for a wide and nearly empty void of planets in the period-radius diagram at the youngest system ages (2-3 Gyr) represented in the CKS sample. We show that the orbital period dependence of the radius valley among the younger CKS planets is consistent with that found among those planets with asteroseismically determined host star radii. Relative to previous studies of preferentially older planets, the radius valley determined among the younger planetary sample is shifted to smaller radii. This result is compatible with an atmospheric loss timescale on the order of gigayears for progenitors of the largest observed super-Earths. In support of this interpretation, we show that the planet sizes that appear to be unrepresented at ages 2-3 Gyr are likely to correspond to planets with rocky compositions. Our results suggest that the size distribution of close-in exoplanets and the precise location of the radius valley evolve over gigayears.
Introduction
By far the most intrinsically common planets known are small (<4 R ⊕ ), close-in (<1 au) planets. NASA's Kepler mission (Borucki et al. 2010) revealed the surprising abundance of these planets; some 30%-60% of Sun-like stars host a small, close-in planet, depending on assumptions about the intrinsic multiplicity and inclination dispersion within planetary systems (Fressin et al. 2013;Petigura et al. 2013;Zhu et al. 2018;He et al. 2019). An enduring mystery posed by small planets is how some accreted sizable atmospheres while others appear to have avoided runaway accretion altogether (e.g., Ikoma & Hori 2012;Lee et al. 2014;Lee & Chiang 2016). Oftentimes Kepler multiplanet systems host planets both with and without atmospheres, in some cases separated from one another by only a hundredth of an astronomical unit (Carter et al. 2012).
Recent progress in understanding small planets has been fueled by improved precision in stellar and planetary parameters. Through homogeneous spectroscopic characterization of >1300 Kepler planet hosts, the California-Kepler Survey (CKS; Johnson et al. 2017;Petigura et al. 2017) revealed that the size distribution of close-in (P < 100 days) small planets is bimodal, with a valley in the completeness-corrected radius distribution between 1.5 and 2 R ⊕ . The radius valley is widely believed to be a signature of atmospheric loss. This belief is bolstered by determinations of planet densities on either side of the valley; planets below the valley, dubbed super-Earths, have densities consistent with a rocky composition, while planets above the valley, known as sub-Neptunes, require atmospheres of a few percent by mass to explain the low measured densities (e.g., Weiss & Marcy 2014;Rogers 2015). In the atmospheric loss model, some fraction of super-Earths are the remnant cores of planets that shed their primordial envelopes, which potentially alleviates the issue of neighboring planets with dissimilar densities (e.g., Lopez & Fortney 2013;Owen & Morton 2016). While exploration of the radius valley among planets orbiting low-mass stars has provided support for an alternative hypothesis (formation in a gas-poor disk without the need for atmospheric loss; Cloutier & Menou 2020), atmospheric erosion remains the leading theory for planets around Sun-like stars.
Atmospheric loss requires energy. Energy deposited into a planet's atmosphere from an internal or external source can heat gas to velocities exceeding the planet's escape velocity. External mechanisms of energy deposition include photoevaporation (heating of the planet's thermosphere by X-ray and extreme ultraviolet radiation; e.g., Owen & Jackson 2012) and impacts by planetesimals or planetary embryos (Liu et al. 2015;Inamdar & Schlichting 2016;Chatterjee & Chen 2018;Wyatt et al. 2020). Internal energy deposition can be provided by the luminosity of a planet's cooling core (e.g., Ginzburg et al. 2016). Notably, planetary evolution models studying the effect of photoevaporation predicted the existence of a radius valley before it was observed (Lopez & Fortney 2013;Owen & Wu 2013;Jin et al. 2014;Chen & Rogers 2016). However, subsequent studies considering the effects of core-powered mass loss were also able to reproduce the bimodal radius distribution of small planets (Ginzburg et al. 2018;Gupta & Schlichting 2019. Photoevaporation and core cooling remain the two leading explanations for the radius valley, and both processes may well be important, but to determine the relative importance of the two effects will require a better understanding of the dependence of the valley on other key parameters.
Determining how empty the radius gap is represents an important step toward understanding its origins. In the initial CKS sample, typical planet radius uncertainties were comparable to the width of the gap so that an intrinsically empty gap would not have been resolved . Van Eylen et al. (2018) studied planets orbiting a subset of Kepler host stars with precise asteroseismic parameters (including ages ranging from ∼2 to 10 Gyr) and found a gap considerably wider and emptier than that found in the initial CKS sample. Including trigonometric parallaxes from Gaia DR2, Fulton & Petigura (2018) were able to improve the median R * errors by a factor of 5 in the CKS sample, but the gap remained populated. Those authors presented simulations that suggest that the gap is not empty (i.e., solely filled in by noisy data), and that there are real planets in the gap. More recently, however, Petigura (2020) showed that a sizable number of planets in and around the gap have poorly determined radii due to high impact parameters, indicating that the gap may be emptier than previously appreciated.
It also appears that the gap, which is a 1D projection of a higher-dimensional manifold, is partially filled in due to a dependence of the gap center on orbital period (or stellar light intensity) and host star mass. The gap center is anticorrelated with orbital period Van Eylen et al. 2018;MacDonald 2019;Martinez et al. 2019;Loyd et al. 2020), which is considered compatible with both the photoevaporation (e.g., Jin & Mordasini 2018;Lopez & Rice 2018;Owen & Wu 2013 and core-powered mass-loss models (e.g., Gupta & Schlichting 2019 but incompatible with formation in a gas-poor disk (Lopez & Rice 2018); at larger orbital periods, only the smallest and least massive cores are susceptible to total atmospheric loss, driving the gap to smaller radii. The length of the radius valley, i.e., its outer boundary in either period or insolation, may also provide clues to its origin, though this parameter remains poorly studied. In the photoevaporation model, the radius valley should not extend beyond orbital periods of 30-60 days, as the incident X-ray and ultraviolet (XUV) flux is believed to be too low to drive substantial mass loss (Owen & Wu 2017). However, the low completeness of the Kepler data set for small planets at these orbital periods presents a challenge for detecting such a transition point.
The gap center is positively correlated with stellar mass (Fulton & Petigura 2018;Wu 2019;Berger et al. 2020a;Cloutier & Menou 2020;Hansen et al. 2021;Van Eylen et al. 2021), although it has been suggested that this trend is due to the relationship between stellar mass and planetary insolation (Loyd et al. 2020). The measured mass dependence of the gap has been used to argue support for photoevaporation (e.g., Wu 2019) but requires that the average planet mass scale approximately linearly with host star mass, an assertion that has not been verified for small planets. By comparison, in the corepowered mass-loss model, the dependence of the radius gap location on stellar mass is a natural consequence of the dependence of planet equilibrium temperature (which partially determines the mass-loss rate in the Bondi-limited regime) on the stellar mass-luminosity relation (e.g., Gupta & Schlichting 2020).
As for metallicity, there is tentative evidence for a wider radius valley for metal-rich stars (Owen & Murray-Clay 2018). Such a dependence could result if the core mass distributions, core bulk densities, or initial atmospheric mass fractions of small planets depend sensitively on the metallicity of the host star, and hence the protoplanetary disk. There is evidence that large Kepler planets (2-8 R ⊕ ) are more common around highermetallicity stars Petigura et al. 2018) and that planets at short orbital periods are preferentially larger around higher-metallicity stars (Owen & Murray-Clay 2018). Both findings are compatible with a scenario in which metalrich stars form more massive cores, on average. It has also been suggested that metal-rich stars host planets with higher atmospheric metallicities, which increases the efficiency of atomic line cooling in photoevaporative flows and decreases mass-loss rates (Owen & Murray-Clay 2018). In the corepowered mass-loss model, the rate at which sub-Neptunes cool and contract is anticorrelated with the opacity of the envelope, which is assumed to be proportional to the stellar metallicity (Gupta & Schlichting 2020). Thus, in both the photoevaporation and core-cooling models, larger sub-Neptunes and a consequently wider radius valley are expected around more metal-rich stars (for fixed mass and age and neglecting any potential scaling between metallicity and core mass distributions).
The characteristic timescale for atmospheric loss among close-in exoplanets has been proposed as a key parameter for assessing the relative importance of photoevaporation and corepowered mass loss. Firm observational constraints on that timescale, however, are lacking. Constraining this timescale through exoplanet population studies may provide a means for discerning the relative importance of proposed mass-loss mechanisms. Core-powered mass loss is believed to operate over gigayear timescales (Ginzburg et al. 2016(Ginzburg et al. , 2018Gupta & Schlichting 2019. By comparison, photoevaporation models predict that the majority of mass loss occurs during the first 0.1 Gyr (e.g., Lopez et al. 2012;Owen & Jackson 2012;Lopez & Fortney 2013;Owen & Wu 2013, corresponding roughly to the length of time a Sun-like star spends as a saturated X-ray emitter (e.g., Jackson et al. 2012;Tu et al. 2015). However, a more recent study found that the majority of the combined X-ray and extreme UV emission of stars occurs after the saturated phase of high-energy emission, implying that XUV irradiation of exoplanet atmospheres continues to be important over gigayear timescales (King & Wheatley 2021). If valid, then observational constraints on exoplanet evolution timescales may not provide a conclusive means for discerning the relative importance of photoevaporation and core-powered mass loss. Nevertheless, there is evidence that the detected fraction of super-Earths to sub-Neptunes increases over gigayears, suggesting that the sizes of at least some planets evolve on these long timescales (Berger et al. 2020a;Sandoval et al. 2021).
A basic prediction of atmospheric loss models is that the radius gap is wider at younger ages and fills in over time; at a fixed value of high-energy incident flux and initial atmospheric mass fraction, photoevaporation models predict that sub-Neptunes with the least massive cores (and smallest core sizes) will cross the gap first, with more massive cores crossing the gap at later times, if at all. As a result, the radius valley is expected to be wider and emptier at early times, progressively filling in with stripped cores of ever larger masses and sizes (e.g., Rogers & Owen 2021). In the core-powered mass-loss model, Gupta & Schlichting (2020) suggested that the average size of sub-Neptunes is expected to decline with age while the average size of super-Earths remains relatively constant, again leading to a wider and emptier radius valley at earlier times.
While the specific theoretical predictions for the age dependence of the radius valley morphology are uncertain, the fundamental prediction from atmospheric loss models that this feature should weaken with increasing age is a firm conclusion. We aim to investigate this hypothesis using the CKS sample. Here we investigate the time evolution of the exoplanet radius gap. In Section 2, we describe our sample selection process, including several filters intended to rid our sample of stars or planets with unreliable parameters. Our analysis procedures are discussed in Section 3, and finally, we interpret our results and summarize our primary findings in Section 4.
Sample Selection
We began with the CKS VII sample published in Fulton & Petigura (2018, hereafter F18). The CKS VII sample is a wellcharacterized subset of all Kepler planet candidates. Stellar characterization for these stars was performed in a homogeneous manner, with spectroscopic T eff , g log , and [Fe/ H] derived from high signal-to-noise ratio (S/N), high-dispersion Keck/HIRES spectra (Johnson et al. 2017;Petigura et al. 2017). Using their spectroscopic T eff and bolometric luminosities computed from Gaia DR2 parallaxes (Gaia Collaboration et al. 2018), extinction-corrected Two Micron All Sky Survey (2MASS) K s magnitudes (Cutri et al. 2003), and theoretical bolometric corrections from the MESA Isochrones and Stellar Tracks (MIST; Choi et al. 2016;Dotter 2016), F18 derived stellar radii from the Stefan-Boltzmann law. They additionally computed ages for the CKS sample using the isoclassify package (Huber et al. 2017), which also depends on the MIST models. We use the F18 median posterior isochrone ages as one source of age in the analysis that follows.
We constructed several filters, many motivated by the cuts outlined in F18, to refine the sample and select those planets and stars with the most reliable parameters. The filters are enumerated as follows.
1. Planet orbital period. We restricted our analysis to planets with orbital periods <100 days. At larger periods, Kepler suffers from low completeness, particularly for small planets. 2. Planet size. We restricted our analysis to planets with sizes <10 R ⊕ . 3. Planet radius precision. We restricted our analysis to planets with fractional radius uncertainties s < R 20% R P P . 4. Planet false-positive designation. We excluded planets identified as false positives in Table 4 of the CKS I paper , which synthesized dispositions from Mullally et al. (2015), Morton et al. (2016), and the NASA Exoplanet Archive (as accessed on 2017 February 1; Akeson et al. 2013). 5. Stellar radius (dwarf stars). We restricted our analysis to dwarf stars with the following condition: The left-hand side of this condition excludes a small number of stars far below the main sequence that may have erroneous parameters. The right-hand side excludes stars that have evolved considerably away from the main sequence. This cut is depicted in Figure 1. We additionally excluded cool stars elevated from the main sequence that would result in unrealistically old ages. This cut was performed by requiring T eff > T isoc , where T isoc is the temperature of a log(age) = 10.25, [Fe/H] = +0.25 MIST v1.1 nonrotating isochrone with an equivalent evolutionary point (EEP) <500. 6. Stellar mass. We wish to isolate the effect of stellar age on the exoplanet radius gap while minimizing the effects of stellar mass as much as possible. We restricted our sample to stars with masses 0.75 < M å /M e < 1.25, where the masses were derived from stellar evolution models in F18. 7. Stellar metallicity. For the same reason we confined our sample in stellar mass, we restricted our analysis to stars with spectroscopically determined metallicities in the range −0.3 < [Fe/H] < +0.3. 8. Isochrone parallax. We removed stars where the Gaia and F18 "spectroscopic" or "isochrone" parallaxes differed by more than 4σ, where the latter quantities were computed in Table 2 of CKS VII. It was speculated by F18 that such discrepancies may be due to flux contamination from unresolved binaries. This cut also removes all stars where the CKS VII isochrone-derived radius, R iso , differed by more than 10% from the radius derived from the Stefan-Boltzmann law. 9. Stellar dilution (Gaia). We used the r 8 column in Table 2 of CKS VII to exclude stars with closely projected sources detected by Gaia that contribute a nonnegligible fraction of the optical flux in the Kepler aperture. We excluded stars where additional sources in an 8″ radius (two Kepler pixels) contribute more than 10% of the cumulative G-band flux (including the target). 10. Stellar dilution (imaging). As in F18, we excluded Kepler Objects of Interest (KOIs) with closely projected stellar companions bright enough to require corrections to the planetary radii of 5% or more. Like F18, we use the radius correction factor (RCF) computed by Furlan et al. (2017) based on high-resolution imaging from several authors, accepting planets for which RCF < 1.05. 11. Unresolved binaries (Gaia). We excluded stars with Gaia renormalized unit weight error (RUWE) values >1.4, where these values were queried from the Gaia archive. 8 The RUWE is a goodness-of-fit metric for a single-star astrometric model (Lindegren et al. 2018). It is a sensitive indicator of unresolved binaries (Belokurov et al. 2020), which are a concern for planet radius studies due to potential flux dilution or misidentification of the planet host. 12. Discrepant photometry. We removed stars with discrepant optical brightnesses, |G − K P | > 1 mag, indicating a potentially erroneous cross-match between the Kepler and Gaia sources. 13. Reddening. We removed stars with reddening estimates of A V > 0.5 mag, where these estimates were sourced from Lu et al. (2021). Stars with high reddening are more susceptible to erroneously determined stellar parameters. 14. Planets with grazing transits. Due to degeneracies inherent to light-curve modeling, planets with grazing transits can have poorly constrained radii. Petigura (2020) showed that there is some level of contamination of the radius valley from planets with grazing transits. Since impact parameters measured from long-cadence photometry are unreliable, we follow Petigura (2020) and exclude planets with R τ < 0.6, where R τ is the ratio of the measured transit duration to the duration of a b = 0, e = 0 transit with the same period around the same star.
⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ( ) - + < < - + 1 T R R T
The overall CKS VII sample contains 1913 planets orbiting 1189 unique stellar hosts. In the analysis that follows, we will refer to the base sample (constructed from the first five cuts enumerated) and the filtered sample (constructed from all of the filters). After applying the filters, the base sample consists of 1443 planets orbiting 871 unique stellar hosts. The filtered sample consists of 732 planets orbiting 466 unique hosts. Later, we find that our analysis is insensitive to many of these restrictions and relax most of them.
Rotation Period Vetting
We supplemented the CKS sample with stellar rotation periods, which we use in Section 3.2 to empirically age rank planet hosts, compiled from the literature. In order to perform an accurate rotation-based selection, it is imperative to have reliable rotation periods. To this end, we performed visual vetting of the full Kepler light curves for each star in the CKS sample. For each star, our period vetting procedure consisted of the following steps. . 3. Perform a Lomb-Scargle (L-S) periodogram analysis of the Kepler PDCSAP light curve using the LombScargle class in the astropy.timeseries package. 4. Phase-fold the PDCSAP light curve on the L-S peak power period and any published period, as well as on the first harmonic and subharmonic of each of the previously mentioned periods. 5. Generate a vetting sheet including all phase-folded light curves, the L-S periodogram, a 120 day segment of the light curve, and the full light curve. 6. Visually examine each vetting sheet, recording the preferred period source and assigning a reliability flag to each period determination (3: highly reliable; 2: reliable; 1: period could not be unambiguously determined; and 0: no periodicity evident). 11
The results of our rotation period vetting are summarized in Table 1. Of the 1189 unique planet hosts in the CKS VII sample, which are predominantly FGK main-sequence stars, we found that approximately 22% have highly reliable rotation periods, 23% have reliable periods, 34% could not have periods determined unambiguously from the light curve, and 21% had no clear periodicity evident in the light curve.
Analysis
Evolution of the P-R Diagram: Isochrone Ages
We first plotted the period-radius (P-R) diagram for CKS planets in four bins of log(age): <9. 25, 9.25-9.5, 9.5-9.75, and 9.75 for both the filtered and base samples (Figures 2 and 3, respectively). 12 This binning scheme was in part chosen because there are few planets with log(age) < 9 or >10. From these figures, we observed a conspicuous void of planets around the radius valley in the youngest age bin (1.8 Gyr). Moreover, the slope of the void appears to be very close to the slope of the radius valley determined in Van Eylen et al. (2018, hereafter V18) from a subset of planets orbiting stars with precise asteroseismic parameters. However, while the slope of the young planet void appears to be consistent with that of the radius valley, the intercept appears to be different. This is evident from the fact that the lower boundary of young sub-Neptunes straddles the V18 line, while the super-Earths are well separated from the V18 valley. In other words, there appears to be a dearth of large super-Earths at younger ages, resulting in an apparent shift in the peak of the super-Earth radius distribution to larger radii at older ages. Our interpretation of this shift is discussed in Section 4.
While age, mass, and metallicity are correlated in the CKS sample, we show that the distribution of masses in each age bin is not changing drastically. The age-metallicity gradient is stronger, and essentially all stars in the youngest age bin are metal-rich. However, it is also clear that stars hosting planets in the radius valley are not exclusively metal-poor but rather have a wide range of metallicities. Additionally, while the [Fe/H] distributions in the two youngest age bins are broadly similar, the distributions of planets in the P-R diagram are markedly different. While these observations offer some degree of assurance that observed features in the P-R diagram are due to age rather than mass or metallicity, we explore the effects of stellar mass and metallicity further in Sections 3.5 and 3.7.
Evolution of the P-R Diagram: Gyrochronology
To this point, we have only considered isochrone ages from F18 in our analysis. While we present a qualitative validation of the F18 ages in Appendix B, there are substantial uncertainties associated with isochrone ages. It is also possible to empirically age rank the CKS sample with a gyrochronology analysis. Recently, Curtis et al. (2020) presented empirical gyrochrones for several open clusters that enable high-fidelity, model-independent age ranking of solar-type stars with welldetermined T eff and rotation periods.
We separated the CKS sample into "fast" and "slow" rotators using the hybrid NGC 6819 + Ruprecht 147 gyrochrone of Curtis et al. (2020), corresponding to an age of ∼2.7 Gyr. To perform this cut, we first converted the gyrochrone from a (B P -R P )-P rot relation into a T eff -P rot relation using the colortemperature polynomial relation presented by those authors (valid for the temperatures considered here). 13 We constructed the young planet sample from the CKS base sample (Section 2) by choosing host stars placed between the 0.12 (Pleiades) and 2.7 Gyr gyrochrones in the T eff -P rot diagram, high-reliability P rot flags, RUWE < 1.4 (to remove unresolved binaries with unreliable rotation periods), and T eff < 6000 K. The T eff < 6000 K cut is motivated by the fact that gyrochrones cluster closely for hotter stars and small (This table is available in its entirety in machine-readable form.) 11 An example vetting sheet is presented in Appendix A, and all sheets are available through the journal. 12 The completeness curve in Figures 2 and 3 was computed for CKS stars using CKS stellar parameters and the methodology of Burke et al. (2015) as implemented in Python at https://dfm.io/posts/exopop/. 13 We opted not to perform the gyrochronology classification in color space because we noted increased scatter in the (B P -R P )-P rot diagram for CKS stars, possibly a result of reddening, metallicity effects, or both.
temperature uncertainties can translate to large uncertainties in age from a gyrochronology analysis. The old rotation-selected planet sample was selected in the same fashion from stars lying above the 2.7 Gyr gyrochrone, except for the high-reliability P rot flag requirement. We found that we assigned the high-reliability flag more often to faster rotators that tend to exhibit higher amplitude and more stable brightness modulations (presumably due to larger, longer-lived spots), while the older, more slowly rotating stars exhibit smaller-amplitude, more sporadic Sun-like variations potentially due to smaller, short-lived spots. Thus, in the slow rotator sample, we accepted stars with either reliable or highly reliable P rot flags.
The distributions of the young and old rotation-selected planet hosts in the T eff -P rot and Hertzsprung-Russell (H-R) diagrams are shown in Figure 4, along with the 1D distributions of their median isochrone ages. We observe that the fast-rotating sample does indeed correspond to stars that lie closer to the zero-age main sequence with median isochrone ages strongly skewed toward younger ages (mostly below 3 Gyr) relative to the CKS sample. The more slowly rotating stars show a distribution of median isochrone ages that is practically indistinguishable from the bulk of the CKS sample but with a significant number of stars in the ∼1-3 Gyr range. However, this is likely to be the result of isochrone clustering on the main sequence and not because those stars are actually young. We also note that only 46% (546/1189) Figure 2. First row: evolution of the Kepler planet population in the P-R diagram for the filtered CKS sample (age bins indicated above each panel). Contours show Gaussian kernel density estimates of planets in the overall CKS sample. The black dashed line indicates the radius valley derived by V18. The gray shaded region indicates the 25% pipeline completeness contour calculated from the CKS sample. Second row: 1D distributions of planet radii for the samples plotted above in each case. The nominal location of the radius gap from Fulton et al. (2017) is indicated by the vertical gray stripe. Third row: our base CKS planet host sample in the T eff -R * plane (gray) with the host stars in the age bins indicated at the top (pink). Stars hosting planets in the radius range 1.6-1.9 R ⊕ are outlined in black. Fourth row: as in the third row, the distribution of planet hosts in the mass-metallicity plane. of the stars in our sample have assigned rotation periods from our period vetting procedure. For the remaining stars, it was not possible to unambiguously assign a period. Those stars are preferentially more evolved relative to the periodic sample, though they are observed across the H-R diagram. Thus, the modest decrement at old ages in the age distribution of the slow rotators relative to the overall CKS sample may be the result of a finite active lifetime for solar-type stars.
After separating the planet hosts into the fast and slow rotator samples, we examined the distributions of the corresponding planet populations in the P-R diagram ( Figure 5). We observe qualitatively similar behavior as to what was found when using isochrone ages to perform age cuts. That is, there is a dearth of exoplanets in the radius valley among planets empirically determined to be younger than ∼2.7 Gyr from a gyrochronology analysis. Among the planets older than ∼2.7 Gyr, the radius valley appears more filled in. Moreover, the slope and boundaries of the radius valley in the left panel of Figure 5 appear to be very close to those derived from the isochrone age-selected sample (as described in Section 3.3).
Measuring the Slope of the Void
As the slope of the radius valley contains information about the mechanism(s) responsible for producing it, we proceeded to characterize the void for four planetary samples described as follows. Each sample is a subset of the base sample, sharing the following cuts: planets orbiting dwarf stars (described in Section 2) with P < 100 days, R P < 10 R ⊕ , and s < R 20 R P P %. The isoc_fgk_1to2 sample also employs the age restriction ( ) < 9 log age 9.25, while the isoc_fgk_lt2 sample is produced from the more inclusive criterion ( ) log age 9.25. As isochrone ages for hotter stars are more reliable than those of cooler stars, the isoc_fg_lt2 sample combines the criteria ( ) log age 9.25 and T eff > 5500 K. Finally, the gyro_ gk_lt3 sample combines the common cuts with the following criteria: T eff < 6000 K, RUWE < 1.4, high-reliability rotation periods (reliability flag of 3), and positions in the T eff -P rot plane between the empirical Pleiades and NGC 6819 + Ruprecht 147 gyrochrones of Curtis et al. (2020). The distributions of these planetary samples in the P-R and insolation-radius planes are depicted in Figure 6.
Following the approach of V18, we used support vector machines (SVMs) to find the decision boundary that maximizes the margins between two distinct classes of planets in the P-R and insolation-radius planes. To label the planets, we found that shifting the V18 radius valley equation downward by 0.07 dex in ( )
Å R R log P 10
provided an unambiguous separation of planets into two classes for the isoc_fgk_1to2 sample. Thus, we used the equation
( ) ( ) = - + Å R R P log 0.09 log d 0.3 P 10 10
to label planets as sub-Neptunes or super-Earths. To implement the (Curtis et al. 2020). In the middle panels, the gray curves show solar-metallicity, nonrotating MIST v1.1 isochrones from log(age) = 9-10 in steps of 0.25 dex. Figure 5. The P-R diagram for exoplanets orbiting stars rotating more rapidly (left) or more slowly (right) than an empirical 2.7 Gyr gyrochrone. The shaded contours represent a 2D Gaussian kernel density estimation for the overall CKS sample. The black dashed lines indicate the margins of the young planet void derived from the isochrone-selected sample in Section 3.3.
SVM classification, we used the sklearn.svm.SVC module in Python with a linear kernel (Pedregosa et al. 2011).
We explored the sensitivity of our results to the regularization parameter, , finding that for < 5, the SVM misclassifies a large fraction of planets and fails to trace the center of the void that is so readily visible by eye (Figure 7). In determining the equation of the void, we ultimately adopt the slope and intercept derived from the = 10 case but recommend = 1000 for determining the upper and lower boundaries of the void. To calculate the uncertainties on the slope and intercept of the radius valley, we performed 10 3 bootstrapping simulations, selecting 50 planets (with replacement) randomly from the young planet samples and recording the slope and intercept resulting from the SVM classification for each bootstrapped sample. Table 2 lists the slopes and intercepts for the young planet void inferred from the SVM bootstrapping simulations. Figures 8 and 9 show the derived radius valley from the bootstrapping simulations, and Figure 10 shows the distributions of slopes and intercepts from this analysis. For the values explored here, the inferred slopes and intercepts of the radius valley are relatively constant. We find in almost all cases that the slope of the valley is consistent with the slope found in V18 at the 1σ level. However, we find an intercept that is systematically smaller than that found by V18 and Martinez et al. (2019) by at least 2σ and, in some cases, as much as 10σ using the quoted uncertainties from those works. While the statistical significance of this difference is highly dependent on the adopted uncertainty (where ours appears to be generally larger), it is clear from Figure 6 that the void we observe is offset from previous determinations of the radius valley. We note that previous works characterized the radius valley among samples with a broader range of ages, while the focus of this analysis is on the younger planets in the CKS sample. In Section 4, we discuss our interpretation of this difference. The level of agreement between the radius valley slopes derived here and in V18 is noteworthy, given that the samples we characterize are ≈30%-100% larger and selected on the basis of age rather than radius precision, which was the impetus for the V18 sample.
The upper and lower boundaries of the radius valley are given by the equation . In the insolation-radius diagram, it is (
( ) ( ) ( ) a b g a = + + Å R R P log log d 1 2 P 10 lower upper 10 2 in the P-R plane or ( ) ( ) ( ) d z d = + + Å Å R R S S log log 1 3 P) ( ) d = + Å Å R R S S log log P 10
10 inc
. Adopted values are in bold.
in the insolation-radius plane. We use the highest parameter explored here for determination of the radius valley boundaries, as it provides the closest match to the data (i.e., the fewest planets inside those boundaries).
Calculation of False-alarm Probability
To determine whether the void we observe in the P-R diagram could be due to chance, we performed simulations to determine the probability of finding n itv or fewer planets in the void from N planets selected at random (without replacement) from the CKS sample. Here N is the total number of planets in each of the samples described in Section 2, i.e., 156, 238, 124, and 190 for the isoc_fgk_1to2, isoc_fgk_lt2, isoc_fg_lt2, and gyro_gk_lt3 samples, respectively. The definition of the radius valley boundaries and hence the true number of planets in the valley, n itv,true , are sensitive to the regularization parameter, , and the specific sample used in the SVM bootstrapping analysis (see Figure 11). For each sample and value, we performed 10 4 simulations, selecting N planets randomly (without replacement) from the overall CKS sample, modeling the planet period and radius uncertainties with normal distributions, and recording the number of planets in the valley, n itv,sim . The false-alarm probability was then computed as the fraction of total trials that satisfied the condition n n itv,sim itv,true . The results of these simulations are tabulated in Table 3. For = 5, we find false-alarm probabilities in the range of 18%-30%, but from Figure 11, it is clear that the SVM margins in this case are so wide as to not provide an accurate description of the void boundaries. The same may be argued for the = 10 case, but even then we find false-alarm probabilities of <10%. Finally, for the = 100, 1000 cases, for which the SVM margins precisely trace the boundaries of the void, we found false-alarm probabilities of =1%.
We also performed Monte Carlo simulations to estimate the probability that the void could be produced from random selection among those planets orbiting hosts rotating more slowly than the empirical 2.7 Gyr gyrochrone (see Figure 4). We started by performing the same generic cuts on the overall CKS sample as were performed on the rotation-selected sample (main-sequence stars, no false positives, RUWE < 1.4, T eff < 6000 K, and planets with P < 100 days, R P < 10 R ⊕ , and radius precision <20%). Then, in 10 4 simulations, we modeled the uncertainties in the planet periods and radii using normal distributions, selected 190 planets at random (without replacement) from the slow rotator sample, and computed the number of planets in the void. Here 190 is the total number of planets in the fast rotator sample. We found that the probability of finding a comparably empty void from the slow rotator sample is <2% when using the gyro_gk_lt3 margins and = 100, 1000. Using the same values but margins derived from the isochrone samples raises the false-alarm probability, but only to ∼2%-6% at most. Taken together, we conclude that the emptiness of the observed void is not due to chance.
Effects of Stellar Mass and Metallicity
Our analysis separates the data set into age bins in order to understand the evolution of planets on a population level. Since age, mass, and metallicity are correlated in the CKS sample, it is difficult to entirely disentangle the effects of each parameter on the distribution of planets in the P-R diagram. We explored how sensitive our analysis is to specific binning schemes by recording the fractional number of planets in the valley over a 2D grid of bin centers and widths in age, mass, and metallicity. We used the definition of the radius valley boundaries expressed in Section 3.3 for this purpose (specifically, the margins given by the = 1000, isoc_fgk_1to2 sample case). Figure 12 shows the results of this exercise; the young planet void is apparent as the light, broad diagonal stripe in the left panel. For small bin widths, the minimum density of the radius valley is achieved for a bin center in the range 9 < log (age) < 9.25. This is unsurprising, as the boundaries of the radius valley were identified in and derived for just such a binning strategy. However, as the bin width in log (age) increases, the minimum density of the valley shifts systematically toward younger bin centers. This suggests that the filling of the radius valley is due to preferentially older planets. Figure 12 also reveals that there is no binning strategy in mass or metallicity that can produce a comparably empty void (in a fractional sense) except in finely tuned regions of parameter space where sample sizes are small. However, while not as pronounced, we do note that the radius valley (as defined in this exercise) appears emptier for lower-mass and metal-rich stars. The latter observation is consistent with the finding of an apparently wider radius valley for metal-rich hosts within the CKS sample (Owen & Murray-Clay 2018), though we note that age and [Fe/H] are anticorrelated.
We also examined the 1D radius and period distributions for CKS planets in the extremes of the stellar age, mass, and metallicity axes ( Figure 13). The purpose of this exercise was to highlight any exoplanet demographic trends as a function of these key stellar parameters. Interestingly, even without completeness corrections or occurrence rate calculations, several of the now well-established trends in the Kepler planet population are evident from Figure , and the rising occurrence of short-period planets (1 day < P < 10 days) of all sizes with increasing metallicity . The other trend that is apparent is the dearth of planets in the radius valley for young stars. The trend of a wider radius valley around more metal-rich stars found by Owen & Murray-Clay (2018) is not immediately obvious, but relative to that study, we use updated planetary radii from F18, do not include completeness corrections, and perform slightly different cuts.
Accounting for Age Uncertainties
In Section 3.4, we assessed the probability that the observed void was due to a chance selection of planets from the overall CKS data set, and in Section 3.5, we explored the sensitivity of the void occupancy to binning schemes in mass, age, and metallicity. Here we attempt to account for stellar age uncertainties in examining the void occupancy as a function of age. As discussed in Appendix B, there may be substantial uncertainties in stellar ages, particularly if those ages originate from isochrones. As a result, when binning in stellar age, there is considerable uncertainty in the degree of contamination by stars with inaccurate ages.
To mitigate the effect of stars with spurious ages, we performed Monte Carlo simulations in which the ages were modeled as normal distributions in log(age) centered on the median values published in F18 with widths taken as the maximum of the lower and upper age uncertainties for each Table 3 False-alarm Probabilities
Sample =5 =10 =100 =1000 isoc_fgk_1to2 30% 3.0% 0.09% <0.01% isoc_fgk_lt2 18% 6.5% 0.08% 0.01% isoc_fg_lt2 15% 14% 0.2% 0.1% gyro_gk_lt3
22% 4.5% 0.03% 0.08% Figure 12. Effects of binning schemes (in age, mass, and metallicity, from left to right) on the fractional occupancy of the radius valley. Note that the color scaling is the same for each panel.
star. While this is not the same as drawing from the empirical posterior probability density functions in age (which are not available), it is a crude proxy. For 50 bin centers in log (age) from 8.25 to 10.25, we then performed 10 3 Monte Carlo simulations in each bin, randomly generating ages as described above, to measure the fraction of planets in the valley as a function of age. We measured the fraction of planets in the valley relative to the total number of planets in each age bin for bin widths of 0.125, 0.25, 0.5, and 1.0 dex. The results of this analysis are shown in Figure 14. The scarcity of planets with log(age) < 9 and >10 leads to large uncertainties in the trend at both extremes, in addition to the larger age uncertainties at younger ages. However, we observe a marginally significant increase in the fraction of planets located in the radius valley in the range 8.75 < log(age) < 9.75. This is in agreement with Figures 2 and 3, which show that the radius valley appears weaker among the oldest planets in the CKS sample. Computing planet occurrence rates in the valley as a function of age might lead to a more robust conclusion on the trend noted here but is outside the scope of the present work.
In What Ways Are Planets in the Valley Different?
In an effort to quantify the parameters that are most important in contributing to the filling of the radius valley, we performed a k-sample Anderson-Darling (A-D) test (Scholz & Stephens 1987) with scipy.stats.anderson_ksamp to test the null hypothesis that the distribution of a given variable for stars hosting planets in the valley was drawn from the same distribution of that parameter among our base CKS sample.
The results of this exercise are summarized in Table 4, and select parameter distributions are shown in Figure 15. In this exercise, we have assumed that the equation for the equations for the radius valley and its boundaries are given by the fourth row of Table 2. This choice is motivated by the fact that higher regularization parameters correspond to tighter boundaries of the valley, offering a cleaner separation between planets in, above, or below the valley. Additionally, for each parameter, we restrict our analysis to those CKS stars/planets for which that parameter is defined (i.e., we exclude targets missing data for a given variable). We also apply the common cuts described in Section 3.3 before performing the A-D tests. In the case of P rot , we also restricted the sample to T eff < 6000 K, where rotation periods are more reliable indicators of age.
Of the nine parameters with the highest normalized k-sample A-D test statistics (and p-values <0.05), all but two pertain directly or indirectly to the star's evolutionary state: P rot from various sources, P rot flag, R var (a measure of the photometric variability amplitude), median posterior age from isochrones, and R * . The other two parameters are fractional R P precision and r 8 (a measure of flux dilution). Thus, of the parameters investigated, those that contribute most to the filling of the radius valley either relate to stellar age or may be associated with erroneous measurements of the planetary radii. Inspection of the parameter distributions (like those shown in Figure 15) reveals that planets in the valley tend to orbit stars that are older, larger, less likely to have a securely detected rotation period, rotating more slowly, and photometrically quieter. Planets in the valley also have lower r 8 values relative to the CKS base sample. Naively, one might expect higher r 8 values among planets in the radius valley, as flux dilution can lead to erroneous planet radius measurements. However, planets in the radius valley are by definition small, so it is perhaps not surprising that there is a preference for stars not affected by crowding.
We also found that the stellar mass and metallicity distributions for stars hosting planets in the radius valley are statistically indistinguishable from those of the CKS base sample. This lends further support to the notion that the feature identified in the CKS data set from age selections is not due to correlations between stellar age, mass, and metallicity.
The parameter that appears to be most important in contributing to the filling of the radius valley is planet radius precision. This suggests that the radius valley may be emptier than is suggested by current data. The fractional stellar radius precision does not, however, contribute to the filling of the valley. This is not surprising, as the typical error budget for a planet's radius in the CKS sample is dominated by the R P /R å uncertainty from light-curve fitting rather than the stellar radius uncertainty (Petigura 2020). In Section 3.8, we examine the possibility that a correlation between planet radius precision and age could conspire to produce the observed void.
Confounding Scenarios
A possibility not yet explored is that the radius valley is inherently empty but, for some reason, planets orbiting stars with younger assigned ages in the CKS sample have more precise radii. In Section 3.7, we established that the planet radius precision is the most important parameter contributing to the filling of the radius valley. We quantified the correlation between fractional planet radius precision and log(age) by computing the Spearman rank correlation coefficient for planets with P < 100 days, R P < 10 R ⊕ , non-false-positive dispositions, and main-sequence host stars (filters 1, 2, 4, and 5 from Section 2). We used the scipy.stats.spearmanr function for this purpose and found a small p-value (2 × 10 −4 ) but a very weak correlation coefficient (ρ < 0.1).
To further investigate the impact of radius precision, we computed the fraction of planets in the valley for young and old samples as a function of fractional radius precision allowed. For a given sample of planets and over a grid of radius precision thresholds, we selected the planets with fractional radius uncertainties smaller the threshold value and computed the ratio of planets in the valley to the total number of planets meeting the radius precision requirement. We performed 10 3 bootstrapping simulations (including modeling of the planet radii as normal distributions) to determine the uncertainties on these trends, which are shown in Figure 16. We found that the radius valley is comparably empty for young and old planets if the fractional radius precision is required to be better than ∼5%. However, this is not unexpected, as our CKS base sample size diminishes steeply below fractional radius uncertainties of 7%. For reference, the median fractional radius uncertainties for the CKS base sample, young isochrone ageselected sample (isoc_fgk_lt2 ), and old isochrone ageselected sample are 5.0%, 4.5%, and 5.2%, respectively.
Finally, we examined the 1D radius distributions for the young and old isochrone age-selected samples with planet radii known to better than 5%. We performed 10 3 bootstrapping simulations (again modeling the planet radii as normal distributions) to determine the uncertainties on these distributions. The results are shown in Figure 17. While radius precision is clearly an important parameter in determining the occupancy of the radius valley, we conclude that it is unlikely to explain the entire deficit observed for the young planet sample. ? Figures 2, 3, and 5 give the impression that the radius valley progressively fills in over time, becoming weaker or disappearing entirely among older planet populations. This interpretation appears to be at odds with the results of V18, who observed a clean gap in the P-R distribution of planets orbiting asteroseismic host stars, which are preferentially older than the stars in our young sample. 14 Notably, planets with ages 3 Gyr in the CKS base sample have a median radius precision of 5.3%, while planets in the V18 sample have a median radius precision of 3.3%. Similarly, in Section 3.8, we found that age-dependent differences in the radius valley filling factor can be resolved at least partially by restricting the analysis to planets with the most precise radii.
Is the Radius Gap Empty
To further investigate this issue, we constructed a new sample, the gold sample, which implements several reliability cuts. In addition to the cuts of the base sample, the gold sample is restricted to planets with a fractional radius precision of <6%, a fractional R P /R * precision of <6%, nongrazing transits (R τ > 0.6), RCF < 1.05, A V < 0.5 mag, RUWE < 1.1, agreement between the F18 isochrone-derived and trigonometric parallaxes, and a KOI reliability score >0.99 from the Q1-Q17 DR25 catalog. We then split this sample into young, gold_lt3, and old, gold_gt3, samples. The young sample includes the restriction that the stellar age inferred from both isochrones and gyrochronology is <3 Gyr, while the old sample requires a planet host to have a median isochrone age >3 Gyr and does not have a rotation period consistent with an age <3 Gyr. For both the gold_lt3 and gold_gt3 samples, we confirmed that the corresponding distributions in R P precision, R P /R * precision, and singletransit S/N (defined as ( ) R R CDPP3 P 2 * ) were not statistically different either from each other or from the overall distributions in the gold sample, yielding p-values > 0.25 in each case from k-sample A-D tests. The gold_lt3 sample in comparison to the V18 and gold samples in the P-R diagram are shown in Figure 18. From that figure, it appears that the reliability cuts have a significant impact on how well defined the super-Earth and sub-Neptune distributions are, as well as how empty the gap appears at all ages, though it is not entirely devoid of planets. Furthermore, it is clear from Figure 18 that the gap in the gold_lt3 sample is offset from the gap in both the V18 and gold samples, indicating that the difference is unlikely to be due to systematic differences in planet radii between the two studies. For reference, we also show the P-R distribution of V18 planets using the CKS radii, which highlights the importance of the precise (R P /R å ) values used by V18 in resolving the gap (for a detailed discussion, see Petigura 2020).
We proceeded to perform the same SVM analysis as was presented in Section 3.3 with one difference: classification of the samples into super-Earths and sub-Neptunes was performed using the threshold R P = 1.8 R ⊕ rather than a period-dependent classification scheme. The reason for this choice is because this scheme clearly works well for the gold_lt3 sample and allows us to test the sensitivity of the analysis to the classification step. The results of our analysis are presented in Figure 19 and Table 5. We find that despite the simplified classification scheme, a negative slope in the P-R diagram is still preferred (though the data are also consistent with no orbital period dependence). Furthermore, we observe that, independent of regularization parameter, there is a persistent offset in the center of the gap for the gold_lt3 sample, compared to both the gold and gold_gt3 samples.
Related to this last point, we emphasize that the gap identified in this work is primarily due to a lack of large super-Earths at young ages, as opposed to a difference in the sub-Neptune size distribution or some combination of the two. This is most apparent in Figures 17 and 18. We note that the young planet samples are always smaller than the control samples, and the dearth of large super-Earths at young ages could be due in part to small number statistics. To assess the probability of this scenario, we performed 10 4 simulations and measured the fraction of outcomes in which the number of >1.5 R ⊕ planets in a control sample was equal to or fewer than the number of >1.5 R ⊕ planets in the young sample. In each simulation, we selected 40 planets at random from the control sample, corresponding to the size of our young super-Earth sample. We accounted for planet radius uncertainties in both the young and control samples by modeling the radii as normal distributions given their published uncertainties. For the control samples, we used the CKS gold and V18 samples, where we used the V18 radius valley equation to select only the super-Earths in each. In both cases, we found that 2% of the simulations resulted in outcomes where the number of >1.5 R ⊕ super-Earths was greater in the young sample than in the control sample. We also compared the young and control super-Earth size distributions with a k-sample A-D test, finding that the null hypothesis can be rejected at the 1% level.
In conclusion, we propose a solution to resolve the apparent tension described in the beginning of this section and to explain all of the observations to date: the radius gap is intrinsically empty, or at least emptier than previously appreciated, but its precise location shifts with the age of the planetary population. Since the radius gap appears to have an orbital period dependence, a gap that is intrinsically empty in the P-R plane will always appear filled in when projected along the radius axis, even if the radii are perfectly known. Similarly, if the location of the gap also depends on host star mass, age, or metallicity, as has been suggested, then the gap will only appear empty in sufficiently narrow projections of parameter space. While this proposed solution would help explain some of our observations, we emphasize that we have not conclusively shown it to be the case. Confirming or rejecting this hypothesis may be possible with (1) a larger sample providing sufficient coverage of the P-R plane across the Figure 17. Small planet size distributions among CKS planets with fractional radius uncertainties better than 5%. Uncertainties (the 16th and 84th percentiles) are determined from bootstrapping simulations with planet radii modeled as normal distributions.
variables of interest and/or (2) a thorough, multivariate investigation of the radius gap in order to find the projection of the data resulting in the emptiest gap. We leave such an investigation to future works and emphasize that planet radius uncertainties (resulting from inaccurate light-curve fits, stellar radius uncertainties, or more pernicious sources, such as flux dilution from unresolved binaries) remain an obstacle to our understanding of the radius gap.
Discussion and Conclusions
We observe a nearly empty void in the P-R plane for close-in (P < 100 days) exoplanets orbiting stars younger than ∼2-3 Gyr. The void was first identified among a sample of planets with median posterior isochrone ages <1.8 Gyr but is also present among planets with stars rotating faster than an empirical 2.7 Gyr gyrochrone. The difference between these two timescales could conceivably be due to systematic offsets Figure 18. Planet distributions in the P-R diagram for the samples described in Section 3.9. In the middle panel, planets in the V18 asteroseismic sample are plotted using the F18 radii. Lines connect each planet in that sample to its radius as determined by V18. between the CKS isochrone ages and ages implied from a gyrochronology analysis. Because the gyrochrone used to perform our sample selection is calibrated to open clusters with main-sequence turnoff ages, the longer timescale may more accurately reflect the lifetime of this feature in the P-R diagram.
We derived equations for the center of this void, which we refer to as the young planet gap, as a function of orbital period, P, and insolation, S inc : For periods in the range of 3-30 days, describing the bulk of our sample, this places the center of the radius valley at 1.87-1.56 R ⊕ . Over this same period range, the lower boundary of the void is in the range of ∼1.6-1.4 R ⊕ , while the upper boundary is at ∼2.1-1.8 R ⊕ . From a subset of the CKS sample created using reliability and precision cuts, we similarly derived equations for the radius valley valid for all ages: The slope of the young planet gap in the P-R diagram is consistent at the 1σ level with the slope of the radius valley measured from the asteroseismic sample in V18 but with an intercept that is smaller by ∼3σ using the uncertainty reported by those authors. The smaller intercept among the "young" planet sample corresponds to a shift in the radius valley toward smaller radii and would be compatible with a prolonged massloss timescale for the sub-Neptune progenitors of the largest observed super-Earths. An alternative explanation could be the late-time formation of secondary or "revived" atmospheres through endogenous or exogenous processes (e.g., Kite & Barnett 2020;Kite & Schaefer 2021). Differentiating between these two hypotheses might be achieved with detailed composition modeling or atmospheric studies of the largest super-Earths.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = - + = + Å - + Å - + Å - + R R P R R S Slog( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = - + = + Å - + - + Å Å - + R R P R R S
The shallow, negative slope of the void is compatible with models of atmospheric loss through photoevaporation (e.g., Owen & Wu 2013Jin & Mordasini 2018;Lopez & Rice 2018) or core cooling (Gupta & Schlichting 2019 but incompatible with the steeper negative slope implied for one model of impact-driven atmospheric erosion (Wyatt et al. 2020). The negative slope we find is also incompatible with the positive slope predicted by models of late-stage formation in a gas-poor disk (Lopez & Rice 2018). However, we note that the void is only marginally inconsistent with being flat when adopting a more conservative regularization parameter in the SVM analysis. The slope of this void in the insolation-radius plane is shallower (by 2σ-3σ) than the slope found by Martinez et al. (2019).
Both rotation-selected and isochrone-selected planet samples show the same qualitative trend: an absence of large super-Earths at young ages. We estimate that the probability of this feature being due to chance is <1% for both the isochrone-selected and rotation-selected samples. We also showed that this feature is relatively insensitive to various data reliability filters and unlikely to be the result of correlations between stellar age, mass, and metallicity. Simulations accounting for age and planet radius uncertainties show an increasing fraction of planets residing in this gap as a function of age (see Figure 14). The occupancy of the radius valley is also clearly dependent on the precision of planetary radii, and we note that the differences between the young and old planetary samples diminish with more stringent precision requirements (Figure 16). However, resolving the discrepancies entirely requires discarding more than half of the CKS sample. A larger sample size and higher-precision planetary radii for the entire CKS sample would help to more securely determine how much of the discrepancy between young and old planet populations is astrophysical and how much is due to noise.
While a more detailed study of planet radius demographics as a function of age, mass, and metallicity is left for future works, we note that our findings are broadly consistent with expectations from both the core-powered mass loss and photoevaporation theories. This is most evident from Figure 20, where the sub-Neptune size trends with age, mass, and metallicity among those CKS planets with the most precise radii are shown in relation to scalings that approximately, though not exactly, mimic those presented in Gupta & Schlichting (2020). The slope in the stellar mass versus planet radius plane is shallower than predicted in core-cooling models but more similar to that predicted by photoevaporation models, provided that planet mass scales approximately linearly with stellar mass (Wu 2019). The gigayear timescale we find for the evolution of the radius valley is more compatible with core-powered mass-loss models than the canonical timescale of 0.1 Gyr from photoevaporation. However, although photoevaporation models predict that the radius gap will emerge on a timescale of 0.1 Gyr, some small fraction of planets are expected to cross the gap on timescales of ∼1 Gyr or more (Rogers & Owen 2021), which is compatible with our observations. The difference in the radius distributions between young and old planets is primarily driven by an absence of large super-Earths (1.5-1.8 R ⊕ ) at young ages, rather than an absence of small sub-Neptunes (see Figure 17). As a result, the precise location of the radius valley is shifted to larger planet sizes at older ages. To better understand the compositions of planets missing from the young planet radius distribution, we compiled data for well-characterized, confirmed exoplanets from the NASA Exoplanet Archive (Akeson et al. 2013). We selected planets with masses known to 25% precision or better, radii with 10% precision or better, orbital periods <100 days, and host stars with 4500 K < T eff < 6500 K to match the CKS sample. We computed the bulk densities of these planets and compared the distribution of planets in the radius-density plane to composition curves from Zeng et al. (2019). Among these well-characterized planets, we observe a clean separation in the radius-density plane (also observed by Sinukoff 2018) between planets that are consistent with rocky compositions and those that require a significant volatile component (such as an H 2 -He atmosphere, H 2 O-dominated ices/fluids, or some combination of the two) to explain their bulk compositions (Figure 21). We also observed that the radius valley identified in Van Eylen et al. (2018) bridges the gap between planets in these two exist for a far larger portion of the Kepler sample, we restrict our analysis here to only those stars that overlap with CKS VII. Figure 23 shows the comparison of ages and other parameters from F18 and B20. For 80% of the stars with age estimates in both catalogs, the age estimates agree to within ∼0.4 dex. The median offset in ages is 0.075 dex, with the F18 ages being systematically older, but this shift is smaller than the typical age uncertainties from either catalog. The age discrepancies should not be due to differences in the adopted stellar models; both F18 and B20 use the isoclassify package (Huber et al. 2017) to compute ages from MIST v1.1 models (Choi et al. 2016;Dotter 2016).
To better understand the origins of the age discrepancies between the two studies, we searched for correlations between ( ) D log age yr 10 1 and other parameters in the data set. We found that ( ) D log age yr 10 1 is most strongly correlated with ΔM * and ΔT eff (see Figure 24). As both F18 and B20 determine mass and age simultaneously from stellar models, and these two parameters are intrinsically related, the ΔM * -( ) D log age yr 10 1 correlation is unsurprising. The correlation with ΔT eff , however, is more informative. Here F18 derived T eff from high-resolution spectroscopy, while B20 derived T eff from isochrones using photometry (specifically Sloan g and 2MASS K s ) and parallaxes as input.
We examined the dependence of T eff -color relations on [Fe/H] and A V and found, when using the CKS spectroscopic parameters, that [Fe/H] can explain most of the dispersion in the T eff -color relations. By contrast, when using the photometric T eff and [Fe/H] from B20, there is no clear metallicity gradient in the T eff -color relations.
We find that ΔT eff is more strongly correlated with the reddening values (sourced from either B20, L21, or Gaia) than it is with any of the metallicity parameters. While reddening might help to explain temperature and age differences for some sources, we note that differences in photometric and spectroscopic temperature scales persist independent of reddening corrections (Pinsonneault et al. 2012).
We next compared the F18 and B20 ages with those determined from precise asteroseismic parameters. Silva Aguirre et al. (2015) determined ages for a sample of 33 Kepler planet candidate host stars with high-S/N asteroseismic observations, achieving a median statistical uncertainty of 14% on age. We compare the ages from F18 and B20 with the asteroseismic ages in Figure 25. We find reasonably good agreement with the asteroseismic ages for both F18 and B20. The residual scatter between the isochrone and asteroseismic ages is 0.11 dex for F18 and 0.22 dex for B20. In both cases, the residuals are comparable to the median age uncertainties from those catalogs.
Gyrochronology ages were computed using the stardate software package (Angus et al. 2019a(Angus et al. , 2019b. 16 The stardate ages were computed in the gyrochronology mode alone rather than in the combined isochrone-fitting and gyrochronology mode. The stardate gyrochronology relations are calibrated in Gaia color space. Using the rotation periods we vetted in Section 2, we noticed increased scatter in the (B P − R P )-P rot plane compared to the T eff -P rot plane, where T eff is the CKS spectroscopic temperature from F18. As such, rather than using the star's actual Gaia colors, which are susceptible to reddening, we converted the F18 spectroscopic T eff and B20 photometric T eff to the predicted Gaia colors using the relation in Curtis et al. (2020), which was calibrated for stars with negligible reddening. Using the vetted P rot and predicted (B P − R P ) colors, we then computed the gyrochronology ages (without uncertainties). Our comparison of the isochrone and gyrochronology ages is shown in Figure 25. We note that there is better agreement between F18 and the gyrochronology ages at young ages (<1 Gyr).
We note that the stardate model has not been updated to include recently determined open cluster rotation period sequences in its calibration. As such, we can compare the CKS sample to empirical gyrochrones from Curtis et al. (2020). This comparison is shown in Figure 26, which shows that the F18 isochrone ages do not always map predictably onto the T eff -P rot plane. For example, in the F18 log(age) bin of 8.75-9 dex (≈0.6-1 Gyr), approximately half of the stars fall below the 1 Gyr gyrochrone, and half lie above it. Similarly, in the F18 log(age) bin of 9.5-9.75 dex (≈3.2-5.6 Gyr), a nonnegligible number of stars fall below the log(age) ≈ 9.4 (≈2.5 Gyr) gyrochrone. However, we note that the vast majority of stars with F18 isochrone ages of log(age) < 9.25 fall below the log(age) ≈ 9.4 gyrochrone. This is in agreement with the comparison made to the stardate gyrochronology ages, in the sense that the majority of stars with F18 isochrone ages 1.8 Gyr appear to be younger than 2.7 Gyr from a gyrochronology analysis. Figure 26. In each panel, contours show the Gaussian kernel density estimate of CKS planet hosts in the T eff -P rot plane. Points depict stars with F18 isochrone ages indicated in the titles of each panel. The solid, dashed, dashed-dotted, and dotted lines indicate polynomial fits to the empirical gyrochrones of the Pleiades (log (age) ≈ 8.1), Praesepe (log(age) ≈ 8.8), NGC 6811 (log(age) ≈ 9), and NGC 6819 + Ruprecht 147 (log(age) ≈ 9.4) clusters, respectively (Curtis et al. 2020). Finally, we also examined the evolution of other physical parameters known to correlate with age, such as the variability amplitude R var , near-UV (NUV) excess, and velocity dispersion. We tracked velocity dispersion using v tan , the velocity tangential to the celestial sphere, and v b , the velocity in the direction of the galactic latitude, sourced from Lu et al. (2021). GALEX NUV magnitudes were obtained from Olmedo et al. (2015). For a crude approximation of the NUV excess, we performed a quadratic fit to the full Kepler Q1-Q17 DR25 sample in the (G − G RP ) versus (NUV − K s ) color-color diagram. The NUV excess was then defined as a star's (NUV − K s ) color minus the quadratic color-color trend. No dereddening was performed. Figure 27 shows the evolution of these parameters as a function of age. For both the F18 and B20 isochrone ages, we observe increasing dispersion in v tan and v b with age, as expected. The strongest expected correlation is observed for R var (sourced from Lu et al. 2021) and F18 age, with R var declining for the first ∼3 Gyr before plateauing. The average NUV excess appears to decline over a similar timescale when using the F18 ages, though that trend is less significant, and there may be residual systematics from the manner in which we computed the excess. Both the R var and NUV excess trends are expected, as starspot coverage, variability amplitudes, and chromospheric activity are known to decline with age. By comparison, when using the B20 ages, the behavior of R var and NUV excess with age is not as expected. We conclude by noting that, while substantial uncertainties remain for isochronal ages, there is qualitative agreement between the CKS ages and the ages (or age indicators) derived from independent methods. In some of the comparisons above, the CKS and GKSPC ages perform comparably well, though it is at the youngest ages (3 Gyr) where the GKSPC ages do not reproduce some expected trends. As the evolution of small planets at early times is a primary focus of this work, we adopt the CKS ages.
Figure 3 .
3Same as Figure 2 but for the base sample.
Figure 4 .
4The T eff -P rot diagram (left column), H-R diagram (middle column), and median isochrone age distributions (right column) for the young (top row) and old (bottom row) rotation-selected samples. The shaded contours in the left and middle columns show Gaussian kernel density estimates of the full CKS sample in those respective planes. In the left column, the solid, dashed, dashed-dotted, and dotted lines indicate polynomial fits to the empirical gyrochrones of the Pleiades (≈0.12 Gyr), Praesepe (≈0.63 Gyr), NGC 6811 (≈1 Gyr), and NGC 6819 + Ruprecht 147 (≈2.7 Gyr) clusters, respectively
Figure 6 .
6Age-selected samples of planets in the P-R (left column) and insolation-radius (right column) planes. Age selections are described in Section 2. Previously determined equations for the radius valley are shown as black lines. Point colors indicate the classification used in the SVM analysis (described in Section 3.3).
Figure 7 .
7Effect of the regularization parameter, , on the support vector classification for young CKS planets (the isoc_fgk_1to2 sample) in the P-R diagram. In each panel, the point colors indicate the planet classification provided in the SVM classification analysis. Points circled in black indicate the support vectors. The solid lines indicate the decision surface of maximal separation, while the dashed lines indicate the margins (as discussed in the text).
The equation for the radius valley in the P-R diagram is of the form
Figure 8 .
8The P-R diagram for planets in the isoc_fgk_1to2 (first row), isoc_fgk_lt2 (second row), isoc_fg_lt2 (third row), and gyro_gk_lt3 (fourth row) samples. Point colors indicate the classifications used in the SVM analysis. The gray line and shaded region show the median and 16th-84th percentile width of the radius valley from the SVM bootstrapping simulations. The dashed lines indicate the median margins from the SVM analysis. The regularization parameter, , is indicated at the top of each panel.
Figure 9 .
9Same asFigure 8but for the insolation-radius plane.
Figure 10 .
10Gaussian kernel density estimation of the distribution of radius valley slopes (α) and intercepts (β) from the SVM bootstrapping simulations with different regularization ( ) parameters. The circles with error bars indicate the values derived in Van Eylen et al. (2018) from planets orbiting asteroseismic stars. The squares with error bars indicate the values derived by Martinez et al. (2019) from an independent spectroscopic analysis of the CKS sample.
Figure 11 .
11Effect of the regularization parameter, , on the boundaries of the radius valley (indicated by the dashed lines) and hence the number of planets in the valley (dark points). The samples from top to bottom are isoc_fgk_1to2, isoc_fgk_lt2, isoc_fg_lt2, and gyro_gk_lt3.
13: larger sub-Neptunes around more massive (Fulton & Petigura 2018; Wu 2019; Cloutier & Menou 2020) and metal-rich (Petigura et al. 2018) stars, the rising occurrence of ultrashort-period (P < 1 day) planets with decreasing stellar mass (Sanchis-Ojeda et al. 2014)
Figure 13 .
13Planet radius (top row) and period (bottom row) distributions among our CKS base sample for host stars of different age (left column), mass (middle column), and metallicity (right column).
Figure 14 .
14Results of Monte Carlo simulations exploring the occupancy of the radius valley as a function of age, accounting for uncertainties in stellar ages. The dark line shows the median trend resulting from the simulations, while the shaded contours show the 68.3rd, 95.4th, and 99.7th percentile ranges. The curves have been smoothed with a Savitzky-Golay filter for clarity.
Figure 15 .
15Comparison of 1D parameter distributions for planets in the valley versus the CKS base sample (as described in Section 3.7).
Figure 16 .
16Fraction of planets in the radius valley as a function of maximum fractional radius uncertainty. For each age-selected sample shown, the fraction of planets in the valley is computed for the subsample of planets with fractional radius uncertainties lower than the value on the ordinate. Solid lines and shaded bands show the median and 16th-84th percentile range from bootstrapping simulations, respectively. In each panel, the fiducial dashed line shows the value of fractional radius uncertainty for which both young and old samples contain more than 150 planets (corresponding to 21 ± 4 expected planets in the valley if selected at random from the CKS base sample).
Figure 19 .
19Smoothed Gaussian kernel density estimates of the distributions of the intercepts (left) and slopes (right) resulting from the SVM bootstrapping analysis of the CKS gold samples presented in Section 3.9. At left, the R 10 parameter indicates the center of the radius gap at an orbital period of 10 days. Each row corresponds to a different regularization parameter, , indicated in the figure legend. At right, vertical lines indicate predictions from the impact erosion (IE; Wyatt et al. 2020), photoevaporation (PE; Lopez & Rice 2018), core-powered mass-loss (CPML; Gupta & Schlichting 2019), and gas-poor formation (GPF; Lopez & Rice 2018) theories.
Figure 20 .
20The 2D Gaussian kernel density estimates of the distribution of CKS planets in the age-radius (left), stellar mass-radius (middle), and metallicity-radius (right) planes. Our base sample is shown with the additional requirement of fractional radius uncertainties <6%. The light dashed lines indicate the nominal location of the radius valley. The dark dashed lines are not fits but are drawn as a visual guide, with their slopes indicated in the upper left corner of each panel.
Figure 21 .
21Well-characterized exoplanets in the radius-density plane. Planet composition curves fromZeng et al. (2019) are shown. The beige line indicates an Earthlike rocky composition (32.5% Fe + 67.5% MgSiO 3 ), and the similarly shaded swath is bounded by the pure Fe (upper bound) and MgSiO 3 (lower bound) curves. Dotted lines indicate composition curves for Earth-like cores with H 2 atmospheres at 700 K for different atmospheric mass fractions, which are indicated above each curve. The vertical pink band indicates the range of planetary radii at the center of the young planet gap identified in this work for orbital periods in the 3-30 days range. The hatched band indicates the equivalent radius range for the same orbital periods of the gap identified inVan Eylen et al. (2018).
Figure 23 .
23Comparison of F18 and B20 isochrone ages, stellar radii, T eff , and [Fe/H] (from left to right). Residuals are show in the bottom row for each panel.
Figure 24 .
24Trends in age discrepancy between the CKS (F18) and GKSPC (B20) catalogs. Errors on the age differences are omitted for clarity.
Figure 25 .
25Comparison of F18 and B20 ages with asteroseismic ages from Silva Aguirre et al. (2015; top panels) and gyrochronology ages from this work.
Figure 27 .
27Validation of isochronal age estimates. We show v tan , v b , R var , and ( ) / F F log 10 NUV Ks , from left to right, as a function of isochronal ages from F18 (top row) and B20 (bottom row). Spearman rank correlation coefficients (ρ) and p-values are printed in the top left corner of each panel. Black points with error bars indicate the mean and standard deviation of the data binned by 0.125 dex in log(age).
Table 1
1Rotation Periods of KOIs in CKS VII SampleKOI
KIC
P rot
P rot Ref.
Flag
A18 P rot
M13 P rot
M15 P rot
W13 P rot
D21 P rot
1
11446443
L
L
1
24.85
L
70.55
L
43.37
2
10666592
L
L
1
19.60
L
70.69
L
46.65
6
3248033
L
L
0
22.77
L
L
L
L
Note. Flag meanings are as follows: 3, highly reliable; 2, reliable; 1, true period could not be unambiguously determined; and 0, no periodicity evident. References:
A18, Angus et al. 2018; M13, McQuillan et al. 2013; M15, Mazeh et al. 2015; W13, Walkowicz & Basri 2013; D21, this work. Only a portion of the table is shown
here to demonstrate its form and content.
Table 2
2Results of SVM Bootstrapping SimulationsSample
α
β
γ
δ
ò
ζ
isoc_fgk_1to2
5
-
-
+
0.09 0.06
0.06
-
+
0.33 0.07
0.06
-
+
0.14 0.01
0.01
-
+
0.06 0.05
0.04
-
+
0.13 0.06
0.12
-
+
0.14 0.01
0.01
Table 4
4Results of k-sample A-D TestsParameter
Ref.
A-D
Test Stat.
A-D p-
value
Sample Size (Valley/
Control)
s R
R
P
P
F18
14.30
0.0010
196/1443
P rot flag
D21
6.08
0.0015
196/1443
P rot
M15
4.82
0.0040
135/1055
R var
M15
4.67
0.0045
180/1334
log(age)
F18
4.37
0.0059
196/1443
R *
F18
3.61
0.011
196/1443
r 8
F18
2.53
0.030
196/1443
P rot
M13
2.17
0.042
36/371
P rot
D21
1.95
0.051
55/592
P rot
A18
0.67
0.17
109/873
S/N 1
D21
0.65
0.18
190/1420
T eff
F18
0.62
0.18
196/1443
R τ
P20
0.30
>0.25
190/1415
s
R
R
F18
0.25
>0.25
196/1443
M *
F18
0.21
>0.25
196/1443
Parallax
F18
0.096
>0.25
196/1443
CDPP3
D21
−0.16
>0.25
190/1420
P rot
W13
−0.19
>0.25
34/335
A V
B20
−0.50
>0.25
188/1390
G mag
DR2
−0.56
>0.25
190/1420
A V
L21
−0.57
>0.25
187/1382
[Fe/H]
F18
−0.63
>0.25
196/1443
RUWE
D21
−0.64
>0.25
190/1418
RCF
F18
−0.94
>0.25
55/423
Note. References: A18, Angus et al. 2018; D21, this work; DR2, Gaia
Collaboration et al. 2018; F18, Fulton & Petigura 2018; L21, Lu et al. 2021;
M15, Mazeh et al. 2015; M13, McQuillan et al. 2013; P20, Petigura 2020;
W13, Walkowicz & Basri 2013.
Table 5
5Results of SVM Bootstrapping Simulations for the CKS Gold SamplesNote. The equation for the radius valley in the P-R diagram is of the formSample
α
β
γ
δ
ò
ζ
gold
5
-
-
+
0.18 0.07
0.09
-
+
0.42 0.07
0.07
-
+
0.15 0.02
0.01
-
+
0.11 0.05
0.05
-
+
0.02 0.07
0.13
-
+
0.15 0.02
0.01
gold
10
-
-
+
0.12 0.06
0.07
-
+
0.37 0.06
0.05
-
+
0.11 0.01
0.01
-
+
0.07 0.04
0.04
-
+
0.11 0.08
0.11
+
-
0.11 0.01
0.01
gold
100
-
-
+
0.05 0.04
0.06
-
+
0.31 0.05
0.04
-
+
0.06 0.01
0.01
-
+
0.03 0.04
0.03
-
+
0.19 0.05
0.1
-
+
0.06 0.01
0.01
gold
1000
-
-
+
0.02 0.05
0.05
-
+
0.28 0.06
0.04
-
+
0.03 0.01
0.01
-
+
0.01 0.04
0.02
-
+
0.24 0.07
0.05
-
+
0.03 0.01
0.01
gold_gt3
5
-
-
+
0.22 0.11
0.09
-
+
0.49 0.1
0.1
-
+
0.16 0.02
0.02
-
+
0.15 0.06
0.04
-
-
+
0.04 0.11
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+
0.16 0.02
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-
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0.13 0.08
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0.4 0.09
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0.12 0.02
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gold_gt3
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-
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+
0.29 0.05
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-
+
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0.2 0.06
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0.27 0.04
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0.34 0.05
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0.1 0.04
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0.33 0.05
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-
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(
)
(
)
a
b
=
+
Å
R R
P
log
log
d
P
10
10
. In the insolation-radius diagram, it is
(
)
(
)
d
=
+
Å
Å
R R
S S
log
log
P
10
10 inc
. Adopted values are in bold.
The Astronomical Journal, 161:265 (29pp), 2021 June David et al.
https://gea.esac.esa.int/archive/
https://archive.stsci.edu/kepler/search_retrieve.html 10 https://exoplanetarchive.ipac.caltech.edu/docs/PurposeOfKOITable. html#cumulative
We note that cross-matching the V18 sample with Silva Aguirre et al.(2015) and F18 reveals that the asteroseismic sample contains host stars with a broad range of ages, from ≈2 to 12.5 Gyr.
https://github.com/trevordavid/radius-gap
https://github.com/RuthAngus/stardate
ORCID iDsTrevor J. David https:/ /orcid.org/0000-0001-6534-6246 Gabriella Contardo https:/ /orcid.org/0000-0002-3011-4784 Angeli Sandoval https:/ /orcid.org/0000-0003-1133-1027 Ruth Angus https:/ /orcid.org/0000-0003-4540-5661 Yuxi (Lucy) Lu https:/ /orcid.org/0000-0003-4769-3273 Megan Bedell https:/ /orcid.org/0000-0001-9907-7742 Jason L. Curtis https:/ /orcid.org/0000-0002-2792-134X Daniel Foreman-Mackey https:/ /orcid.org/0000-0002-9328-5652 Benjamin J. Fulton https:/ /orcid.org/0000-0003-3504-5316 Samuel K. Grunblatt https:/ /orcid.org/0000-0003-4976-9980 Erik A. Petigura https://orcid.org/0000-0003-0967-2893composition regimes. Meanwhile, the young planet gap identified in this work appears to correspond only to planets in the rocky composition regime. Thus, assuming the atmospheric loss hypothesis is correct, the planets that eventually fill the young planet gap may correspond to the large end of the size distribution of stripped cores. This is an important point in the context of disentangling correlations between stellar mass, age, and metallicity in the CKS sample. The evidence for a wider radius valley among metal-rich stars is driven mostly by larger sub-Neptunes, on average, with one explanation being the decreased cooling efficiency of planets with higher-metallicity envelopes(Owen & Murray-Clay 2018). By comparison, we observe a sub-Neptune size distribution that is relatively constant below 3 Gyr, while the average size of super-Earths appears to increase over this same time frame(Figure 20). These observations are not easily explained by the anticorrelation between age and [Fe/H] in the CKS sample or the naive expectation of more massive cores around metal-rich stars from core accretion models. Given that a planet's size is correlated with its mass, one physical interpretation for this observation is that the largest, most massive cores lose their atmospheres at later times.It is also worth noting that a prolonged mass-loss timescale for some super-Earths might help to explain the rising occurrence of long-period super-Earths with decreasing metallicity observed byOwen & Murray-Clay (2018). Those authors noted that such planets are difficult to explain in the photoevaporation model and might have instead formed after the protoplanetary disk dispersed, akin to the canonical view of terrestrial planet formation in the solar system. However, we note that metallicity and age are correlated in the CKS sample, with the median age of the metal-poor sample in Owen & Murray-Clay (2018) being approximately 0.4 dex older than the metal-rich sample. If mass loss, regardless of the mechanism, proceeds over gigayear timescales, then one might expect a rising occurrence of super-Earths with increasing age (and hence decreasing metallicity).In a companion paper,Sandoval et al. (2021)found tentative evidence that the fraction of super-Earths to sub-Neptunes rises with system age from ∼1 to 10 Gyr. That work accounted for uncertainties in stellar ages, planetary radii, and the equation for the radius valley itself. The result is in agreement with a previous finding byBerger et al. (2020a), who found that, among planets orbiting stars more massive than the Sun, the fraction of super-Earths to sub-Neptunes is higher among older stars (>1 Gyr) than it is for younger stars (<1 Gyr). Collectively, the present work and the studies mentioned above provide evidence for the evolution of small planet radii over gigayear timescales.The code and data tables required to reproduce the analyses and figures presented in this paper are made publicly available. 15 This paper is dedicated to the memory of John Stauffer, a valued mentor whose energy and determination were an inspiration to many. We thank the anonymous referee for a thorough and insightful review, as well as Eric Ford, Christina Hedges, David W. Hogg, and Josh Winn for helpful discussions. T.J.D. is especially grateful to Chelsea Yarnell for her irreplaceable support throughout the COVID-19 pandemic. This paper includes data collected by the Kepler mission, funded by the NASA Science Mission directorate. This work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC).Funding for the DPAC is provided by national institutions, inAppendix A Rotation Period Vetting SheetsThe rotation period vetting sheets (as described in Section 2.1) are available in a Zenodo repository at doi:10.5281/zenodo. 4645437. An example sheet is shown inFigure 22.Appendix B Stellar Age ValidationAs we are concerned with the time evolution of the exoplanet radius gap, our study hinges on the accuracy of the stellar ages. Main-sequence stars, which constitute the majority of Kepler planet hosts, typically have large age uncertainties; this is because the changes in a star's observable properties over its main-sequence lifetime are small relative to the typical measurement uncertainties in those properties. However, the high precision of Gaia parallaxes and photometry has enabled the determination of relatively precise stellar ages from isochrones. The median lower and upper uncertainties on log (age) for stars in the CKS sample are 0.12 and 0.14 dex, respectively.A true assessment of the accuracy of stellar ages is not possible; essentially all methods for stellar age determination are model-dependent, and benchmarks to calibrate these methods are lacking(Soderblom 2010). However, because the Kepler field is so well studied, it is at least possible to determine the degree of agreement between isochrone ages published by different authors. It is also possible to determine the agreement between ages determined from isochrones versus those determined from gyrochronology or asteroseismology.To validate the ages used in this study, we compared the isochrone age estimates from F18 with those published in the Gaia-Kepler Stellar Properties Catalog (GKSPC; hereafter B20,Berger et al. 2020aBerger et al. , 2020b, the asteroseismic ages determined in SilvaAguirre et al. (2015), and the gyrochronology ages determined here. We note that while GKSPC agesFigure 22. Example rotation period vetting sheet. The Kepler light curve is phase-folded on periods determined from the literature (first row), as well as the first harmonic (second row) and subharmonic (third row). The period determinations of different authors (indicated at top) are presented in a column-wise fashion and color-coded for convenience. In the fourth row, the first 120 days of the light curve (left) and an L-S periodogram (right) are shown. In the fifth row, the full Kepler light curve is shown.
The Gaia archive website is https:// archives.esac.esa.int/gaia. This work made use of the gaiakepler.fun cross-match database created by Megan Bedell. the institutions participating in the Gaia MultiLateral Agreement (MLA). jupyter (Kluyver et al. 2016), matplotlib. numpy (van der Walt et al. 2011), pandas (pandas Development Teamthe institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www. cosmos.esa.int/gaia. The Gaia archive website is https:// archives.esac.esa.int/gaia. This work made use of the gaia- kepler.fun cross-match database created by Megan Bedell. Facilities: Kepler, Gaia, GALEX. Software: astropy (Astropy Collaboration et al. 2013, 2018), jupyter (Kluyver et al. 2016), matplotlib (Hunter 2007), numpy (van der Walt et al. 2011), pandas (pandas Development Team 2020;
Wes Mckinney, ; Waskom, scikit-learn (Pedregosa et al. 2011), scipy. Wes McKinney 2010), seaborn (Waskom et al. 2017), scikit-learn (Pedregosa et al. 2011), scipy (Jones et al. 2001).
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"Relations between entropy rate, entropy production and information geometry in linear stochastic systems",
"Relations between entropy rate, entropy production and information geometry in linear stochastic systems"
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"Adrian-Josue Guel-Cortez \nCentre for fluid and complex systems\nCoventry University\nCoventryUK\n",
"Eun-Jin Kim \nCentre for fluid and complex systems\nCoventry University\nCoventryUK\n"
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In this work, we investigate the relation between the concept of "information rate", an information geometric method for measuring the speed of the time evolution of the statistical states of a stochastic process, and stochastic thermodynamics quantities like entropy rate and entropy production.Then, we propose the application of entropy rate and entropy production to different practical applications such as abrupt event detection, correlation analysis, and control engineering. Specifically, by utilising the Fokker-Planck equation of multivariable linear stochastic processes described by Langevin equations, we calculate the exact value for information rate, entropy rate, and entropy production and derive various inequalities among them. Inspired by classical correlation coefficients and control techniques, we create entropic-informed correlation coefficients as abrupt event detection methods and information geometric cost functions as optimal thermodynamic control policies, respectively. The methods are analysed via the numerical simulations of common prototypical systems.
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10.1088/1742-5468/acbc24
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2302.03415
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587250c9de542fabff371c465da913cfecb2dc7a
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Relations between entropy rate, entropy production and information geometry in linear stochastic systems
August 2017
Adrian-Josue Guel-Cortez
Centre for fluid and complex systems
Coventry University
CoventryUK
Eun-Jin Kim
Centre for fluid and complex systems
Coventry University
CoventryUK
Relations between entropy rate, entropy production and information geometry in linear stochastic systems
August 2017Information GeometryStochastic ThermodynamicsLinear Stochastic Systems Submitted to: J Stat Mech
In this work, we investigate the relation between the concept of "information rate", an information geometric method for measuring the speed of the time evolution of the statistical states of a stochastic process, and stochastic thermodynamics quantities like entropy rate and entropy production.Then, we propose the application of entropy rate and entropy production to different practical applications such as abrupt event detection, correlation analysis, and control engineering. Specifically, by utilising the Fokker-Planck equation of multivariable linear stochastic processes described by Langevin equations, we calculate the exact value for information rate, entropy rate, and entropy production and derive various inequalities among them. Inspired by classical correlation coefficients and control techniques, we create entropic-informed correlation coefficients as abrupt event detection methods and information geometric cost functions as optimal thermodynamic control policies, respectively. The methods are analysed via the numerical simulations of common prototypical systems.
Introduction
Information geometry or the application of differential geometry to the information sciences has brought to light new tools for the analysis of dynamical systems [1,2]. For instance, the information length (IL) [3,4], given by the time integral of information rate, describes the total amount of statistical changes that a time-varying probability distribution takes through in time. IL appears as an important tool for the analysis of the non-equilibrium processes by providing a possible link between stochastic processes, complexity and geometry. In fact, IL has already been applied with success to different scenarios such as the quantification of hysteresis in forward-backward processes [5,6], correlation and self-regulation among different players [6], phase transitions [7], and prediction of sudden events [8]. Yet, the thermodynamic significance of IL and information rate seem to be less understood.
To find a connection between IL and thermodynamics, we can use stochastic thermodynamics (ST) [9,10]. As it has already been popularised, ST makes use of stochastic calculus to draw a correspondence between micro/mesoscopic stochastic dynamics and macroscopic thermodynamics. In other words, we use ST to describe the interaction of a micro/mesoscopic system with one or multiple reservoirs. For instance, the dynamics of a Brownian particle suspended in a fluid in thermodynamic equilibrium is described by a Langevin/Fokker Planck equation. ST introduces time information uncertainty relations in thermodynamics [11], information theory to causality, modelling and control [12], optimal protocols [16], or fluctuation relations [14,9]. In addition, it has inspired works in neuroscience [17], system dynamics [18] and control theory [19].
The main aim of this paper is to elucidate the link between thermodynamic quantities (e.g., entropy rate [20,21]) and information rate (detailed in §IV) while exploring their application to abrupt event detection, correlation analysis, and control engineering. Even though some relations between information geometry and stochastic thermodynamics have been derived already (for instance, see [22,23,24]), this work focuses on generalising a relation initially proposed in [3,4] for the Ornstein-Uhlenbeck (OU) process governed by a first-order linear stochastic differential equation. Note that, in comparison to [24] which determines inequalities between information geometry and thermodynamic observables. The relation reported in [3] connects information geometry to the "entropy balance equation" by explicitly using the values of entropy rate and entropy production. We generalise this relation to a set of linear Langevin equations using the corresponding Fokker-Planck equation [25] for multivariable linear processes. By utilising the exact time-varying probability density function (PDF), we calculate the values of the entropy rate, entropy production rate, and information rate. Then, we establish an inequality and an equality between information rate, entropy production and entropy rate for general linear stochastic systems and fully decoupled systems, respectively. Our results suggest that a weighted value of entropy production plus the square of the entropy rate is generally an upper bound of the information rate and that such a relation is fully determined by the structure of the system's harmonic potential.
Furthermore, after establishing an information-thermodynamic relation, we propose normalised correlation coefficients based on information rate and entropy production [27], and compare them in the analysis of abrupt events. Specifically, we perform the numerical simulation of a second-order stochastic process where the abrupt events are stimuli in the form of impulse-like functions applied to the system's mean value and covariance matrix. The results suggest that the information rate coefficient performs better when detecting/predicting abrupt events. Moreover, we conduct abrupt events analysis for systems with more than two random variables via the Euclidean norm of the information rate and entropy production as approximate correlation coefficients.
Finally, we explore the design of optimal "static" feedback control algorithms and the corresponding effects on entropy rate and entropy production for the minimisation of the number of statistical fluctuations via the solution of cost functions using IL and entropy production.
The rest of the paper is organised as follows. Section 2 describes the mathematical model and preliminary results to be used throughout the paper. Section 3 details the calculation of the entropy rate, entropy production and entropy flow in multivariable stochastic systems. Section 4 introduces the concept of information rate and IL and presents their relation to the thermodynamic quantities of entropy production and entropy rate. In Section 5, new normalised correlation coefficients in terms of entropy rate and entropy production are introduced. Section 6 demonstrates the application of the main results through the numerical simulation of different toy models. Section 7 states the conclusions of the paper and future work. Finally, from Appendix A to Appendix D, we present the detailed steps for the derivation of the main results in the paper.
Model
Consider the following set of Langevin equations
dx i dt = f i (x; t) + ξ i (t).(1)
Here, f i : R n → R is a function that maps the variables of the vector x ∈ R n := [x 1 , x 2 , . . . , x n ] to a real value at a given time t ∈ R; ξ i is a short-correlated random noise satisfying ξ i (t) = 0 and ξ i (t)ξ j (t ) = 2D ij δ(t − t ), where D ij is the amplitude of the correlation function ξ i (t)ξ j (t ) . In this work, f i (x; t) is a linear function defined as
f i (x; t) := A i x(t) + B i u(t) = n j=1 a ij x j (t) + p j b ij u j (t),(2)
where it becomes clear that u(t) ∈ R p is a vector of continuous time dependant functions such that u(t) := [u 1 (t), u 2 (t), . . . , u p (t)] ; A i and B i are the i-th row vectors of the constant matrices A ∈ R n×n and B ∈ R n×p used to couple the different system states x j (t) and deterministic forces u j (t), respectively. Hence, we consider that the set of particles described by (1) are driven by a harmonic potential (A i x(t)), deterministic force (B i u(t) due to x-independent but possibly time-dependent u(t)), and a stochastic forcing term ξ i (t). Note that we have used a state space representation [28] of the term f i as we are concerned with applications of stochastic thermodynamics/information geometry to dynamical control systems, causality and abrupt events analysis. The Fokker-Planck equation corresponding to the Langevin equation (1) is given by
∂p(x; t) ∂t = − i ∂ ∂x i (f i (x; t)p(x; t)) + i j D ij ∂ 2 p(x; t) ∂x i ∂x j .(3)
Equation (3) describes the time evolution of the PDF p(x; t) : R n → R for x := {x 1 , x 2 , . . . , x n } in (1). Equation (3) can be written in terms of a probability current as follows
∂p(x; t) ∂t = − i ∂ ∂x i J i (x; t),(4)
where J i is the i-th component of the probability current J defined by
J i (x; t) = f i (x; t)p(x; t) − j D ij ∂ ∂x j p(x; t).(5)
In general, the solutions of Equation (4) require numerical simulations. However, thanks to the linearity of f i (x; t) in (1), we have an analytical time-dependent solution of Equation (4) given by the following Proposition.
Proposition 1 Given an initial multivariable Gaussian PDF p(x; t 0 ), the solution of p(x; t) in (3) is Gaussian and described by [29] p(
x; t) = 1 det(2πΣ) e Q(x;t) ,(6)where Q(x; t) = − 1 2 (x − µ(t)) Σ −1 (t) (x − µ(t))
; µ(t) ∈ R n and Σ(t) ∈ R n×n are the mean and covariance value of the random variable x.
According to Proposition 1, the dynamics of the PDF at every instant of time are governed solely by the value of µ(t) and Σ(t). In the case of linear stochastic systems, the value of µ(t) and Σ(t) can be obtained by solving the following set of differential equations [30]μ
i (t) = n j a ij µ j (t) + p j b ij u j (t),(7)Σ ij (t) = n k a ik Σ kj (t) + n k a jk Σ ki (t) + 2D ij (t).(8)
Equations (7) and (8) can be rewritten in the following matricial forṁ
µ(t) = Aµ(t) + Bu(t),(9)Σ(t) = AΣ(t) + Σ(t)A + 2D(t).(10)
In Equation (10), D is an square matrix whose elements are D ij (t).
Entropy rate
Given a time-varying multivariable PDF p(x; t), its entropy rate is defined as [20]
S(t) = d dt S(t) = − R nṗ (x; t)ln (p(x; t)) d n x.(11)
By substituting (3) in (11), we obtain
d dt S(t) = R n i ∂ ∂x i J i (x; t) ln (p(x; t)) d n x = − R n i J i (x; t) Å ∂ ∂x i ln (p(x; t)) ã d n x.
(12) Now, after substituting (5) in (12), we have
d dt S(t) = − R n i J i (x; t) f i (x; t) D ii − J i (x; t) D ii p(x; t) − j =i D ij ∂ ∂x j p(x; t) D ii p(x; t) d n x.(13)
From (13), the entropy production rate of the system corresponds to the positive definite part
Π = R n i J i (x; t) 2 D ii p(x; t) d n x,(14)
while the entropy flux (entropy from the system to the environment) is
Φ= R n i J i (x; t)f i (x; t) D ii − j =i D ij J i (x; t) ∂ ∂x j p(x; t) D ii p(x; t) d n x.(15)
In this paper, we focus on the case when D ij = 0 if i = j to simplify (15) as
Φ = R n i Å J i (x; t)f i (x, t) D ii ã d n x.(16)
From Equations (14)-(16), we can define the entropy production Π J i and the entropy flow Φ J i from the J i probability current as
Π J i = R n J i (x; t) 2 D ii p(x; t) d n x, Φ J i = R n J i (x; t)f i (x; t) D ii d n x.(17)
Finally, entropy rate can be expressed as follows
dS dt = Π − Φ,(18)
where, from (17), clearly Π = i Π J i and Φ = i Φ J i . Equation (18) is a well known expression for irreversible processes [31]. Notice that (14)- (15) require that D ii > 0. If D ii = 0, we have Π J i = 0 and
Φ J i = ≠ ∂f i (x, t) ∂x i ∑ = a ii .(19)
Entropy rate for Gaussian dynamics
By using p(x; t) given in equation (6), we derive the expressions for entropy production (14) and entropy flux (16) in terms of µ, Σ, A and D as follows.
Relation 1
The value of entropy production Π and entropy flow Φ in a Gaussian process whose mean µ and covariance Σ are governed by equations (9)-(10) are given by
Π =μ D −1μ +Tr A D −1 AΣ + Tr Σ −1 D +2Tr(A),(20)Φ =μ D −1μ +Tr A D −1 AΣ + Tr(A).(21)
Proof 1 See Appendix A.
From Relation 1, we readily obtain the entropy rate (for further details, see (A.12) in
Appendix A) dS dt = 1 2 d dt ln|Σ| = 1 2 Tr Ä Σ −1Σ ä .(22)
Clearly, equation (22) can also be obtained directly from the time derivative of the entropy S in a Gaussian Process
S = n 2 ln(2π) + 1 2 ln |Σ| + 1 2 n.(23)
Again, if D ii = 0 ∀i = 1, 2, . . . , n, we have Π = 0 and Φ = Tr(A).
Expressions similar to (20)- (22) but with an emphasis in distinguishing between variables that are even and odd under time reversal can be found in [31].
Relation between Entropy rate and information rate
As noted previously in Section 1, we aim to elucidate the link between entropy production Π, entropy rateṠ, and information rate. We recall that for a time-varying multivariable PDF p(x; t), we define its IL L as
L(t) = t 0 Ç R n p(x; τ ) [∂ τ ln p(x; τ )] 2 d n x å dτ = t 0 Γ(τ )dτ,(25)
where Γ(τ ) = » R n p(x; τ ) [∂ τ ln p(x; τ )] 2 d n x is called the information rate and Γ 2 (τ ) the information energy. The value of Γ 2 (τ ) can also be understood as the Fisher information where the time is the control parameter [4]. Since Γ gives the rate of change of p(x; t), its time integral L quantifies the amount of statistical changes that the system goes through in time from the initial PDF p(x; 0) to a final PDF p(x; t) [25].
When p(x; t) is a Gaussian PDF, the information rate Γ of the joint PDF takes the compact form [3,25,26]
Γ 2 =μ Σ −1μ + 1 2 Tr Ä (Σ −1Σ ) 2 ä .(26)
As it is mentioned in [3], if the PDF of Equation (1) is described by an univariate Gaussian PDF (i.e. the Ornstein-Uhlenbeck (OU) process) the information rate Γ is related to the entropy rateṠ and the entropy production Π via
Γ 2 = D Σ Π +Ṡ 2 .(27)
Equation (27) can easily be confirmed after some algebra with the following expressions
Π =μ 2 D + a 2 Σ D + D Σ + 2a =μ 2 D +Σ 2 4ΣD , S = 1 2Σ Σ , Γ 2 =μ 2 Σ + 1 2 ÇΣ Σ å 2 ,(28)
where µ, Σ, D and a are the scalar version of µ, Σ, D and A, respectively, in Proposition 1. Extending (27) to the case of a n-th order Gaussian process is one of the main contributions of this work. Such a result is given by the following relations.
Relation 2
Given an n-th order Gaussian process whose mean and covariance are described by Equations (9)-(10), a relationship between entropy production Π, entropy rateṠ, and information rate Γ is given by
0 ≤ Γ 2 ≤ E u := Tr(Σ −1 )Π Tr(D) +Ṡ 2 − 2g(s),(29)
where s = [Ṡ J 1 ,Ṡ J 2 , . . . ,Ṡ Jn ] , g(s) := n i<jṠ J iṠ J j andṠ J i is the contribution to entropy rate by the current flow J i , i.e.
S J i = − R n ∂ ∂x i J i (x; t) ln (p(x; t))d n x = Π J i − Φ J i .(30)
Proof 2 See Appendix B.
Relation 2 provides an inequality between information rate Γ (an information metric), entropy rateṠ and entropy production Π where the entropy rateṠ J i of each current flow J i is explicitly taken into account. In the relation, Π is normalised by the product between the amplitude of the environmental fluctuations and the inverse of the system dynamics covariance due to its nature as an extensive quantity [3]. If Σ is constant, the normalised entropy production dominates the square of the information rate. On the other hand, when the system fluctuations change, the flow of information between the system and its environment is considered by the termṠ 2 − 2g(s) in (29). In addition, from Relation 2 we have
L(t) ≤ L u (t) := t 0 » E u (τ )dτ.(31)
Since minimising L u will minimise L, we can obtain both a minimum entropy production and a minimum statistical variability behaviour through L u . For unstable systems, we can avoid the computation of the term g(s) involving the contribution to entropy rate by each current flow J i by using the following relation.
Relation 3
Given the same conditions as in Relation 2, but considering that the eigenvalues ϕ i ∈ C of the matrix A satisfy the following inequality
{ϕ i } > 0 ∀i = 1, 2, . . . , n.(32)
Then, the following result holds
0 ≤ Γ 2 ≤ I u := 1 n Tr(Σ −1 )Π u Tr(D) +Ṡ 2 ,(33)
where Π u ≥ Π (an upper bound of entropy production) defined by
Π u := Tr(μμ )Tr(D −1 )+ 1 4 Tr(Σ −1 )Tr(Σ) 2 Tr(D −1 ).(34)Proof 3 See Appendix C.
Now, we investigate the case when a relation between Γ,Ṡ and Π can be expressed in the form of equality. If and only if A in (9)-(10) is a diagonal matrix, i.e. we have a set of linearly independent stochastic differential equations (this can be after applying decoupling transformations [32]), the following result holds.
Relation 4 Given a n-th order Gaussian process where all its random variables are independent, we have
Γ 2 := i D ii Σ ii Π i + iṠ 2 i ,(35)
where Π i andṠ i are the entropy production and entropy rate from the marginal PDF p(x i , t) of x i , respectively.
Proof 4 See Appendix D .
Again, the value of Π i in (35) is given by Equation (17) whileṠ i is given by (30). From (35), it is inferred that a first order system is a special case of Relation 4. In addition, Equation (35) tells us that the geodesic (length-minimising curve between the initial and final PDF) of L(t) can be computed utilising the entropy rate and entropy production values (for further details on the geodesic problem, see [33]). More importantly, since Relation 4 permits us to equate the effects of IL geodesic dynamics on the system stochastic thermodynamics, Equation (35) can be used as part of a cost function employed to design controls that lead to both high energetic efficiency and minimum variability system closed-loop responses. (for further details, see [3,4]).
On correlation coefficients
In this section, we define different types of correlations coefficients in terms of entropy production and information rate utilising mutual information and Pearson correlation coefficient, which will be used for abrupt event detection and causality analysis as discussed in Section 6.
Mutual information
The mutual information between two continuous random variables x i and x j with a joint PDF p(x; t) at time t is defined as
I ij (t) := R 2 p(x; t) ln Å p(x; t) p(x i , t)p(x j , t) ã d 2 x = S i (t) + S j (t) − S(t).(36)
Here, p(x i , t) and p(x j , t) are the marginal PDFs of the random variables x i and x j , respectively. Hence, the sub-index i in the entropy S refers to the entropy from the marginal PDF of x i and its value is simply
S i (t) = 1 2 + ln » 2πΣ ii (t) .(37)
Mutual information represents the amount of information of a random variable that can be obtained by observing another random variable. Hence, it is a measure of the mutual dependence between the two variables [27]. To measure correlations between two random variables in a process, we can utilise common normalised variants of the mutual information, for instance, the total correlation formula [34,35] ρ
I (t) := 2 I ij (t) S i (t) + S j (t) ,(38)
where x i and x j are treated symmetrically. Equation (38) is a weighted average of the asymmetrical uncertainty coefficients C ij (t) and C ji (t), defined as
C ij (t) := I ij (t) S i (t) , C ji (t) := I ij (t) S j (t) ,(39)
weighted by the entropy of each variable separately [36]. The uncertainty coefficient (39) gives a value between 0 and 1, indicating no association or complete predictability of x i from x j (given x j , what fraction of x i we can predict), respectively. Thus, (38) gives an average of the predictability between x i and x j . The total correlation formula (38) is as an alternative to the well-known Pearson correlation coefficient
ρ := Σ ij Σ ii Σ jj ,(40)
when dealing with non-linear relationships between the random variables [37,38,39]. Pearson correlation coefficient ρ and mutual information I are related to each other through the following expression Figure 1: Venn diagram describing the meaning of (42). A similar diagram can be made for (43).
I ij = − 1 2 log 1 − ρ 2 .(41)
Information rate and entropy production correlation coefficients
In analogy to (38) to (40), we define new normalised correlation coefficients between two variables x i and x j in terms of information rate and entropy production as follows
ρ Γ (t) := Γ i (t) + Γ j (t) − Γ(t) Γ(t) ,(42)ρ Π (t) := Π i (t) + Π j (t) − Π(t) Π(t) .(43)
Here, Π i and Π j are the contributions from the variable x i and x j to the entropy production Π (see Equation (D.2)). The values of Γ i and Γ j are the information rates from the marginal PDFs of x i and x j , respectively. For instance, given the marginal PDF p(x i , t) of the random variable x i the value of Γ i is defined as follows
Γ i := R p(x i ; t) [∂ τ ln p(x i , t)] 2 dx i .(44)
Equations (42)- (43) are not defined exactly as the Pearson correlation coefficient (40) or the normalised correlation coefficient of the mutual information (38). Instead, they are expressed analogously to the information quality ratio, a quantity of the amount of information of a variable based on another variable against total uncertainty [40]. Hence, ρ Γ /ρ Π is said to quantify the predictability of information rate/entropy production of a variable based on another variable. In other words, these coefficients give a weight of how much information one variable can add to the information rate or entropy production of another variable. Also, importantly, these quantities can work for any nonlinear relation. A graphical description of Equation (42) in the form of Venn diagram is shown in Figure 1.
Applications
So far, we have introduced a complete set of tools that establish a connection between information geometry and thermodynamics for linear systems. In this section, we apply these to the different linear stochastic toy models and scenarios. Firstly, we start by exploring a simple second order stochastic model §6.1 representing a harmonically bound particle. Secondly, we will explore a simple decoupled third order stochastic model §6.1.3 commonly used to represent the simplified dynamics of an optical trap in the three-dimensional space. Thirdly, we will investigate higher order models to elucidate the effects of system's dimension on entropy production §6.2. Fourthly, we will provide the analysis of abrupt events/perturbations in the dynamics of a second and fourth order stochastic system §6.3. Finally, we will explore two optimisation problems for the design of thermodynamic control algorithms §6.4.
6.1. Second and third order process 6.1.1.
Harmonically bound particle Consider the following Langevin equation describing a harmonically bound particle
ñẋ 1 (t) x 2 (t) ô = ñ 0 1 −ω 2 −γ ô ñ x 1 (t) x 2 (t) ô + ñ ξ 1 (t) ξ 2 (t) ô ,(45)
where the parameters ω and γ are related to the system's natural frequency and damping, respectively. First, to explore Relations 1 and 2, in Figure 2 we plot the changes on entropy ratė S computed by using equation (22) and compared them with the value of Π−Φ obtained from equations (20)- (21), confirming the expected relationṠ = Π−Φ. Second, to briefly demonstrate that E u > Γ 2 , we also show the difference between Γ 2 (using equation (26)) and As can be concluded from Figure 2a, for an undamped harmonic oscillator with γ = 0, the value of E u − Γ 2 tends to decrease with time, meaning that they become equal over time. Here, Γ,Ṡ > 0 ∀t ≥ 0 because the system is permanently oscillating. Once we increase γ, the system goes to the equilibrium giving Γ,Ṡ → 0 and E u ≥ 0 due to Π = Φ. In general, for any A whose eigenvalues have negative real part and u(t) = 0, entropy production Π and entropy flow Φ in the long-time limit take the following values
lim t→∞ Π(t)= Tr A D −1 AΣ(∞) + Σ −1 (∞)D + 2A , lim t→∞ Φ(t)= Tr A D −1 AΣ(∞) + A ,(46)
where
Σ(∞) = 2 lim t→∞ { t 0 e A(t−τ ) De A (t−τ ) dτ }.
As we see, the transient behaviour and longtime limit of entropy production and entropy flow are fully determined by the value of e At , which in turn (obviously) depends on the eigenvalues of the matrix A. In §6.4, we discuss how such eigenvalues can be modified through a control algorithm (for example, using a full-state feedback control method [41]). In system (45) 6.1.2. Correlation coefficients Using a bivariate PDF that satisfies (45), we will compare the different correlation coefficients defined in Section 5. To this end, Figure 3 shows the time evolution of ρ Γ , ρ Π , ρ I and ρ; the phase portraits of x 1 vs x 2 , Γ 1 vs Γ 2 and Π 1 vs Π 2 . The parameters used in this simulation are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, Σ(0) = 0.01I and D = 0.001I where I is the identity matrix.
To facilitate the analysis, the phase portrait shows time snapshots with the complete contour of the PDF where times are marked by vertical dashed lines. The time evolution of the Pearson correlation coefficient ρ and the total correlation coefficient ρ I may be the easiest coefficients to interpret since their values are related to the shape of the bivariate PDF. For instance, Figure 3a shows that ρ has negative sign because of the negative slope on the "linear" correlation between x 1 and x 2 . Similarly, the total correlation coefficient ρ I is negative with a maximum of −0.34 at some 0.68135 < t < 1.9323. Recall that an advantage of the mutual information coefficient ρ I over the Pearson coefficient ρ is that ρ I does not assume whether the association of the random variables is linear or not while ρ does. This causes ρ to be zero even when the variables are still stochastically dependent (for further details, see [37,38,39]).
Understanding the meaning of ρ Γ and ρ Π is a bit more complicated as its behaviour cannot be easily related to the shape of the PDF through time. This is because even though the values of ρ Γ and ρ Π clearly take into account the correlation effects through Σ, they are also sensitive to the mean values of x 1 and x 2 . Hence, they should be interpreted by comparing the time evolution of the pair of quantities whose correlation is studied. For instance, Figure 3b shows that ρ Γ > 0 in the time intervals where the time evolution of Γ 1 and Γ 2 have the same tendency or similar behaviour. ρ Γ reaches its maximum value when Γ 1 and Γ 2 are almost similar in amplitude and rate of change. The different behaviour in the time evolution of Γ 1 and Γ 2 at 0.26 t 2 depicts an anticorrelation which is corroborated by a value of ρ Γ < 0. Regarding the thermodynamic effects, Figure 3 shows that Π 1 and Π 2 are always anticorrelated as ρ Π < 0 at all t. The monotonically decrement of ρ Π represents a continously increasing loss of predictability of Π 2 by Π 1 .
Three-dimensional decoupled process
We now consider fully decoupled linear stochastic systems (i.e. where A is a diagonal matrix). A practical example of a three-dimensional linear decoupled process corresponds to the simplified version of the mathematical description of an optical trap shown in Figure 4. The model consists of a set of three independent overdamped Langevin equations [42] given by equation (47). Here, x and y represent the position of the particle in the plane perpendicular to the beam propagation direction and z represents the position of the particle along the propagation direction. The stiffnesses of the trap in each of these directions are κ x , κ y and κ z , respectively. γ is the particle friction coefficient. ξ 1 , ξ 2 and ξ 3 are independent delta-correlated noises, i.e. ξ i (t) = 0, ξ i (t)ξ i (t ) = 2D ii δ(t − t ) and ξ i (t)ξ j (t ) = 0 ∀i = j with i = 1, 2, 3.
ẋ (t) y(t) z(t) = − κx γ 0 0 0 − κy γ 0 0 0 − κz γ x(t) y(t) z(t) + ξ 1 (t) ξ 2 (t) ξ 3 (t) .(47)
Since (47) is a fully decoupled linear stochastic model, it permits us to show the applicability of Relation 4. To this end, in the left plot of Figure 5, we showṠ, Π − Φ, and Γ computed from equations (22), (20) minus (21), and (26)- (35), respectively. In Figure 5 we can see that both entropy rateṠ → 0 and information rate Γ → 0 as the system goes to equilibrium. The right plot of Figure 5, a plot depicting the value of the information rate upper bound E u (Relation 2) minus Γ 2 showing E u → 0 at equilibrium. The exact value of E u (t) in equilibrium is
lim t→∞ E u = Tr(Σ −1 (∞))Π(∞) Tr(D) − 2g(H(∞)) = 0,(48)
where
Σ −1 (∞) = −D −1 A, Π(∞) = 0. H(∞) = 0.(49)
Equation (49) applies only to systems with diagonal and stable A (proof at Appendix E). Since Π → 0, any decoupled linear system is reversible at equilibrium. where I is the identity matrix.
Higher order systems and the upper bounds of information rate Γ
As discussed in Section 4, it is possible to relateṠ, Π and Γ using the inequalities from Relations 2 and 3. These inequalities become highly relevant when the order of the stochastic models increase, for instance, when using toy models in control engineering scenarios [19]. In this section, we take the case when A is a randomly chosen Hurwitz matrix whose size varies from 2 − 50, i.e. we choose linear stochastic systems that contain from 2 to 50 random variables. Figure 6 shows the phase portrait of Γ(t f ) 2 vṡ S(t f ), and the phase portrait of E u (t f ) − Γ(t f ) 2 vs Π(t f ). Figure 6 is also separated in sub-figures 6a and 6b showing the cases when matrix A is diagonal and non diagonal, respectively. Note, t f refers to the time close enough the system's equilibrium; in our simulations t f = 300. The phase portraits contain numbers to indicate the value at t = t f ; the number also indicates the order of the stochastic system. Regarding the portraits of Γ(t f ) 2 vsṠ(t f ), for every Hurwitz A (i.e. diagonal and non diagonal matrix with negative real part eigenvalues) lim t→∞ E(t) = lim t→∞Ṡ (t) = 0 as expected. Meanwhile, for the same processes Π > 0 at equilibrium (see equation (46)). When looking at the phase portraits of E u (t f ) − Γ(t f ) 2 vs Π(t f ), we see that E u (t f ) > 0 when A is non diagonal due to Π(t f ) > 0 for some t (see Figure 6a). On the other hand, as demonstrated in Equations (48) and (49) E u (t f ) = Π(t f ) = 0. Again, meaning that every fully decoupled linear system is reversible at equilibrium.
Abrupt events analysis
When the temperature changes abruptly in a system like (45), the value of D (noise amplitude) is affected, contributing to the uncertainty in the control of the particle's position. To bring light to the analysis and study of abrupt events, we use our toy models and simulate an abrupt change in the system's temperature by using the following 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 impulse like function for the ii-element of the noise amplitude matrix D and on the input function u(t)
D ii (t) = D 0 + 1 |a| √ π e −( t−tp a ) 2 ,(50)u(t) = 1 |a| √ π e −( t−tp a ) 2 .(51)
Here, the second term on RHS of (50) and (51) takes a non-zero value for a short time interval around t p and a changes the amplitude of the impulses.
6.3.1. Harmonically bound particle Again, we start our analysis by considering system (45). . The rest of the simulation parameters are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, and Σ(0) = 0.01I where I is the identity matrix. 7, the coefficient ρ I is the most sensitive to noise amplitude perturbations, as it shows an abrupt change before the peak of the perturbation at t = 4 followed by a couple of extra abrupt changes at t ≥ 5. To explain the sensitiveness of ρ I , in 7a, we have included time evolution of S 1 + S 2 , S, Σ 22 , I 12 , S 1 and S 2 instead of a phase portrait as in 8a. The divergences in ρ I can be explained if we notice that the values of entropy (specially S 2 ) go from negative to positive values after the perturbation occurs. Let us recall that differential entropy S can be negative and it is understood as a relative privation of information. When negative, its value means we have less disorder/uncertainty or more information/order. The increment in temperature given by the perturbation increases the level of uncertainty in the system and thus entropy, then when it decreases entropy decreases as well. Such changes lead to S, S 1 +S 2 and I 12 to be zero provoking divergences in ρ I at different instants of time. Meanwhile, since it precedes the aforementioned perturbation (see Figure 7b), the value of Γ predicts the abrupt event (corroborating the previous results shown in [8]). Regarding the perturbation in u(t) shown in Figure 8, the coefficients ρ and ρ I are no longer useful because they are not sensitive to changes in the mean value of the PDF (see Figure 8a). In contrast, an abrupt event in the mean value is well captured by ρ Γ and ρ Π . Figures 8b and 8c show that the values of ρ Γ and ρ Π change abruptly at the time t = 4 when perturbation occurs. Figure 8b presents negative ρ Γ at t ≈ 4 due to the large difference between Γ 2 and Γ 1 . For the similar reasons, ρ Π also presents a high decrement at t ≈ 4. Here, the coefficients are able to detect the perturbation over the mean value but they are no longer able to predicted it.
Controllable canonical form
The analysis of abrupt events in high order systems can be done by an offline method, such as the application of the Euclidean norm to each marginal or joint information rate/entropy production of the random variables in the system. Recall that the Euclidean norm of any time dependant function ϑ(t) is defined as follows
ϑ(t) := Å t f 0 ϑ(τ ) 2 dτ ã 1 2 .(52)
As a demonstration, using Equation (52), here we study the effects of abrupt events in the noise amplitude matrix D(t) and the force input u(t) of the popular controllable canonical form of the state-space realization of a linear system given bẏ
x(t) = 0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 1 −d n −d n−1 −d n−2 · · · −d 1 x(t) + 0 . . . 1 u(t) + ξ(t).(53)
Here, x := [x 1 , x 2 , . . . , x n ] ∈ R n , u ∈ R and ξ := [ξ 1 , ξ 2 , . . . , ξ n ] ∈ R n is a vector of random variables with ξ i (t) = 0, and ξ i (t)ξ j (t ) = 2D ij δ(t − t ). Model (53) provides us with a structure where the input enters a chain of integrators making it to move every state in the Langevin equation (i.e. they are fully controllable). We consider the case when (53) . The rest of the simulation parameters are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, and Σ(0) = 0.01I where I is the identity matrix. Figure 9a depicts the time evolution of Γ i , Π i andṠ i ∀i = 1, 2, 3, 4. It also includes the time evolution of the three dimensional space (Γ, Π,Ṡ). Figure 9b shows the norms of Γ i , Π i ,Ṡ i ∀i = 1, 2, 3, 4 and Γ, Π,Ṡ in the form of a spider plot. For instance, the value of the norm of the information rate Γ computed from the joint PDF p(x; t) and from the marginal PDF p(x 2 , t) is ||Γ|| ≈ 10000 and ||Γ 2 || ≈ 31.6, respectively. As we can see, the effects on Γ i , Π i andṠ i ∀i = 1, 2, 3, 4 by the random variables is hierarchical with regards to their amplitude (for example |Ṡ 4 | > |Ṡ 3 | > |Ṡ 2 | > |Ṡ 1 | at almost all the time) and the equilibrium of (Γ, Π,Ṡ) is (0, 0, 0). When we add a perturbation in u which affects directly to x 4 (see Equation (53)), we obtain the results shown in Figure 10. Such an abrupt event causes an notable increment in the norms of the states (See Figure 10b) which still maintains the same hierarchical order in the states (x 4 is the most affected in comparison with x 1 ) due to the system's structure as expected. The direct effect of the abrupt event on each variable's time evolution is shown in Figure 10a. Recall that u is applied directly to x 4 . Again,Ṡ is unperturbed since the event affects only the mean value of the PDF. On the other hand, if we separately include a 11 illustrates the norms of Γ, Π,Ṡ in the form of bar plots (after applying perturbations to each D ii ). The plots indicate a domino effect in the marginal PDFs as follow. When only D 11 is perturbed no clear effects can be seen in the rest of the variables but when D 33 is perturbed x 3 , x 2 and x 1 increase their values. Same happens after perturbing D 44 , again this is due to the structure interaction of the system we are studying. This implies that norms provide an approximate value of the dependence between the variables of the random process. Figure 11: Abrupt event analysis using the norms of Π,Ṡ, Γ at the marginal PDF p(x i , t) and the joint PDF p(x; t). Each plot depicts a perturbation on a given D ii according to (50) at t p = 6 with a = 0.1 and D 0 = 0.01 in system (53). The rest of the simulation parameters are µ(0) = [0, 0, 0, 1] and Σ(0) = 0.1I where I is the identity matrix.
Minimum variability control design
The design of "efficient" machines remains still a common practical problem in different research areas. Note that here, efficiency can be related to the existence of minimum variability or entropy production as it was highlighted in previous works [5,33,43,44]. The generation of efficient processes, as we describe here, can be accomplished by designing optimal protocols through work and heat minimisation [45], the Wasserstein distance [46] or inverse engineering (for a complete review, see [47]) to drive our system while being subject to thermodynamic constraints. In this context, as a final application, here we explore the use of IL to design a classical control technique in a given linear stochastic process. From the field of control engineering, we will take the full-state feedback controller given by
u(t) = −kx(t),(54)
where k ∈ R 1×n . Through this control, we obtain the following closed-loop systeṁ
x(t) = A cl x(t) + ξ(t),(55)
where A cl = A − Bk. The full-state feedback control permits us to manipulate the system's mean value via changing the eigenvalues of A. As discussed previously, such eigenvalues also modify the time evolution of Σ. In systems like (45), the value of Σ can as well be manipulated by the temperature of the environment whose value is related to the elements D 11 and D 22 of the noise amplitude matrix D.
Taking the aforementioned details into consideration, we propose the following optimisation problems for the design of minimum variability controls min k,D
J 1 = t f 0 Γ(τ )dτ, s.t.μ = A cl μ Σ = A cl Σ + ΣA cl + 2D µ(0) = m, Σ(0) = S k l,i ≤ k i ≤ k u,i , 0 ≤ D ii ≤ D max ∀i = 1, 2, . . . , n,(56)
and min k,D
J 2 = ||Γ(t) 2 − Γ(0) 2 ||, s.t.μ = A cl μ Σ = A cl Σ + ΣA cl + 2D µ(0) = m, Σ(0) = S k l,i ≤ k i ≤ k u,i , 0 < D ii ≤ D max ∀i = 1, 2, . . . , n.(57)
In Equation (56), J 1 is a cost function that considers the minimisation of IL from t = 0 to t = t f to obtain the "minimum" statistical changes in the given period of time. On the other hand, Equation (57) considers a cost function J 2 equal to the norm of Γ(t) 2 − Γ(0) 2 . The objective of J 2 is to keep Γ 2 constant through time (with the least amount of fluctuations) to approximately follow the "geodesic", a problem well described in [33]. Both optimisation problems are subject to the dynamics of the mean and covariance of the PDF given certain initial conditions for them. The problems also consider upper and lower limits to k i and D ii ∀i = 1, 2, . . . , n given by k l,i , k u,i and 0, D max , respectively. Note D ii ≥ 0 because the temperature cannot be negative. The values of k l,i and k u,i are determined such that the following stability condition is satisfied
|sI − A cl | = 0 ∀s ∈ C s.t. s > 0.(58)
Using ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, Σ 11 = Σ 22 = 0.01, Σ 12 = Σ 21 = 0,
D max = ∞, k l,1 = −1, k l,2 = −2, k u,1 = k u,2 = ∞ and u(t) = −[k 1 k 2 ] ñ x 1 x 2 ô
in system (45), we have explored the solution of Equations (56) and (57) via the MATLAB Toolbox FMINCON [48]. The solutions give us the set values of k and D that give at least a local minimum. Note that our goal here is to see the implications of a solution to such problems instead of rigorously finding the global optimal solution. Figure . This values produce an abrupt change in Γ, a quasi-logarithmic change in L with a maximum value slightly over 5 and a slow almost critically damped change in the system dynamics towards the equilibrium. In addition, the control quickly drives Π andṠ to zero. Even though the control action imposes a slow evolution of the mean value, the information rate quickly decreases. Such behaviour is desirable for systems where minimum information variability is more important than the speed under which we reach the equilibrium. The solution of the optimisation problem (57) depicted in Figure 13 shows that a geodesic solution is obtained when entropy production Π and entropy rateṠ are zero. This is imposed by the resultant control parameters k 1 = −0.8431, k 2 = −2, D 11 = 0 and D 22 = 0 which generate a harmonic oscillatory behaviour of the mean value µ = [µ 1 , µ 2 ] and small changes in the time evolution of the covariance matrix elements Σ 11 , Σ 12 and Σ 22 to keep Γ constant at all t. Note that, D 11 = D 22 = 0 just means the absence of any external stochastic noise. However, there is stochasticity in the system due to the stochasticity in the initial condition (that is, our initial PDFs have a finite width). This optimal result is similar to the one shown in [49] where authors find that the optimal energy landscape corresponds to an harmonic oscillator at all times.
Conclusions
We have derived relations between information rate, entropy production and entropy rate for linear stochastic Gaussian processes changing according to the structure of the harmonic potential. Given such information-thermodynamic connection, we explore the applicability of these quantities to the design of thermodynamic optimal control algorithms and the detection of abrupt events in models of different nature. The results demonstrate that information rate ρ Γ and entropy production ρ Π correlation coefficients predict and detect abrupt events in the first and second moments of the stochastic dynamics, respectively. For higher-order systems, the norm of the information/thermodynamic quantities represents a fair approximation of the correlation between all the system random variables. Besides, the control applications show that it is possible to obtain the geodesic of the information length via a simple full-state feedback control algorithm.
Since the proposed results permit us to establish a clear connection between information geometry and thermodynamic quantities, we can undoubtedly create cost functions that lead to energetically efficient (minimum entropy production) and organised (with minimum information variability) behaviours through IL. In the future, we plan to explore further connections with the area of control theory for applications in more complex scenarios. For instance, we can extend our results to non-linear systems by employing approximation methods such as the Laplacian assumption [52,53,50]. Then, we can utilise modern control techniques such as the model-predictive-control [54,51] to find the solution to the proposed optimisation problems J 1 and J 2 . This could bring potential benefits in the research areas of population dynamics [55] or inference control [56,57].
Appendix A. Derivation of Entropy rate for Gaussian Process
We start by applying the definition of entropy (14) production and entropy flux (16) giving us
Π J i = 1 D ii f i (x, t) 2 +D ii AE Å ∂Q(x) ∂x i ã 2 ∏ +2 ≠ ∂f i (x, t) ∂x i ∑ , (A.1) Φ J i = 1 D ii f i (x, t) 2 + ≠ ∂f i (x, t) ∂x i ∑ . (A.2)
Before continuing, it is useful to note that [58] ∂Q
∂x k = − 1 2 i δx i Σ −1 ki + j δx j Σ −1 jk = − i δx i Σ −1 ki = −δx Σ −1 k (A.3) where δx i = x i − µ i , δx := x − µ = [δx 1 , . . . , δx n ] and Σ −1 k is the k-th column of the inverse matrix Σ −1 of Σ. Besides, f i (x) 2 =x A i A i x + u B i B i u + 2u B i A i x, (A.4)
where we recall that A i is the i-th arrow of the matrix A. Therefore [58] AE
D ii Å ∂Q(x) ∂x i ã 2 ∏ = D ii δx Σ −1 i (Σ −1 i ) δx = D ii Tr(∆ i Σ), (A.5)
and
f i (x) 2 D ii = 1 D ii Tr(Γ i Σ) + µ Γ i µ + u Ω i u + 2u ϕ i µ (A.6) where ∆ i = Σ −1 i (Σ −1 i ) , Γ i = A i A i , Ω i = B i B i , and ϕ i = B i A i . Furthermore, we have that ∂f i (x) ∂x i = a ii .
Then
Π J i = 1 D ii Tr(Γ i Σ) + µ Γ i µ + u Ω i u + 2u ϕ i µ + D ii Tr(∆ i Σ)+2a ii , Φ J i = 1 D ii Tr(Γ i Σ) + µ Γ i µ + u Ω i u + 2u ϕ i µ + a ii . (A.7)
Finally, since
n i µ Γ i µ D ii = µ Å A 1 A 1 D 11 + · · · + A n A n D nn ã µ = µ A D −1 Aµ, (A.8) n i Tr(∆ i Σ) D ii = Tr ÇÇ Σ −1 1 (Σ −1 1 ) D 11 + · · · + Σ −1 n (Σ −1 n ) D nn å Σ å = Tr Σ −1 D −1 (Σ −1 ) Σ . (A.9)
Now, after applying the same reasoning to all the terms in the right hand side of (A.7) and simplifying, we get Since the stochastic simulation uses its solutions to compute the averages f i and (∂ x i Q) 2 in (A.1). Figure A1 includes a comparison of the analytical vs numerical values of f 1 , f 2 , (∂ x 1 Q) 2 , (∂ x 2 Q) 2 and Π showing a perfect agreement between both approaches as expected.
Π =μ D −
Appendix B. Proof of Relation 2
For any real matrix A in system (9)-(10), we can rewrite the second term in the right hand side of (26) as follows Now, sinceṠ = n iṠ J i where againṠ J i is the contribution of the current flow J i to the total entropy rateṠ, we see that each eigenvalue λ i =Ṡ J i . Note that, for simplicity, we have changed the argument of the function g from H to a new variable vector s = [Ṡ J 1 ,Ṡ J 2 , . . . ,Ṡ Jn ] , i.e. now g : R n → R. This ends our proof.
Appendix C. Proof Relation 3
To derive the result shown in Relation 2, we first consider the following preliminary results [60,61,62] Tr From the right hand side of (C.6), we define the part inside the curly brackets as Which gives us the expression in our result (29). The value of Π u can be proved to be an upper bound of Π from the following reasoning From (C.8), the main result follows straightforwardly using (C.6) leading to our main result in Relation 2.
Appendix D. Proof of Relation 4
If A is an n × n diagonal matrix, then Σ andΣ are also diagonal and the following expressions hold
Γ 2 = i Γ 2 i = i μ 2 i Σ ii + 1 2 ÇΣ ii Σ ii å 2 , (D.1) Π = i Π i =μ D −1μ + 1 4 Tr Ä Σ −1Σ 2 D −1 ä = i Çμ 2 i D ii +Σ 2 ii 4Σ ii D ii å , (D.2) S = iṠ i = 1 2 iΣ ii Σ ii . (D.3)
By rearranging Equations (D.2) and (D.3) to form Γ 2 , we have
Γ 2 i = D ii Σ ii Π i +Ṡ 2 i . (D.4)
which leads to our result.
Appendix E. Proof of equation (49) Given that A is a diagonal and stable matrix, we havė In this manner, the entropy production and entropy rate are
Π(∞) = i Çμ i (∞) 2 D ii +Σ ii (∞) 2 4Σ ii (∞)D ii
E u (from Relation 2). Our simulations were done for fixed value of ω by varying the value of γ (Figure 2a) and vice-versa (Figure 2b).
Figure 2 :
2Computational experiment of a second order Langevin equation using D 11 = D 22 = 0.01, x(0) = 1, y(0) = 1, Σ 0 11 = Σ 0 22 = 0.1 and Σ 0 12 = Σ 0 21 = 0.
, the bigger the value of γ the quicker we arrive to equilibrium. On the other hand, increasing the value of ω with γ > 0 increments the oscillations on the transitory response (seeFigures 2a and 2b)[8,25].
Figure 3 :
3Time evolution of ρ Γ , ρ Π , ρ I , ρ, Γ 1 , Γ 2 , Π 1 and Π 2 in comparison with the evolution of the snapshots of the phase portraits of x 1 vs x 2 . The parameters used in this simulation are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, Σ(0) = 0.01I and D = 0.001I where I is the identity matrix.
Figure 4 :
4Particle of mass m in a three dimensional optical trap.
Figure 5 :
5Computational experiment of a three-dimensional optical trap using κ x = 10, κ y = 3, κ z = 1, γ = 1, x(0) = 1, y(0) = 0.1, z(0) = 0.5, Σ = 0.1I and D = 0.01I
) A is non diagonal.
Figure 6 :
6Entropy rateṠ(t f ) vs the square of information rate Γ(t f ) 2 and the values of E u (t f ) − Γ(t f ) 2 vs Π(t f ). The simulations use randomly chosen stable linear systems from order n = 2 to n = 50. The red numbers indicate the order of the system and its position the value at t f = 300. In all simulations the system starts out of the equilibirum with initial conditions µ = [1, 1, . . . , 1] , Σ = 0.1I and D = 0.01I where I is the identity matrix varying from size 2 to 50.
Figure 7 :
7Figures 7 and 8 show the computer simulation results and time evolution of abrupt events in D and u, respectively. The noise amplitude is perturbed via the element D 22 of the matrix D and the input force u only affects the state x 2 . Figures 7 and 8 are divided in three panels, 7a/8a which includes the time evolution of the correlation coefficients ρ and ρ I (Figure 8a also includes the phase portrait of x 1 vs x 2 ); 7b/8b shows the time evolution of ρ Γ , Γ, Γ 1 and Γ 2 ; 7c/8c the time evolution of ρ Π , Π, Π 1 and Π 2 . From Figure Abruptevent numerical experiment u(t) = 0, D 11 (t) = 0.001 and D 22 (
is of 4th order. The values of the parameters are d 4 = [d 1 , . . . , d n ] = [−1.5165, −5.2614, −6.7985, −4.2206] . InFigures 9 to 11, we use the notation x i ∀i = 1, 2, 3, 4 and x to refer to the values of Π,Ṡ and Γ computed from marginal PDF p(x i , t) and from the joint PDF p(x; t), respectively.
Figure 8 :
8Abrupt event numerical experiment D 22 (t) = D 11 (t) = 0.001 and u(t)
Figure 9 :
9Behaviour with no abrupt event D ii (t) = D 0 = 0.01, u(t) = 0. The rest of the simulation parameters are µ(0) = [0, 0, 0, 1] and Σ(0) = 0.1I where I is the identity matrix.
Figure 10 :
10Behaviour with abrupt event D ii (t) = D 0 = 0.01, u(t) = 1 |0.1| √ π e −( t−tp 0.1 )
Figure 12 :
12Full-state feedback control and temperature setting minimising J 1 . A local minima is at k = [2.1229, 4.4453] and D = [0.2684, 0; 0, 2.1181]. The system parameters and initial conditions are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, Σ 11 = Σ 22 = 0.01, Σ 12 = Σ 21 = 0. time evolution of Γ, L, Π, x 1 , x 2 and the spaces (p(x, t), x 1 , x 2 ) and (x 1 , x 2 ) after applying the values of k and D that give a solution to the optimisation problem (56). As a result, the value of k contains k 1 = 2.1229 and k 2 = 4.4453 and D, contains D 11 = 0.2684 and D 22 = 2.1181
Figure 13 :
13Full-state feedback control and temperature setting minimising J 2 . A local minima is at k = [−0.8431, −2] and D = [0, 0; 0, 0]. The system parameters and initial conditions are ω = 1, γ = 2, µ 1 (0) = 0.5, µ 2 (0) = 0.7, Σ 11 = Σ 22 = 0.01, Σ 12 = Σ 21 = 0.
1μ +Tr A D −1 AΣ + Tr Σ −1 D +2 Tr(A), (A.10) Φ =μ D −1μ +Tr A D −1 AΣ + Tr(A), (A.11) S = Tr Σ −1 D +Tr(to the result given in Relation 1. To provide extra confidence in the validity of expressions (A.10)-(A.12). Consider as an example, the stochastic simulation [59], i.e. the multiple numerical solution of the stochastic differential equation, of (45) using the parameters µ(0) = [1, 0.5] , a 11 = 0, a 12 = 1, a 21 = −2, a 22 = −3, D = 0.01I and Σ(0) = 0.1I to compute the value of entropy production Π via (A.1). Then, let us compare it with the "analytical" solution obtained using (A.10) and the solution of equations (9)-(10).
Figure A1 :
A1AD + 4Σ −1 DΣ −1 D = Tr(2A 2 + 4Σ −1 AD + 2Σ −1 DΣ −1 D) + Tr(2Σ −1 AΣA + 4Σ −1 AD + 2Σ −1 DΣ −1 D)= 2 Tr (Σ −1 D + A) 2 + 2 Tr(Σ −1 (AΣA D −1 +2A+DΣ −1 )D).Equation (B.1) can be written in terms of Entropy production Π and entropy rateṠ using the following results. First, from the fact that Π ≥ 0 and Σ −1 , D 0 we getTr(Σ −1 )Π Tr(D) = Tr(Σ −1 ) Tr μμ D −1 + AΣA D −1 + 2A + DΣ −1 Tr(D) ≥ Tr(Σ −1μμ ) + Tr(Σ −1 (AΣA D −1 +2A+DΣ −1 )D). (B.2)Now, taking λ i as the eigenvalues of the matrix H := Σ −1 D + A, we havėS 2 = Tr(H) 2 = ( i λ j = Tr(H 2 )+2 n i<j λ i λ j = Tr(H 2 )+2g(H). Stochastic simulation (numerical) vs analytical solution of the entropy production Π for (45) using the parameters µ(0) = [1, 0.5] , a 11 = 0, a 12 = 1, a 21 = −2, a 22 = −3, D = 0.01I to produce Σ(0) = 0.1I. The results show an agreement between the analytical and numerical results as expected. Finally, using (B.1)-(B.3) in (26) we get 2g(H) ≤ Γ 2 + 2g(H) ≤ Tr(Σ −1 )Π Tr(D) + Tr(H 2 ) + 2g(H), 2g(H) ≤ Γ 2 + 2g(H) ≤ Tr(Σ −1 )Π Tr(D) +Ṡ 2 , 0 ≤ Γ 2 ≤ Tr(Σ −1 )Π Tr(D) +Ṡ 2 − 2g(H). (B.4)
) 2 ) 2 − 1 ) 2 ≤− 1 )
22121Tr(Σ) 2 = Tr(Σ −1 ) Tr(Σ) Tr(Σ −1 ) Tr(Σ)≥ Tr(Σ −1Σ ) 2 ≥ Tr((Σ −1Σ ) 2 ) (C.4)Then, by applying the previous results to the definition of Γ 2 in(26), we have0 ≤ Γ 2 = Tr(μ Σ −1μ Tr(Σ) 2 +Ṡ 2 .Now, multiplying both sides of inequality (C.5) by Tr(D −1 D) and factorising Tr(Σ −1 ) from its right hand side, we have 0 ≤ nΓ 2 ≤ Tr(Σ −1 ){Tr(μμ ) Tr(D Tr(Σ) 2 Tr(D −1 D)} + nṠ Tr(Σ −1 ){Tr(μμ ) Tr(D Tr(Σ) 2 Tr(D −1 )} Tr(D) + nṠ 2 .
Π u ≥− 1 )
u1Tr(μ D −1μ ) + Tr(Σ −1 ) Tr(ΣAΣA +2ΣAD+D 2 ) Tr(D −1 ) ≥μ D −1μ +tr Σ −1 Σ(AΣA D −1 +2A+Σ −1 D)DD −1 =μ D −1μ + tr(AΣA D −1 + 2A + Σ −1 D).(C.8) Note that for Π u ≥ Π we need A 0. A similar result can be found starting from the definition of Π u in (C.7) as follows Tr(Σ −1 )Π u Tr(D) ≥ Tr(Σ −1 ){Tr(μμ ) Tr(D −1 D) Tr(Σ) 2 Tr(D −1 D)}. (C.9)
the result mentioned in equation(49).Σ −1 (∞) = −D −1 A, Π(∞) = 0. s(∞) = 0. (E.9)
with t p = 4. The rest of the simulation parameters are µ(0) = [0, 0, 0, 1] and Σ(0) = 0.1I where I is the identity matrix. perturbation in each element D ii of the noise matrix D, similar results occur.Figure
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|
[] |
[
"Static chaos in spin glasses against quenched disorder perturbations",
"Static chaos in spin glasses against quenched disorder perturbations",
"Static chaos in spin glasses against quenched disorder perturbations",
"Static chaos in spin glasses against quenched disorder perturbations"
] |
[
"Vicente Azcoiti [email protected] \nDepartamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain\n\nUniversidad Carlos III\nButarque 1528911Leganes, Madrid (Spain\n",
"Eduardo Follana [email protected] \nDepartamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain\n\nUniversidad Carlos III\nButarque 1528911Leganes, Madrid (Spain\n",
"Felix Ritort [email protected] ",
") ",
"Vicente Azcoiti [email protected] \nDepartamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain\n\nUniversidad Carlos III\nButarque 1528911Leganes, Madrid (Spain\n",
"Eduardo Follana [email protected] \nDepartamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain\n\nUniversidad Carlos III\nButarque 1528911Leganes, Madrid (Spain\n",
"Felix Ritort [email protected] ",
") "
] |
[
"Departamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain",
"Universidad Carlos III\nButarque 1528911Leganes, Madrid (Spain",
"Departamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain",
"Universidad Carlos III\nButarque 1528911Leganes, Madrid (Spain",
"Departamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain",
"Universidad Carlos III\nButarque 1528911Leganes, Madrid (Spain",
"Departamento de Fisica Teorica\nFacultad de Ciencias\nDepartamento de Matematica Aplicada\nUniversidad de Zaragoza\n50009ZaragozaSpain",
"Universidad Carlos III\nButarque 1528911Leganes, Madrid (Spain"
] |
[] |
We study the chaotic nature of spin glasses against perturbations of the realization of the quenched disorder. This type of perturbation modifies the energy landscape of the system without adding extensive energy. We exactly solve the mean-field case, which displays a very similar chaos to that observed under magnetic field perturbations, and discuss the possible extension of these results to the case of short-ranged models. It appears that dimension four plays the role of a specific critical dimension where mean-field theory is valid. We present numerical simulation results which support our main conclusions.MA/UC3M/03/95 DFTUZ/95/04 cond-mat/9502045
|
10.1088/0305-4470/28/14/008
|
[
"https://arxiv.org/pdf/cond-mat/9502045v1.pdf"
] | 2,826,775 |
cond-mat/9502045
|
164ca121cd5a4f371750f9ac38181bf4797bf51c
|
Static chaos in spin glasses against quenched disorder perturbations
arXiv:cond-mat/9502045v1 11 Feb 1995 May 24, 2018
Vicente Azcoiti [email protected]
Departamento de Fisica Teorica
Facultad de Ciencias
Departamento de Matematica Aplicada
Universidad de Zaragoza
50009ZaragozaSpain
Universidad Carlos III
Butarque 1528911Leganes, Madrid (Spain
Eduardo Follana [email protected]
Departamento de Fisica Teorica
Facultad de Ciencias
Departamento de Matematica Aplicada
Universidad de Zaragoza
50009ZaragozaSpain
Universidad Carlos III
Butarque 1528911Leganes, Madrid (Spain
Felix Ritort [email protected]
)
Static chaos in spin glasses against quenched disorder perturbations
arXiv:cond-mat/9502045v1 11 Feb 1995 May 24, 2018Short title: Static chaos in spin glasses PACS. 75.24 M-Numerical simulation studies. PACS. 75.50 L-Spin glasses.
We study the chaotic nature of spin glasses against perturbations of the realization of the quenched disorder. This type of perturbation modifies the energy landscape of the system without adding extensive energy. We exactly solve the mean-field case, which displays a very similar chaos to that observed under magnetic field perturbations, and discuss the possible extension of these results to the case of short-ranged models. It appears that dimension four plays the role of a specific critical dimension where mean-field theory is valid. We present numerical simulation results which support our main conclusions.MA/UC3M/03/95 DFTUZ/95/04 cond-mat/9502045
Introduction
A long debated problem in spin glass theory concerns the correct description of the statics of the low temperature phase [1]. There is wide consensus on the fact that the mean-field theory is well understood in its essentials, while the nature of the equilibrium states for short-ranged models is still a controversial subject. Two compeeting pictures or approaches have been proposed: the mean-field picture and the droplet model. The mean-field theory has revealed enormously complex as a comprehensive approach to understand short range models. Consequently, the search for different approaches like droplet models [2] has been encouraged. These models, being phenomenological, try to capture the main aspects underlying the equilibrium and non-equilibrium properties of short-ranged models. Unfortunately, the mean-field way and these phenomenological approaches are far from being complementary and much effort has been devoted during the past years to discern what is the correct picture. Numerical simulations have played a prominent role in this task even though the main question still remains unsolved. The main problem relies on the large amount of computer time needed in order to reach the equilibrium.
Despite of the fact that both pictures are in fact contradictory in their essentials, there are however some common predictions in both approaches. Since it is very difficult to decide what is the correct picture, the strategy of searching for common features in both pictures can be useful to shed light on this controversy. Static chaos appears as a good starting point for this program. By static chaos we understand the sensitivity of the low temperature phase of spin glasses against static perturbations, like changes in the temperature or changes in the magnetic field. Mean-field theory [3] and droplet models [2] predict that spin-glasses, in the most general case, are chaotic. In mean-field theory, the mechanism of chaos is due to the small free energy differences between the different equilibrium states. These are of order O(1/N) and a small perturbation completely reshuffles the Boltzmann weights w α ∼ exp(−Nβf α ) of the different equilibrium states (α and f α stand for equilibrium state and its free energy respectively). In droplet models, the application of a perturbation causes a reorganization of the spin-spin correlations at long distances. In both pictures the system is much sensitive to the applied perturbations.
A nice example of chaos concerns the sensitivity of spin-glasses against magnetic field perturbations [3,4]. The chaos exponent (to be defined in the next section) for this type of perturbation has been computed in mean-field theory [3] and numerically measured in short-ranged models [4]. Surprisingly, this chaos exponent does not depend on the dimensionality of the system [4]. Even though we do not know a theoretical derivation of this result, it appears to be enough sound in order to be considered. Droplet models can give an explanation for this result under the assumption that 3 is the lower critical dimension in Ising spin glasses (also a long debated problem [5]). In the context of droplet models, the chaos exponent for magnetic field perturbations is related to the thermal exponent θ which measures the free energy cost of the droplet excitations. The result θ = d−3 2 implies that the chaos exponent is 2/3 and does not depend on the dimension. This has to be compared to the known results, θ = −1(exact) in d = 1 [6], θ ≃ −0.48 in d = 2 [7] and the exact result for the chaos exponent (2/3) in the Gaussian approximation to mean-field theory [3] Apparently the simple expression previously reported for θ correctly matches the small d regime to the infinite dimension result.
Regarding other type of perturbations, the situation is less clear. For instance in the case of temperature changes, it remains unclear how much chaotic is the system (see [8] for recent results). If chaos exists then it is certainly small and the possibility that chaos is marginal [3] cannot be excluded. Numerical results in the case of short-range models [9] show that chaos in temperature is also very small, as in the mean-field case.
The perturbations previously commented share the common property that they add energy to the system. This work is devoted to the study of a perturbation which does not add extensive energy to the system. In particular we will study chaoticity against changes of the realization of the quenched disorder. Because of the self-averaging property we expect that changes in the realization of the disorder (keeping the form of the disorder distribution) shoud not add extensive energy to the system. Therefore chaoticity appears because of a complete reshuffling of the free energies of the configurations. We will show that the system displays chaos very similarly as for the case of magnetic field perturbations. Criticality of chaos against perturbations of the quenched disorder has been studied by other groups [10]. The perturbation we are interested in differs from others by the fact that we change the sample realization without moving the system to a new point in the phase diagram.
The paper is organized as follows. Section 2 is devoted to the study of chaos in meanfield models. In section 3 we discuss on the results for the short-ranged models. Section 4 presents the scaling approach we have used to obtain the chaos exponents and shows the numerical results. Finally we present our conclusions in section 5.
Chaos in mean-field theory
We consider the models described by the Hamiltonian
H[σ] = − (i,j) J ij σ i σ j − h i σ i(1)
The couplings in (1) are Gaussian symetrically distributed random variables with zero mean and 1/N variance, where N is the number of spins. The perturbation we consider consists in changing randomly the sign of a fraction r of the couplings, i.e. for each coupling we change its sign with probability r. On average, a total number Nr of the couplings J ij are changed to −J ij . In this way we keep the new configuration of J ′ s in the same ensemble of disorder realizations without moving the system in the phase diagram. This is different from other type of perturbations in which, for instance (see [10]), the J ij are changed by a small amount δ · x ij where x ij is a random number and δ is small. In this case the variance of the distribution is increased (it grows proportionally to δ 2 ) and we add energy to the system. In what follows we will consider, for simplicity, the case of zero magnetic field.
Denoting by R the set of couplings which change sign, then we can write the perturbed Hamiltonian as,
H r [σ] = − (i,j) J ij σ i σ j + 2 (i,j)∈R J ij σ i σ j(2)
The sum runs over nearest neighbors in a lattice of dimension d. The mean-field case can be obtained in several ways. In the infinite-range model or SK model, all the spins interact one to the other. Alternatively, one can consider the finite connectivity random lattices (with fixed or average number of neighbors [11]
H 12 [σ, τ ] = H[σ] + H r [τ ](3)
We define the usual spin-glass correlation functions,
G(x) = σ 0 τ 0 σ x τ x(4)
where (..) means average over the quenched disorder and .. corresponds to the thermal average over the full Hamiltonian H 12 . The degree of coherence of the two systems is measured by the overlap function,
P (q) = δ(q − 1 N i σ i τ i ) (5) At large distances G(x) behaves like G(x) ∼ exp(−x/ξ(r))(6)
where ξ(r) is the chaos correlation length, which is finite for a finite perturbation r (we identify the perturbation with the fraction r of changed couplings). The chaos correlation length diverges when r → 0 if the unperturbed system stays in the spin-glass phase, including the critical point. This is because in the limit r → 0, G(x) converges to the usual spin-glass correlation function which has infrared singularities due to the existence of zero modes. The chaos correlation length diverges like,
ξ(r) ∼ r −λ (7)
where λ is the chaos exponent which can be exactly computed in some particular cases. From this definition it is clear that static chaos is absent when the system stays at a temperature above the spin-glass transition (the paramagnetic phase). But this assertion is true only if ξ(r) smoothly converges to the finite correlation length at that temperature when r → 0. This is the case in mean-field theory but should not be necessarily true in finite dimensions [12]. The exponent λ can also depend in principle on the temperature. We will show in mean-field theory that λ is constant in the low T phase. Furthermore, at the critical point, we expect λ to depend on the critical exponents, even though this is not always the case 1 . Now we face the problem of computing the exponent λ in mean-field theory. We follow the standard procedures (see [4] for details) and we apply the replica method to the full Hamiltonian H 12 eq.(3),
βf = lim n→0 log(Z n J ) nN(8)
Introducing Lagrange multipliers for the different order parameters one gets a saddle point integral
Z n J = dP dQ dR exp(−NA[P QR]) (9) with A[P QR] = β 2 2 a<b (P 2 ab + Q 2 ab ) + β 2 2 a,b (R 2 ab ) − log(T r στ exp(L))(10)
where a, b denote replica indices which run from 1 to n, and
L[σ, τ ] = β 2 a<b (Q ab σ a σ b + P ab τ a τ b ) + β 2 √ 1 − 2r a,b (R ab σ a τ b )(11)
There is one stable solution to the equations of motion,
Q ab = P ab = Q SK ab R ab = 0(12)
where Q SK ab is the solution for the unperturbed system. The order parameter R ab measures the degree of correlation eq.(5) between the two systems via the relation,
P (q) = 1 n 2 a,b δ(q − R ab )(13)
The stability of the solution R = 0 means that there is chaos against coupling perturbations. This is indeed very similar to the case of chaos in a magnetic field. Now we can compute, in the Gaussian approximation, the correlation function G(x) of eq.(6). The computations can be easily done in Fourier space. We define,
C(p) = x G(x)e ipx(14)
In order to find C(p) we need to compute the spectrum of fluctuations in the direction R ab around the stable solution (eq. (12)). The full expression has been reported in [3]. Its singular part is given by
C(p) = qmax 0 dq qmax 0 dQ p 2 + 1 + α(q)α(Q) (p 2 + 1 − (1 − 2r)α(q)α(Q)) 3(15)
with
α(q) = β(1 − q max + qmax q dq x(q))(16)
where β is the inverse temperature and q(x) is the order parameter function associated to the spin-glass. This expression yields the singular behavior of the correlation function in the spin-glass phase [4],
C(p) ∼ p −4 , p → 0(17)
The chaos correlation length ξ(r) is given by the minimum eigenvalue of the stability matrix,
λ min = 2β 2 r(18)
This yields,
ξ(r) ∼ λ − 1 2 min ∼ r − 1 2 (19)
This result is valid at and below the critical point. We expect it to be valid also in other mean-field models like, for instance, finite connectivity random lattices. In this case, where analytical calculations become much more involved, we expect to obatin the same results. This will be nicely corroborated by our numerical simulations in section 4.
Chaos in short-ranged systems
Now we face the problem of extending our results to finite dimensions. This is a non trivial task and we will present a derivation only for the one-dimensional case.
The chaos exponent can be exactly computed in one dimension since we know how to construct the ground state in this case. The exponent λ is a zero-temperature exponent because there is no phase transition at finite T . The Hamiltonian reads,
H = − i J i σ i σ i+1(20)
The perturbation consists in changing the sign of a fraction r of the couplings in a random way, and we assume that the J ′ s are distributed around J = 0 with a finite weight at J = 0 (this is essential for the scaling arguments). When a fraction of the couplings is changed, the new ground state is constructed inverting domains close to the deffects. The energy excess of these deffects scales like rL, where L is the length of the spins chain. On the other hand, domain excitations (in this case these excitations are inversions of compact domains) scale like L θ with θ = −1, where θ is the thermal exponent introduced in droplet models (see [6] for a derivation.). This gives the chaos correlation length,
ξ ∼ r − 1 2(21)
Domains of lenth L above this characteristic length ξ are destroyed by the bond deffects. Below the characteristic length, the domains are nearly insensitive to the perturbation. This result is valid for a distribution of couplings with finite weight at J = 0 2 . We have found therefore for the chaos correlation length the same result as in mean-field theory . Unfortunately we cannot do more in order to compute this exponent in other dimensions because we do not know the structure of the ground state. Anyway, we can try to estimate the energy the excitations after applying the perturbation. In the more general case, a perturbation of the type we are studying here will modify the ground state energy by a quantity proportional to the number r of created defects (because the total fraction of frustrated and unfrustrated bonds is finite). In addition, we can suppose that this energy will scale like the size of the system L α with α ≤ d. In principle, the exponent α is unknown and we do not know how to estimate it. If we assume α = d+1 2 (and using θ = d−3 2 ), this gives the dimension independent result ξ ∼ r − 1 2 . Unfortunately we are unable to estimate the exponent α and a numerical computation of this exponent in 2 dimensions would be very interesting.
Numerical results
In this section we will discuss on our Monte Carlo simulations in order to test the results obtained in the previous sections for the chaos exponents. Furthermore, we will present simulations in four dimensions. Our results are compatible with the fact that the chaos exponents in finite dimensions are compatible with the mean-field ones.
We have simulated two types of mean-field models (the Sherrington Kirkpatrick -SKmodel [14] and the random finite connectivity lattice model [11]) and a four-dimensional (4d) Ising spin glass for which the existence of a finite T phase transition is well established [15]. Monte Carlo simulations implement the Metropolis algorithm (for the mean-field models) and the heat-bath algorithm (in the 4d case). Special attention has been payed in order to thermalize the samples.
The finite-size scaling approach
In order to measure the chaos exponents, we have performed a finite-size scaling analysis [4]. The idea is to compute the overlap between two copies of the system, one copy with an initial realization of the disorder, the other one with the perturbed realization. The overlap is defined as,
q = N i=1 σ i τ i(22)
We define the chaos parameter a(r),
a(r) = σ i τ i 2 r σ i τ i 2 r=0(23)
i.e. we normalize the correlation between the unperturbed and the perturbed system to the autocorrelation of the unperturbed system. In this way a(0) = 1 by definition. The system is chaotic if the quantity a(r) (in the thermodynamic limit) jumps to 0 as soon as r is finite. This means that,
lim r→0 lim N →∞ a(r) = 0 while a(0) = 1(24)
It is crucial to perform the limits in the order previously indicated. Since a is an adimensional quantity, we expect it will scale like
a ≡ f (L/ξ)(25)
where ξ is the chaos correlation length of eq.(6). In the mean-field case we find for the spin-glass phase (using eqs. (17) and (19))
a ≡ f (Nr 2 ) .(26)
and at the critical point we get (using the singular behavior C(p) ∼ p −2 for p → 0)
a ≡ f (Nr 3 ) .(27)
In the case of short-range models we can derive the scaling behavior using equation
(21) a ≡ f (rL 2 ) (28)
Since only one exponent (the chaos exponent) must be fitted, these scaling relations are highly predictive. Comparing equations (26) and (28) we observe that d u = 4 plays the role of a specific critical dimension. The situation is the same as in the case of magnetic field perturbations [4], where the value of this dimension only depends on the behavior of the propagator C(p) in the limit p → 0. In this case we expect the scaling functions f (x) in equations (26) and (28) to coincide except by the presence of some logarithmic corrections. A numerical test of this prediction is shown in the following subsections.
Numerical results in mean-field models
The SK model is defined by the following hamiltonian
H = i<j J ij σ i σ j (29)
with the J ij distributed according to the function p(J ij ). In the thermodynamic limit, the only relevant feature of the p(J ij ) is its variance (we restrict to distributions with zero mean). To speed up the numerical computations we have taken a binary distribution of couplings, i.e. the J ′ s can take the values ± 1 √ N with equal probability. We have simulated the SK model at the critical temperature T = 1 and below the critical temperature. We have computed the chaos parameter a for different values of r (tipically r runs from 0 to 0.5). Simulations were done for lattice sizes ranging from N = 32 to N = 1000. Figures 1 and 2 show the scaling laws eq.(26) and (27) at the critical point T = 1 and below the critical point T = 0.7 respectively. Data do nicely fit the predictions. We have also simulated the random finite-connectivity lattice model(FC model). In this model each point of the lattice is connected (in average) to a finite number c of neighbors 3 . In this case the FC model model is defined by,
H = i<j J ij σ i σ j (30)
where the J ij are distributed according to,
P(J ij ) = c N p(J ij ) + (1 − c N )δ(J ij )(31)
and p(J ij ) is given by,
p(J ij ) = 1 2 δ(J ij − 1) + 1 2 δ(J ij + 1)(32)
The parameter c is the average connectivity of the lattice. This model can be exactly solved, the only difference with respect to the SK model being that there appear an infinite set of order parameters, which can be absorbed in a global order parameter [11]. The model has a phase transition at a temperature β c given by,
1 = (c − 1) ∞ −∞ p(J)tanh 2 (β c J) = (c − 1)tanh 2 (β c )(33)
3 One can also consider the case in which the connectivity is fixed and equal to c This expression implies that to have a phase transition, we need c > 2. To compare with the results of the 4d case, we have simulated the FC model with c = 8, in order to have the same number of nearest neigbors than the 4d model. The transition temperature is in this case T c ≃ 2.76. We have simulated this model at the critical temperature and below that temperature, at T = 2.0. The results for the chaos parameter a are shown in figures 3 and 4. The agreement with the scaling predictions (eqs. (26) and (27)) is also fairly good.
Numerical results in four dimensions
We have also done numerical simulations of the Ising spin glass model in four dimensions with the purpose of analysing the dimensionality effects on the chaos exponent. We have considered the Ising spin glass at d = 4 because it is widely accepted that there is a finite T phase transition in this case 4 .
We have simulated the model (eq.(1)) with a nearest neighbour interaction, periodic boundary conditions and using a discrete binary distribution of couplings as in eq.(32). We expect to obtain the same results as in the case of a continuous distribution of couplings with p(J = 0) finite. The model has a transition at T c ∼ 2.05 [15]. We have done simulations at T = T c and T = 1.7. The results are shown in figures 5 and 6.
At the critical point we obtain a chaos exponent λ ∼ 2 3 . It is not clear to us how to obtain this exponent in terms of the critical exponents and if it represents a new critical chaos exponent.
The results in figure 6 show that eq.(28) is in pretty good agreement with the data. Now we will show that four dimensions is well compatible with the upper critical dimension for the criticality of chaos. In order to get this result, we will compare the different values of the chaos parameter a for different sizes with the corresponding values of the FC model with c = 8. We compare with the FC model, instead of the SK model, because we expect that logarithmic corrections, if present, should be smaller in the FC model than in the SK model. Both are mean-field models even though the FC model resembles the finite d model much more than does the SK model. This fact should reflect in the nature of the corrections to the universal mean-field behavior. It is clear that in order to compare the FC model with the four dimensional model we have to put the system in equivalent points within the phase diagram. We expect the universal function f (x) to depend on the temperature (which is an external parameter) in the following way
a ≡ f (A(T )(L/ξ) d )(34)
In four dimensions, the scaling function f still depends on the temperature via the universal amplitude A(T ). It is reasonable to assume that the dependence of the amplitude A(T ) on the temperature enters through the spin-glass order parameter q(T ). More concretely, below but close to T c we expect,
A(T ) ∼ q 2 (T )(35)
because the argument of the scaling function f of equation (34) scales like the singular part of the free energy which in mean field theory scales like Q 2 ab (see eq. (10)). Consequently we have to normalize the adimensional ratio (L/ξ) d to the corresponding value of the Edwards-Anderson order parameter for that temperature. For N = 256 the FC model gives q(T = 2.0) ≃ 0.11 and the 4d model at L = 4 gives q(T = 1.5) ≃ 0.25, the ratio of both numbers being 2.5. Simulation data for both models are shown in figure 7. If one considers the SK model then one observes that data fits well but not so nicely as in the case of the FC model.
Conclusions
We have investigated the sensitivity of spin glasses against the application of a particular static perturbation. In particular, we have studied the nature of the static chaos when a perturbation to the realization of the quenched disorder is applied to the system. This can be done in several ways. In our case we have considered a perturbation wich, in average, does not add energy to the system. Due to the self-averaging property we expect that a change in the sign of a finite fraction of the total number of couplings in the system should not change its mean statistical properties (and in particular, its energy). This makes the new perturbed system to stay in the same point in the phase diagram. The existence of strong chaos for this type of perturbation proves that the reshuffling of the Boltzmann weights of the different states is complete. This differs from the case where the perturbation consists in applying a magnetic field to the system or where its temperature is changed. In these cases extra energy is supplied to the system.
We have solved the mean field theory and we have extracted the chaos exponent for this type of perturbation. The analytical solution of this problem is very similar to that of chaos against magnetic field perturbations where the chaos correlation length can be exactly computed [3]. This is in contrast to what happens when the temperature is changed. In the last case the system is much robust against the perturbation and a high degree of correlation between the configurations at both temperatures is preserved [8].
We have observed that the mean-field chaos exponent 1/2 in the spin-glass phase is exact also at one dimension. A finite-size scaling approach to the criticality of chaos shows that d = 4 plays the role of an upper critical dimension for the chaos problem. Finite-size scaling studies are very powerful in order to get the chaos exponents. This is because we only need to determine one free parameter to make the data corresponding to different sizes to collapse in a unique scaling function. We have performed numerical simulations of mean-field models which are in agreement with the theory. Simulations in four dimensions are in very good agreement with the fact that 4 plays the role of an upper critical dimension for the criticality of chaos (see figure 7). Furthermore, the fact that the mean-field chaos exponent is also exact in one dimension suggests that mean-field theory is probably correct at any dimension. This is indeed very similar to what happens in the case of magnetic field perturbations.
Finally we would like to point out two possible extensions of this work. Firstly it would be interesting to make dynamical studies of the relaxation of the overlap function against this type of perturbation (as done for the remanent magnetization after application of a magnetic field). We expect to see aging effects as in the case of magnetic field perturbations. Secondly, it would be interesting to extend the study of chaos to the metastable states using the TAP formalism. Most probably, similar chaotic properties will be observed in the structure of the metastable states.
Acknowledgements
E. F. acknowledges Gobierno de Navarra through a predoctoral grant for financial support. We thank also the CICY T institution for partial financial support.
Figure Captions
Fig. 1
1Chaos in the SK model at the critical point T c = 1.
Fig. 2
2Chaos in the SK model at T = 0.7 in the spin glass phase.
Fig. 3
3Chaos in the FC model with c = 8 at the critical point T c ≃ 2.76.
Fig. 4
4Chaos in the FC model with c = 8 at T = 2.0 in the spin glass phase.
Fig. 5
5Chaos in the 4d Ising spin glass at the critical temperature T c ≃ 2.05. We obtain λ ∼ 2/3 for the chaos exponent.
Fig. 6
6Chaos in the 4d Ising spin glass at T = 1.7 in the spin-glass phase. The mean-field chaos exponent λ = 1 2 fits data very well.Fig. 7Chaos in the 4d Ising spin glass at T = 1.7 compared to the FC model with c = 8.The abcisa x corresponds to Nr 2 (with N = L 4 in four dimensions). This scaling suggests that four dimensions is the upper critical dimension for the criticality of chaos.
) .
)Once we have defined the perturbation we construct a full Hamiltonian H 12 [σ, τ ] defined in a space of two sets of variables {σ i , τ i ; i = 1, ..N}. The Hamiltonian H 12 is the sum of the unperturbed Hamiltonian H[σ] plus the perturbed Hamiltonian H r [τ ],
For the problem of chaos in temperature there is a new chaos exponent independent of the usual critical exponents[13].
For a distribution of couplings with zero weight at J = 0 one finds ξ ∼ r −1
In the three dimensional case there is still much controversy on the existence of a finite T transition[5]
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K H Fischer, J A Hertz, Spin Glasses. Cambridge University PressK. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge University Press 1991);
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[] |
[
"An Evolution Model of Complex Systems with Simultaneous Cooperation and Competition",
"An Evolution Model of Complex Systems with Simultaneous Cooperation and Competition"
] |
[
"Xiu-Lian Xu \nCollege of Physics Science and Technology\nYangzhou University\n225002YangzhouChina\n",
"Chun-Hua Fu \nCollege of Physics Science and Technology\nYangzhou University\n225002YangzhouChina\n",
"Hui Chang \nCollege of Physics Science and Technology\nYangzhou University\n225002YangzhouChina\n",
"Da-Ren He \nCollege of Physics Science and Technology\nYangzhou University\n225002YangzhouChina\n"
] |
[
"College of Physics Science and Technology\nYangzhou University\n225002YangzhouChina",
"College of Physics Science and Technology\nYangzhou University\n225002YangzhouChina",
"College of Physics Science and Technology\nYangzhou University\n225002YangzhouChina",
"College of Physics Science and Technology\nYangzhou University\n225002YangzhouChina"
] |
[] |
Systems with simultaneous cooperation and competition among the elements are ubiquitous. In spite of their practical importance, knowledge on the evolution mechanism of this class of complex system is still very limit. In this work, by conducting extensive empirical survey to a large number of cooperation-competition systems which cover wide categories and contain the information of network topology, cooperation-competition gain, and the evolution time, we try to get some insights to the universal mechanism of their evolutions.Empirical investigations show that the distributions of the cooperation-competition gain interpolates between power law function and exponential function. Particularly, we found that the cooperation-competition systems with longer evolution durations tend to have more heterogeneous distributions of the cooperation-competition gain. Such an empirical observation can be well explained by an analytic model in which the evolution of the systems are mainly controlled by the Matthew effect, and the marginal heterogeneity of the initial distribution is amplified by the Matthew effect with similar speed in spite of the diversity of the investigated systems.
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10.1016/j.physa.2011.06.024
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[
"https://export.arxiv.org/pdf/1101.0406v1.pdf"
] | 119,250,859 |
1101.0406
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5a386e53a1ba4673a484552e1e4c0de5a10e6f42
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An Evolution Model of Complex Systems with Simultaneous Cooperation and Competition
Xiu-Lian Xu
College of Physics Science and Technology
Yangzhou University
225002YangzhouChina
Chun-Hua Fu
College of Physics Science and Technology
Yangzhou University
225002YangzhouChina
Hui Chang
College of Physics Science and Technology
Yangzhou University
225002YangzhouChina
Da-Ren He
College of Physics Science and Technology
Yangzhou University
225002YangzhouChina
An Evolution Model of Complex Systems with Simultaneous Cooperation and Competition
1cooperation-competitioncooperation sharingheterogeneity PACS: 8975-k8723Kg8975Da
Systems with simultaneous cooperation and competition among the elements are ubiquitous. In spite of their practical importance, knowledge on the evolution mechanism of this class of complex system is still very limit. In this work, by conducting extensive empirical survey to a large number of cooperation-competition systems which cover wide categories and contain the information of network topology, cooperation-competition gain, and the evolution time, we try to get some insights to the universal mechanism of their evolutions.Empirical investigations show that the distributions of the cooperation-competition gain interpolates between power law function and exponential function. Particularly, we found that the cooperation-competition systems with longer evolution durations tend to have more heterogeneous distributions of the cooperation-competition gain. Such an empirical observation can be well explained by an analytic model in which the evolution of the systems are mainly controlled by the Matthew effect, and the marginal heterogeneity of the initial distribution is amplified by the Matthew effect with similar speed in spite of the diversity of the investigated systems.
Introduction
Simultaneous cooperation and competition is the common feature for a number of complex systems, including the commercial [1][2], ecological [3][4][5], social [6][7][8], biochemical [8], economical [8][9][10] and some technological systems [11][12][13][14][15][16][17][18]. Revealing the universal rules and corresponding mechanisms controlling the evolution of such cooperation-competition (C-C) systems attracts much attention from the scientists in physics. In literature, several models have been proposed to explain the empirically observed statistical properties of this kind of C-C systems [4][5][6][7][8][9][10]. For example, in Ref. [4], to explain the commonly observed exponential truncation in the degree distribution of the mutualistic ecological networks, Guimaraes Jr. and coworkers developed a model by introducing the "build-up" mechanism. The resulted degree distribution based on their model interpolates between the power law and exponential decay, suggesting that some ecological factors confined the appearance of large hubs (or the accretion of the degree distribution heterogeneity) [4]. In Ref. [10], by considering the Matthew effect (rich gets richer) in the growth rate of the so-called "competition gain", the author developed an evolution model which can successfully reproduce the observation of wealth condensation in economic systems.
The success of the above models demonstrates that it is important to simultaneously consider the cooperation and competition in modeling the evolution of the cooperation-competition systems.
However, all these previous work focused on the specific system(s). One interesting question is whether the evolution mechanism derived based on these specific systems is also applicable to other cooperation-competition systems, or equivalently, can we derived a certain universal rule which controls the evolution of the whole class of cooperation-competition systems? Answering to this question is very difficult since it requires large number of empirical data covering wide categories. To our knowledge, such kind of work is still lacking up to now.
In this work, as an effort to get some insights to the above question, we try to collect a number of cooperation-competition systems with large diversity. The collected systems include not only the data of network topology but also the data of cooperation-competition gain (in this work, we call it cooperation sharing (CS)). More importantly, many of the systems also contain the data of time during of the system evolution, which makes it possible to directly extract the information of the evolution dynamics by empirical survey. Our empirical results show that the distributions of the cooperation sharing for all the studied systems interpolate between power law function and exponential function. Particularly, the degree of heterogeneity of the distribution for each system is strongly correlated with the time duration of the system evolution. The systems with longer evolution durations tend to have more heterogeneous distributions of the cooperation sharing. Such an empirical observation can be well explained by an analytic model in which the evolution of the systems are mainly controlled by the Matthew effect, and the marginal heterogeneities of the initial distributions are modified by the Matthew effect with the similar speed in spite of the large diversity of the investigated systems. Our results revealed a general feature for this kind of cooperation-competition systems, contributing to the understanding of the complex systems with simultaneous cooperation and competition.
The rest parts of the article are organized as follows. In section 2 we will introduce the collections of the cooperation-competition systems investigated in this work, and some empirical results are provided. Based on the empirical results, an evolution picture of the cooperation-competition systems is proposed. In section 3, an analytic discussion on the model will be presented. In section 4 we will give the comparisons between the analytic results derived based on our model and the empirical observations. In the last section, a summary and some discussions will be presented.
Data collection and empirical investigations
In the cooperation-competition systems we studied, elements cooperate and compete in some groups [11][12][13][14][15][16][17][18]. The groups, which are the platform of the cooperation and competition among the elements, can be places, organizations, events, or activities. The elements are the participants of the cooperation-competition, and they can be human beings or abiotic things that are subjected to influences of human beings. The same element may be involved in a number of groups simultaneously. For example, to describe the cooperation and competition among the movie actors, the movie actors who are casted in a certain movie are considered as elements of the C-C system, while the movies are considered as groups [18]. Similarly, when studying the evolution of the human languages, the languages, which coexist in a geographical region, can be considered as elements, and the geographical regions can be considered as groups. During the long time evolution, the languages compete for being used by more people [18]. In each group of a C-C system, some elements make concerted effort to accomplish a task, which often create a type of product. The product may induce several kinds of resources. The elements, when they cooperate, also compete for a larger piece of the resources. For example, some Hollywood actors work together to produce a movie, which should bring box office income (a countable resource) and famousness (an uncountable resource). Meanwhile, they compete for playing more important roles in the movie, which usually means higher salaries and famousness. In another example, some languages cooperate for creating an "oral and written communication" of some different kinds of people in a group (region). The produced resource of the cooperation is the populations who live together in the region and use some of the languages. In a historical period, the languages compete for being used by more people (that is why some languages got extincted, but some other languages have been used by more and more people and spread to more and more geographical regions). The number of people who use a certain language can be regarded as a countable resource, which the element (the language) shares.
In this work, we use h i to denote the group number in which the ith element takes part. It can be expressed as i = ∑ [11,12]. In Ref. [13], Fu et al. defined the CS as the part of a countable resource which the element shares. Considering that the competition intensity should depend on the group size, in this work the total CS of element i in group l is defined as W li =T l z il , where the z il denotes the countable resource shared by element i in group l. The normalized total cooperation sharing (NTCS) of the ith element, ω i , can be defined as [14,16,18].
( )/ [ ( ) ] i li l lj l l j l T z T z ω = ∑ ∑ ∑
In constructing the real world C-C systems, the data were collected based on the following rules:
(1) All the elements can be unambiguously assigned to the related groups; (2) The information of the amount of CS shared by the elements in each group is available; (3) The evolution duration of each group (this concept will be introduced bellow) is available. We totally constructed more than 60 systems, among which only 12 systems have all the reliable information. For example, Hollywood movie actor cooperation may be a famous C-C system. Unfortunately, it is difficult to get the CS data. Therefore, the 12 systems can be considered to be randomly selected and reasonably cover the essential features of the C-C systems. The 12 C-C systems are listed in Table 1 (MD) (e.g., cocktails) system, and 12) the world language distribution system (WLD). More details on the data collections and constructions of these C-C systems can be found in Ref. [18]. In most cases, C-C among the elements occurs right after the birth of the groups; therefore, we can define the evolution duration as the time duration between the group birth and the group termination or data collection. We define the "group birth" as the time when the first element joins the group, and the "group termination" as the time when the cooperation task is accomplished and the elements disband.
The description of C-C dynamics certainly will be in a logarithmic time scale because the evolution durations of different systems must be very different and cover many orders of magnitudes.
We emphasize that for the current purpose, it is sufficient that the data of the evolution duration are reliable within one order of magnitude, although the exact evolution duration for all the groups may be obtained. In some systems there are many groups, and the groups show different evolution durations. It is meaningless to list the evolution durations for all the groups. As listed in Table 1, we use the longest and the shortest evolution durations, τ max and τ min , among the groups of the C-C system, as well as the average of them in the current work, which are sufficient to capture the order of the magnitude and the error range of the evolution durations. More detailed discussions were given in Ref. [18].
In Ref. [18] we presented the details of the definitions and interpretations for the 12 real world C-C systems. The NTCS distributions for all these systems were also reported. The results showed that all the distributions obey the so-called "shifted power law (SPL) functions" which can be expressed as ( ) (
) P x x γ α − ∝ + [12]
. When 0 α = , the SPL function is reduced to a power law function.
When 1 α → , we can prove that the SPL function is reduced to an exponential decay function in the condition that x is normalized (i.e., 0<x i <1 and , M denotes the total number of x) [18].
Therefore an SPL interpolates between a power law and an exponential decay. The parameter α characterizes the degree of deviation from a power law. We emphasize that in general an SPL is not a power law, and γ is not the power law scaling exponent. It is not strange to observe a very large γ as will be discussed later. We will show that both the γ and α characterize the heterogeneity of the distributions, and that, very possibly, the γ and α keep a general correlation which encodes the information of the evolution mechanism of the C-C systems. The definitions of the elements and the groups, the empirical evolution durations, and the parameters of the NTCS distribution function for each of the C-C systems are listed in Table 1. . The longest and the shortest group evolution durations, τ max and τ min , and the averaged evolution duration τ are also listed. From Table 1, one can see that the Nα (N is the element number) and γ basically show a monotonic relation. This implies a possibility that the correlation between the Nα and γ obeys a general function, which may encode the information of the evolution mechanism of the systems.
In the next two sections, we will discuss this further. Note that in this work, we always use the Nα when discussing the SPL parameter α. As mentioned above, the parameter α characterizes the degree of deviation of the distribution ( ) ( , as used in this work. Fig. 1 shows the SPL functions best fitting the NTCS distributions for the 12 real world C-C systems. The corresponding parameters α and γ of the SPL functions are listed in Table 1.
More details of the SPL fitting were provided in Ref. [18]. For clarity, all the fitting lines are translated to have a common point without changing their slopes (i.e., γ values of the SPL functions) on the double-logarithmic plane. Note that the lengths of the fitting lines are meaningless, and are used to distinguish two lines with similar slopes.
In Table 1 and Fig. 1, the systems rank in descent order of the γ values. Interestingly, one can observe that the averaged evolution duration τ basically shows a monotonic increase with the deceasing of the γ values. Considering that the γ values describe the distribution heterogeneities of the CS, the monotonic dependence of the γ values on the evolution duration τ indicates that the cooperation-competition systems with longer evolution durations tend to have more heterogeneous distributions of the CS. This empirical observation strongly supports the following picture of the system evolution: the time duration of the system evolution plays dominant role in controlling the distribution heterogeneity of the CS of the C-C systems, and all these C-C systems follow the similar evolution mechanism, Consequently, the final distribution heterogeneity (described by γ [17,19])
only depends on the system evolution duration. Inspired by the previous work in the specific C-C systems [4,10], we propose that the evolution of the system is dominated by the Matthew effect.
Suppose that all the systems start the competition and cooperation at a common time point, the parameter γ of each system will get increasingly smaller when it evolves for longer time (i.e., τ becomes longer). This picture resembles the case of running. The heterogeneity (difference between the athletes) is small in a short distance running, while it becomes larger and larger when the distance becomes longer. As is well known [19], on the x-P(x) plane, a line with a larger slope indicates a weaker heterogeneity of the distribution. The most homogeneous distribution shows a line with an infinitely large slope (an ideal vertical line), which means that all the elements possess a common x value. According to the empirical observations that γ monotonically depends on τ, we suppose that such a vertical line corresponds to τ=0, as shown in Fig. 1. Thus, in our model we assume that at the beginning (t=0), all the systems show even NTCS distributions as shown by the line indicated by τ=0 in Fig. 1. Right after the start of time evolution (τ becomes larger but is still close to the starting point), the systems suffer a random perturbation so that CS of the elements show marginal differences. The corresponding fitting lines of the NTCS distributions for all the 12 systems should be almost on the same position as denoted by the line indicated by τ ≈0 in Fig. 1. As the time develops the elements in all the systems simultaneously cooperate and compete. The Matthew effect amplifies the differences of the element CS and the distribution heterogeneity. The system, which has a longer τ, finally shows a fitting line with smaller slope γ in Fig. 1, corresponding to a larger CS distribution heterogeneity. In order to show the difference of the four lines with the largest slopes, the lines are shown in the inset of Fig. 1 where the region of
x-axis is greatly magnified. We will show that such a simple evolution picture is reasonable and supported by empirical observations.
Analytic model of the system evolution
The above proposed evolution picture can be easily implemented in an analytic model. In the model, at τ=0 all the elements in any of the systems possess a common NTCS value ω 0 . The value can be an arbitrary small number. Without losing generality, in Fig. 1 we choose ω 0 =10 -4 .
When τ becomes larger but still close to zero, the NTCS values of the elements show marginal differences due to certain random perturbations. We sort the N elements according to their NTCS values ω i (i=1,⋅⋅⋅,N) by descent order with i being the ranking number, i.e., by Barabasi and Albert, in which the authors analytically derived a power law evolution description of network degree with a negative exponent based on a "rich gets richer" mean field dynamic equation. It is known that the ranking number i should obey a similar power law [19].
Consequently, the NTCS evolution function can be written as
/ 0 0 ( ) ( ) t c i i W t N ω ω − = −(1)
where W i (t) denotes the net CS at time t. The normalized form of W i (t) is given by
1 / 0 0 1/ [ ] i i i t c i N W W W N x dx N ω ω ω − = − ∑ ∫ .(2)
In the next section we will show that although the value of the constant c in Eqs. 1 and 2 does not influence the main results, the value of 100 years is chosen in this work (i.e., c=c s =100 years), with which the analytical results can best reproduce the empirical data. When τ/c s < 1, from Eqs.1 and 2 we get
/ [ ( )] s c s i s c i N N c N τ τ τ ω τ τ − − = + − .(3)
where τ is the time when the system reaches the final state. The probability for finding an element whose NTCS is smaller than ω i is given by
/ ( ) 1 [ ( )] s c s i i i s c N i N P N c N τ τ τ ω ω ω τ τ − − − ′ < = = − + − .(4)
Therefore,
/ ( 1 / ) ( ) ( ) ( ) ( ) ( ) s s c c i i s s i i i i s P c c N P c N τ τ γ ω ω τ τ ω ω ωα ω τ τ τ − − + − ′ ∂ < − = = + ∝ + ∂ − .(5)
Since τ/c s < 1, we have γ(τ)=(1+c s /τ)>2. As can be seen in Table 1, 10 of the 12 real world systems (No.1-No.10) belong to this type [18]. For these systems, one gets
Similarly, when τ/c s = 1, we get
(1 / ) 1 ( ) ( ) ( ) (ln 1) s c i i i P N N τ γ ω ω ω α − + − ∝ + = + − .(9)
For this kind of systems, the analytic expression When τ/c s > 1, we obtain a similar form as
(1 / ) / ( ) ( ) ( ) s s c s i i i c s c P c N N τ γ τ τ ω ω ω α τ − + − − ∝ + = + − .(10)
In this case, we have γ(τ)=(1+c s /τ)<2. In Table 1 one can see that only 2 real world systems, Table 1.
Comparison between analytic and empirical results
The analytic expression Table 1 one can see that 10 of the 12 real world systems (No. show τ/c s < 1 and γ(τ)=(1+c s /τ)>2 [18]. For these systems the analytical conclusion gives
Summary and discussions
This article shows that the CS distributions in all the C-C systems interpolate between the power law function and exponential function, which can be described by the SPL function.
Both the distribution parameters of SPL function, α and γ, signify the heterogeneities of the distributions. Based on the empirical investigations obtained for 12 real world C-C systems, we proposed a very simple C-C evolution model where Matthew effect dominates so that different systems show different heterogeneities of CS distributions only due to the different evolution durations. Based on this model we analytically obtained the correlation functions between the α, γ and the evolution duration τ, which are in good agreement with the empirical observations.
The relationship between the α and γ, i.e.,
( ) 2 N γ τ α τ = + ( )
, is also in agreement with our previous investigation obtained in a very different way [17], which may suggest that the relationship is general and captures the essence of the evolution dynamics of the C-C systems.
People may raise questions about the model idea proposed above. For example, one may feel hard to accept that Matthew effect is the only dominating factor in the evolution of the C-C systems. It is natural to also consider the opposite factors which tend to make the CS distributions even. To investigate the effects of such even factors, we tried to add random perturbations, which tend to destroy the distribution heterogeneities therefore mimics the effects of the even factors, during the evolution of the systems. The resulted CS distribution shows enhancement of the "middle class" elements, which obviously deviates from empirically observed SPL distributions (data not shown). In comparison, all the empirical investigations on the 12 real world C-C systems support the Matthew effect dominated evolution model proposed in the current work.
In the current model, the even factor is introduced only at the beginning of the competition.
This resembles the situations in Chinese history during which many dynasties alternated. Quite often, when an old dynasty suffered perdition due to too large wealth heterogeneity, a new dynasty emerged; this destroyed almost all the heterogeneities. The randomly arising heterogeneity then was magnified in the new dynasty. Differently, a modern government knows well that, in order to keep the society stable, it is very important to maintain the wealth heterogeneity below a threshold.
Therefore a modern government often adopts some policies to shrink the wealth heterogeneity, which resembles the introduction of the even factors during the system evolution discussed in the last paragraph. So, we argue that the model considering the even factors during the system evolution may be valid only for the C-C systems with a "wise manipulator", which are rarely observed in nature. This is why it is difficult to obtain empirical supports for the model considering both the Matthew effect and the even factors. This discussion supports that the current model should describe the common feature of most real world C-C systems.
In this work, although we tried our best to collect as many data as possible, the difficulties in collecting all the necessary information, especially the cooperation sharing and evolution duration, renders the collected data still far from sufficient. Apparently, collecting more data with all the necessary information will be highly useful to more robustly test the above proposed evolution model in the future work.
with b i j = 1 if element i is involved in group j and b i j = 0 otherwise. The so-called "group size", T j , denotes the number of the elements which take part in group j. T j
, which include 1 )
1Chinese university matriculation (CUM), 2) the 2004 Athens Olympic Game (OG), 3) notebook PC selling at Taobao website (PCST), 4) information technique (IT) product selling (ITPS), 5) journal impact factor system (JIF), 6) author academic level system (AAL), 7) the 200 richest Chinese magnates in 2004 (RCM), 8) the USA county population system in 1900 (UCO19), 9) the USA county population in 2000 (UCO20), 10) Beijing restaurant system (BR), 11) mixed drink
.
As the time develops, Matthew effect magnifies the differences betweenthe ω values so that a large ω i gets larger. A simple but appropriate description of the NTCS evolution may be a power law form of ω i with a negative exponent which is proportional to the time.Such an assumption of the evolution function is supported by the "B-A model" proposed in Ref.[20] FIG. 1. (color online) The NTCS distribution fitting lines of 12 real world C-C systems. The two lines indicated by τ=0 and τ≈0 are explained in the text. The inset shows the zoom-in of the region around x=10 -4 .
(
No.11) and (No.12), are of this type, and that the systems show a very long τ value [18]effective. The systems should show very small α values, as shown in
only common conclusion for all the three cases discussed in the last section. Fig. 2 shows the comparison of this rule with the empirical data. In the figure the vertical coordinates of the solid circles denote the empirical data of τ. The vertical coordinates of upper and lower positions of the error bars are drawn according to the empirically obtained longest and the shortest evolution durations, τ max and τ min , respectively. The solid line is the least-square fitting of all the data. The slope of the fitting line, -1.11, is very close to the analytic value (-1.0). Different c s values can shift the fitting line upward or downward without changing its slope. When we chose the c s value of 100 years as mentioned in the last section, the line representing the analytic results will be very close to the fitting line of the empirical data.
.
Fig. 3 and 4 show the comparison between the theoretical and the empirical results. Similarly, in Fig. 3 the vertical coordinates of the solid circles denote the empirical data of τ. The vertical coordinates of the upper and lower positions of the error bars are drawn according to τ max and τ min , respectively. The solid line is the least-square fitting of the data. The slope of the fitting line (-1.10) is very close to the analytic value (-1.0). If we chose the c s value of 100 years, again the line representing the analytic results will be almost identical to the fitting line of the empirical data. In Fig. 4 the crosses denote the empirical data of the 10 systems. The solid line is drawn based on the analytic results, i.e., ( ) . Obviously, all the empirical data are in good agreement with the analytic results. FIG. 2. The empirical relation between τ and γ.
FIG. 3 .
3The empirical relation between τ and Nα. FIG. 4. The empirical relation between γ and Nα.
Table 1．
1．The interpretations and the numbers of groups and elements. Two parameter values, γ and Nα (Ndenotes element number) of the NTCS distribution functions ( ) (
)
P x
x
γ
α −
∝ +
is invalidated since α does not relate to τ.1
( ) 1
γ τ
τ −
− ∝
is still valid, whereas the relation
( ) 2
N
γ τ
α τ
=
+
( )
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|
[] |
[
"The Vela Pulsar in the Near-Infrared ⋆",
"The Vela Pulsar in the Near-Infrared ⋆"
] |
[
"Yu A Shibanov \nIoffe Physical Technical Institute\nPolitekhnicheskaya 26194021St. PetersburgRussia\n",
"A B Koptsevich \nIoffe Physical Technical Institute\nPolitekhnicheskaya 26194021St. PetersburgRussia\n",
"J Sollerman \nAlbaNova\nDepartment of Astronomy\nStockholm Observatory\nSE-106 91StockholmSweden\n",
"P Lundqvist \nAlbaNova\nDepartment of Astronomy\nStockholm Observatory\nSE-106 91StockholmSweden\n"
] |
[
"Ioffe Physical Technical Institute\nPolitekhnicheskaya 26194021St. PetersburgRussia",
"Ioffe Physical Technical Institute\nPolitekhnicheskaya 26194021St. PetersburgRussia",
"AlbaNova\nDepartment of Astronomy\nStockholm Observatory\nSE-106 91StockholmSweden",
"AlbaNova\nDepartment of Astronomy\nStockholm Observatory\nSE-106 91StockholmSweden"
] |
[] |
We report on the first detection of the Vela pulsar in the near-infrared with the VLT/ISAAC in the Js and H bands. The pulsar magnitudes are Js = 22.71 ± 0.10 and H = 22.04 ± 0.16. We compare our results with the available multiwavelength data and show that the dereddened phase-averaged optical spectrum of the pulsar can be fitted with a power law Fν ∝ ν −αν with αν = 0.12 ± 0.05, assuming the color excess EB−V = 0.055 ± 0.005 based on recent spectral fits of the emission of the Vela pulsar and its supernova remnant in X-rays. The negative slope of the pulsar spectrum is different from the positive slope observed over a wide optical range in the young Crab pulsar spectrum. The near-infrared part of the Vela spectrum appears to have the same slope as the phaseaveraged spectrum in the high energy X-ray tail, obtained in the 2 − 10 keV range with the RXTE. Both of these spectra can be fitted with a single power law suggesting their common origin. Because the phase-averaged RXTE spectrum in this range is dominated by the second X-ray peak of the pulsar light curve, coinciding with the second main peak of its optical pulse profile, we suggest that this optical peak can be redder than the first one. We also detect two faint extended structures in the 1. ′′ 5 − 3. ′′ 1 vicinity of the pulsar, projected on and aligned with the south-east jet and the inner arc of the pulsar wind nebula, detected in X-rays with Chandra. We discuss their possible association with the nebula.Send offprint requests to: Yuri Shibanov, [email protected] ⋆ Based on observations collected at the European Southern Observatory, Paranal, Chile (ESO Programme 66.D-0568).
|
10.1051/0004-6361:20030652
|
[
"https://export.arxiv.org/pdf/astro-ph/0305273v1.pdf"
] | 18,476,578 |
astro-ph/0305273
|
410384e1fb12de8054a708e41c2281a4c3683015
|
The Vela Pulsar in the Near-Infrared ⋆
May 2003 February 5, 2022
Yu A Shibanov
Ioffe Physical Technical Institute
Politekhnicheskaya 26194021St. PetersburgRussia
A B Koptsevich
Ioffe Physical Technical Institute
Politekhnicheskaya 26194021St. PetersburgRussia
J Sollerman
AlbaNova
Department of Astronomy
Stockholm Observatory
SE-106 91StockholmSweden
P Lundqvist
AlbaNova
Department of Astronomy
Stockholm Observatory
SE-106 91StockholmSweden
The Vela Pulsar in the Near-Infrared ⋆
May 2003 February 5, 2022Received 30 January 2003 / Accepted 25 April 2003arXiv:astro-ph/0305273v1 15 Astronomy & Astrophysics manuscript no. h4279 (DOI: will be inserted by hand later)Infrared: general -pulsars: individual: Vela pulsar -stars: neutron
We report on the first detection of the Vela pulsar in the near-infrared with the VLT/ISAAC in the Js and H bands. The pulsar magnitudes are Js = 22.71 ± 0.10 and H = 22.04 ± 0.16. We compare our results with the available multiwavelength data and show that the dereddened phase-averaged optical spectrum of the pulsar can be fitted with a power law Fν ∝ ν −αν with αν = 0.12 ± 0.05, assuming the color excess EB−V = 0.055 ± 0.005 based on recent spectral fits of the emission of the Vela pulsar and its supernova remnant in X-rays. The negative slope of the pulsar spectrum is different from the positive slope observed over a wide optical range in the young Crab pulsar spectrum. The near-infrared part of the Vela spectrum appears to have the same slope as the phaseaveraged spectrum in the high energy X-ray tail, obtained in the 2 − 10 keV range with the RXTE. Both of these spectra can be fitted with a single power law suggesting their common origin. Because the phase-averaged RXTE spectrum in this range is dominated by the second X-ray peak of the pulsar light curve, coinciding with the second main peak of its optical pulse profile, we suggest that this optical peak can be redder than the first one. We also detect two faint extended structures in the 1. ′′ 5 − 3. ′′ 1 vicinity of the pulsar, projected on and aligned with the south-east jet and the inner arc of the pulsar wind nebula, detected in X-rays with Chandra. We discuss their possible association with the nebula.Send offprint requests to: Yuri Shibanov, [email protected] ⋆ Based on observations collected at the European Southern Observatory, Paranal, Chile (ESO Programme 66.D-0568).
Introduction
The Vela pulsar was firmly identified in the optical range by the detection of optical pulsations with the radio pulsar period (Wallace et al. 1977;Manchester et al. 1978). The identification has been further confirmed by measuring the proper motion and parallax of the radio pulsar and its optical counterpart (Legge 2000;Caraveo et al. 2001), by broad-band photometry revealing peculiar colors of the counterpart typical for the optical emission of rotation powered pulsars (Lasker 1976;Mignani & Caraveo 2001 and refs. therein), and by the high polarization of the optical emission (Wagner & Seifert 2000). The Vela pulsar is an intermediate age, ∼ 10 4 yr, isolated neutron star (NS). Its parameters are listed in Table 1.
In comparison with older pulsars detected in the optical range (see, e.g., Mignani et al. 2000), the Vela pulsar with V= 23. m 6 is brighter by at least 1 − 2 stellar magnitudes. However, available spectral information on its optical emission has been limited to broad-band UBVRI photometry , which suggests a flat optical spectrum typical for young Crab-like pulsars. This is in contrast to the middle-aged, ∼ 10 5 yr, pulsars PSR B0656+14 and Geminga whose broad-band spectra are less monotonous, and to the strong excess in the near infrared (IR) part of the spectrum of PSR B0656+14 (Koptsevich et al. 2001). This may be an evidence of spectral evolution of the optical emission with pulsar age. The young Crab pulsar shows no excess in the IR. Thus, the extension of the spectrum of the Vela pulsar towards the IR is useful to determine whether the optical properties of this pulsar are closer to those of younger or middleaged NSs, and to get additional constraints on the pulsar spectral evolution with age.
The Vela pulsar has also been studied in the high energy range, from soft X-rays to γ-rays (e.g., Pavlov et al. 2001b;Harding et al. 2002 and refs. therein). Its multiwavelength spectrum is presumably nonthermal with different slopes in different high energy ranges. An exception is the soft X-ray range where a strong excess over a power law 1 background is believed to be due to the thermal emission from the surface of the NS (Ögelman et al. 1993;Pavlov et al. 2001b). Different slopes imply different nonthermal radiation mechanisms at work in the magnetosphere of the pulsar (e.g., synchrotron, curvature, inverse Compton scat- Taylor et al. 1993, unless specified otherwise tering radiation, etc.). These mechanisms are involved in different ways in the two competing models for the non-thermal emission of pulsars, the "polar cap" model (e.g., Daugherty & Harding 1996) and the "outer gap" model (e.g., Cheng et al. 1986;Romani 1996). It is not yet clear which of these competing models best represents the observations, and which radiation mechanisms are actually responsible for the observed emission in each band. In this respect, observations in the IR are important to get additional constraints on these mechanisms and models. For instance, in the polar cap model IR photons, as well as optical ones, can be produced by inverse Compton scattering of softer photons by primary and/or pair cascade relativistic particles in the magnetosphere.
A symmetrical Crab-like pulsar wind nebula (PWN) with a torus and jet structure has recently been detected in X-rays with the Chandra observatory around the Vela pulsar (Helfand et al. 2001;Pavlov et al. 2001b;Pavlov et al. 2001a). Being fainter than the Crab PWN, the Vela nebula has not yet been detected in the optical range, perhaps because the pulsar field is crowded by background stars. Some of the structures of the Crab PWN, identified in the optical, appear brighter in the near-IR range (Sollerman & Flyckt 2002), showing a PL spectrum with a negative slope close to that observed in X-rays. There are indications that the X-ray Vela PWN has a PL spectrum of a similar slope (Gotthelf & Olbert 2002). In this context, deep imaging of the Vela pulsar field in the near-IR might be more promising than imaging in the visual range for detection of the Vela PWN and for studying the mechanisms of the interaction between the relativistic pulsar wind and the ambient matter.
Here we report on the first detection of the Vela pulsar in the near-IR J s and H bands, obtained with the VLT. The observations, data reduction, astrometry, and photometry are described in Sect. 2. The results are discussed in Sect. 3 in conjunction with the data available in other spectral bands, and summarized in Sect. 4.
Observations and data analysis
ISAAC observations and data reduction
The field of the Vela pulsar was observed during three nights, December 14 and 15, 2000, and January 5, 2001
with the Infrared Spectrometer And Array Camera 2 (ISAAC) attached to the Antu Telescope (UT1) of the European Southern Observatory's Very Large Telescope (ESO VLT). A log of the observations is given in Table 2. In the SW (Short Wavelength) imaging mode, the Rockwell Hawaii HgCdTe 1024×1024 array detector was used. The pixel size on the sky was 0. ′′ 147 and the field-of-view was 2. ′ 5×2. ′ 5. The observations were performed in the J s and H bands in jitter mode, with a jitter box size of 20 ′′ . The J s filter is being established as the new standard broadband filter at ≈ 1.24µm by most major observatories (Keck, Gemini, Subaru, ESO), and is photometrically more accurate than the classical J because it is not cut off by atmospheric absorption (Simons & Tokunaga 2002;Labbè et al. 2003). The detector integration times (DITs) were 45 s and 13 s in the J s and H bands, respectively. Each observation was built up by a number of DITs per exposure (NDITs), where we used 4 NDITs for the J s band and 6 NDITs for the H band. The number of exposures (NEXP) in each observational block (ObsID) was 13 in J s and varied from 12 to 26 in the H band. Total exposure times (NDIT×DIT×NEXP summed over all ObsIDs) were thus 7020 s and 8268 s in the J s and H bands, respectively. Fig. 2. Small panels show the 6 ′′ ×6 ′′ field of view centered at the pulsar position in the J s and H bands. Images with the pulsar counterpart subtracted are also shown in the lower panels for each band, and the corresponding contour plots are given. Reference frame for the contour plots shows the image scale only. The pulsar and nearby objects o1, o2 and o3 are marked in the H band contour plot. The crosses in the J s contour plots show the pulsar position as derived from the HST/WFPC2/F555W image with an uncertainty negligible for this image scale. The arrow in the J s contour plot shows the pulsar proper motion direction, see Sect 2.2 for details.
The data were reduced with the Eclipse 3 and NOAO IRAF software. Dark and flatfield images were prepared using standard Eclipse recipes. Then each ObsID was considered separately. Image offsets were determined using the geomap routine and four field stars. The sky background level was determined and the images were summed using the routine jitter with the parameters RejectHalfWidth = 7, RejectMin = 2, RejectMax = 4, which were chosen based on the resulting image statistics. The parameters of the fully reduced images for each ObsID are summarized in Table 2. Finally, offsets between these images were determined and the images were combined. Parts of the resulting images 4 containing the pulsar are shown in Fig. 1 of the contour maps correspond to the levels (in counts) above the background l n = S + nσ, where S is the mean sky value near the pulsar, σ is the sky standard deviation per pixel, and n = 1, 2, . . . , 6.
Astrometry and morphology of the pulsar field
For astrometrical referencing of the VLT images we used the HST/WFPC2 image obtained on January 15, 2000 ). The pulsar is clearly detected in this image. Positions of 11 reference stars from the image were used to construct the coordinate transformation between the HST and VLT images with the IRAF routines geomap/geoxytran. The rms errors of the transformation were less than one third of the ISAAC pixel size in both RA and Dec. The pulsar position in the ISAAC images at the epoch of the VLT observations was calculated using the pulsar pixel coordinates in the HST image and the PSR Fig. 2. 10 ′′ ×10 ′′ vicinity of the Vela pulsar in the J s band. Objects o2, o3, and four stars in the upper left corner of the image are subtracted (cf. Fig. 1). The arrow shows the pulsar proper motion direction. A faint thin elongated object oriented approximately perpendicular to the proper motion is seen within the region bounded by the ellipse. It coincides with the central part of the inner arc of the PWN detected in X-rays (see Sect. 3.4 for details).
pulsar proper motion ). This position is marked by a cross in the contour plots of the J s images in Fig. 1. The pulsar counterpart is clearly detected with the offsets −0. ′′ 01(5) 5 and 0. ′′ 01(2) in RA and Dec, respectively, from this position. The errors account for the centering uncertainties in the ISAAC images, coordinate transformation and the pulsar proper motion uncertainties.
Three point-like objects, o1, o2, and o3, are detected in the 6 ′′ ×6 ′′ vicinity of the pulsar. They are marked in the H-band contour plot in Fig. 1, and their offsets from the pulsar position are listed in Table 3. The faintest object o1 is detected only in the H band.
An extended structure is seen in the 1. ′′ 5 vicinity south of the pulsar. The structure is more pronounced in the H band, but can also be marginally detected in the J s image, although with a different shape. The examination of the structure in each ObsID shows that its shape varies from one ObsID to another. To discriminate the pulsar from the extended structure, the IRAF/DAOPHOT software was used to construct a point spread function (PSF) in both bands using 10 field stars. The pulsar spatial profile was fitted with this PSF and subtracted from the images. The subtracted images are presented in Fig. 1.
The extension of the structure in the J s band is aligned with the south-east counter-jet from the pulsar detected in X-rays with the Chandra observatory (Helfand et al. 2001;Pavlov et al. 2001a). The X-ray counter-jet is directed opposite to the pulsar proper motion marked by an arrow in the J s contour plot in Fig. 1. It extends up to ∼ 100 ′′ from the pulsar. The ISAAC structure can be a near-IR signature of the X-ray jet in the 2 ′′ 5 Hereafter the numbers in parentheses are uncertainties referring to the last significant digit quoted, for example, 0.01(5) = 0.01 ± 0.05, 22.04(18) = 22.04 ± 0.18. vicinity of the pulsar. However, visual inspection of the 2 ′′ vicinity of the pulsar in the Chandra/HRC image does not reveal any such structure. It may be hidden in the complicated pulsar PSF profile of the HRC image.
We found in the ISAAC images also a hint of a faint thin elongated structure, overlapping with the central part of the inner arc of the Vela PWN in the Chandra/ACIS images (Pavlov et al. 2001a). The structure is seen within the ellipse in Fig. 2. It is aligned approximately perpendicular to the pulsar proper motion direction, as is the X-ray arc, and its offset from the pulsar is 3. ′′ 1 along this direction. The structure is detected at only ∼ 2σ level and only in the J s band (see Sect. 2.3 for details). However, inspection of each separate ObsID image (see Table 2) shows that the structure is absent only in the first J s image, which has the worst seeing, but it is present in the two other images.
To search for the detected extended structures in the images in adjacent bands we examined also the archival RI band images of the pulsar field, obtained with the VLT/FORS 6 on April 12, 1999 (Wagner & Seifert 2000), and in the F675W (overlaps with R) and F814W (overlaps with I) bands, obtained with the HST/WFPC2 7 on March 19, 2000 and on March 15, 2000, respectively ). The reduced images are shown in Figs. 3 and 4. The pulsar is reliably detected in all bands, while the extended structures and o1 are not seen in any of them. The extended structure near the pulsar could not be seen even after the pulsar PSF was subtracted in the RI bands (Fig. 3). The object o2 is seen in both HST bands (integrated exposure time 2600 s), but it is only barely visible in the VLT I band and not detected in the short R band exposure (300 s). This object and the extended structures are also not visible in the HST/WFPC2/F555W image . This means that the detected extended structures, as well as o1, are red objects. To conclude whether these objects are associated with the pulsar nebula or they are background objects, additional observations are required. A change in brightness of these objects would strongly support their association with the highly variable structure of the PWN, as has been observed in Xrays (Pavlov et al. 2001a). Note that wisp structures have been detected and studied close to the Crab pulsar in Xrays and in the optical (Hester et al. 2002), and in the IR (Sollerman & Flyckt 2002). In X-rays and in the optical, the observations show that the wisps vary in flux on a time scale of about one day.
Photometry
The photometric zeropoints for the ISAAC observations, J s0 = 24.81(4) and H 0 = 24.56 (7), were derived using images of the standard star FS13, observed on December 15, 2000. The J band catalog magnitude of FS13 (Hawarden et al. 2000) was used for the J s band. Differences in zeropoints between different ObsIDs were estimated comparing magnitudes of 6 field stars. The differences proved to be no larger than 1% in the J s and 2% in the H bands, and were accounted for only in the resulting photometric error budget. The average Paranal atmospheric extinction of 0.06 mag airmass −1 was used in both bands 8 . The fluxes of point-like objects were measured in apertures with diameters closest to mean seeing values (see Table 2), i.e., 4 pix in J s and 3 pix in H. Aperture corrections for the magnitudes were determined using 3 field stars. These stars, as well as the aperture diameters, were chosen to minimize the uncertainties of aperture corrected magnitudes of the faint pulsar, in both filters simultaneously. An additional check of the photometry was performed using the IR-survey 2MASS 9 . Since the region of the Vela pulsar is not yet released in the archive catalog, we performed photometry of 10 stars present in the J and H bands of the 2MASS "Quicklook Images" and in our ISAAC images. Their comparison shows that magnitude discrepancies are (J s ) ISAAC − J 2MASS = 0.15(12) and H ISAAC − H 2MASS = 0.03(10). The discrepancy in the J band can be partly attributed to the difference in throughput of the J and J s filters. Although photometry on the 2MASS "Quicklook Images" is not recommended because of their poor quality, we regard the obtained coincidence at the 1σ level as a confirmation of the accuracy of our photometric referencing of the ISAAC data.
The pulsar magnitudes, determined with aperture photometry, are J s = 22.61(9) and H = 21.90(13). In addition, the pulsar profile was fitted with the PSF constructed using IRAF/DAOPHOT. The PSF-fitted magnitudes are J s = 22.71(10) and H = 22.04(16). They differ by 1σ from the aperture photometry magnitudes. This difference may be attributed to the contamination of the pulsar counts measured with the aperture photometry by the extended structure near the pulsar. We therefore consider the PSFfitted magnitudes more reliable. Using these magnitudes and the flux zeropoints by van der Bliek et al. (1996), the measured fluxes from the pulsar are F Js = 1.39(12) µJy and F H = 1.64(25) µJy. We performed aperture photometry of the nearby objects o1, o2, and o3, which are marked in Fig. 1. The results of the photometry are summarized in Table 3.
We also measured the surface brightness of the extended source in the 1. ′′ 5 vicinity of the pulsar. It was measured on the images with the pulsar subtracted over the area (2.4 arcsec 2 , the same in both bands) which covers the brightest parts of the structure. The surface brightnesses are 22.98(5) mag arcsec −2 or 1.08(5) µJy arcsec −2 , and 21.51(7) mag arcsec −2 or 2.66(16) µJy arcsec −2 , in the J s and H bands, respectively. The surface brightness of the second structure projected at the PWN inner arc (see Fig. 2) in the J s band is 25.8(7) mag arcsec −2 or 0.08(4) µJy arcsec −2 , and the respective upper limit in the H band is 24.44 mag arcsec −2 or 0.18 µJy arcsec −2 .
Since no photometric standards were observed during the night of the VLT observations in the RI bands, the photometric equations were determined from the Landolt standards 10 , observed on the night before: Table 3. Photometry of the Vela pulsar (PSR) and the nearby objects o1, o2, and o3, marked in Fig. 1. The offsets of the objects from the pulsar position are given in the second column. Each spectral band cell for the PSR and o1 consists of two pairs of magnitude/flux values, divided by lines: upper pair are measured values, lower pair are dereddened values. Dereddening was performed with E B−V = 0.055(5). Each pair consists of the magnitude (upper value) and the flux in µJy (lower value). Only measured magnitudes/fluxes are presented for o2 and o3. All magnitudes are measured via aperture photometry, except for the pulsar magnitudes in the J s and H bands, which are measured with PSF fitting (see Sect. 2.3 for details). No magnitudes for HST bands are calculated. Empty cells mean that the object is not measurable in this band. (5) 18.17 (87) 21.53(4) 5.84 (21) -5.361 (59) 22.70 (7) 2.52 (16) -2.522 (48) Here R and I are the Cousins magnitudes, r and i are the instrumental magnitudes. The ambient conditions monitor 11 shows that the average atmospheric extinction coefficient during the observations of the standards was in the range 0.140−0.155 mag/airmass, and at the time of the pulsar field observations it was 0.150 mag/airmass. Since the difference between them is negligible compared to the uncertainties of the equations presented above, we used these equations for the photometric referencing of our observations without corrections for the extinction variations between the nights. The aperture corrections were done using a PSF constructed from 4 field stars. The measured pulsar magnitudes, R = 23.46(11) and I = 22.90(13), correspond to the fluxes F R = 1.24(12) µJy and F I = 1.65(19) µJy, using the magnitude-flux conversion zero-points provided by Fukugita et al. (1995). The flux/magnitudes of the pulsar and o3 are shown in Table 3.
Pipeline-provided zeropoints and pivot wavelengths were used for the flux calibration of the HST observations (2.51 × 10 −18 erg cm −2 s −1Å−1 / 7995Å and 2.90 × 10 −18 erg cm −2 s −1Å−1 / 6717Å in the F814W and F675W bands, respectively). Aperture photometry was performed for the pulsar and the objects o2 and o3. Aperture corrections were derived from a nearby relatively bright star. The measured pulsar flux (see Table 3) in the F814W band is consistent with the published one , while the flux in the F675W band is apparently ≃25% higher. Both fluxes are compatible with the less accurate VLT fluxes in the RI bands described above. We measured also the fluxes of the nearby objects o2 and o3 and estimated 3σ upper limits of the object o1 in the HST bands. 11 Available at http://archive.eso.org/asm/ambient-server
In Table 3 we also present dereddened magnitudes and fluxes for the pulsar and o1 using E B−V = 0.055(5) (A V ≈ 0.18, R = 3.1). This corresponds to the column density N H = 3.3(3) × 10 20 cm −2 , derived from the combined PL + NS-atmosphere spectral fit of the Vela pulsar X-ray data obtained with the Chandra observatory (Pavlov et al. 2001b). The extinction value is consistent with the new distance to the Vela supernova remnant (SNR) of 250±30 pc (Cha et al. 1999), and with the highest value over the Vela SNR N H ≃ 6 × 10 20 cm −2 (A V ≃ 0.32) found for its southern part (Lu & Aschenbach 2000).
The colors of the brightest stellar object in the pulsar vicinity, o3, suggest that it could be a main sequence K5-K7 star at a distance of ≃ 9 kpc, assuming A V ≈ 2. This is a much higher extinction than we have adopted for the pulsar, but it is consistent with the maximum possible Galactic extinction in the Vela direction, A V ≈ 4 (N H ≃ 7.5 × 10 21 cm −2 ; e.g., Schlegel et al. 1998). The colors of the fainter object o2 are roughly consistent with a cooler and even more distant main sequence star (of spectral type M0) at ≃ 10 − 11 kpc (A V ≈ 3). The object o1 is too red to be consistent with any ordinary Galactic star. It may be associated with the pulsar nebula. It may also be a background extragalactic object, as could also the objects o2 and o3. We discuss this further in Sect. 3.4.
Discussion
Multiwavelength spectrum of the Vela pulsar
In Fig. 5 including the radio (EPN 12 ; S. Johnston, 2002, private communications), the optical ), X-rays from the Chandra (Pavlov et al. 2001b), hard X-rays from the RXTE (Harding et al. 2002) and OSSE (Strickman et al. 1996), and γ-rays from the COMPTEL (Schönfelder et al. 2000) and the EGRET (Kanbach et al. 1994). Here we present unabsorbed fluxes dereddened with the color excess E B−V = 0.055(5) applied to both the optical and X-ray regions. The VLT data in the RI bands are omitted since they are compatible with the more accurate HST fluxes in the respective F675W and F814W bands (cf. Table 3). While the pulsar flux generally decreases with increasing frequency, one can resolve several nonthermal spectral components with different slopes in different spectral ranges. They are presumably of the pulsar magnetospheric origin. An excess in soft X-rays is attributed to the thermal emission from the surface of the NS (Ögelman et al. 1993;Pavlov et al. 2001b).
Phase averaged spectra of the pulsar in the optical and X-rays
As was noted by Pavlov et al. (2001b), , Harding et al. (2002), and seen from Fig. 5, the optical emission of the Vela pulsar is likely to be of nonthermal origin and the optical data are roughly compatible with the low energy extension of the X-ray PL spectral component dominating in the 2 − 10 keV range. This may suggest a similar nature of the optical and the high energy X-ray emission. However, as seen from Fig. 5, because of the limited statistics of the available Chandra data in the 2 − 10 keV range, the extension of the PL component inferred from the Chandra X-ray fit is much less certain than the near-IR and optical data. At the same time, the RXTE data, being compatible with the Chandra results, appear to be less uncertain. A difference in normalization of the PL components detected with Chandra and RXTE can be seen in Fig. 5. It may be due to the fact that only the pulsed component is detected by RXTE (Harding et al. 2002 from the pulsar including an "off-pulse" component, show that the pulsed fraction in the RXTE range can be as high as 80% (Sanwal et al. 2001) and the contribution of the off-pulsed component may not strongly affect the spectral shape. Excluding the 5th most energetic RXTE band, which is likely related to a flatter spectral component dominating in the OSSE and COMPTEL ranges (see Fig. 5), the RXTE spectrum can be fitted with a PL with the spectral index α RXTE ν ≃ 0.41(9). Although this fit is statistically inconsistent (χ 2 = 3.7 per dof), it reflects the spectral slope in the RXTE range: the best fit line lies within a narrow stripe-shaped region shown at the right panel of Fig. 6. Its extension to the optical range overlaps with the optical spectrum. A similar behavior was observed for the middle-aged pulsar PSR B0656+14 (Koptsevich et al. 2001). But the slope of the optical broad-band spectrum as a whole appears to be significantly flatter, α opt ν = 0.12(5) (χ 2 = 0.7 per dof, see the left panel of Fig. 6), and it is not possible to fit all optical and RXTE data with a single PL. We note also deviations from the single power law IR-Optical fit at about 1σ level seen in the J s and U bands.
Such behavior of the phase-averaged optical spectrum suggests that it can be a combination of several spectral components dominating at different phases of the pulsar light curve, as it is seen in the RXTE range (Harding et al. 2002). This can only be proven by deep time-resolved photometry. To our knowledge, no such data have been obtained yet for the Vela pulsar. The most recent "white-light" time-resolved photometry reveals three peaks in the pulsar light curve in the optical range (Gouiffes 1998). In contrast to that, up to 5 peaks were registered in the RXTE bands, and their PL spectra have significantly different indices and intensities. The second RXTE peak consists of two components, soft and hard, and coincides with the second optical peak (Harding et al. 2002). The second hard peak dominates the whole phase-averaged spectral flux, except for the 5th RXTE band, where the first peak with a positive spectral slope contributes significantly, providing a smooth connec-tion to the spectral data in the OSSE range (cf. Fig. 5). The first RXTE peak coincides with the first γ-ray peak.
The measured fluxes in the J s and H bands may imply that the pulsar spectrum could be steeper in the IR than in the optical, as was also observed for the middle-aged pulsar PSR B0656+14 (Koptsevich et al. 2001). Deeper observations of Vela in the H band are needed to state this possible similarity with greater confidence. What is more obvious is that the spectral slope in the near-IR is compatible with the RXTE slope. Combining the first four RXTE bands with the J s and H bands gives a consistent PL fit with α IR−RXTE ν = 0.417(6) (χ 2 = 1.3 per dof) shown by a dashed line in Figs. 5 and 6. The optical bands show a flux excess over this fit.
To better match the whole optical range we combined this nonthermal component with the thermal NS atmosphere component describing the soft X-ray part of the Chandra data (Pavlov et al. 2001b, combined HRC+ASIC fit). The combined model spectrum is shown by solid lines in Figs. 5 and 6. However, the atmosphere component decreases the residuals significantly only in the U and the softest RXTE bands, by contributions from the Rayleigh-Jeans and Wien tails of the thermal emission, respectively. The rest of the optical bands still show a significant excess over the combined atmosphere + PL model. Based on that, we can speculate that the 2nd peak is responsible for the phase averaged emission in both the RXTE and near-IR ranges, while the excess in BVRI is mainly produced by another phase and/or spectral component.
Deeper Chandra observations of the Vela pulsar in the high energy tail of its X-ray emission are needed to perform more accurate phase averaged spectral analysis of the optical and X-ray data. The high spatial resolution of Chandra should avoid the possible uncertainty of the RXTE fluxes which do not properly account for the offpulsed component of the pulsar emission. In this context, we can assume that the contribution of this component just increases the RXTE fluxes by a factor of ≃ 1.6 (+0.2 in Log scale) in all bands to match the Chandra best PL fit line (double-dot-dashes in the right panel of Fig. 6). In this case, the low energy extension of the RXTE PL component would only overlap with the upper part of the B band error-bar, and would suggest a spectral break of the nonthermal component near the UB bands. On the other hand, if the current Chandra best PL fit is closer to reality, the break between the optical and X-ray slopes of the nonthermal component would be near 50 eV, i.e., in the EUV range. We consider these as alternative hypotheses to be tested by future observations.
Optical spectrum of the Vela pulsar and spectral evolution of the pulsar optical emission
The near-IR data extend significantly the broad-band optical spectrum of the Vela pulsar towards longer wavelengths. This allows a detailed comparison with the prop- The youngest Crab (Sollerman 2003) is at the top, the oldest Geminga (Komarova et al. 2002) is at the bottom.
erties of other pulsars observed in the near-IR. In Fig. 7 we compare the optical-IR part of the spectrum of the Vela pulsar with the available phase-averaged optical and near-IR spectra of other pulsars of different ages. Our photometry of the Vela pulsar in the HST/F675W and VLT/R bands does not confirm a dip in this range . Moreover, changing the extinction from A V = 0.4 (Manchester et al. 1978) to the most recent value A V = 0.18 changes the average spectral index in the optical from α ν = −0.2(2) to α opt ν = 0.12(5). This is different from the most recently estimated positive slope of the spectrum of the younger (τ ≈ 10 3 yr) Crab pulsar, which shows a monotonous flux increase from the IR to the FUV range (Sollerman et al. 2000;Sollerman 2003).
Spectra of the middle-aged pulsars PSR B0656+14 and Geminga have dips near around UB bands, and the spectrum of PSR B0656+14 has a strong increase in the near-IR. We see signs of these features also in the Vela spectrum, although at a low significance level. The spectrum of PSR B0656+14 increases significantly towards the IR (Koptsevich et al. 2001), contrary to the spectrum of the Crab pulsar.
Extended structures
Our IR observations allowed us to detect faint extended structures ≃1. ′′ 5 SE and ≃ 3. ′′ 1 NW of the pulsar. These are projected on the SE counter-jet and on the inner arc of the PWN detected in X-rays (Helfand et al. 2001;Pavlov et al. 2001a). We also see a point-like object o1 ≃1. ′′ 2 NW of the pulsar, projected on the NW X-ray PWN jet. All these objects are red and have no reliable counterparts in the optical bands. It is difficult to conclude whether these features are associated with the PWN or they are background objects.
Observations of the much more energetic Crab PWN show that some of its structures are brighter in the IR than in the optical range. For instance, the knot structure, which is only 0. ′′ 6 SE of the Crab pulsar, is more luminous in the IR and has a much steeper spectrum (α knot ν ≃ 0.8) than the Crab pulsar itself (Sollerman 2003). This is consistent with the spatially averaged PL X-ray spectrum of the Crab PWN (Gotthelf & Olbert 2002). Some of the Crab wisps are also better resolved in the IR. Based on that, and on the faintness of the Vela PWN as compared with the Crab one, it would not be too surprising if the Vela PWN could be detected more easily in the near-IR than in the optical.
In this context, the measured flux in the H band, and our 3σ detection limits in the J s , HST/F814W and HST/F675W bands of the point-like object o1 resembling the Crab knot, suggest α o1 ν > ∼ 1.8. Thus, its spectrum is much steeper than that of the Crab knot and also steeper than a spatially averaged X-ray spectrum of the Vela PWN with α PWN ν ≃ 0.5 (Gotthelf & Olbert 2002). We are not aware of any reasonable physical mechanism for such a strong change of the spectral index from X-rays to the optical range, and we therefore believe that o1 is likely to be a distant highly absorbed extragalactic object.
On the contrary, as seen from Fig. 8, the surface brightness in J s and its upper limit in H of the IR inner arc shown in Fig. 2 are compatible with the low energy extension of the time and spatially averaged X-ray spectrum of the Vela-PWN inner and outer arcs. The X-ray arc spectrum can be fitted with a PL with α ν ≃ (0.3 − 0.5) and unabsorbed energy integrated brightness ≃ (3 − 6) × 10 −14 ergs s −1 cm −2 arcsec −2 in the 0.1 − 10 keV range (Kargaltsev et al. 2002;Mignani et al. 2003). We consider this, and the positional coincidence of the inner IR and Xray arcs as arguments in favor of the first detection of a counterpart of the Vela PWN in the near-IR range. The Fig. 8. Time and spatially averaged unabsorbed spectrum of the surface brightness of the Vela PWN inner and outer arc regions in X-rays fitted by a PL (Kargaltsev et al. 2002) together with optical (Mignani et al. 2003) and radio (Lewis et al. 2002) upper limits, and the brightness of the suggested inner arc counterpart in the J s band and its upper limit in the H band marked by a box. Filled regions show uncertainties of the X-ray fit (dashed line) and its extension into the optical range. The ellipse outlines the brightness of the near-IR extended structure projected at the SE X-ray counter-jet in ≃1. ′′ 5 vicinity of the Vela-pulsar.
IR brightness of the structure is also consistent with the deepest optical upper limit of 0.057 µJy obtained recently in the HST/F555W band (Mignani et al. 2003). If it is a real counterpart of the inner arc with the spectral slope described above, only slightly deeper observations, presumably at longer wavelengths, would allow a detection of this PWN structure. The extended IR source closest to the pulsar, and apparently projected on the SE X-ray counter-jet, is an order of magnitude brighter than expected from an extrapolation of the X-ray spectrum into the near-IR/optical range (Fig. 8). Its IR spectrum is also much steeper, α ν ≃ 2.5, than the spectrum of the PWN further away from the pulsar. As in the case of o1, this does not argue in favor of it being associated with the PWN. It could, however, be that the inner jet structure is brighter and has a steeper spectrum, because of possible instabilities of the relativistic particle flow from the pulsar as well as higher radiative losses at shorter distances from the pulsar. Although the upper limit on the optical flux from the IR source agrees with an extrapolation of the flux in the J s and H bands (see Fig. 8), we emphasize that the optical and near-IR images are from different epochs. As mentioned in Sect. 2.2, the emission of the regions close to the Crab pulsar varies on a short time scale, and future comparison between optical and near-IR emission in the vicinity of the Vela pul-sar would benefit from simultaneous observations in these wavelength ranges.
In X-rays the Vela PWN shows high variability of its jet and arc-like structures in position, intensity, and hardness ratio (Pavlov et al. 2001a). Thus, further deep observations of the pulsar field in the near-IR and in the optical would be useful to search for the variability and to prove or reject the association of the detected extended structures SE and NW of the pulsar and o1 with the PWN. Observations of the Vela pulsar in the KL bands would be valuable to investigate the possible increase of its flux towards the IR range. Finally, time resolved photometry and spectral information on the emission of different optical peaks of the pulsar pulse profile would be crucial to understand to which extent nonthermal optical radiation of the Vela pulsar is of the same origin as the nonthermal spectral component seen in the high energy tail of its X-ray spectrum, or whether it is generated by different radiation mechanisms.
Fig. 1 .
1Large panel: 30 ′′ × 30 ′′ overview of the Vela pulsar field extracted from the 2. ′ 5 × 2. ′ 5 frame of the VLT/ISAAC image in the J s band. The region bounded by the 6 ′′ -box is enlarged in the small panels. The same region in different bands is shown in Figs. 3 and 4. The region bounded by the 10 ′′ -box is shown in
Fig. 3 .
36 ′′ × 6 ′′ vicinity of the pulsar, as in small panels ofFig. 1, obtained with the VLT/FORS1 in R and I bands(Wagner & Seifert 2000). Left panels show original images. Right panels show images with the modeled pulsar profile subtracted.F675W F814WFig. 4. 6 ′′ × 6 ′′ vicinity of the pulsar, as in small panels of Figs. 1, and inFig. 3, obtained with the HST/WFPC2 in the F675W and F814W bands.
R
− r = 26.583(09) + 0.052(17)(r − i) I − i = 25.664(13) − 0.068(23)(r − i)
Fig. 5 .
5Dereddened multiwavelength spectrum of the Vela pulsar obtained with different telescopes as marked in the plot. Diamond-shaped fillings represent 1σ confidence regions of the NS-atmosphere + PL fit (dot-dashed line) of the Chandra data; double-dot-dashed and dotted lines show the contributions of nonthermal PL and thermal atmosphere components, respectively. The dashed line shows the PL fit of the near-IR and RXTE data and the solid line represents the sum of this PL component with the atmosphere component. The optical and RXTE ranges are shown enlarged in Fig. 6.
Fig. 6 .
6Blow-up of the optical and X-ray parts of the Vela pulsar spectrum presented in Fig. 5. The scale in both panels is the same. Left panel: IR-UV part of the pulsar spectrum. IR and optical bands are marked. Diamondshaped, stripe-shaped and solid fillings represent 1σ confidence regions of the NS-atmosphere + PL, IR-RXTE, and IR-Optical fits, respectively. The best fits and contributions of different spectral components are shown by different types of lines explained in the right panel. Right panel: X-ray part of the spectrum. Fluxes in the RXTE bands are marked by crosses. The flux in the hardest RXTE band likely belongs to a flat spectral component detected with the OSSE/COMPTEL and was not used in the IR-RXTE fit. Line types are the same as in the left panel and Fig. 5.
Fig. 7 .
7Comparison of the optical spectra of four pulsars.
Table 1 .
1Parameters of the Vela pulsar (PSR B0833−45;
).Observed
Derived
PṖ
µα cos(δ), µ a
δ
π a
l, b b
DM c
τ d
BĖ
d a
ms 10 −15
mas yr −1
mas
cm −3 pc
Myr
G
erg s −1
pc
89
125
−37.2 ± 1.2
28.2 ± 1.3
3.4 ± 0.7
263. • 6
−2. • 8
68.2
0.011 3.38 × 10 12 6.9 × 10 36 294 +76
−50
a Proper motion, parallax, and parallax-based distance (Caraveo et al. 2001) c Dispersion measure
b Galactic coordinates
d Dynamical age
Table 2 .
2Log of VLT/ISAAC observations of the Vela pulsar. Refers to the first image of the ObsID. b Full width at half maximum of the stellar profile.Band Date
Time a Exposure Airmass Seeing b
UT
UT
s
arcsec
Js
14.12.00 7:45
2340
1.072
0.7
15.12.00 6:33
2340
1.084
0.6
7:18
2340
1.069
0.5
H
15.12.00 8:03
1014
1.076
0.5
8:29
1092
1.094
0.5
8:54
936
1.118
0.7
05.01.01 6:20
2028
1.073
0.5
7:05
2028
1.103
0.4
7:51
1170
1.150
0.4
a
. Contour maps of these images are also presented in this figure. Isophotes See http://www.eso.org/projects/aot/eclipse/. //www.ioffe.ru/astro/NSG/obs/vela-ir/.3 4 Images
are
available
in
FITS
format
at
http:
See http://www.eso.org/instruments/isaac/imaging stan dards.html8
we have combined our IR data with the available phase-averaged multiwavelength fluxes of the Vela pulsar10
15
20
25
Log ν [ Hz ]
-4
-2
0
2
4
6
Log F
ν [ µJy ]
Chandra
atm+PL
RXTE
OSSE
COMPTEL
EGRET
VLT+HST+NTT
-8
-4
0
4
Log E [ keV ]
PSR B0833-45
). But the Chandra observations, representing the total flux14.5
15
15.5
Log ν [ Hz ]
0
0.2
0.4
0.6
Log F
ν [ µJy ]
H
J
F 8 1 4 W
F 6 7 5 W
F 5 5 5 W B U
-3
-2.5
-2
Log E [ keV ]
18
18.5
19
-1.4
-1.2
-1
Atm (Chandra) + PL (Chandra)
Atm (Chandra)
PL (Chandra), α ν = 0.5
PL (IR-RXTE), α ν = 0.42
Atm (Chandra) + PL (IR-RXTE)
PL (IR-Optical), α ν = 0.12
0.5
1
1.5
Vela pulsar
Hereafter PL, Fν ∝ ν −αν .
See http://www.eso.org/instruments/isaac/ for details on the instrument, filters and observational technique.
Based on ESO programme 63.P-0002. 7 Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.
http://irsa.ipac.caltech.edu/ 10 Fields of PG1323−085, PG1633+099, and PG1657+078(Landolt 1992).
European Pulsar Network (EPN) archive available at http://www.mpifr-bonn.mpg.de/div/pulsar/data.
Acknowledgements. We are grateful to Soroush Nasoudi-Shoar for initial help with data reductions, to Simon Johnston for providing us with unpublished data on the radio spectrum of the Vela pulsar, to Alice Harding and Mark Strickman for tabulated results of the RXTE observations of the Vela pulsar, to Stefan Wagner for providing us with unpublished VLT data in the I band, to George Pavlov and Roberto Mignani for discussions and for access to the paper on the search for the optical counterpart of the Vela PWN prior to publication, and to the referee Stephen Eikenberry for comments which allowed us to clarify better several points in text. ABK and YAS are grateful to Stockholm Observatory and the Royal Swedish Academy of Sciences, and ABK to the University of Washington, for hospitality. This work has been partially supported by the RFBR grants 02-02-17668, 03-02-17423, and 03-07-90200, the Royal Swedish Academy of Sciences and the Swedish Research Council. PL is a Research Fellow at the Royal Swedish Academy of Sciences supported by a grant from the Wallenberg Foundation.SummaryHere we provide a summary of our most important results.1. We have, for the first time, detected the Vela pulsar in the near-IR in the J s and H bands. 2. Our IR fluxes combined with the available broadband optical data confirm the nonthermal origin of the pulsar emission in IR-optical range. The combined phase-averaged unabsorbed IR-optical spectrum is fitted with a single PL with a negative slope. This is in contrast to a positive slope of the unabsorbed spectrum of the younger Crab pulsar. 3. The IR-optical spectrum and the phase-averaged PL spectral component detected in the high energy tail of the pulsar X-ray emission cannot be fitted with a single PL. This suggests either a spectral break in the NUV-EUV range, or the presence of an additional spectral component with a flatter spectrum dominating in the optical range. In the latter case, the IR and the Xray spectrum can be fitted with a single PL suggesting the same origin of the nonthermal pulsar emission in the second pulse of the pulsar pulse profile in both the X-ray and IR ranges. 4. We detected two faint objects in the 1. ′′ 5 vicinity of the pulsar. They are projected on the SE counter-jet and the NW jet of the Vela PWN detected in X-rays. Both of them are extremely red and have no counterparts in the optical range. Their IR fluxes are apparently inconsistent with the expected IR brightness of the PWN obtained by extrapolation of its X-ray spectrum into the IR range. Finding variability of the objects would strongly support their association with the highly variable PWN. 5. A thin extended structure aligned with the inner arc of the X-ray PWN is marginally seen in the J s band. Its brightness is consistent with the X-ray PWN spectrum. However, its reality and association with the PWN structure need to be confirmed at higher significance level by deeper observations.
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|
[] |
[
"Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction",
"Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction"
] |
[
"Grégoire Ferré \nCapital Fund Management\n23-25 rue de l'Université75007ParisFrance\n"
] |
[
"Capital Fund Management\n23-25 rue de l'Université75007ParisFrance"
] |
[] |
We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings. *
|
10.1051/m2an/2023042
|
[
"https://arxiv.org/pdf/2111.10819v2.pdf"
] | 244,477,866 |
2111.10819
|
f35d1f7878f28ddf499218e5c625acfa8218297a
|
Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction
April 12, 2022
Grégoire Ferré
Capital Fund Management
23-25 rue de l'Université75007ParisFrance
Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction
April 12, 2022
We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings. *
Introduction
Rare events play a crucial role in many scientific contexts, from networks dynamics to molecular systems and climatic hazards. By rare we mean that the frequency of appearance of such events is low compared to the impact of its realization, relatively to the time scale of observation. If one is interested in sampling these phenomena to estimate probabilities of occurrence or average impact, standard methods generally lead to a high relative variance, hence poor quality of estimation. This can be understood heuristically, as few samples realize large values of the estimator. Hence, many samples are required to obtain an accurate estimation of the quantity of interest. This is typically not affordable in practice, since one single simulation of a realistic system may be already computationally expensive.
A natural goal is thus to reduce the relative variance of a naive Monte Carlo estimator, and we focus here on the case of diffusions. Many techniques have been designed so far, in particular genetic algorithms relying on a selection mechanism [6,1], and importance sampling, which modifies the system to enhance the probability of observing the chosen rare event [4]. We consider here the latter technique, which has been applied to various systems of interest, see for instance [7,24,19] and references therein.
However, there are infinitely many ways to bias a system in order to enhance the probability of rare events, many of them may not prove efficient in order to achieve variance reduction -a wrong bias may even deteriorate the properties of the estimator [18]. In the specific case of diffusion processes we know that, in general, there exists a unique optimal bias leading to a zero-variance estimator [14,30,24], and that this control is the solution to a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) depending on the parameters of the system [16].
Since a close form of the optimal control is not available in most cases, it needs to be approximated. As for any evolution PDE, finite difference techniques can be used [5,29], with error estimates available for instance in Lebesgue-like norms. More recently, a series of works also address the approximation of Hamilton-Jacobi problems through a more sampling-oriented point of view [30,22,31,19].
A major issue with all these techniques is that, although they may perform well in some practical cases, one is generally not ensured that the approximation of the optimal control indeed reduces variance. For the case of diffusions, it is proved for the time consuming approach proposed in [30], under a stringent condition of locally uniform convergence of the approximation. More recently, the authors in [20] provide general conditions for an abstract importance sampling estimator to be efficient. Although powerful, the generality of the latter approach makes it difficult to apply to diffusion processes.
The purpose of this paper is to provide a new criterion for assessing the performance of an approximate control in terms of variance reduction. For this we rely on the important work [25], which introduces a notion of solution to the Feynman-Kac and Hamilton-Jacobi equations that rely on a stochastic interpretation of the generator. This allows to consider non-smooth problems -we may call such solutions «stochastic viscosity solutions». This theory uses a finite entropy condition in relation with the Girsanov theorem and extended generators based on martingale techniques. Our main contribution is to adapt this notion of solution to an approximation context: this leads us to introduce the notion of stochastic viscosity approximation of order k (or k-SVA). Such approximate controls solve the Hamilton-Jacobi equation along trajectories tilted by the control itself up to a small process of order k in the temperature. In addition, a low temperature consistency should hold -these two conditions being related to low temperature variance and bias respectively. We naturally introduce these conditions through the Girsanov path change of measure, and then prove that an importance sampling estimator based on such a control is k-log efficient (in particular its relative variance ratio decays exponentially at order k − 1), which is our main result.
In order to illustrate that our method can be applied to actual approximation schemes, we consider the approach proposed in [19] to bias the dynamics along the instanton, or reacting path of the dynamics. We prove, under geometric conditions, that the estimator proposed in this work is a stochastic viscosity approximation of order 1, hence it is 1-log efficient. We then turn to the next order approximation presented in [11], showing that under similar conditions it provides a stochastic viscosity approximation of order 2, hence it is 2-log efficient and with logarithmic relative variance decaying linearly to zero. A simple numerical application confirms our theoretical findings.
Our results are organized as follows. Section 2 presents the general setting of the work with our assumptions, together with reminders on importance sampling and instanton trajectories. In Section 3, we present our main definition of stochastic viscosity approximation, and the associated variance reduction property. The case of the approximation expanding around the instanton path is presented in Section 4, with an illustrative numerical application. Finally, we discuss our results and further research directions in Section 5. Some technical proofs are gathered in the Appendix.
Controlled diffusions and Hamilton-Jacobi-Bellman equations
This section is devoted to generalities on controlled diffusions and importance sampling. We first present in Section 2.1 our setting together with the basic assumptions on the dynamics, before turning in Section 2.2 to the notion of importance sampling. Section 2.3 recalls some elements on the optimal control associated with our problem. We close in Section 2.4 with some facts on small temperature asymptotics that are not necessary for presenting stochastic viscosity approximations, but that will be useful in Section 4 to build such an approximate solution for a given system.
Dynamics and free energy
This section introduces some notation used throughout. We consider a diffusion process (X ε t ) t 0 evolving in R d with d ∈ N * , and satisfying the following stochastic differential equation (SDE):
dX ε t = b(X ε t ) dt + √ εσ dBt,(1)where b : R d → R d , σ : R d → R d×m and (Bt) t 0 is a m-dimensional
Brownian motion for some m ∈ N * . In the above equation, ε > 0 is a parameter that may vary. Conditions on b and σ are provided in Assumption 1 below, and at this stage the initial condition of (X ε t ) t 0 is an arbitrary probability measure. The dynamics (1) induces a semigroup of operators (P ε t ) t 0 on the space of continuous bounded functions
C b (R d ) defined by ∀ ϕ ∈ C b (R d ), ∀ t 0, P ε t ϕ : x ∈ R d → Ex[ϕ(X ε t )],(2)
where Ex denotes expectation over all realizations of (1) when the process is started at x ∈ R d . We will also use the notation Er,x when the process is started at position x at time r > 0. By continuity of the trajectories, the semigroup is strongly continuous on C b (R d ) and thus has a generator Lε with domain D(Lε) defined by
∀ ϕ ∈ D(Lε), Lεϕ = lim t→0 P ε t ϕ − ϕ t ,
where the limit is in supremum norm. When tested over smooth enough functions, Itô calculus shows that Lε is represented by the differential operator Lε defined by
∀ ϕ ∈ C 2 (R d ), Lεϕ = b · ∇ϕ + ε 2 D : ∇ 2 ϕ,(3)
where D = σσ T denotes the diffusion matrix and · is the scalar product on R d . Moreover, ∇ 2 denotes the Hessian operator, while for two matrices A, B belonging to R d×m , we write A : B = Tr(A T B).
The typical problem we want to address in this paper is the computation of the following free energy, or Feynman-Kac density
Aε = Ex 0 e 1 ε f (X ε T ) , and Zε = ε log Aε,(4)
where T > 0 is a final time and x0 ∈ R d a the fixed initial condition for (1) at time t = 0. The quantity Aε serves as a smoothed version of the probability:
P(X ε T ∈ S), for a Borel set S ⊂ R d . This probability, which is interesting for various applications, coincides with (4) when f (x) = 0 for x ∈ R d and f (x) = −∞ for x / ∈ R d . For technical reasons we rather use stronger assumptions gathered below, which still encompass interesting cases. Assumption 1. The following regularity requirements hold:
1. The matrix D = σσ T is positive definite. 2. The field b belongs to C ∞ (R d ) d and is globally Lipschitz, i.e. there is C lip such that ∀ x, y ∈ R d , |b(x) − b(y)| C lip |x − y|.
The function f belongs to C ∞ (R d ) and is upper bounded.
Under Assumption 1 it is standard that (1) has a unique strong solution for all times [23] and that Aε is finite for all ε > 0 (because f is upper bounded). The semigroup (P ε t ) t is a strongly continuous semigoup with elliptic (non-degenerate) generator Lε for which the representation (3) holds for smooth functions. These very restrictive conditions are imposed in order to avoid non-essential technical details, but can be relaxed depending on the situation of interest. For instance, the second point of the assumption ensures well-posedness of the SDE (1) and the associated PDE, but could be replaced by a local Lipschitz assumption together with Lyapunov condition on the generator [28]. Similarly, this global Lipschitz condition is used in Section 4 to obtain standard error estimates, but could also be replaced there by a local Lipschitz condition toegether with an appropriate control. Finally, the upper boundedness of f could be weakened under a Lyapunov condition to allow upper-unbounded functions, see the technique used in [13].
Estimators and importance sampling
In a high dimensional context, it impossible to estimate the integral in (4) by numerical quadrature over a discretized grid, which is why we typically resort to stochastic sampling of the expectation. We know that the estimator of Aε in (4) based on generating independent samples of the random variable
Aε = e 1 ε f (X ε T )
is unbiased but typically has high (relative) variance. This is the case in particular in the low ε regime, which is the one of interest in physical applications. A possible way to reduce the variance is to control the process (1) with a function g (for now arbitrary), and to consider the so-called tilted process ( X ε t ) t∈[0,T ] defined through the stochastic differential equation
d X ε t = b( X ε t ) dt + D∇g(t, X ε t ) dt + √ εσ dBt.(5)
We generically denote by ( X ε t ) t∈[0,T ] a process tilted by a function g to be made precise, and belonging to the class of functions described below.
Assumption 2. A real-valued function g is said to be a biasing or importance sampling function if
g ∈ C 1,2 ([0, T ]× R d )
and the Novikov condition holds, namely
Ex 0 e 1 2 T 0 |σ∇g| 2 ( X ε t )dt < +∞.(6)
Under Assumption 2, the Girsanov change of measure between the paths ( [23,Chapter 3,Proposition 5.12] or [28]) and the details in Appendix A) turns Aε into
X ε t ) t∈[0,T ] and ( X ε t ) t∈[0,T ] (seeAε = Ex 0 exp 1 ε f ( X ε T ) + g(0, x0) − g(T, X ε T ) + 1 ε T 0 (∂tg + Hεg)(t, X ε t ) dt ,(7)
where we introduced the Hamilton-Jacobi-Bellman nonlinear differential operator Hε defined by:
∀ g ∈ C 2 (R d ), Hεg = b · ∇g + ε 2 D : ∇ 2 g + 1 2 |σ∇g| 2 .
One can easily check that Hε is actually the logarithmic transform of Lε, namely
∀ g ∈ C 2 (R d ), Hεg = ε e −g/ε Lεe g/ε .(8)
This logarithmic transform is a standard tool for studying partial differential equations related to control problems, see for instance [16,27,10].
In what follows, for a function g to be specified and satisfying Assumption 2, we denote by
Gε = exp 1 ε g(0, x0) − g(T, X ε T ) + 1 ε T 0 (∂tg + Hεg)(t, X ε t ) dt
the Girsanov weight of the path change of measure. Thus (7) rewrites:
Aε = Ex 0 e 1 ε f ( X ε T ) Gε .
As a consequence of the Girsanov's theorem, the estimator of Aε based on independent realizations of the random variable
Aε = e 1 ε f ( X ε T ) Gε(9)
provides an unbiased estimator of Aε, i.e. it holds Ex
0 [ Aε] = Ex 0 [Aε] = Aε.
The goal of importance sampling is to find the best biasing function g to minimize the relative variance of the tilted estimator Aε. In order to provide a metric for the search of the best tilting g, we recall that the relative variance of the tilted estimator (see for instance in [30] and references therein) is defined by
Ex 0 e 1 ε f ( X ε T ) Gε 2 − Ex 0 e 1 ε f ( X ε T ) Gε 2 Ex 0 e 1 ε f (X ε T ) 2 = Ex 0 e 2 ε f ( X ε T ) (Gε) 2 Ex 0 e 1 ε f (X ε T ) 2 − 1.
We thus consider the ratio:
ρ(ε) = Ex 0 e 2 ε f ( X ε T ) (Gε) 2 Ex 0 e 1 ε f (X ε T ) 2 ,
which controls the evolution of the relative variance as ε → 0. Since ρ typically grows exponential as ε → 0, we are interested in the evolution of ρ at log-scale, and we thus introduce the following definition.
Definition 1. The relative log efficiency of an estimator is defined as
R(ε) = ε log ρ(ε).(10)
We say that an estimator is k-log efficient if there is k 0 such that
R(ε) = O(ε k ) when ε → 0.
In words, k-log efficiency describes a class of estimators with a small relative variance at exponential scale, in the low temperature regime. Although the notation k suggests that k is an integer (which will be the case in our examples), this does not need to be the case in general. Note also that, when 0 < k < 1, the relative log efficiency goes to zero but the relative variance is not even bounded (since R(ε) has an additional ε factor), it only grows at a subexponential rate. We are thus mostly interested in the case k 1 when the relative variance is indeed under control.
The optimal control
There is actually a simple way to find a solution to the problem of minimizing the relative variance of Aε. Consider for this the Girsanov formula (7): all the terms are stochastic except g(0, x0), which is deterministic. In order to reduce the variance to zero, one can therefore choose g = gε solution to
(∂tgε + Hεgε)(t, x) = 0, ∀ (t, x) ∈ [0, T ) × R d , gε(T, x) = f (x), ∀ x ∈ R d .(11)
If we assume that the solution to the above equation is well-defined, having ( X ε t ) t∈[0,T ] biased with gε indeed makes the random variable Aε deterministic, hence the associated estimator has zero variance. Actually, under Assumption 1, standard parabolic estimates 1 show that there is a unique smooth solution gε to (11) for all ε > 0. 1 Recall the logarithmic transform (8). Then ψε = e gε/ε is solution to the linear adjoint Fokker-Planck equation
(∂tψε + Lεψε)(t, x) = 0, ∀ (t, x) ∈ [0, T ) × R d , ψε(T, x) = e f (x) ε , ∀ x ∈ R d .(12)
Since f is upper bounded by Assumption 1 and the coefficients are smooth with non-degenerate Laplacian, the above PDE has a welldefined smooth solution by standard parabolic estimates. Moreover, since e f /ε > 0 the solution ψε is also positive and one can thus check that gε = ε log ψε is solution to (11).
We can assume that gε satisfies Novikov's condition (6), which can be checked in practical situations, but we actually do not need this property for the results presented in Section 3. Moreover, gε is associated with the following probabilistic representation:
gε(t, x) = ε log ψε(t, x),(13)
where ψε is defined by
ψε(t, x) = Et,x e 1 ε f (X ε t ) .(14)
We can prove by standard Itô calculus on ψε together with the logarithmic transform (8) that gε as defined by (13) is the solution to (11), see [17, Chapter 2, Section 3]. As a natural consequence of (13)- (14), we have
ψε(0, x0) = Aε,
and similarly
gε(0, x0) = Zε.(15)
In other words, the value of the optimal control at the starting point of the dynamics provides the free energy of the system. We shall recall this important feature in Section 3 when defining our notion of approximate solution.
Remark 1 (Stochastic viscosity solutions).
It is interesting at this point to observe that the condition for gε to solve (11) is too stringent to minimize the variance of Aε. It is indeed sufficient that
(∂tgε + Hεgε) (t, X ε t ) = 0, gε(T, X ε T ) = f ( X ε T ),(16)
for almost all t ∈ [0, T ] and almost surely with respect to the process ( X ε t ) t∈[0,T ] defined in (5) when the bias is gε. This notion of solution based on a stochastic representation is actually presented in details in the recent paper [25]. While [25,Theorem 5.9] is motivated by considering non-smooth coefficients and solutions for the HJB equation through extended generators, it motivated the current study for its relation with variance reduction. In other words, it suggests that the optimal control only needs to be well approximated around appropriately titled trajectories (and not everywhere in space) for relative variance to be reduced. This is the basis of the notion of stochastic viscosity approximation that we propose in Section 3.1.
Low temperature asymptotics
We conclude this section with some low temperature asymptotics concerning the dynamics. As mentioned above, these results are not needed for our presentation of stochastic viscosity approximations in Section 3. However, they will be crucial in Section 4 to propose examples of such approximations.
The zero temperature HJB equation reads
(∂tg0 + H0g0)(t, x) = 0, ∀ (t, x) ∈ [0, T ) × R d , g0(T, x) = f (x), ∀ x ∈ R d .(17)
This is a first order nonlinear PDE (recall H0ϕ = b · ∇ϕ + |σ∇ϕ| 2 /2) for which a smooth solution often does not exist, because of the lack of diffusive regularization. However, a viscosity solution can most often be described through the relation [17, Chapter 2, Theorem 3.1]:
∀ (t, x) ∈ [0, T ] × R d , g0(t, x) = lim ε→0 gε(t, x).
Although this will not be strictly necessary for our study, this relation provides intuition on what we will show next: the viscosity solution of (17) is the low temperature limit of the finite temperature solution. Our proposal in Section 3 is a combination of this small temperature (viscosity) technique with the notion of stochastic solution mentioned in Remark 1.
In some cases, mostly when the function g0 is smooth, we can describe the solution through its characteristics system. The same methodology applied in a large deviations context leads to the instanton dynamics, which is described by a forward-backward system [19,11] with R d -valued position (φt) t∈[0,T ] and momentum (θt
) t∈[0,T ] solution to φ t = b(φt) + Dθt, φ0 = x0, θt = −(∇b)(φt) · θt, θT = ∇f (φT ).(18)
This dynamics describes the path realizing fluctuations in the low temperature limit. In regular cases, the following assumption holds.
Assumption 3. The dynamics (18) admits a unique solution (φt, θt) t∈[0,T ] , which is of class C 1 ([0, T ]) d × C 1 ([0, T ]) d .
There are various ways to introduce the instanton dynamics (18). Let us provide a maybe unusual, PDEoriented motivation for this object, by introducing the function
∀ (t, x) ∈ [0, T ] × R d ,ĝ 1 ε (t, x) = f (φT ) − 1 2 T t θs · Dθs ds + θt · (x − φt).(19)
Note that we keep the ε dependency onĝ 1 ε although it does not depend on ε, because such an approximation may depend on ε in general, as shown in Section 4. This function is made of three parts: a constant term, a time-dependent only term, and a linear term in the variable (x − φt). We may thus callĝ 1 ε a time-inhomogeneous first order polynomial in the variable (x − φt).
To understand better this function, we first see thatĝ 1 ε (T, x) = f (φT )+∇f (φT )·(x−φT ), which is a linearization of f around the instanton's terminal point φT . Let us now compute the value of the HJB operator ∂t + Hε overĝ 1 ε for any (t, x) ∈ [0, T ] × R d :
∂tĝ 1 ε + Hεg 1 ε (t, x) = 1 2 |σθt| 2 +θt · (x − φt) − θt ·φt + b(x) · θt + 1 2 |σθt| 2 = |σθt| 2 − ∇b(φt) · (x − φt) − θt · (b(φ) + Dθt) + b(x) · θt = θt · b(x) − b(φ) − ∇b(φt) · (x − φt) = O (x − φt) 2 ,
where we used (18) for the second line, while the last inequality is derived by Taylor expansion. We actually obtain from the above computation that
(∂tĝ 1 ε + Hεĝ 1 ε )(t, φt) = 0, ∀ t ∈ [0, T ), g 1 ε (T, φT ) = f (φT ).(20)
In other words,ĝ 1 ε solves (11) along the instanton. Moreover, we recall that the instanton provides the limit of the free energy Zε as ε → 0. Indeed, we can easily prove in the setting of Assumptions 1 and 3, by expanding in the Girsanov formula (7), that it holds
lim ε→0 Zε = Z0 = f (φT ) − 1 2 T 0 θs · Dθs ds.(21)
In words, the limiting free energy is the maximum value of f that can be reached under quadratic penalization of the momentum. The free energy Zε can actually be expanded in power of ε to any order, see the discussion in [11, Section 3.2]. Finally, we notice the equalitŷ
g 1 ε (0, x0) = Z0,(22)
which is compatible with the relation (15) between free energy and optimal control in the zero temperature limit. Since our discussion in Remark 1 suggests that importance sampling only requires approximation of HJB around some trajectories and (φt) t∈[0,T ] describes the most likely fluctuation path as ε → 0, we understand that the ansatz (19) can serve as a first guess for approximating the solution gε of the finite temperature equation (11) along paths relevant with respect to the free energy consistency (22), and hence reducing the variance. We will see in Section 4 that this is indeed the case.
Remark 2 (About our assumptions). Before presenting our results, it is interesting to note that Assumptions 1, 2 and 3 are of two kinds:
• The conditions in Assumptions 1 and 2 are of technical nature. They ensure the well-posedness of the quantities at hand and sufficient regularity of the optimal control. They can easily be relaxed provided relevant estimates are made available.
• Assumption 3 is of geometrical nature, and has a crucial importance. It is easy to build examples for which all quantities are smooth, but where this assumption does not hold [18], typically for symmetry or convexity reasons. An example is to set d = 1, b = 0, σ = 1, f (x) = tanh(|x|) and x0 = 0: in this case the instanton may maximize the final value of f either by going to the right or to the left for the same kinetic cost, so there is no uniqueness of the instanton. This is a situation similar to that appearing when solving first order PDEs with the method of characteristics.
Stochastic viscosity approximations
We now present the main results of the paper: the definition of stochastic viscosity approximations in Section 3.1, and the associated variance reduction property in Section 3.2.
Definition
The goal of this work is to find criteria for a function g to reduce the variance of the estimator (9). For this, we saw in Section 2.3 that the optimal solution to the variance reduction problem is the solution to a HJB equation, whose associated estimator has zero variance. Since the solution to this equation is in general unknown, it is natural to try to approximate it. There exist a series of works dedicated to approximating Hamilton-Jacobi PDEs, typically by considering decentered finite difference schemes and assessing convergence through a Lebesgue or Sobolev distance to the exact solution [5,29,3,26,2]. Such convergence criteria rely on non-local quantities, and they are a priori not related to the problem of variance reduction. In other word, there is no continuous mapping between the estimation of the HJB equation and relative variance of the associated estimator. Moreover, we are interested in situations where the dimension d of the system is very large, so that finite difference schemes are typically not applicable. In this context of optimal stochastic control in high dimension, a series of more specific techniques have also been designed, for instance [30,22,31,9]; yet the same issue of proving that variance reduction is achieved for a given approximation remains -it may even deteriorate the associated estimator [18].
In order to provide an element of answer in this direction, and this is the main contribution of this paper, we now introduce a notion of approximation to the HJB problem (11) that is consistent with the goal of reducing the relative variance of the estimator Aε in the low temperature regime. Instead we follow the stochastic solution approach discussed in Remark 1 (see in particular [25] and references therein), which suggests for g to be an approximate solution of (11) around tilted trajectories. Let us now make this idea more precise with the following definition.
Definition 2 (Stochastic viscosity approximation). Let Assumption 1 hold, and letĝε be a function satisfying
Assumption 2 with ( X ε t ) t∈[0,
T ] the process tilted byĝε through (5). We say thatĝε is a k-stochastic viscosity approximation of (11) if there exist k, ε0 > 0, an R d -valued adapted process (Q ε t ) t∈[0,T ] (residual process), and a real-valued random variable Q ε f (residual terminal condition) such that the following conditions hold: • Free energy consistency: for ε ∈ (0, ε0],ĝ
ε(0, x0) = gε(0, x0) + O(ε k ).(23)
• Approximate HJB equation along tilted paths: for any ε ∈ (0, ε0],
(∂tĝε + Hεĝε)(t, X ε t ) = ε k Q ε t , gε(T, X ε T ) = f ( X ε T ) + ε k Q ε f .(24)
for almost all t ∈ [0, T ] and almost all realization of ( X ε t ) t∈[0,T ] 2 .
• Bound: There is a constant CQ 0 independent of ε such that the random variable
Q ε = Q ε f + T 0 Q ε t dt (25) satisfies ∀ ε ∈ (0, ε0], log E e Q ε CQ.(26)
Let us provide some comments on Definition 2. The main condition (24) is forĝε to be an approximate solution of (11) (including the boundary conditions) along trajectories tilted byĝε itself. By approximate we mean that evaluating the Hamilton-Jacobi operator along such stochastic trajectories is an appropriately bounded stochastic process (which we may call residual process) scaled by ε k . We thus understand the naming stochastic viscosity approximation, which we abbreviate as SVA.
Let us now turn to condition (23) by first recalling that, as shown in Section 2.3,
gε(0, x0) = Zε.
We thus require in (23) a consistency in terms of free energy: the initial value of the approximate control should already approximate well the deterministic quantity we are interested in, independently of sampling issues. This condition makes sense: while (24) ensures small variance, (23) ensures unbiasedness of the deterministic limit up to terms of order ε k . Although this condition may not seem crucial for sampling at first sight (because the control is ∇g so we may add a constant), it is actually important (think of the terminal condition for instance). Actually this kind of free energy approximate argument was already used in [12] for studying errors arising in the time step discretization of ergodic Feynman-Kac semigroups.
Remark 3.
• We ask forĝε to be of class C 1,2 for computing derivatives in the classical sense, but using the notion of stochastic derivative (see Remark 1) we could consider less regular approximations (typically almost everywhere differentiable functions, a situation that may arise for piecewise defined functions). This is a nice feature of SVAs.
• We also emphasize that Definition 2 applies way beyond the scope of Assumption 1, typically as soon as the stochastic process (1) and its associated Fokker-Planck equation (12) make sense.
• We claim by no mean that the bound (26) is optimal in any sense: it is simply convenient for the proofs below, and seems reasonable to check at least in simple situations.
• At this stage, Definition 2 is not related to the low temperature asymptotics presented in Section 2.4, and does not require Assumption 3 to hold. However, we will use these asymptotics in Section 4 to build stochastic viscosity approximations.
Relation to variance reduction
As explained in Section 2.3, we have defined our notion of stochastic viscosity approximation precisely in order to make the tilted estimator Aε close to deterministic by reducing the random terms in the Girsanov weight to a small-noise factor. Using Definition 1 of log-efficiency, we can state more precisely the variance reduction property of a k-stochastic viscosity approximation.
Theorem 1.
Assume thatĝε is a k-SVA of gε for some k 1. Then the associated estimator Aε is k-log efficient.
This is our main result associated to the definition of stochastic viscosity approximation. Contrarily to Lebesgue-norm criteria that may not ensure good variance reduction properties, we here draw a very clear connection between SVA and log-efficiency of the associated estimator. Let us present the proof, which is rather straightforward.
Proof. Let us consider first the quantity
Ex 0 e 2 ε f ( X ε T ) (Gε) 2 = e 2 εĝ ε(0,x0 ) Ex 0 e 2 ε f ( X ε T )−ĝε(T, X ε T )+ T 0 (∂ tĝε +Hεĝε)(t, X ε t ) dt .
The approximate HJB equation (24) (including the terminal condition) in Definition 2 implies that
Ex 0 e 2 ε (f ( X ε T )−ĝε(T, X ε T )+ T 0 (∂ tĝε +Hεĝε)(t, X ε t ) dt = Ex 0 e 2 ε ε k Q ε F +ε k T 0 Q ε t dt .
Together with the free energy consistency (23) and recalling the expression (25) of the residual Q ε , we obtain
Ex 0 e 2 ε f ( X ε T ) (Gε) 2 = e 2 ε (gε(0,x0)+O(ε k )) Ex 0 e ε k−1 Q ε .
Since we assume that k 1, the function x ∈ R+ → x ε k−1 is concave for ε 1 (thus we may replace ε0 by min(ε0, 1) if necessary). As a result, we can use Jensen's inequality combined with (26) to obtain, for ε ∈ (0, ε0]:
Ex 0 e ε k−1 Q ε Ex 0 e Q ε ε k−1 e ε k−1 C Q .
Finally, by (15)
, it holds E e 1 ε f (X ε T ) 2 = e 2 ε gε(0,x 0 ) , so that we obtain the bound ρ(ε) e 2 ε (gε(0,x 0 )+O(ε k )) e ε k−1 C Q e 2 ε gε(0,x 0 ) = e C Q ε k−1 +O(ε k−1 ) .
This implies that
R(ε) = ε log ρ(ε) = O(ε k )
as ε → 0, which is the desired result.
Note that we cover the case k 1 for simplicity and because this is the main situation of interest (since relative variance is a priori not even bounded for k < 1), but one can prove a similar result for k ∈ [0, 1) under stronger assumptions, for instance by assuming that Q ε is bounded.
Instanton expansion as a stochastic viscosity approximation
In this section, we consider the approximationĝ 1 ε presented in Section 2.4. Since this function satisfies the HJB equation around the instanton that defines the limit free energy, it is a good guess for approximating gε in the low temperature regime. We actually show that this choice, motivated by earlier works [19,11], is indeed a 1-SVA of the optimal control under appropriate assumptions. It is thus associated with a 1-log efficient estimator. This provides a positive variance reduction property for this importance sampling scheme under geometric conditions, which had not been shown so far -a fact we prove in Section 4.1. In Section 4.2 we consider the next order expansion proposed in [11], which is a 2-SVA of the optimal control, hence improving on variance reduction. We finally propose in Section 4.3 a simple numerical application to illustrate the validity of our predictions. In all of Section 4, we let Assumption 3 hold.
Instanton bias
Recall the approximation (19) proposed in Section 2.4:
∀ (t, x) ∈ [0, T ] × R d ,ĝ 1 ε (t, x) = f (φT ) − 1 2 T t θs · Dθs ds + θt · (x − φt),
where the instanton (φt, θt) t∈[0,T ] is given by (18). This function is well-defined thanks to Assumption 3, and again ( X ε t ) t∈[0,T ] is the dynamics biased byĝ 1 ε as in (5). Moreover, we keep the ε-dependency in the notation ofĝ 1 ε for generality although it does not depend explicitly on this parameter. We recall that we understandĝε as a time-inhomogeneous first-order polynomial in the variable (x − φt).
Compared to the previous work [11], we add a constant and a time-only dependent components (the first two terms in (19)). This is not crucial, but makes all the approximation argument more elegant. Indeed, as will be made precise in the proof below, the constant term in (19) allows the approximation to satisfy boundary conditions consistent with that of gε, while the time dependent term compensate the loss of energy in the Hamilton-Jacobi operator and allows the free-energy consistency (23) to hold.
Before proving thatĝ 1 ε is a 1-stochastic viscosity approximation of gε, we first show that a trajectory tilted byĝ 1 ε is a perturbation of the instanton -very much in the spirit of the expansions that can be found in [17,Chapter 2].
Lemma 1. Under Assumptions 1 and 3, there exist an adapted process
(Y ε t ) t∈[0,T ] and a constant CT > 0 such that ∀ t ∈ [0, T ], X ε t = φt + √ εY ε t ,(27)
and for almost all t ∈ [0, T ] the following bound holds
∀ ε > 0, |Y ε t | CT |σBt|,(28)
almost surely with respect to realizations of the Brownian motion.
The proof 3 of Lemma 1 can be found in Appendix B.1. It is now a simple matter of Taylor expansion to prove thatĝ 1 ε is a stochastic viscosity approximation of gε under some technical conditions.
Theorem 2. Let Assumptions 1 and 3 hold and suppose that
• There exists a small enough δ f 0 such that:
∀ y ∈ R d , sup x∈R d y · ∇ 2 f (x)y δ f |y| 2 .(29)
• There is a small enough δ b 0 such that
∀ y ∈ R d , ∀ t ∈ [0, T ], sup x∈R d y · θt · ∇ 2 b(x) y δ b |y| 2 .(30)
Then the functionĝ 1 ε defined in (19) satisfies Assumption 2 and is a 1-SVA of the HJB problem. As a result, it defines a 1-log efficient estimator.
The proof of Theorem 2 is presented in Appendix B.2. We may interpret the geometric conditions (29)-(30) as complementary to Assumption 3. First, (29) asks for f to be «mostly concave». Typically, we may expect problems to occur when f has several pronounced maxima, so that the maximum value of f can be reached in several ways. A similar condition (30) holds for the drift b: if we further assume that ∇f has nonnegative coordinates, then θt has nonnegative coordinates at least for small times, and so (30) is a concavity condition on b. Obviously, these conditions are quite drastic and could be relaxed if one proves additional stability properties of the process or has other information on the drift b (for instance bounded second derivative).
Riccati bias
One motivation for the current work was to better understand the approximation proposed in [11]. The argument of the authors was to considerĝ 1 ε as the first term of a time-dependent Taylor expansion in the argument (x − φt), which naturally suggests to continue the series with higher orders. Based on this earlier work [11, eq. (21)], we introduce the following quadratic approximation, for a R d×d -valued dynamics (Kt) t∈[0,T ] ,
g 2 ε (t, x) = f (φT ) − 1 2 T t θs · Dθs ds + ε 2 T t D : Ks ds + θt · (x − φt) + 1 2 (x − φt) · Kt(x − φt).(31)
By going one order further in the expansion performed in the proof of Theorem 2 (see [11,Section 3] for details) it can be checked that a natural definition for (Kt) t∈[0,T ] is to solve the following Riccati system:
K t + (∇b) T Kt + K T t ∇b + ∇ 2 b · θt + K T t DKt = 0, KT = ∇ 2 f (φT ).(32)
By imposing (φt, θt) t∈[0,T ] to be solution to (18) and (Kt) t∈[0,T ] to be solution to (32) (assuming well-posedness of these systems, which is true at least for small times) while assuming appropriate bounds on ∇ 3 f and ∇ 3 b, we can show thatĝ 2 ε is a 2-SVA of gε. The details of these computations, left to the reader, follow the same Taylor expansion strategy as for Theorem 2. Sinceĝ 2 ε defines a 2-SVA, the associated estimator is 2-log efficient. This entails that the second order approximation proposed in [11] indeed improves on variance reduction compared to the first order one earlier proposed in [19].
Numerical application
We now illustrate our theoretical findings by proposing a simple situation in which all our conditions are satisfied and our results apply. We then perform a numerical simulation that confirms our expectations.
The system is a simple one dimensional Ornstein-Uhlenbeck process, i.e. d = 1, σ = 1, b(x) = −x. In order to propose a smoothed version of a probability like P(X ε T > 1), we set f (x) = −(x − 2) 4 /4. This weight function gives larger importance to those trajectories ending around 2. We also set x0 = −1, so tilted trajectories typically relax towards 0 before being pushed away to larger positive values. This system is non-trivial while satisfying Assumption 1. In this case the solution to the instanton equation (18) is actually explicit, while the Riccati equation (32) is stable and can be integrated with a simple Euler scheme. For more general dynamics, the instanton can be computed with the algorithm described in [19, Section III A] (which we actually use in practice). Finally, f (x) 0 and b (x) = 0 for all x ∈ R, so the conditions (29) and (30) are satisfied and Theorem 2 applies. As a result,ĝ 1 ε defines a 1-SVA of the system. Similarly, since the system is quite simple, we can show thatĝ 2 ε is a 2-SVA.
In order to perform numerical simulations, we set T = 5 and discretize the SDE, the instanton and Riccati equations with a time step ∆t = 5×10 −3 . We numerically assess that this value is small enough to neglect the bias due to time discretization compared to the relative variance we are interested in (not shown). We draw 5 × 10 6 trajectories to estimate the relative variance ratio R(ε) defined in (10) for a series of values of ε. The left panel in Figure 4.3 shows the evolution of the instanton and Riccati terms with time. While momentum adds an upward drift, the negative Riccati term forces the process to remain around the instanton. On the right panel we show the evolution of R(ε) with ε for the different estimators. In spite of noise, we see that R(ε) is roughly constant without biasing, meaning relative variance grows exponentially in this case. On the other hand, the linear and quadratic decay at log scale for the first and second order biases respectively are in accordance with the results of Theorem 2 and the discussion in Section 4.2.
Discussion
This work is concerned with the issue of variance reduction for the Monte Carlo estimation of exponential-like expectations, as they often arise in statistical physics and other areas. Our main goal was to design a framework to assess whether a given approximation of the optimal Hamilton-Jacobi control indeed reduces relative variance as temperature becomes small, and at which scale.
To achieve this, we introduce the notion of stochastic viscosity approximation (SVA), inspired by recent developments on the theory of Feynman-Kac partial differential equations [25]. We believe our definition is (10) with ε in logarithmic scale, for the unbiased estimator and the dynamics biased byĝ 1 ε andĝ 2 ε .
meaningful: a SVA approximately solves the Hamilton-Jacobi-Bellman equation along tilted paths, while a zerotemperature consistency should hold. This is precisely what is needed in the Girsanov theorem to make stochastic terms small, hence an associated powerful variance reduction property. This definition comes in sharp contrast with standard approximation techniques criteria, which typically control a non-local error to assess convergence as a mesh size decays to zero. Here we show that even an approximation that does not depend on temperature or any parameter going to zero can have a vanishing log-efficiency in the zero temperature limit! Since this study was in part motivated by earlier heuristics on importance sampling [19,11], we specifically study these approaches. We show, under geometrical conditions, that the first and second order techniques proposed in [19,11] define stochastic viscosity approximations of order one and two respectively, thus providing variance reduction properties for these heuristics. This also shows that our definition can indeed be applied to actual approximation schemes. A simple numerical example illustrates our results.
As for all new definition, connection between stochastic viscosity solutions and existing works should be explored further. In particular, concerning variance reduction, it seems interesting to understand better the link with the large deviation criteria designed in [20] as well as the more involved Isaacs equations [8]. Moreover, it seems natural to revisit existing approximation techniques and check whether they match our definition and, as already noted in [25], the relation with viscosity solutions of HJB problems should be made clearer, appart from the variance reduction problem.
Finally, although our work gives ground to the schemes based on instanton expansions as the ones presented in Section 4, such schemes do not apply as such when the solution to the instanton problem is not unique, which is the case in many situations. Two ways can be followed to overcome this issue. The first is to extend the approximation to a situation with several instantons, to build a more general expansion of the HJB equationthis is the subject of ongoing investigation of the author. The second, more pragmatic, is to use tools such as the region of strong regularity [15] to update the computation of the instanton at certain times, in the flavour of [21]. Whatever approach is chosen, we believe the notion we introduced in this work is valuable as it quantitatively shows that, when the goal of control is variance reduction, one should focus on approximating the HJB equation along relevant low temperature paths.
In order to obtain (26) and conclude, it remains to bound Q ε
Figure 1 :
1Left: Instanton, momentum and Riccati matrix (here scalar values). Right: Evolution of R(ε) as defined in
It is likely that a weaker probabilistic sense is sufficient.
We mention that the estimate(27) does not make sense if the instanton equation(18) has several solutions. This is the main limitation of the techniques proposed in Section 4.
Although we impose a constraint on ∇ 2 f over the whole space, we see that it is essentially important to control this quantity around φ T . One difficulty however comes from the fact that φ T itself depends on the function f . This makes a general condition for f to satisfy the bound hard to state at this level of generality. The same is true for the condition on ∇ 2 b.
AcknowledgementsThe author is particularly grateful towards Charles Bertucci for reading a first version of the manuscript and providing a series of useful comments. He also thanks Tobias Grafke and Hugo Touchette for interesting discussions on large deviations, as well as Djalil Chafaï and Gabriel Stoltz for stimulating general discussions and encouragements.A From Girsanov to Hamilton-Jacobi-BellmanFirst write the Girsanov formula for the path change of measure[23]between the processes (X ε t ) t 0 and ( X ε t ) t 0 :Then use Itô formula over a trajectory of ( X ε t ) t 0 using the generator (3) (which can be used sinceIntegrating in time and dividing by ε, the above equation becomesInserting the above equation into (33) leads to(7).B Proofs of Section 4In order to avoid overloaded notation, we write g =ĝ 1 ε in all this section.B.1 Proof of Lemma 1The proof follows a standard Gronwall argument. Let ε > 0 be arbitrary. Since the solution to(18)is unique, we can define for all t ∈ [0, T ] the adapted processSincewhere we used(18). Now, integrating in time and taking the absolute value leads toRecalling that Y ε 0 = 0 and that b is globally Lipschitz with constant C lip (by Assumption 1), we obtainGronwall lemma then implies that ∀ t ∈ [0, T ], |Y ε t | |σBt| e T C lip , which holds for any continuous realization of the Brownian motion, hence almost surely, and for any ε > 0. We thus have the bound (28), while(27)follows from the definition (34) of Y ε t , which concludes the proof.B.2 Proof of Theorem 2We first notice that g(0, x0) = Z0.Since Zε = Z0 + O(ε) this readily implies that g(0, x0) = gε(0, x0) + O(ε). Therefore, the position-independent term in g automatically provides the free energy consistency(23). Moreover, since ∇g only depends on time, it clearly satisfies the conditions of Assumption 2. We now check the approximate HJB equation(24)with bound (26) by relying on Lemma 1. Consider the terminal condition first. Using the terminal value of θT in (18), we haveTaylor's integral theorem applied at first order to f around φT shows, using Lemma 1, that we have almost surelyAccording to(29), it holdswhich we will use to control the residual process 4 . Let us now check the dynamical part in(24). First, we easily computeTherefore, using Lemma 1 and equation(18)we obtainFinally, by defining the processwe have (∂tg + Hεg)(t, X ε t ) = εQt. t . We apply again Taylor's theorem at first order between X ε t and φt for all time t 0 and continuous realization of ( X ε t ) t∈[0,T ] , now to the R d×d -valued function ∇b, to obtainFinally, the condition (30) ensures thatand the total residual Q ε defined in (25) satisfies:Using(28)in Lemma 1 then leads towhich satisfies the integration condition (26) provided δ f and δ b are small enough (compared to the noise σ).In the end, all the conditions of Definition 2 are satisfied andĝ 1 ε is a 1-SVA of the HJB problem(11). According to Theorem 1, the associated estimator is 1-log efficient, which concludes the proof.
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[
"R V Shchelushkin \nDepartment of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n",
"Arne Brataas \nDepartment of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway\n"
] |
[
"Department of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway",
"Department of Physics\nNorwegian University of Science and Technology\nN-7491TrondheimNorway"
] |
[] |
We consider spin and charge flow in normal metals. We employ the Keldysh formalism to find transport equations in the presence of spin-orbit interaction, interaction with magnetic impurities, and non-magnetic impurity scattering. Using the quasiclassical approximation, we derive diffusion equations which include contributions from skew scattering, side-jump scattering and the anomalous spin-orbit induced velocity. We compute the magnitude of various spin Hall effects in experimental relevant geometries and discuss when the different scattering mechanisms are important.
|
10.1103/physrevb.71.045123
|
[
"https://export.arxiv.org/pdf/cond-mat/0409130v1.pdf"
] | 119,377,235 |
cond-mat/0409130
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c4406e92ef5bf36a544005000d67047e48b9c01c
|
6 Sep 2004
R V Shchelushkin
Department of Physics
Norwegian University of Science and Technology
N-7491TrondheimNorway
Arne Brataas
Department of Physics
Norwegian University of Science and Technology
N-7491TrondheimNorway
6 Sep 2004(Dated: March 23, 2022)Draft Spin Hall effects in diffusive normal metals.numbers: 7210-d7215Gd7350Jt
We consider spin and charge flow in normal metals. We employ the Keldysh formalism to find transport equations in the presence of spin-orbit interaction, interaction with magnetic impurities, and non-magnetic impurity scattering. Using the quasiclassical approximation, we derive diffusion equations which include contributions from skew scattering, side-jump scattering and the anomalous spin-orbit induced velocity. We compute the magnitude of various spin Hall effects in experimental relevant geometries and discuss when the different scattering mechanisms are important.
I. INTRODUCTION
Spin flow in nanostructures has recently attracted considerable interest in the scientific community 1,2,3 . The vision of magnetoelectronics and spintronics is to inject, manipulate and detect spins in nanostructures which can give new functionality in electronic devices. The spin flow can be controlled by e.g. external electric or magnetic fields.
A Hall voltage builds up perpendicularly to the current flow under an applied magnetic field in normal metals due to the Lorentz force. The Hall voltage increases with applied magnetic field. Magnetoelectronic circuits are often realized by using ferromagnets that can spin polarize the current flow. In ferromagnets, there is an anomalous Hall voltage proportional to the magnetization, e.g. a transverse charge potential, even in the absence of an applied magnetic field. The anomalous Hall effect is caused by the spin-orbit interaction, which correlates the momentum of the electron with its spin. This causes an dependence of the electron flow with the relative angle between its direction and the non-zero magnetic order parameter in ferromagnets. 4,5,6 In ferromagnetic metals, spin-orbit interaction is also a source of crystalline magnetic anisotropy energies since the spin-orbit interaction couples the magnetization with the crystal structure.
The scattering mechanisms responsible for the anomalous Hall effect are skew scattering 7 and side-jump scattering 8 as well as the anomalous velocity operator due to spin-orbit interaction and impurity scattering. A schematic picture of the skew scattering mechanism is shown in Fig. 1. After scattering off the impurity potential, there is a spin-dependent probability difference, represented by small angles, of the electron trajectories. This leads e.g. to a slightly larger chance that electrons with spin up moves upwards and electrons with spin down moves downwards after scattering. The side-jump mechanism is also caused by the combined spin-orbit and impurity scattering, see Fig. 2. After scattering off the impurity, a small "side-jump", develops between the trajectories of electrons with spin up and down far away from the scattering center. Additionally, the spin-orbit interaction does not commute with the electron momentum operator. This leads to an anomalous velocity operator that can contribute to the Hall effects, as well.
Spin-orbit scattering is also important in normal metals. It is well known that it causes a loss of spin coherence. Hirsch predicted the existence of a novel spin Hall effect 5 analogues to the anomalous Hall effect in ferromagnets and developed a phenomenological theory for the effect. In the absence of spin-orbit scattering, electrons with spin-up and spin-down scatter equally on non-magnetic impurities. However, as seen above, for nonzero spin-orbit interaction, when a current passes through the sample an imbalance between left-moving and right-moving particles is established, and an accompanying transverse spin accumulation potential builds up in the system.
Zhang computed this spin potential in the diffusive transport regime, and found that it should be measurable 9 . He considered longitudinal transport in a thin normal metal film, and computed the resulting transverse spin Hall voltage. In this regime, the spin-accumulation is weak and he found that the spin Hall voltage is governed by the anomalous velocity operator. By using the framework developed by Hirsch and Zhang, effects of contact resistances on the spin Hall effect have also been considered 10 .
Spin-orbit scattering is also important in nand p-doped semiconductors, where Rashba-type spin-orbit coupling 11 leads to interesting spin Hall effects 12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 .
The most interesting case is spin transport in a two-dimensional electron system (2DES). The study of this effect is controversial. The debate 16,17,19,20,21,22,24,27 is focused on whether or not the spin Hall conductivity has a universal value σ sH = e/(8πh) and in what regime this result is applicable. Our study is complementary to these studies of spin Hall effects in semiconductors. In our normal metal case, the extrinsic 28 spin Hall effects arise due to the spin-orbit at impurities. In semiconductors, spin-orbit interaction can be important even in ballistic systems 29 , in the absence of impurities, for systems with broken spatial inversion symmetry.
In this work, we derive the spin Hall effects in the presense of spin diffusion from the Eilenberger equation 30 in presence of spin-orbit coupling 31 , magnetic impurities and non-magnetic impurities. We use the Keldysh Green function technique in the quasiclassical approximation. Our calculations go beyond the assumptions in Ref. 9, which only included effects of the anomalous velocity operator, in that we rigorously derive diffusion equations that also include effects of skew scattering and side-jump scattering. Our results agree with the results by Zhang in the limit he considered, where skew scattering and sidejump scattering can be disregarded. We also consider spin and charge transport in normal metals in another transport regime, phenomenologically treated by Hirsch, when normal metals are biased by ferromagnets, in which spins accumulate in the normal metal even in the absence of the spin-orbit interaction. We demonstrate that in this regime, skew scattering and side-jump scattering cannot be disregarded, and compute the spin Hall and spin-orbit induced charge Hall effects. In our analysis, we also consider transport through ferromagnet-normal metal interfaces beyond the assumptions in Ref. 10 by using the boundary conditions obtained within magneto-electronic circuit theory, which is necessary in dealing with noncollinear spin and magnetization directions. Finally, our formalism also explicitly incorporates the effects of scattering off magnetic impurities, which can reduce the spin Hall effects.
Our paper is organized in the following way. In the next section, Section II, we outline our starting point microscopic Hamiltonian, and explain the diffusion equations for spin and charge flow that we have obtained rigorously by using the non-equilibrium Keldysh Green function approach in the quasi-classical approximation. We compute from the diffusion equations the spin Hall effects in some relevant experimental geometries in Section III. Our derivation of the charge and spin diffusion equation is an important part of our work, and its details are given in Section IV. Finally, we give our conclusions in Section V.
II. MODEL AND TRANSPORT EQUATIONS
We consider quasi-particles in a normal metal that interact with non-magnetic and magnetic impurities and include the spin-orbit interaction. The Hamiltonian of the system is
H = −h 2 2m ∇ 2 + V imp +V so +V sm ,(1)
where we below will introduce the various terms. Impurity scattering is represented by the short-range potential
V imp = i γ i δ(r − r i ),(2)
where r i is the coordinate of the i -th impurity center and γ i is the strength of the scattering potential. It is assumed that the scatterers are disordered so that
< V imp (r) >= 0 and < V imp (r)V imp (r ′ ) >= δ(r−r ′ )nγ 2 ,
where n is the impurity density and γ 2 is the average fluctuation of the scattering strengths. The spin-orbit interaction is described by the Hamil-
tonian,V so = 1 2 ᾱ hk 2 F (σ × ∇V imp ) p + h.c. ,(3)
where α is the dimensionless spin-orbit coupling constant, k F is the Fermi wave number and h.c. denotes the hermitian conjugate. Magnetic impurites are introduced bŷ
V sm = V sm (r)σ · S(r),(4)
where V sm (r) is the strength of the coupling of the itinerant electron spin to the spin of the magnetic impurity S(r).
We are interested in the transport properties of diffusive system where the system size is much larger than the mean free path. A rigorous method to obtain the correct diffusion equation is to start from a microscopic description using the non-equilibrium Keldysh formalism. We consequently employ the Keldysh approach with two approximations. First, we consider the quasiclassical approximation, which is valid on length scales much larger than the Fermi wavelength, L ≫ λ F . Second, we use the diffusion approximation which is valid when the system size is much larger than the mean free path,
L ≫ l = v F τ .
The full derivation of the diffusion equation is an important part of the present paper, but it is technically complicated and in order to make the paper more easily accessible we delay its derivation to the interested readers in Section IV. First, we show and explain the spin and charge diffusion equations as well as the expression for the corresponding currents that we obtain. We introduce charge and spin-distributions µ c and µ s , so that the charge density and spin-density are
n(r) = N o µ c (r), s(r) = N o µ s (r),
where N o is the density of states. After considerably algebra outlined in Section IV, we find that the resulting diffusion equation for the charge distribution functions is simply
∇ 2 µ c = 0 .(5)
Similarly, the spin-distribution for small spin-orbit interactions, α ≪ 1, is governed by
D∇ 2 µ s = 1 τ so + 1 τ sm µ s ,(6)
where D = 1 3 v 2 F τ is the diffusion coefficient in terms of the Fermi velocity v F and the elastic scattering time τ , τ so is the spin-flip relaxation time due to the spin-orbit interaction,
1 τ so = 8α 2 9τ(7)
and τ sm is the spin-flip relaxation time due to magnetic impurities,
1 τ sm = 8πn sm N 0 S(S + 1)v 2 sm 3 .(8)
Here we have expressed the strength of the magnetic impurity potential from Eq. (4) in the momentum representation v sm , n sm is the concentration of the magnetic impurities, and S is the spin of the impurity. Thus, both the diffusion equations for charge and spin have the familiar forms used exensively in the literature for spin and charge transport. Skew scattering, side-jump scattering and effects of the anomalous velocity operator are all contained in the expressions for the current. We find that the total 2 × 2 current in spin-space can be expressed aŝ
j =ĵ o +ĵ av +ĵ ss +ĵ sj ,(9)
whereĵ o is the ordinary current without spin-orbit interaction,ĵ av is the current due to the anomalous velocity operator,ĵ ss is the current due to skew scattering and j sj is the current due to side-jump scattering. The full derivation of these currents is given in Section IV. The charge and spin currents can be obtained by the trace of Eq. (9) with the unit matrix and the Pauli matrices, respectively. In the limit of weak spin-orbit interaction (to lowest order in α), which is relevant for most normal metals, we compute the following contributions to the current:
eĵ o = −σ 1 2 1 ∇µ c + ∇(µ sσ ) ,(10)eĵ av = σ αh 6mD [∇ × µ s +σ × ∇µ c ] ,(11)eĵ sj = σ α 3 (σ∇)µ s ,(12)eĵ ss = −σασ(∇µ s ) ,(13)
where σ = e 2 N 0 D is the conductivity. The anomalous current,ĵ av , contributes both to the spin and charge current. Contributions from skew scattering and side-jump scattering,ĵ ss andĵ sj , only affect the spin-current. The current contributionĵ sj depends on the divergence of the spin accumulation in the direction of the current for any spin, which is the side-jump mechanism. The current contributionĵ ss arises from skew scattering. If we compare our results with Zhang 9 , we see that in addition to the contributions from the anomalous velocity operator we include terms representing skew scattering and side-jump scattering. We demonstrate below that these additional contributes could correctly be disregarded in the geometry in Ref. 9, but that they the dominate spin Hall effects in other systems. We also derive expressions for the total current in case of arbitary α (see Section IV).
III. EXPERIMENTAL IMPLICATION.
Let us now employ our theory to calculate the magnitude of the spin Hall effect in experimental relevant geometries. For simplify, we will consider the cases with small spin-orbit interaction (α ≪ 1), where the small α expressions for the current, (10), (11), (12) and (13), are valid.
A. Thin metallic film.
We consider first a pure normal metal as considered in Ref. 9 and shown in Fig. 3. A thin film normal metal of length L and width d is attached with perfect contacts with zero resistance to a left reservoir with local chemical potential µ L and a right reservoir with local chemical potential µ R . In a pure normal metal system, there is no spin-accumulation in the limit α → 0. That means that the spin-accumulation is small, being induced by the spin-orbit interaction. From (11), (12) and (13) we thus see that contributions from the skew scattering and sidejump scattering to the current are of a higher order in the spin-orbit scattering than the anomalous current and can be disregarded. In the case of pure normal metals, in the weak spin-orbit interaction limit, the current can thus simply be expressed in terms of
j ≈ĵ 0 +ĵ av ,(14)
where the anomalous current simplifies to
eĵ av ≈ σ αh 6mDσ × ∇µ c .(15)
The effect of the spin-orbit interaction is consequently to induce a transverse spin Hall potential. The magnitude of the spin Hall effect depends on the system size and geometry. If the system size is smaller than the spindiffusion length, a spin accumulation cannot build up within the system, and consequently the spin Hall effect vanishes. Therefore, we consider the situation when the system size is much larger than the spin-diffusion length, L ≫ l sf = √ Dτ s , where the total spin-flip relaxation time has contributions both due to spin-orbit scattering and magnetic impurity scattering, 1/τ s = 1/τ so + 1/τ sm . The solution of the diffusion equation is similar to the treatment in Ref. 9. At distances larger than the spindiffusion length from the reservoirs, the spin potential only depend on the transverse y-coordinate. In this regime, the general solution of the diffusion equation (6) has the form
µ s = c 3 e y/l sf + c 4 e −y/l sf ,(16)
where c 3 and c 4 are constants to be determined. We can determine the constants c 3 and c 4 from the boundary condition that there is no particle or spin flow across the transverse boundaries, e.g. the 2×2 current in spin-space must satisfy:ĵ
y (x, y = ±d/2) = 0 .(17)
Using the simplified equations (14) and (15) determining the current, we find c 3 and c 4 and the spatially dependent transverse spin potential:
µ (A) s,z (y) = 3 8 µ R − µ L k F L τ sm τ sm + τ so sinh(y/l sf ) cosh(d/2l sf ) .(18)
This expression exactly agrees with the result of Zhang 9
µ s,z ≡ µ ↑ − µ ↓ = l sf E x C h /C sinh(y/l sf )/cosh(d/2l sf )
when we insert the Drude conductivity C = e 2 τ (k F ) 3 /6π 2 m, the anomalous Hall conductivity C h = e 2 α o (k F ) 3 /6π 2 (the dimensionless α = α oh k 2 F ) and the electric field in the x-direction E x = (µ R − µ L )/L. In addition, we use the identification between the spin-orbit spin-flip relaxation time and elastic scattering time, Eq. (7). Formula (18) expresses the transverse spin Hall effect for thin metallic films with small spin-orbit interaction. There are accumulation of spins directed perpendicular to the film. As we see, the spin-Hall effect vanishes when scattering by magnetic impurities dominates the spinorbit scattering, τ sm ≪ τ so , as expected. When magnetic impurity scattering is weak, and in the limit of a wider film than the spin-diffusion length d ≫ l sf , we make the observation that the magnitude of spin Hall effect is "universal", µ
(A) sH ≡ µ (A) s,z (d/2) − µ (A) s,z (−d/2): µ (A) sH ≈ 3 2 µ R − µ L k F L ,(19)
e.g. the spin Hall voltage does not depend on the spinorbit interaction constant α. By "universal", we mean that the spin Hall potential does not depend on the strength of the spin-orbit scattering potential. This implies that, as long as the scattering off magnetic impurities is weak, light metals (e.g Cu, Al) will give a similar spin Hall voltage as heavy metals (e.g Pt). Note that the spin Hall potential depend on the Fermi wave vector of the metal, k F , and thus weakly depends on the type of normal metal through this dependence. The reason for the "universality" is that although the spin-Hall potential is proportional to the spin-orbit scattering it only builds up within the spin-diffusion length which is inversely proportional to the spin-orbit scattering strength. This section also illustrates the differences between the extrinsic spin-orbit scattering off impurities in normal metals, as treated here, and spin-orbit scattering induced by the Rashba term and impurities in the twodimensional electron gas formed in semiconductor heterostructures. In the two-dimensional electron gas, there is currently a controversy whether the spin Hall conductivity can reach a universal value in dirty systems, σ sH = e/(8πh), independent of the spin-orbit scattering strength. In normal metals, we see from (15) that the spin Hall conductivity is not universal, but the spin Hall voltage (19) can be "universal" when scattering off magnetic impurities is weak.
B. Thin metallic film in tunneling contacts to ferromagnet and normal metal.
We will now consider another transport regime, in which there is a finite spin-accumulation present in the normal metal even in the absence of spin-orbit scattering. The spin-accumulation can be achieved by sandwiching the normal metal between two ferromagnets when the system is driven out of equilibrium, see Fig. 4. There are two extreme regimes depending on the ratio between the system length and the spin-diffusion length. The first regime, when the length is large, (L ≫ l sf ), does not provide novel physics since it is similar to the previous example. In this case any spins injected from the ferromagnets will be lost, and the system resembles a pure normal metal where a spin Hall voltage can build up transverse to the current direction. There is no charge Hall effect produced by the spin-orbit interaction in the middle of the system in this case. Therefore, we consider the second, more interesting regime, when the normal metal is short, (L ≪ l sf ). In this case both spin and charge Hall potentials can build up transverse to the current direction governed by the spin-orbit scattering.
In order to further simplify the computation of the diffusion equations (5), (6) we assume that the normal metal is narrow, e.g. d ≪ l sf as well. The diffusion equations then simplify to ∇ 2 µ c = 0 and ∇ 2 µ s = 0.
In addition, we need the boundary condition of the spin and charge flow through the tunnel contacts from the ferromagnetic reservoirs into the normal metal wire. These boundary conditions follow from magnetoelectronic circuit theory, so that the interface transport can be described by spin-dependent conductances G ↑ and G ↓ for spin aligned and antialigned to the magnetization and a mixing conductance of reflection G ↑↓ for spins in the normal metal that are non-collinear to the magnetization direction. 32 For most systems, ReG ↑↓ ≫ ImG ↑↓ , and this will be assumed in the following, which simplifies the expressions for charge and spin currents 32 . The charge current along the transport, x, direction through the tunneling contact can be written as
eI c = (G ↑ + G ↓ ) µ F c − µ c (0, y) − (G ↑ − G ↓ )m · µ s (0, y)(20)
in terms of the local quasi-equilibrium chemical potential in the ferromagnet µ F c and the spin and charge chemical potentials in the normal metal close to the ferromagnetic interface µ s (x = 0, y) and µ c (x = 0, y). Here m is the magnetization of the ferromagnet (|m| = 1). For a tunnel contact 2ReG ↑↓ = G ↑ + G ↓ . 32 The spin-current along the transport, x, direction is then
eI s = m[(G ↑ − G ↓ )(µ F c − µ c (0, y)) + (G ↑ + G ↓ )µ F s ] − 2ReG ↑↓ µ s (0, y).(21)
We assume that the spin-orbit interaction is weak, and expand the charge and spin accumulations to first order in α:
µ c (x, y) = µ (0) c (x, y) + αδµ c (x, y) ,(22)µ s (x, y) = µ (0) s (x, y) + αδµ s (x, y),(23)
where the zeroth order terms are the charge and spin accumulations in the absence of spin-orbit interaction and the corrections are caused by the spin-orbit interaction. First, we compute the zeroth order terms that correspond to α = 0. The boundary conditionĵ 0 y (x, y = ±d/2) = 0 of no current out of the transverse edges dictates that the general solutions µ x L . Now we can equate the current through the tunnel contact with the current in normal metal at x = 0. This is a boundary condition on the ferromagnet T 1 (see Fig.4). For the charge current
− G o c 1 = (G ↑ + G ↓ )(µ L − c 0 ) − (G ↑ − G ↓ )m · c 0 ,(24)
where the conductance of the normal metal is G o = σwd/L, w is the width of the film. For spin-current
− G o c 1 = −(G ↑ + G ↓ )c 0 + m[(G ↑ − G ↓ )(µ L − c 0 ). (25)
At the normal metal-normal metal tunnel contact T 2 (see Fig.4) we use the same expressions for currents (20), (21) with G = G ↑ = G ↓ , and the boundary conditions are
G(c 0 + c 1 − µ R ) = −G o c 1 ,(26)G(c 0 + c 1 ) = −G o c 1 .(27)
After straightforward calculations we find
µ (0) s (x) = 1 − G G + G o x L G ↑ tot − G ↓ tot G ↑ tot + G ↓ tot (µ L − µ R )m,(28)µ (0) c (x) = c 0 − 4 G o 1 G ↑ tot + 1 G ↑ tot (µ L − µ R ) x L ,(29)
where the totale spin-dependent conductances of the system are given by the resistances in series 1/G ↑(↓) tot = 1/G ↑(↓) + 1/G + 1/G o . c 0 can be expressed similarly, but is not shown here since it does not govern the spinorbit induced correction to the currents which will be considered next.
Next we introduce spin-orbit interaction which will produce current correction δĵ(x, y) so that the full current is expressed asĵ =ĵ 0 + δĵ. The boundary condition δĵ y (y = ±d/2) = 0 gives
∂ ∂y δµ c | y=±d/2 = −h 3mD ∂ ∂x δµ (0) s,z(30)
∂ ∂y δµ s,z | y=±d/2 =h 3mD
∂ ∂x δµ (0) c(33)
where we have introduced the total polarization of the conductance of the system p tot ≡ (G ↑ tot − G ↓ tot )/(G ↑ tot + G ↓ tot ). The derivative of x-component of the correction to the spin potential is governed by side-jump current, ycomponent by skew scattering current and z-component by anomalous current. In the limit d ≪ L we can expand the corrections to the chemical potentials in the small parameter (d/L), so that the solutions can be represented as
δµ c = δµ (0) c + d L δµ (1) c (34) δµ s = δµ (0) s + d L δµ (1) s(35)
Due to simplified diffusion equations we write
δµ (i) c = c (i) 0 + c (i) 1 x + c (i) 2 y,(36)δµ (i) s = c (i) 0 + c (i) 1 x + c (i) 2 y.(37)
where i = 0, 1. We are interesting for coefficients c
∂ ∂x δµ c | x=0 = G ↑ + G ↓ G o δµ c (0, y) + G ↑ − G ↓ G o mδµ s (0, y), ∂ ∂x δµ s | x=0 = G ↑ − G ↓ G o δµ c (0, y)m + G ↑ + G ↓ G o δµ s (0, y) − 4 3 ∂ ∂x δµ (0) s,x e x ,
and similarly for the contact T 2 . If we consider these conditions for δµ c ∼ = δµ
(0) c , δµ s ∼ = δµ (0) s we derive c (0) 2 = c (0) 2 = 0. Conditions (30)-(33) give c (1) 2 =h m z 3mD G p tot G + G o µ L − µ R d c (1) 2,x = − 2 3 m y G p tot G + G o µ L − µ R d c (1) 2,y = 2m x G p tot G + G o µ L − µ R d c (1) 2,z = −h 3mD 4 G o G ↑ tot G ↓ tot G ↑ tot + G ↓ tot µ L − µ R d ,
Finally, we have for the spin-orbit induced charge Hall effect
δµ (B) c (x, y) ≈ αm z G p tot G + G o µ L − µ R k F L y l ,(38)
and for the spin Hall effect
δµ (B) s,x (x, y) ≈ − 2 3 αm y µ L − µ R L G p tot G + G o y,(39)δµ (B) s,y (x, y) ≈ 2αm x µ L − µ R L G p tot G + G o y,(40)δµ (B) s,z (x, y) ≈ − 4α k F lG o µ L − µ R L G ↑ tot G ↓ tot G ↑ tot + G ↓ tot y,(41)
where l is the electron mean free path. As we see from the equation (38) the spin-orbit induced charge Hall effect is non-zero only for nonzero magnetization of the ferromagnet in the z-direction (direction which is perpendicular to the film). At the same time the spin-Hall effect of the spins along z, (41), is independent of the magnetization direction. So, assuming m = {0, 0, 1} the magnitude of Hall effect is
µ (B) H ≈ −α d l G p tot G + G o µ R − µ L k F L ,(42)
and the spin Hall effect
µ (B) sH ≈ α d l 4 G o G ↑ tot G ↓ tot G ↑ tot + G ↓ tot µ R − µ L k F L .(43)
The spin-orbit induced charge Hall effect vanishes when the polarization goes to zero, as should be expected. In comparison, we give the magnitude of spin Hall effect in pure normal metal regime in the limit of d ≪ l sf . As directly follows from eq. (18)
µ (A) sH (d ≪ l sf ) = α d l µ R − µ L k F L .(44)
To evaluate the expressions (42), (43) and compare them to (44) we assume the conductance at the tunnel barrier T 2 equals to the sum of spin-dependent conductances at the barrier T 1 , so G = G ↑ + G ↓ and take a typical value of the polarization of the conductance of the system p ≡ (G ↑ − G ↓ )/(G ↑ + G ↓ ) = 1/2. The limit G ≫ G o is not interesting because in this case our system will be similar to a pure normal metal attached to reservoirs, which was considered in the previous section. Another limit G ≪ G o is also less interesting due to the small induced voltage across the normal metal and consequently vanishing Hall effects. So, we consider the most interesting case G = G o . In this case
µ (B) sH µ (A) sH (d ≪ l sf ) = 3 7 ,(45)µ (B) H µ (A) sH (d ≪ l sf ) = 1 7 .(46)
We see from these expressions that both the charge and the spin Hall effects are comparable to the spin Hall effect in the pure normal metal regime attached to leads with perfect normal metal contacts in the regime L ≫ l sf . Although the magnitudes of charge and spin Hall effects have a similar structure in both regimes, they have a different origin in principle. The effects in this section are due to skew scattering, side-jump scattering and anomaloys velocity, while the effects in the previous section are due to the anomalous velocity operator only. We treated in this section the regime d ≪ L. We expect that the spin Hall and the charge Hall effects increase their magnitudes µ sH with increasing of d until saturation when d ∼ l sf similary to the case when the spin Hall effect in pure normal metal attains the "universal value" (19).
IV. MICROSCOPIC DERIVATION OF DIFFUSION EQUATION
We will in this section derive the diffusion equations rigorously from the microscopoic Hamiltonian with the Keldysh Green's function technique in the quasiclassical limit. The nonmagnetic impurity potential (2) in terms of incident k and scattered k ′ wave vec-
tors is V imp (k, k ′ ) = i γ i exp −i (k − k ′ ) r i . Conse- quently, the spin-orbit interaction in this representation isV so (k, k ′ ) = −i(α/k 2 F )σ · (k × k ′ ) V imp (k − k ′ ).
Thus, in normal metals with dilute impurities the electrons interact with the potential V (k, k ′ ) ≡ V imp + V so :
V (k, k ′ ) =M V imp (k, k ′ ),(47)
where we have introduced the 2 × 2 matrix in spin-spacê
M ≡1 − i α k 2 Fσ (k × k ′ ) .(48)
In addition, the electrons interact with magnetic impurities (4) to be discussed below. Our transport theory is based on the Keldysh formalism 30,33 . In this formalism the Green's function has the form
G = Ĝ RĜK 0Ĝ A ,(49)
where the retarded, advanced and Keldysh Green's functions areĜ
R = −iθ(t 1 − t 2 ) [Ψ(x 1 ), Ψ + (x 2 )] + ,(50)G A = +iθ(t 2 − t 1 ) [Ψ(x 1 ), Ψ + (x 2 )] + ,(51)G K = −i [Ψ(x 1 ), Ψ + (x 2 )] − .(52)
Here Ψ is the fermion annihilation operator, Ψ + is the fermion creation operator, both in the Heisenberg picture, and the coordinate x i denotes both spatial position and time, x i = (x i , t i ). Note that the fermion annihilation and creation operators are 2-component vectors in spin-space. The self-energy has the same triangular matrix structure as the Green's function,
Σ = Σ RΣK 0Σ A .(53)
We denote 4 × 4 matrices in Keldysh space by the symbol "check" (ˇ) and 2 × 2 matrices in spin space by the symbel "hat" (ˆ). Next we define the center-of-mass and relative variables x = 1 2 (x 1 + x 2 ), r = x 1 − x 2 , and Fourier transform with respect to the relative coordinate r in order to obtain the Green's function in the mixed representatioň
G(p, x) = dre −ip·rǦ (x + r/2, x − r/2) .(54)
We will also use the ξ-integrated (quasiclassical) Green's functionǧ(n, r) = (i/π) dξǦ(p, r), where ξ = p 2 /2m− µ, and n is a unit vector along the momentum at the Fermi surface (n = k F /|k F |). Let us first consider the current produced by the normal velocity operator of the electrons,ĵ N . This current is expressed aŝ j N = (e/m)ℜ Ψ + pΨ , where p is the momentum operator. Introducing the Green's functions, the current due to the normal velocity operator iŝ
j N (x 1 ) = − e 2m lim x1→x2 (∇ 1 − ∇ 2 )Ĝ K (x 1 , x 2 ) . (55)
Inserting the Fourier representation of the Green's function in the quasiclassical approximation gives for the ordinary current in the mixed representation 30
j N = eN o 2 dε dn 4π v F n ĝ K (n, r) ,(56)
where ... denotes averaging over impurities and N o is the density of states at the Fermi level. The total current also has contributions caused by the anomalous velocity operator. This contribution to the current can be expressed 30,34 aŝ
j av = eN o 2 dε dn 4π dn ′ 4π v so (n, n ′ )ǧ(n ′ , r) K . (57)
The anomalous current caused by the spin-orbit interaction isv
so (r) ≡ dV so (r) dp = ᾱ hk 2 Fσ × ∇V imp (r),(58)In Fourier space, v so (k, k ′ ) = ᾱ hk 2 Fσ × k | ∇V imp | k ′ =N V imp (k, k ′ ).
(59) where we have introduced the 2 × 2 matrix in spin-spacê
N ≡ i ᾱ hk Fσ (k − k ′ ) .(60)
The challenge is now to find the average, v so (n, n ′ )ǧ . Note that this is different than the average Green's function appearing in the contribution from the normal velocity operator (56). In the anomalous current (58), a product of the spin-orbit scattering potential and the Green's function, both of which depends on the impurity configuration has to be evaluated, and in general one should expect that v soǧ = v so ǧ . Using the Dyson equation ǧ =ǧ o +ǧ oΣ ǧ and v soǧ =NM −1 Vǧ we find the result
v soǧ =NM −1 Σ ǧ .(61)
The self-energy partΣ can be expressed in the Born approximation asΣ = VǧV and has one contribution due to scattering off non-magnetic impurties and two contributions due to the spin-orbit interaction,Σ(n) = Σ i (n) +Σ (1) so (n) +Σ (2) so (n) +Σ sm (n), wherě
Σ i = − ih 2τ ǧ(n ′ ) n ′ ,(62)Σ (2) so = − ihα 2 2τ σ(n × n ′ )ǧ(n ′ )σ(n × n ′ ) n ′ (63) Σ (1) so = −h α 2τ ǧ(n ′ )σ(n × n ′ ) + h.c. n ′ .(64)
Scattering by magnetic impurities (4) do not cause additional terms in the expressions for the current density, but gives an additional contribution the the electron self-energyΣ
sm = − ih 2τ sm 1 3 iσ i ǧ(n ′ ) n ′σi ,(65)
where the spin-flip relaxation time due to magnetic impurity scattering is
1 τ sm = 8π 3 n sm N o S(S + 1) dn ′ 4π |V sm (n − n ′ )| 2 ,(66)
n sm is the concentration of the magnetic impurities and S is the quantum spin number of the impurity. The self-energy due to scattering off magnetic impurities will leads to simple additional spin-flip relaxation terms in the spin diffusion equation. In normal metals, the retarded component of Green's function equalsĝ R =1, andĝ A = −g R = −1. By evaluating the Keldysh component Eq. (61) the anomomalous current is given aŝ
j av = − ieNoh 8τ dε dn 4π dn ′ 4πNM −1 [(2 + α 2 )ĝ K (n ′ ) + α 2σ (n × n ′ )ĝ K (n ′ )σ(n × n ′ ) − iα{ĝ K (n ′ )σ(n × n ′ ) + σ(n × n ′ )ĝ K (n ′ )}] + h.c.(67)
We consider transport in the diffuse transport regime. In the diffusive regime, characterized by v F τ ≪ L (L is the system size) the Green's function is almost isotropic 30 , and we can then expand the Green's function in form of isotropic and nonisotropic parts,
dεĝ K (r, n) =μ o (r) +ĵ 1 (r) n.(68)
After integrating of the formula (67) we have the result
j av = eN o 8τ k F (σ ×ĵ 1 −ĵ 1 ×σ)f 1 + i e i jk ǫ ijk [Tr(σ jĵ1,k )f 2 + Tr(σ kĵ1,j )f 3 ] (69)
where Tr(...) denotes the sum of diagonal elements of a matrix and the spin-orbit interaction strength dependent functions f i (α) are
f 1 = −α + α 2 9 + α(1 − α 2 ) (n ′ z ) 2 − α (n z ) 2 (n ′ x ) 2 1 + α 2 |n × n ′ | 2 dn 4π dn ′ 4π
(70)
f 2 = 2α 9 + 2α 2 2α(n i ) 2 (n ′ j ) 2 (n ′ k ) 2 − (n k ) 2 (n ′ j ) 2 1 + α 2 |n × n ′ | 2 dn 4π dn ′ 4π
(71)
f 3 = 2α 3 + 2α 2α(n i ) 2 (n ′ k ) 4 + 2α 2 (n k ) 2 (n ′ i ) 2 (n ′ k ) 2 − (n ′ k ) 2 1 + α 2 |n × n ′ | 2 dn 4π dn ′ 4π(72)
In the limit of weak spin-orbit interaction (α ≪ 1) we thus find the current contribution due to the anomalous veloicty operator
j av = αeN o 8τ k F ĵ 1 ×σ −σ ×ĵ 1(73)
We have now found the full expression for the current density in terms of the distribution functions and we will now proceed to compute the diffusion equation. The starting point in calculating the Green's function is the equation of motion for the impurity averaged quasiclas-sical Green's function, the Eilenberger equation.
hv F ∇ ǧ + i Σ , ǧ − = 0,(74)
where v F is Fermy velosity. We will in the following omit the impurity average symbol and only implicity write averaging over momentumk ork ′ . We calculate Keldysh components of the various self-energy commutators. For scattering off non-magnetic impurities we obtain the wellknown result:
i Σ i ,ǧ K =h τ ĝ K − ĝ K(75)
For spin-orbit scattering, which gives rise to side-jump and skew-skattering, using σ(n × n ′ )σ(n × n ′ ) n ′ =1 we findΣ
(2)R so = − ihα 2 2τ1 ,Σ (2)A so = ihα 2 2τ1 (76) and i Σ (2) so ,ǧ K =h α 2 τ 2 3ĝ K + σ(n × n ′ )ĝ K (n ′ )σ(n ′ × n) .
(77) Similarly, using σ(n × n ′ ) = 0, we findΣ
(1)R so = Σ (1)A so = 0 and i Σ (1) so ,ǧ K = ihα τ ĝ K (n ′ )σ(n × n ′ ) −σ(n × n ′ )ĝ K (n ′ )
(78) In the diffusive transport regime, we use the representationμ
o = µ c1 + µ sσ .(79)
Employing (76), we find after averaging over momentum, the expression forΩ
(i) so ≡ i dε Σ (2) so ,ǧ K ,
where ε is the energy spectrum of the system:
Ω (1) so = ihα 3τ n(ĵ 1 ×σ) + n(σ ×ĵ 1 ) ,(80)Ω (2) so = 2hα 2 3τ µ sσ + (nσ)(nµ s ) +ĵ 1 n . (81)
Similarly, for magnetic impurity scattering, from (65) we findΣ
R sm = − ih 2τ sm1 ,Σ A sm = ih 2τ sm1 ,(82)Σ K sm = − ih 2τ sm µ c1 − µ sσ ,(83)
andΩ sm =h τ sm µ sσ +h 2τ sm j 1 n .
After substitution of (75), (81), (80), (84) into Eilenberger equation (74) and averaging over dn we find
1 3 v F ∇ĵ 1 + 8α 2 9τ + 1 τ sm µ sσ = 0 .(85)
Next, we find a second equation by averaging over dn the (74) multiplated by n
v F τ ∇μ o + (1 + 2α 2 3 )ĵ 1 + iα 3 (ĵ 1 ×σ) + (σ ×ĵ 1 ) = 0 .(86)
Eq. (86) can be solved to givê
j 1 = −v F τ [ 1 1+2α 2 /3 ∇μ o + K 1 ∇(µ sσ ) − K 2 (σ∇)µ s + (K 2 − K 1 )σ(∇µ s )] ,(87)
where
K 1 = 4α 2 (3 + 2α 2 ) 2 − 4α 2 ,(88)K 2 = 2α(3 + 2α 2 ) (3 + 2α 2 ) 2 − 4α 2 .(89)
We now use the representation (68) to find from (56) the current caused by the normal velocity operator
j N (r) = eN o 2 dn 4π v F n (μ o +ĵ 1 n) = eN o v F 6ĵ 1 . (90)
After substitution (87) into equation above we find finally that there are three contributions to the current density from the normal velocity operator:
j ss = − αeN o D 2α 2 + 2α + 1σ (∇µ s )(93)
Also using expression forĵ 1 (87) we can rewrite the formula for anomalous current (69) likê j av (r) = − eNovF 4kF { 2α 2 2α 2 +2α+1 ∇ × µ s +σ × ∇µ c f 1 (α) + i e i jk ǫ ijk [(K 1 (α)f 2 (α) − K 2 (α)f 3 (α)) ∇ k µ s,j + (K 1 (α)f 3 (α) − K 2 (α)f 2 (α)) ∇ j µ s,k ]}
Diffusion equations for both spin and charge distribution functions can be derived directly from the equation (85) after substitution (87):
∇ 2 µ c = 0 (95) (2α 2 + 3) 2 D (2α 2 + 3) 2 − 4α 2 ∇ 2 µ s = µ s τ so + 4α 2 D (2α 2 + 3) 2 − 4α 2 ∇(∇µ s ).
(96) In the limit of weak spin-orbit interaction (α ≪ 1) we obtain the simplified diffusion equations (5) and (6) and the simplified expression for the current density (9), (10), (11), (12), and (13).
V. CONCLUSIONS
We have derived diffusion equations for spin and charge flow in normal metals and the associated expression for the spin and charge currents. The total current consists of four contributions: Ordinary current, anomalous current, side-jump current and skew scattering current. These macroscopic diffusion equations, allows computation of charge Hall voltages and spin Hall voltages in pure normal metals or hybrid ferromagnet-normal metal systems.
We have considered two experimental relevant geometries and calculated Hall and spin-Hall voltages in the case of weak spin-orbit interaction. In pure normal metals with no ferromagnetic contacts, there is no charge Hall effect due to the spin-orbit interaction, and the spin Hall effect is caused by the anomalous current, in agreement with the observation in Ref. 9. In this geometry, we have made the additional observation that the spin Hall voltage reaches and "universal" value independent of the strength of the spin-orbit interaction, when spin-flip scattering due to spin-orbit scattering dominates spin-flip scattering due to magnetic impurities. When a spin-polarized current is injected into a normal metal film, both a Hall voltage and a spin Hall voltage exist. The magnitude of the Hall voltage is governed by sidejump, skew scattering and anomalous currents when the system is shorter than the spin-diffusion length. For systems longer than the spin-diffusion length, the Hall voltage vanishes, and the spin Hall effect resumes the value dominated by the anomalous current. In the intermediate regime, the competition between skew scattering, side-jump scattering and anomalous velocity operator determines the spin Hall and charge Hall voltages.
FIG. 1 :
1Schematic picture of the skew scattering mechanism. An incident electron with spin up (down) scatters preferrably with a postive (negative) angle. FIG. 2: Schematic picture of the side-jump mechanism. The trajectory of the outgoing electrons is shifted to the upper (lower) side the scattering center at large distances for spin up (down) states.
FIG. 3 :
3The thin metallic film with a contact to reservoir.
FIG. 4 :
4The thin metallic film with tunneling contacts T1 and T2 to ferromagnet and normal metal.
j ; j = 0, 1. The boundary condition on the contact T 1 gives
j
N (r) =ĵ o (r) +ĵ sj (r) +ĵ ss (2 )(1 + 2α 2 )(3 − 2α 2 ) ∇(µ sσ )] (91) j sj = α(3 + 2α 2 )eN o D (3 + 2α 2 ) 2 − 4α 2 (σ∇)µ s(92)
AcknowledgmentsWe would like to thank Andy Kent, Jan Petter Morten and Oleg Jouravlev for stimulating discussions. This work has been supported in part by the Research Council of Norway, NANOMAT Grants No. 158518/143 and 158547/431, and through Grant No. 153458/432.
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[] |
[
"Simulation of reconstructions of the polar ZnO (0001) surfaces",
"Simulation of reconstructions of the polar ZnO (0001) surfaces"
] |
[
"H Meskine \nDepartment of Process & Chemical Engineering\nUniversity Of Strathclyde\nJames Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom\n",
"P A Mulheran \nDepartment of Process & Chemical Engineering\nUniversity Of Strathclyde\nJames Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom\n"
] |
[
"Department of Process & Chemical Engineering\nUniversity Of Strathclyde\nJames Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom",
"Department of Process & Chemical Engineering\nUniversity Of Strathclyde\nJames Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom"
] |
[] |
Surface reconstructions on the polar ZnO(0001) surface are investigated using empirical potential models. Several possible reconstructions based around triangular motifs are investigated. The quenching of the dipole moment in the material dominates the energetics of the surface patterns so that no one particular size of surface triangular island or pit is strongly favoured. We employ Monte Carlo simulations to explore which patterns emerge from a high temperature quench and during deposition of additional ZnO monolayers. The simulations show that a range of triangular islands and pits evolve in competition with one another. The surface patterns we discover are qualitatively similar to those observed experimentally.
|
10.1103/physrevb.84.165430
|
[
"https://arxiv.org/pdf/1006.1826v1.pdf"
] | 118,372,111 |
1006.1826
|
34709dfddc33f791fbe4db1fe6f39282b94423d5
|
Simulation of reconstructions of the polar ZnO (0001) surfaces
9 Jun 2010
H Meskine
Department of Process & Chemical Engineering
University Of Strathclyde
James Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom
P A Mulheran
Department of Process & Chemical Engineering
University Of Strathclyde
James Weir Building75, G1 1XQMontrose, GlasgowUnited Kingdom
Simulation of reconstructions of the polar ZnO (0001) surfaces
9 Jun 2010
Surface reconstructions on the polar ZnO(0001) surface are investigated using empirical potential models. Several possible reconstructions based around triangular motifs are investigated. The quenching of the dipole moment in the material dominates the energetics of the surface patterns so that no one particular size of surface triangular island or pit is strongly favoured. We employ Monte Carlo simulations to explore which patterns emerge from a high temperature quench and during deposition of additional ZnO monolayers. The simulations show that a range of triangular islands and pits evolve in competition with one another. The surface patterns we discover are qualitatively similar to those observed experimentally.
I. INTRODUCTION
Zinc oxide has a wide range of applicability from electronics to catalysis. 1 For example it is used as part of ZnO/Me/ZnO (Me=metal) multilayer functional glass coatings designed to filter heat-generating infra-red solar radiation. This is usually achieved by incorporating a thin low emissivity metal layer a few nanometres thick. 2 In the case where silver is used to construct a ZnO-Ag-ZnO sandwich, it has been shown 3,4 that the lower Ag(111)/ZnO(0001) interface may fail for reasons not yet fully understood, leading to a sizable cost increase in the manufacturing of these devices. To understand this effect, one needs to characterise the interfacial structure that arises from the growth of Au on Zn(0001). A prerequisite for this is a fundamental understanding of the Zn(0001) surface that templates this growth.
Zinc oxide (zincite) has the well-known wurtzite structure with lattice parameters at room temperature and ambient pressure of a = 3.25Å, c = 5.207Å, and u = 0.3825, and space-group P 6 3 mc (no. 186 in crystallographic tables). 5,6 The structure may be understood as two interpenetrating hexagonal lattices, with each Zn (resp. O) sitting at the centre of a distorted O (resp. Zn) tetrahedron. The crystal when cut along the (0001) or (0001) planes is known to be a type III polar material according to the Tasker classification. [7][8][9] That is to say that the unit cell is comprised of alternative negative and positive charged layers. This ultimately leads to a diverging electrostatic potential and should make the two polar surfaces of ZnO energetically unfavourable. This, however, is not the case as both the O-terminated and Znterminated polar surfaces show remarkable stability. 10 Consider a slab of the material with bulk-terminated polar surface as used in typical computations (see Figure 1). Since the polar ZnO(0001) and ZnO(001) surfaces occur naturally, there must be a mechanism to quench the dipole moment that exists normal to the slab surface. In order to quench this macroscopic dipole moment a transfer of charge across the slab of (1 − 2u)σ ≈ 0.235 × σ is necessary, where σ is the surface charge density. 10 This may be understood in terms of the electrostatic energy change when charge is moved from one surface to the other in the direction of the internal electric field of the unreconstructed slab. Once sufficient charge has been moved, the counter electric field thus established cancels the one due to the bulk structure.
There are several mechanisms which may compensate the charge at the surface and counteract the macroscopic dipole moment of the semi-infinite crystal. Three, not necessarily incompatible, mechanisms have been considered in the literature: (i) adsorption of charged species e.g. hydroxilation, (ii) modification of the surface region by reconstruction, and (iii) direct charge transfer.
Until recently, the exact nature of such a mechanism in ZnO was not well understood, but recent theoretical and experimental studies [11][12][13] may have resolved the issue. A combination of surface microscopy techniques and Density Functional Theory (DFT) have indicated that, depending on the atmospheric environment, mechanisms (i) and (ii) may be active in quenching the polarisation of the ZnO(0001) surfaces. For the Zn terminated orientation and in hydrogen-rich conditions, the surface is best passivated by adsorption of hydroxyl groups, while under low hydrogen partial pressure the surface tends to form triangular reconstructions that appropriately compensate the charge imbalance created by the surface cut. This work seems to rule out the third mechanism (iii) which had previously been proposed, 14 involving charge transfer between O-terminated and Zn-terminated surfaces.
The above theoretical studies have relied on DFT which, while accurate, has a high computational cost and does not allow for a comprehensive search of the phase space. For example, in the case of the triangular reconstruction, STM scans show a range of triangle sizes while DFT studies only allow comparison of the energetics of single, relatively small configurations. In this work we present a study which combines fast, albeit less accurate, empirical potentials with Monte-Carlo simulations to study the structure and energetics of the polar ZnO(0001) surface, focusing on the surface reconstruction mechanism to quench the dipole. This approach is justified by the dominant role electrostatics plays in the surface resconstructions. 13 The rest of the paper has the following structure. Section II describes the methodology employed, discussing the empirical potentials, surface relaxation calculations and Monte Carlo (MC) simulations. The results are presented in Section III, firstly for the energetics of various surface reconstructions, and then for the surface patterns that emerge from the MC simulations. The implications of the results are discussed in the following section, and our conclusions are given in the final Section V.
II. METHODOLOGY
A. Empirical potentials and surface slab calculations
Previous work by Catlow and coworkers 8,15 has shown that empirical potentials are well suited to describe the details of the structure of the polar surfaces of most oxides. The parameters for a Buckingham potential are fitted following ref. 16 to reproduce a variety of properties of Zinc Oxide. The potential was kept as simple as possible as is appropriate for the desired level of computation. For details of the validation of the interatomic potential parameters we refer the reader to ref. 17 . The total energy is computed by summing all pair interactions of the form
E(r ij ) = q i q j 4πǫ 0 r ij + A exp(−r ij /ρ) − C/r 6 ij(1)
where r ij = r i − r j is the distance between two ions with charges q i and q j . In this work we use formal ionic charges ±2e in all our computations. The first term of Eq. 1 is the long-range Coulomb pair interaction, while the second and third terms correspond respectively to the repulsive and attractive terms of the short-range Buckingham pair potential.
The polarisability effects are described by a core-shell model, where the oxygen ion and its electronic cloud are modelled by a massive core and a mass-less shell carrying different charges (but with total charge −2e) and linked by a spring with energy
E spring = 1 2 k i δ 2 i ,(2)
where k i is the spring constant for ion i and δ i is the core-shell distance. The empirical parameters A, ρ, C and k i are determined by fitting to available experimental properties, such as the elastic constants. The fitting was performed with the GULP code 18 using 8 potential parameters and various parameters from observable data.
The computed bulk properties are compared to some relevant experimental values in Table I, and the values of the parameters used in this work are given in Table II. Surface structure calculations were performed using three-dimensional periodic slabs with a large vacuum gap normal to the (0001) surface. Each slab contains one Zn-terminated surface and one O-terminated surface (see Fig. 1). Several reconstructions were created at the surfaces, where overall charge neutrality was ensured by removing the oppositely charged species from the other side of the slab. The outermost three surface layers on each side of the slab were allowed to relax. The energy of different surface reconstructions were computed and an energy hierarchy constructed by comparing their surface energies. In order to estimate the surface energy of a given surface structure we use the total energy of the bulk unit cell as a reference state, in which case the surface energy is given by
γ = (E slab − N cell × ε bulk ) /2A(3)
where A = (N s a) 2 sin γ is the surface area of a slab (see Fig. 1) containing N s × N s × N z lattice units, E slab is its relaxed total energy, and ε bulk is the energy of a bulk unit cell. The quantity N cell is the effective number of unit cells in the slab calculated by dividing the number of atoms in the slab by the number of atoms in a bulk unit cell ( four in the case of ZnO). The factor of two accounts for the fact that the surface energy is the average surface energy of both sides of the slab. It is important to emphasise again that the surface energy calculated in this way depends on slab thickness, unless we have perfect quenching of the dipole moment by creating a net charge transfer of 0.235×σ from one surface to the other in the reconstruction. For this reason, we will compare the energies of various surface structures using the same slab thickness N z = 6 (twelve bilayers).
B. Monte-Carlo simulations
While a large number of structures may be explored using the above empirical model, it is impractical to find the lowest energy reconstruction using more and more elaborate guesses of the surface structure. In order to explore the large phase space of possible surface reconstructions we used Monte Carlo (MC) simulations with bulk lattice positions in a slab. The ions in the three upper-most bilayers of the slab are allowed to hop within their own bilayer and into the bilayers directly above or below. The simulations were started from different initial configurations and the system left to evolve according to the Metropolis algorithm. The initial configurations of the three uppermost bilayers were formed either by ion removals or by addition of ions to the clean slab. The overall charge neutrality was again ensured by adding/removing the oppositely charged species from the other side of the slab at the start of each simulation. As several studies 7 have shown that the surface layer relaxation is less than 0.1Å, for the sake of simplicity (and computational efficiency) we have neglected the effect of lattice relaxation on surface energy during these MC simulations. A periodic slab model is used throughout, with N z = 6 bulk unit cells along the c axis and a large vacuum of L z = 30Å added to form the super-cell. We verify that L z is large enough by ensuring that the surface energy does not depend on L z . Successive configurations are generated by a series of nearest neighbour hops of either species selected at random at the uppermost Zn-terminated surface, with acceptance probability e −∆E/kBT where T is the temperature and ∆E is the difference between total energies of the successive trial configurations. The bottom Oterminated surface reconstruction remains fixed in the simulations. The energy of a given atom in the slab is simply the sum of all its pair interaction with the other ions in the slab, with the pair interaction of ion i given by
ε i = j E(r ij ).(4)
Here, also for computational efficiency, we neglect the shell model component of the potential E spring in the MC work only. Since in this system we have no mechanism to quench any dipole across the slab, we ensure that the simulation is started from configurations with only a small residual dipole. The initial configuration of the MC simulation is then disordered by running the simulation at very high temperature leading to a fully disordered arrangement of the surface species, after which the temperature is lowered abruptly. The simulations are run at high temperature for a large enough number of steps that the initial ordering disappears.
The most expensive step of the simulation is the energy evaluation which includes long-range terms. The Coulomb sum being conditionally convergent in a periodic system, we make use of the Ewald sum 20-22
E R = 1 2 n ′ i,j q i q j erfc(αr ij,n ) r ij,n(5)E K = 2π V k =0 e −k 2 /4α 2 k 2 |S(k)| 2(6)E 0 = − α √ π i q 2 i + 2π V M 2 z(7)
where q i is the formal charge of on i, r i is the position of the ion within the periodic slab, S(k) = i q i e ikri is the structure factor, and M z is the z coordinate of the total dipole moment in the slab M = i q i r i . The parameter α is determined using the requirement that the Ewald sum is accurate yet efficient (see for example 23 ). It is worth noting that this expression is the more computationally efficient 3D version of the Ewald sum, not the two-dimensional version. If the vacuum slab is chosen large enough, only a correction due to the residual surface dipole is necessary. The above total energy is computed once at the beginning of the run, and updated in the course of the simulation by only computing the energy difference between trial configurations. This considerably speeds up the computation of the energy and scales as N 1/2 , where N is the number of particles in the system.
III. RESULTS
A. Surface reconstructions and energy hierarchy
On the zinc-terminated surface, it was experimentally shown that the triangular reconstructions are a single bilayer high. We therefore form the reconstructions by removing Zn and O atoms from the top-most bilayer and refer in what follows to the excess zinc vacancy concentration Θ vac . Recall that we remove the opposite charge species from the O-terminated surface, keeping overall charge neutrality and imposing an electric field across the slab which compensates the bulk dipole moment for Θ vac = 0.235. The stability of the surface reconstruction is assessed by studying the surface energy as a function of this excess Zn vacancy concentration. Table III shows the surface energies for various zinc vacancy con- centrations and their corresponding surface reconstructions illustrated in Figure 2 . Note that the reconstruction in the √ 48 × √ 48 surface unit cell as shown in Fig. 2 is not the most stable in this work, contrary to the prediction from density functional theory. This may be explained by a simple electrostatic argument, where the formal ionic representation of the species tends to overestimate the contribution of the Coulomb interaction, and thus favours smaller surface reconstructions.
We have computed the surface energies of several more surface reconstructions, for varying values of the excess zinc concentration, as well as for isolated vacancies. The results are summarised in Fig. 3 and are plotted in reference to the surface energy of the bulk-terminated slab with N z = 6. We reiterate that the surface energy of the bulk-terminated slab is ill-defined due to the presence of a large surface dipole. It is only the difference in surface energies of various reconstructions that is of interest. The most stable reconstruction is the small 4 × 4 cluster with the smallest triangular vacancy (see Figure 2), with the more exotic reconstruction on the √ 48 × √ 48 surface cell lying nearby. The isolated vacancies consistently have a larger surface energy. Relaxation of the surface layers has a small effect on the surface stability, but does not impact the overall ordering of the various reconstruction energies.
In conclusion, the surface hierarchy obtained using the simple empirical potential shows no preference for large triangular reconstructions. Instead, small clusters which locally quench the surface dipole are preferred. The small impact of surface relaxation reflects the dominance of dipole moment quenching. This means that surface relaxation can be neglected in the MC simulations which follow, in which we allow the system to explore configuration space to see if larger reconstructions emerge naturally in our model with formal ionic charges.
B. Monte-Carlo Simulations
We have performed the MC simulation of a high temperature quench on a slab with N z = 6, N s = 32 and with the surfaces initially tessellated with the 4 × 4 triangular reconstructions giving N = 22, 528 ions in total. Representative snapshots of the species distribution in the three uppermost layers of the Zn-terminated surface are shown in Fig. 4(a-c). We show the three uppermost bilayers separately (bilayer 1-3) at (a) the start of the simulation, (b) at the end of the high temperature run, and (c) after quenching. The figure clearly shows the effect of the high temperature, with the ordered triangular reconstructions completely disappearing from the first bilayer. The second and third bilayers are occupied as well, with no apparent ordering. After the system is cooled and left to evolve, large triangular reconstructions in the lowest bilayer begin to nucleate via aggregation of small triangular units. The smallest such unit is the one unit-cell triangle (three O ions surrounding one Zn ion) and is consistent with the earlier empirical potential result. The second bilayer only shows a few isolated a−side triangles, while the third layer is now completely empty. These observations are qualitatively identical for a large class of system size and parameters.
In Fig. 5 we show the evolution of the surface energy and the dipole moment normal to the surface in the simulation. During the high temperature anneal, both the surface energy and dipole moment are large in magnitude. This is due to the almost random placement of the surface layer ions into the 3 accessible bilayers, which leads to an obvious loss of bonding energy. The dipole moment also increases in magnitude since there is an excess of oxygen ions over zinc in the surface, so displacing them on average by one bilayer changes the total moment in the system. Upon quenching, the surface energy and dipole moment quickly reduce in magnitude, and in fact reach slightly lower values than in the starting configuration. At the start of the simulation, the dipole moment is −162.29 e.Å, and is not zero since Θ vac = 0.25 rather than the ideal Θ vac = 0.235. After the temperature is raised, the dipole moment is roughly −300 e.Å, since the topmost ions are now randomly distributed. By the end of the quench, after N MC = 4 × 10 5 , its value is slightly lower, −236.15 e.Å. This is achieved by the few ions occupying the second bilayer. For the surface energy, the quenched value is 0.36 eV/Å 2 , as opposed to 0.59 eV/Å 2 after heating.
The quenched MC simulation clearly shows that the system can evolve to a structure with preferred triangular motif and no regular tessellation of the surface. It is of interest to see if this effect also emerges during a simulation where the total number of ions in the surface layers increases over time, mimicking epitaxial growth. Starting from the end of a quenched simulation, we perform the MC annealling but now add pairs of Zn 2+ /O 2− ions at separate, randomly chosen sites in the three uppermost layers of the slab. Results from a simulation performed on a N z = 6, N s = 32 slab (N = 8, 800 ) are shown in Fig. 6. Again, the simulation shows that triangular reconstructions form spontaneously, and grow larger by nucleation from characteristic smaller aggregates. We also see that the growth on the 2nd bilayer proceeds before the 1st bilayer is complete, leading to a surface with multiple ad-islands and pits of various size.
In Fig. 5 we also show how the surface energy and dipole moment change during the deposition simulation. The addition of the ions allows the system to find structures with decreasing magnitude of dipole moment, since thereby lowering the electrostatic energy. An interesting feature shown by Fig. 5 is that both surface energy and dipole moment oscillate as more ions are added, and reach even lower values as we deposit more and more ions. Thus the system finds a steady state with lower energy and dipole moment by making use of the larger number of degrees of freedom made available by the deposited ions.
IV. DISCUSSION AND CONCLUSION
It is observed from the STM results 11 that not one but several triangular features co-exist at the surface of Zn-terminated ZnO(0001). This behaviour also emerges in our Monte-Carlo simulations, for both the quenched structure in Fig. 4 and the deposition structure in Fig. 6. The reason for this can be traced to the small energy differences between the various surface reconstructions shown in Table III; there is no substantially preferred reconstruction, provided θ vac is close to 0.235 locally. Therefore the patterns that emerge in the MC simulations result from the competitive growth of energetically comparable triangular reconstructions. For this reason, we do not observe any long time coarsening of the structures in the simulation, as can be seen from the surface energy evolution shown in Fig. 5 during the quenched phase before deposition starts, and neither do we observe the system being restored to its regular tessellated starting configuration.
Our quenched MC simulation in Fig. 4 produces an interesting surface morphology that is reminiscent of the experimental STM images 11,13 . Furthermore, the simulation with increasing surface coverage of Fig. 6 also includes other morphological features such as co-existing ad-islands and pits that are found experimentally. Therefore, we believe that the simulations capture some of the main physical processes that give rise to these surface reconstructions. However, it is important to note that the reconstructions found by the simulation are much smaller than those observed experimentally. The largest triangle observed from the MC simulation has a side of the order of 20Åthus only reproducing the smallest clusters observed in the STM scans. It is possible that this is due to the lattice sizes and simulation durations that are accessible. Another explanation is that our use of formal charges (±2e) on the ions is not justified in this system. Using formal charges probably overestimates the strength of the Coulomb interaction, thereby tending to make the triangular clusters at the surface more compact. Work to augment the current model using a charge equilibration (or QEq) scheme 24 is therefore planned.
In conclusion, we believe that the models presented here do help to explain the Zn-terminated surface reconstructions observed experimentally. Whilst the accuracy of our models cannot compete with DFT, we are not restricted to studying individual reconstructions. The use of empirical potentials allows us to explore the phenomenology of the surface reconstructions more freely, and we find that a broad range of characteristic triangular motifs naturally emerge in our simulations, qualitatively consistent with the STM results.
FIG. 6: In a way similar to 4 we show snapshots of the three upper bilayers as the deposition of a single monolayer (ML) is carried out. The top row shows the three bilayers before the deposition is started, the middle one after 0.5 ML was deposited, and the lower one after a full ML was deposited onto the surface. Note that the triangular features of the reconstruction are essentially conserved throughout the deposition.
FIG. 1 :
1Bulk crystal structure of wurtzite zinc-oxide with bulk lattice parameters a = 3.25Å, c = 5.207Å, and u = 0.3825Å. In the bulk each ion is four-fold coordinated, while the surface atoms have only three-fold coordination. (colour online)
and an additional m = 3 inner pit within the larger triangle. The pits and terraces are created by removing Zn and O atoms. The topmost layer atoms are shown as large spheres (light red for O, dark blue for Zn) and the next layer atoms are shown by smaller spheres. The "bulk" atoms in the slab (everything below layer 2) are shown only by their bonds. Upper left we show the smaller triangular pits in a 3 × 3 and 6 × 6 surface unit cell.
FIG. 3 :
3Change in surface energy as a function of excess zinc vacancy on the surface expressed in fractions of a monolayer, where ∆γ = γnvac − γ clean is the net surface energy compared to the surface energy of the bulk-terminated surface. The blue line refers to bulk position of the atoms while the red curve correspond to optimised position for the three bilayers nearest the surface. The surface energy change when creating isolated vacancies is shown by the black curve. Note that the small spheres refer to the topmost bilayer, while the larger ones refer to the bilayer immediately below it.
FIG. 4 :
4Snapshot of Zn 2+ and O 2− ions in the three topmost layers at different stages of the evolution of the MC simulation. The initial configuration (a) is evolved at high temperature until the initial reconstructions are melted giving rise to a random distribution of species in all three layers (b). The temperature is then abruptly at which point triangular structures begin to spontaneously in bilayer 1, while the top bilayers are gradually emptied.(c).
TABLE I :
IComparison of some of the computed bulk properties to available experimental data. The experimental data are taken from standard tables19 .ZnO (wurtzite) This work Experiment
a(Å)
3.27
3.25
c(Å)
5.18
5.207
u
0.3819
0.3825
ε T
11
4.22
9.26
ε T
33
4.59
11.0
C11(GPa)
222.22
209.7
C33(GPa)
220.14
210.9
TABLE II :
IIInteratomic potential parameters used for the Buckingham potential. These were obtained by fitting to the experimental parameters of Tab. The spring Constants are in eV.Å −2 : kO = 15.52 , kZn = 8.57.A(eV) ρ(Å) C(eV×Å 6 ) r cutof f
Zn−O s 499.6 0.359
0.0
0−10Å
O s −O s 22764.0 0.149
27.88
0−12Å
TABLE III :
IIISummary of the surface energies of several slabs, all with Nz = 6, and various total number of ions N . For the relaxed structures. the total energies are computed using GULP with the potential parameters listed in table II. The three outermost bilayers are allowed to relax while the rest of the slab is kept fixed. The vacancy concentration Θvac refers to the ratio of excess zinc vacancies in the uppermost surface layer created by removing zincs and oxygen from both sides of the slab.Θvac(ML) N
γ(eV /Å
2 )
rigid relaxed
Bulk
-
-
-
2 × 2
0.500
88 4.210 4.162
4 × 4
0.250
352 0.255 0.086
√
48 ×
√
48
0.208
1034 0.306 0.245
6 × 6
0.055
856 2.248 1.447
U. Ozgur et al., Journal of Applied Physics 98, 041301 (2005).2 H. J. Glasser, Large Area Glass Coating (Von Ardenne Anlagentechnik, GmBH, 2000).
Acknowledgement: This work was supported by the UK's Engineering and Physical Sciences Research Council grant EP/C524349 and by the University of Strathclyde.
The simulation is started from a 4 × 4 tiled surface with a small residual surface dipole. We proceed to run the simulation at high temperature (T high ) for a large enough number of steps, then (a) quench to a lower temperature (T low ). The surface energy and dipole moment both settle at a lower value than initially. (b) Successive deposition of Zn/O pairs are then executed. Surface energy and dipole moment computed during a full Monte-Carlo run with , Ns = 32. 1FIG. 5: Surface energy and dipole moment computed during a full Monte-Carlo run with , Ns = 32. The simulation is started from a 4 × 4 tiled surface with a small residual surface dipole. We proceed to run the simulation at high temperature (T high ) for a large enough number of steps, then (a) quench to a lower temperature (T low ). The surface energy and dipole moment both settle at a lower value than initially. (b) Suc- cessive deposition of Zn/O pairs are then executed, up to 1
then (c) 2 ML, allowing for a large enough number of steps between events to reach a new steady-state (d). ML. ML, then (c) 2 ML, allowing for a large enough number of steps between events to reach a new steady-state (d).
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. D Fincham, Molecular Simulation. 131D. Fincham, Molecular Simulation 13, 1 (1994).
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|
[] |
[
"Concept-based Adversarial Attacks: Tricking Humans and Classifiers Alike",
"Concept-based Adversarial Attacks: Tricking Humans and Classifiers Alike"
] |
[
"Johannes Schneider [email protected] \nInstitute of Information Systems\nUniversity of Liechtenstein\n\n",
"Giovanni Apruzzese [email protected] \nInstitute of Information Systems\nUniversity of Liechtenstein\n\n"
] |
[
"Institute of Information Systems\nUniversity of Liechtenstein\n",
"Institute of Information Systems\nUniversity of Liechtenstein\n"
] |
[] |
We propose to generate adversarial samples by modifying activations of upper layers encoding semantically meaningful concepts. The original sample is shifted towards a target sample, yielding an adversarial sample, by using the modified activations to reconstruct the original sample. A human might (and possibly should) notice differences between the original and the adversarial sample. Depending on the attackerprovided constraints, an adversarial sample can exhibit subtle differences or appear like a "forged" sample from another class. Our approach and goal are in stark contrast to common attacks involving perturbations of single pixels that are not recognizable by humans. Our approach is relevant in, e.g., multistage processing of inputs, where both humans and machines are involved in decision-making because invisible perturbations will not fool a human. Our evaluation focuses on deep neural networks. We also show the transferability of our adversarial examples among networks.
|
10.1109/spw54247.2022.9833874
|
[
"https://arxiv.org/pdf/2203.10166v1.pdf"
] | 247,594,517 |
2203.10166
|
8164ff72e08935751685099f8d17e7bba7be2bab
|
Concept-based Adversarial Attacks: Tricking Humans and Classifiers Alike
Johannes Schneider [email protected]
Institute of Information Systems
University of Liechtenstein
Giovanni Apruzzese [email protected]
Institute of Information Systems
University of Liechtenstein
Concept-based Adversarial Attacks: Tricking Humans and Classifiers Alike
Index Terms-Adversarial AttacksSemantic AttacksDeep Learning
We propose to generate adversarial samples by modifying activations of upper layers encoding semantically meaningful concepts. The original sample is shifted towards a target sample, yielding an adversarial sample, by using the modified activations to reconstruct the original sample. A human might (and possibly should) notice differences between the original and the adversarial sample. Depending on the attackerprovided constraints, an adversarial sample can exhibit subtle differences or appear like a "forged" sample from another class. Our approach and goal are in stark contrast to common attacks involving perturbations of single pixels that are not recognizable by humans. Our approach is relevant in, e.g., multistage processing of inputs, where both humans and machines are involved in decision-making because invisible perturbations will not fool a human. Our evaluation focuses on deep neural networks. We also show the transferability of our adversarial examples among networks.
I. INTRODUCTION
Adversarial samples are typically generated to be nonrecognizable by humans [1], [2]. This is commonly achieved by adding perturbations resulting from a specific optimization process to the inputs. While this is arguably a preferred adversarial scenario, in this work, we aim at a different objective: generating adversarial samples by modifying "higher-level concepts". This leads to adversarial samples that are, possibly with some effort, recognizable through human inspection.
An attacker might aim to construct adversarial samples that are not only classified differently from their ground truth but also contain specific attributes and are therefore perceived differently by a human. This likely holds in a multi-stage classification process with a human in the loop. For example, a seller might list an item for sale on an online marketplace (e.g. a house or a car), and upload images or other information on such item. Such a marketplace might employ existing artificial intelligence (AI) systems relying on deep learning (DL) to obtain a price estimate [3]. An attacker, i.e., a malicious seller, may want to simultaneously trick (i) such an AI system to output higher price estimates, and (ii) potential buyers by making them believe that the price is appropriate. The attacker might want to enhance visual aspects, increasing the item's appeal (e.g., making a house appear under lighting conditions impossible in reality or employing perspective distortions). Thus, the optimization objective for an attacker might be to change a given (original) sample towards a target sample (e.g., an image of an expensive house with appealing lighting conditions). Put differently, adversarial and original samples tend to exhibit visible differences that are likely to aim to deceive or nudge a human observer-a theme that recently attracted much attention (e.g., [4]). The predicted class of the resulting adversarial sample should generally be identical to the class of the target sample but could be any class. The adversarial sample should also contain specific characteristics (also called "concepts") of the target sample-at least to some extent, since we constrain the adversarial sample to be similar to the given (original) sample. For example, consider an online seller of fashion articles whose website is crawled by a search engine classifying images into brands. The fashion seller might want to fool humans by altering the images of its articles. Potential buyers should perceive it as more appealing but should not complain that the shown product differs from the actual physical product. Aside from tricking humans, the seller might also aim to mislead an AI of the search engine, i.e., conduct search engine poisoning [5]. The search engine might utilize image recognition to identify brands of items. The seller might want the search engine to confuse the image of his/her product with that of a well-known, commonly searched, expensive brand.
Optimizing for non-recognizability by humans might not necessarily be the best option to disguise an adversarial sample if humans check (some) inputs and outputs of an AI to detect attacks. Misclassifications due to imperceptible adversarial perturbations (e.g., [2]) are "astonishing" for humans, who will likely suspect that an adversarial attack is taking place and will quickly react (e.g., sanitizing the input, applying defenses, or re-training the AI system). However, if the misclassified sample looked somewhat confusing, though still well-recognizable, a human is less likely to suspect an attack. Legal consequences can differ if such a sample could occur in reality. In this case, it is difficult to prove that the sample was created to deceive the classifier, since altering depictions of articles are common practice (in marketing). Rather the AI system might be said to perform poorly. This paper discusses how to generate adversarial samples by modifying higher-level representations of inputs. We focus on how to create adversarial samples similar to a given target or desired sample, i.e., exhibiting characteristics that should be present in the adversarial sample. Technically, our work is based on encoding and decoding between different spaces, i.e., a space derived from representations of classification models, a space derived from visual representation as perceived by humans, and linear interpolation. To demonstrate the effectiveness of our method, we evaluate common DL models against our adversarial examples and assess their transferability across different DL models.
II. BACKGROUND AND RELATED WORK
Machine Learning (ML) represents a valuable asset for modern tasks [6]; in particular, Deep Learning has become a true technology enabler [7]. However, many efforts have shown that all such methods are vulnerable to adversarial attacks (e.g., [8]), giving birth to a new research area commonly referred to 'adversarial machine learning' [9]. Let us summarize this field and outline the unique traits of our paper.
A. Adversarial Machine Learning
Attacks against ML leverage the so-called 'adversarial samples': the principle is to apply a perturbation to an input sample so as to trick an ML model into producing an incorrect output [10]. Previous efforts showed that similar attacks could break any ML model, including those based on traditional classifiers (e.g., SVM [11]) and DL ones (e.g., [2]). Noteworthy is also that adversarial attacks can be successful in diverse domains (and, hence, on different data types), such as computer vision [2], speech analysis [12], or cyber detection [13]. Even real ML systems have been defeated (e.g., [14], [15]).
Adversarial attacks are denoted with a threat model describing the relationship of the attacker with the target system, usually in the form of goal, knowledge, capability and strategy [9]. Common terms associated to such threat models are 'whitebox' and 'black-box' [10]: the former envision attackers with complete knowledge of the target ML system, whereas in the latter the attacker knows nothing. Depending on such assumptions, the attacker can opt for different strategies, such as crafting the 'perfect' perturbation-e.g., the well-known 'Projected Gradient Descent' (PGD) method [16]-or exploit the transferability property of adversarial examples [17]. Although all adversarial attacks share the same objective (i.e., compromising the decisions of a ML system), the attacker may have more specific goals. Such goals can include minimizing the perturbations [2], reducing the amount of interactions with the target system [18], inducing a 'targeted' output [19]; but also more advanced goals are possible (e.g. [20]).
With respect to previous works, we consider a unique threat model: such uniqueness is given by the different goals and strategies of our adversarial samples. Indeed, we assume that the attacker wants to introduce perturbations that can be spotted by a human; to do so, the attacker leverages high-level concepts provided by the first layers of deep neural networks. This differs considerably from previous work (e.g., [2], [21]). Our attacks can be considered as introducing perturbations during the "preprocessing" phase of an AI system (e.g., [22], [23]); however, our proposal differs in the method used to generate such perturbations. Indeed, we leverage techniques (i.e., [24]) within the explainable AI (XAI) domain [25]: some papers use similar techniques to 'explain' adversarial attacks (e.g., [26]), but no paper uses such techniques to 'generate' adversarial samples-to the best of our knowledge.
In particular, our perturbations involve techniques (i.e., [24], [27]) within the explainable AI (XAI) domain [25]: some papers use similar techniques to 'explain' adversarial attacks (e.g., [26]), but no paper uses such techniques to 'generate' adversarial samples-to the best of our knowledge. Our work also relates to natural adversarial samples or out-of-distribution samples [28], [29], since our resulting samples can look natural but different from typical, frequent samples.
B. Artificial Sample Generation
Synthtetically generating samples from latent spaces, as done in this work, is a rich topic in research. It has been extremely successful in the last few years using generative adversarial networks [30] as well as variational autoencoders which can (re)construct images from a latent space. In creative domains, e.g., the arts, the latent representation used to generate new samples is often randomly chosen [31]. In contrast, we focus on latent representations originating from actual samples or interpolations between their latent representations. In security, outputs of the last layer of ML models have been used to reconstruct training samples [32]. Their the goal is not to generate novel samples but rather to retrieve as much information on a specific training sample as possible. Traversing the latent space to generate samples between two classes is commonly used to understand the decisions of classifiers, which is relevant to XAI. For instance, the authors of [33] construct contrastive explanations by using a variational autoencoder, and a linear SVM-the latter served as a separate classifier from the model to understand. They traversed the latent space feeding latent codes into a CNN to assess when a sample changed its class and can be considered contrastive. With respect to [33] we assume a different setting. Specifically, we do not use a linear SVM (but rather parts of the model to be attacked). We potentially alter (given) original and target samples to be assessed before embedding them into the latent space, i.e., we consider multiple embedding spacesthe auto-encoding space, and the space of the classifier, i.e., activations of layer L t .
III. PROBLEM DEFINITION
A. Requirements
Our method requires complete knowledge of a DL model M = (L 0 , L 1 , ..., L k−1 ) being a sequence of k layers, a target layer L t ∈ M , an original sample X O and a dataset X = {X}, i.e., (input) samples that stem from the data distribution the model should handle, e.g., training data without any label information. We denote by M -t and M t-the submodel consisting of the first layers up to layer t (including it) and from L t+1 up to the last layer L k−1 (including it). The output of the model M (X) is a probability p(Y |X) for each class Y , the output of a model M -t (X) are the activations of layer L t for the input X. We denote Y G X as the ground truth class of X and Y M X as the predicted class by model M . Such settings can represent either 'white-box' (if M is the attacked model) or a 'black-box' (if M is a surrogate of the attacked model [18]) assumptions: as we will show ( §V), our adversarial examples can transfer between similar M . Finally, we assume an attacker that cannot 1 affect the training procedure of M .
B. Objectives
We consider three objectives to construct an adversarial sample X A based on the original sample X O for a given target sample X T . 1) Being "mis-classified": The adversarial sample should be classified as the class of X T , i.e., Y M X A = Y G X T . 2) Mimicking (concepts of) a target sample: The adversarial sample X A should be similar to X T as measured by a given encoding function f , i.e., the L2-norm ||f (X A ) − f (X T )|| should be small. The encoding function f might measure semantic differences by transforming to the feature space of the classifier using f (X) := M -t (X), the identity function f (X) := X or an encoder (of an autoencoder trained on X ) or, possibly, f is given by human judgment assessing to what extent X O and X A are similar with respect to pre-defined characteristics.
The identity function f (X) = X constrains visual differences in image space (recognizable by humans), while f (X) := M -t (X) constrains differences of information relevant to the classifier. It can happen that two samples X, X appear identical for the classifier (at a higher abstraction level), i.e. ||M -t (X) − M -t (X )|| = 0, although they appear very differently for a human. We use the
identity function f (X) = X, i.e., ||X O − X A || < η.
That is, we constrain differences in image space. 3) Remaining similar to the original sample: The adversarial sample X A should be close to the original, i.e., ||f (X O ) − f (X A )|| < η for a given η and encoding function f . We consider the same encoding function f as described priorly. Additionally, one might require that X A is a sample of the distribution the model M was trained for. Assume P provides the likelihood of a data sample then we might aim for max P (X A ). The likelihood function P could be approximated using the training data. Our method perturbing only upper layer activations tends to produce "likely" samples X A , i.e., samples often (but not always) appear to be realistic and they do not contain "invisible" (unnatural) modifications that trigger a misclassification (unlike traditional attacks).
IV. METHOD Our method aims to compute an adversarial sample X A that bears semantic similarity to a target sample X T , while obeying constraints on the maximum perturbation. One way to achieve this is to linearly interpolate between the original sample X O and the target sample X T , i.e.
X A := b · X O + (1 − b) · X T by minimizing b ∈ [0, 1] given the constraint ||X O − X A || < η.
To find a (locally) optimal coefficient b binary search can be used as stated in Algorithm 1.
E(X A ) := b · E(X O ) + (1 − b) · E(X T ) {encoding of adversarial sample} 6: X A := R (E(X A )) {(decoded) adversarial sample} 7: if ||X O − X A || < η then b 0 := b 8:
else b 1 := b 9: end while 10: return X A As we shall discuss ( §V-A), the outcome of linear interpolation of samples tends to be easily recognizable as being artificially constructed. Therefore, one might perform the interpolation in a latent space given by an encoder E stemming from an autoencoder AE = (E, R ), i.e., E(X
A ) := b · E(X O ) + (1 − b) · E(X T ).
The AE, i.e., the encoder E and decoder R are trained on training data X . The adversarial sample can be obtained using the decoder, i.e., X A := R (E(X A )). However, we propose to interpolate from an input X O of X O that contains only information on X O captured at layer L t , i.e., by M −t (X O ). We obtain X O using a decoder R, i.e., X O = R(M −t (X O )). The decoder R is trained by using activations M −t (X) as inputs and reconstructing X, i.e., the decoder R minimizes ||X − R(M −t (X))|| for X ∈ X [24], [27]. The process to obtain an adversarial sample X A is illustrated in Fig. 1. To understand the process, conceptually, we can think of two spaces: a classifier embedding space, and a latent embedding space from the autoencoder. To alter the class of a sample we can move in either space and use a decoder to get a sample based on the latent representations. When using the classifier embedding space, specific concepts of the original sample (and target sample) might be ignored, because they are not relevant for classification as discussed in the context of explainability [24], [27]. In our work, we also aim to obtain samples similar to a given target sample. Thus, linearly interpolating between encodings of the original sample and the target sample in classifier space seems not the right way, since the encoding of the target sample is stripped of information that we do not necessarily want to discard. Our objective is to be visually similar to the target sample. It seems more appealing to interpolate in an autoencoding space which allows to accurately reconstruct samples. A viable approach is to either use the original sample's encoding directly, or an encoding of the original sample.Samples with irrelevant information or rare characteristics of a sample tend to be poorly reconstructed. Their reconstruction can appear "blurry" resembling more of an average. For example, if a T-shirt has a very unique pattern, shape or color, none of which is very relevant for classification, the reconstruction might fail to reconstruct it accurately, resulting in a blurry outline (shape), "average" pattern (e.g., uniform color), and an average color (e.g., grey tone) [27]. Thus, generally, the reconstruction appears more of an average. Therefore, common samples close to the average will be reconstructed better, whereas 'outliers' are poorly decoded.Overall, removing non-classification relevant information from X O , i.e., using X O = R(M −t (X O )), pushes X O to denser areas, where reconstructions are better, i.e., difference ||X A − X O || is lower, and it is not negatively impacting the objective to maximize, i.e., p(Y G D |X A ). V. EVALUATION We perform a qualitative and quantitative evaluation focusing on image classification using CNNs. We assess multiple methods based on linear interpolation with the target sample X T and original sample X O embedded in a (latent) space with appropriate reconstruction. We investigate four scenarios using linear interpolation:
• No encoding: Interpolation from X O to X T in image space. • Classifier encoding M -t : Interpolation from M -t (X O ) to
M -t (X T ) in space given by M -t ; as decoder we use R; • Encoder E from the autoencoder AE = (E, R ) and a version of X O maintaining information relevant for classification: We transform X O using R(M -t (X O )) and interpolate from E(R(M -t (X O ))) to E(X T ) in the latent space given by E; as decoder we use R . • As the prior scenario but interpolating from E(X O ) to E(R(M -t (X T ))). Setup. We consider common DL methods. For the classifier we used different variants of VGG, i.e., VGG-11 and VGG-13, and ResNet-10. In particular, for the model M to attack we focused on a V GG−11. We also used the adversarial samples on evaluation classifiers M Ev ∈ {V GG − 13, ResN et − 10} to see if adversarial samples are transferable. For VGG-11 we reconstructed after a ReLU unit associated with a conv layer. We used the second last layer corresponding to the last conv-layer of V GG − 11 as target layer L t , i.e., t = 10. All encoders, i.e., E and M -t , share the same architecture, i.e., encoders are models M -t of a VGG-11 network and decoders R and R are given by ClaDec [24]. ClaDec use a standard decoder design relying on 5x5 deconvolutional layers.
We employed two datasets namely Fashion-MNIST and MNIST. Fashion-MNIST consists of 70k 28x28 images of clothing stemming from 10 classes. MNIST of 60k digits objects in color; for both datasets, 10k samples are used for testing. As maximal difference of ||X A − X O || we used η = 15 for Fashion-MNIST and η = 25 for MNIST. The thresholds can be chosen arbitrarily: larger thresholds yield adversarial samples that more likely fool a classifier, but also look more different to the original sample X O . As datapreprocessing, we scaled all images to 32x32, performed standardization and autoencoded all images using a separately trained autoencoder to "smoothen" outliers. For MNIST this makes very little difference, since autoencoders tend to almost perfectly reconstruct all samples. For Fashion-MNIST it has not impact for somewhat common samples, but it helps for outliers. For outliers, clothes with a seemingly random dotted pattern cannot be reconstructed well and are transformed to more common samples. For such outliers our method would otherwise fail to work, since the encoding and decoding using AE = (E, R ) even for the unmodified original sample X O would result in an error larger than the permitted threshold η, i.e.,
||X O − R (E(X O ))|| > η.
We train all models for reconstruction using the Adam optimizer for 64 epochs, i.e., the autoencoder for data and the decoder R from classifier representations M -t . The classifiers to be attacked were trained using SGD for 64 epochs starting from a learning rate of 0.1 that was decayed twice by 0.1. We conducted 3 runs for each reported number, e.g., we trained all networks (classifiers, encoders, decoders) 5 times. We show both averages and standard deviations. The baseline performance on each dataset matches the state-of-the-art without data augmentation: for MNIST we achieved mean accuracy above 99% and for Fashion-MNIST above 90%. For each class c we used all samples in the test data of classes not c as original samples. For each original sample, the target sample X T was chosen randomly from samples of class c. Thus, we computed about 9k adversarial samples for each class (and each run). Fig. 2 show adversarial samples for all four evaluated methods for interpolation. It can be observed that interpolation in image space (X O to X T ) is easily recognizable, i.e., both the original and target sample are well-recognizable when looking at the resulting adversarial sample. For example, in the last row (and last column) of each panel, it is easy to recognize the shoe and the T-shirt and digits 2 and 5. As such, the attack is easy to disguise. Methods employing encoding and decoding can yield non-interpretable images, as shown in the last row of each panel. This often happens if visual differences between the target and original samples are large. It also depends on the embedding, i.e., how classes are positioned in the encoded space. When comparing interpolation (M -t (X O ) to M -t (X T )) in latent space using the classifier M -t to those using the encoder E, it can be seen that the encodings using the classifier M -t tend to lead to non-desirable differences between the adversarial, target and original sample. For instance, in Fig. 2 for the shoe in the 3rd row, it increases the height of the heel both the target sample and the original sample, which have no or lower heel size, i.e., the adversarial sample disagrees with both. Adversarial samples from "E(X O ) to E(R(M -t (X T )))" and "E(R(M -t (X O ))) to E(X T )" look similar, overall the former are somewhat better. This can be seen, for example, for the shoe in the 3rd column having a small black area (which also exists for the target sample, i.e., the sandal), and it has lower brightness in the upper part of the shoe compared to the original boot. This indicates a slight move towards the target sample, which suffices to classify the adversarial sample as the target sample, i.e., as a sandal. Thus, while differences between the reconstructions of the two methods appear more subtle, they are often sufficient to lead to different classification outcomes. However, it also becomes evident that some of the reconstructions appear to be instances of the other class. That is, they might be called "forged samples" that contain some but not very pronounced features of the original sample. For example, the first row in Fig. 2. There are also samples containing elements of both target and original samples and seem somewhat ambiguous with respect to what class they belong to. For instance, the third row and 4th column for Fashion-MNIST depicts a sample that looks like a dress based on the upper part, but in the lower part it has elements of a pant, i.e., a dark area indicating a separation as for pant legs. For MNIST the second and third rows contain such samples, i.e., the nines (3rd and 4th column) do appear as nines but the loop of the "9" is not very pronounced and the samples can be easily mistaken as 1. Similarly, in the next column the 3rd and 4th column still look more like the original sample. Still, they contain elements of the target sample, i.e., a vertical line in the lower part, and the top part of the 5 only has a very short horizontal bar. In conclusion, some generated samples differ only subtly from the original, some are mixtures containing elements of both, and some appear more like samples from the target class. The outcome depends on the maximal allowed perturbations and learnt embeddings, e.g., whether original and target classes are embedded near each other, which also depends on their visual similarity.
B. Quantitative Results
From the quantitative results in Table I, we can see that our samples are well-transferable between classifiers, i.e., an adversarial sample obtained using model M also works on a different classifier without any changes to it. Furthermore, results are consistent for both datasets, i.e., the ordering of methods with respect to metrics, accuracy and difference to target sample ||X A −X T || is the same. We conducted statistical tests showing that differences between any two methods are significant at p < 0.001 for accuracy and ||X A − X T || except the last two methods where we only have p < 0.01. We can also see that interpolating directly between the original and the target sample yields the lowest loss to the target sample, i.e., ||X A − X T ||, and the lowest correct classifications by M , i.e., Acc(M ). However, as shown in our qualitative evaluation, interpolations in the image space are often easily recognizable and, therefore, inadequate. Moreover, operating in the latent space given by layer L t (i.e., the encoder M -t ) yields poorest results among all methods (i.e., largest difference to the target sample X T ) and also leads to fewest samples X A being classified as X T (i.e., correct classification are highest). This is because reconstructions from M -t are generally poor since the latent representations only accurately represent information relevant to classification. Thus, the constraint ||X A −X O || is violated without much movement towards the target sample X T . For X T that can be considered rare or outliers it might happen that even the reconstruction R(M -t (X T )) of the latent representation M -t (X T ) of X T shows large differences to X T . Using encodings E(X T ) of the target sample leads to better results than using those of R(M -t (X T )) maintaining only classification relevant information. This is expected, since using R(M -t (X T )) we adjust X A towards a sample that differs from the proposed target X T , i.e., we move towards the wrong "target", as also shown with examples in our qualitative evaluation. Compared to other adversarial attacks, the model accuracy might seem fairly high, i.e., Acc(M ) is commonly above 30% for adversarial samples. This is not unexpected, since often the original sample X O is difficult to transform to the target class, i.e., that of X T . For instance, it is non-trivial to make a shoe appear like a T-shirt or turn a 2 into a 5 (see last row in Fig. 2). In such cases, the adversarial samples might be of low quality.
VI. DISCUSSION AND FUTURE WORK
We altered dense representations in an auto-encoded space and in a space resulting from layer activations of a classifier. Adversarial and original samples tend to exhibit visible differences that aim to deceive or nudge a human. The parameter η allows controlling the degree of the differences. However, very small η likely do not result in adversarial samples classified differently from the original sample for our technique based on linear interpolation. In future work, we might also consider attacks not based on a target sample, but directly manipulating layer encodings X O := M -t (X O ) using conventional attack methods (e.g., [36]) to yield perturbations p. These attacks will result in misclassifications on the upper classifier M t-(X O ), i.e., M t-(X O ) = M t-(X O + p), and adversarial samples X A := R(X O + p) through reconstruction. Our optimization procedure using binary search converges to a local optimum, which could be improved by using multiple search intervals.
To derive our results, we first preprocessed data using a separate autoencoder. Generally, our method ( Fig. 1) using encoders and decoders introduces noise that reduces the possible addition of perturbation through optimization towards adversarial samples. Therefore, high-quality encoders and decoders and, in turn, sufficient training data for them are relevant and a limitation of our technique. While public datasets are available for many real-world objects, for more specialized datasets such as MRI images, this might not hold.
One might also approach our problem as a two-step problem: Create a sample that fools a human and then alter this sample using conventional adversarial attacks. However, this defies one of our motivations, i.e., an adversarial sample should be "realistic" in the sense that it is a sample that might occur in normal usage. Additionally, joint optimization of both objectives possibly leads to better outcomes.
Our method requires complete knowledge of a DL model, which can be associated with 'white-box' attackers. We observe that, in some domains, our assumptions are viable: for instance, many deep learning systems are trained on the ImageNet dataset [37], which is publicly available [38]. In these settings, even a 'black-box' attacker can be successful because they can create a surrogate model and transfer the successful adversarial examples to the original model [17]. Our evaluation showed that our adversarial examples can be leveraged for similar strategies ( §V).
VII. CONCLUSIONS
In this paper, we investigated adversarial attacks to construct adversarial samples that might present visible differences to a given (original) sample. Our adversarial samples should be similar to a target sample though they are still constrained to be similar to the original sample. Considering that existing systems rely on the cooperation of AI and humans, our adversarial samples will represent an attractive strategy for well-motivated attackers.
Fig. 1 :
1Semantic attack using an original sample X O and a target sample X T . We first remove information not relevant for classification from the original sample through encoding and decoding before reconstructing an interpolation between the autoencoded representation of the original and the target sample.
Fig. 2 :
2Original, target and adversarial samples for different en-/decodings and interpolation for Fashion-MNIST(left) and MNIST(right). Yes/No indicates, whether the model got fooled by X A , i.e. it outputs the class of X T for X A A. Qualitative Results
TABLE I .
IResults for MNIST and FashionMNISTDataset
Interpolation
||X A − X T ||
||X A − X O ||
Acc(M )(VGG-11))
Acc(M Ev )(VGG-13)
Acc(M Ev )(Res.-10)
MNIST
X O to X T
12.42±1.25
24.73±0.149
0.08±0.073
0.11±0.075
0.09±0.081
M −t (X O ) to M −t (X T )
24.38±1.71
24.71±0.15
0.44±0.117
0.41±0.134
0.42±0.124
E(X O ) to E(R(M −t (X T )))
19.87±1.794
24.85±0.11
0.28±0.081
0.26±0.079
0.27±0.084
E(R(M −t (X O ))) to E(X T )
20.41±1.837
24.73±0.172
0.21±0.078
0.2±0.077
0.2±0.079
Fashion-
MNIST
X O to X T
20.83±1.317
14.95±0.043
0.42±0.14
0.44±0.15
0.41±0.132
M −t (X O ) to M −t (X T )
27.23±1.44
14.84±0.037
0.64±0.052
0.62±0.056
0.62±0.049
E(X O ) to E(R(M −t (X T )))
25.84±1.436
14.85±0.03
0.57±0.059
0.58±0.057
0.56±0.055
E(R(M −t (X O ))) to E(X T )
25.22±1.365
14.92±0.048
0.53±0.065
0.53±0.065
0.51±0.06
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|
[] |
[
"OmniHorizon: In-the-Wild Outdoors Depth and Normal Estimation from Synthetic Omnidirectional Dataset",
"OmniHorizon: In-the-Wild Outdoors Depth and Normal Estimation from Synthetic Omnidirectional Dataset"
] |
[
"Jay Bhanushali \nIndian Institute of Technology Madras\nTouchlab, ChennaiIndia\n",
"Praneeth Chakravarthula \nPrinceton University\nPrincetonUSA\n",
"Manivannan Muniyandi \nIndian Institute of Technology Madras\nTouchlab, ChennaiIndia\n"
] |
[
"Indian Institute of Technology Madras\nTouchlab, ChennaiIndia",
"Princeton University\nPrincetonUSA",
"Indian Institute of Technology Madras\nTouchlab, ChennaiIndia"
] |
[] |
Understanding the ambient scene is imperative for several applications such as autonomous driving and navigation. While obtaining real-world image data with perpixel labels is challenging, existing accurate synthetic image datasets primarily focus on indoor spaces with fixed lighting and scene participants, thereby severely limiting their application to outdoor scenarios.In this work we introduce OmniHorizon, a synthetic dataset with 24,335 omnidirectional views comprising of a broad range of indoor and outdoor spaces consisting of buildings, streets, and diverse vegetation. Our dataset also accounts for dynamic scene components including lighting, different times of a day settings, pedestrians, and vehicles. Furthermore, we also demonstrate a learned synthetic-to-real cross-domain inference method for in-thewild 3D scene depth and normal estimation method using our dataset. To this end, we propose UBotNet, an architecture based on a UNet and a Bottleneck Transformer, to estimate scene-consistent normals. We show that UBot-Net achieves significantly improved depth accuracy (4.6%) and normal estimation (5.75%) compared to several existing networks such as U-Net with skip-connections. Finally, we demonstrate in-the-wild depth and normal estimation on real-world images with UBotNet trained purely on our Om-niHorizon dataset, showing the promise of proposed dataset and network for scene understanding.
|
10.48550/arxiv.2212.05040
|
[
"https://export.arxiv.org/pdf/2212.05040v2.pdf"
] | 254,535,869 |
2212.05040
|
f0bcd437c9faaa21113a1c191966c58ad36f0ebb
|
OmniHorizon: In-the-Wild Outdoors Depth and Normal Estimation from Synthetic Omnidirectional Dataset
Jay Bhanushali
Indian Institute of Technology Madras
Touchlab, ChennaiIndia
Praneeth Chakravarthula
Princeton University
PrincetonUSA
Manivannan Muniyandi
Indian Institute of Technology Madras
Touchlab, ChennaiIndia
OmniHorizon: In-the-Wild Outdoors Depth and Normal Estimation from Synthetic Omnidirectional Dataset
Understanding the ambient scene is imperative for several applications such as autonomous driving and navigation. While obtaining real-world image data with perpixel labels is challenging, existing accurate synthetic image datasets primarily focus on indoor spaces with fixed lighting and scene participants, thereby severely limiting their application to outdoor scenarios.In this work we introduce OmniHorizon, a synthetic dataset with 24,335 omnidirectional views comprising of a broad range of indoor and outdoor spaces consisting of buildings, streets, and diverse vegetation. Our dataset also accounts for dynamic scene components including lighting, different times of a day settings, pedestrians, and vehicles. Furthermore, we also demonstrate a learned synthetic-to-real cross-domain inference method for in-thewild 3D scene depth and normal estimation method using our dataset. To this end, we propose UBotNet, an architecture based on a UNet and a Bottleneck Transformer, to estimate scene-consistent normals. We show that UBot-Net achieves significantly improved depth accuracy (4.6%) and normal estimation (5.75%) compared to several existing networks such as U-Net with skip-connections. Finally, we demonstrate in-the-wild depth and normal estimation on real-world images with UBotNet trained purely on our Om-niHorizon dataset, showing the promise of proposed dataset and network for scene understanding.
Introduction
The task of monocular depth estimation from omnidirectional images has gained substantial attention in recent years [5,17,21,30,42]. Monocular depth estimation presents unique set of challenges including handling distortions from the equirectangular projections. Moreover, the quality and diversity of the datasets play an important role in the robustness of depth estimation [46]. Although there has been a growing interest for realisitic scene representations from real-world [4,43], obtaining real omnidirectional im- ages with accurate per-pixel depth data is challenging. On the other hand, most existing synthetic datasets are skewed towards indoor spaces with limited depth range [45] which unforunately do not generalize well for in-the-wild depth estimation in outdoor scenarios with diverse scene components and large depth ranges [1,14]. While there are synthetic scene simulators [11] and datasets [33] for generating outdoor scenarios, these datasets specifically cater to the task of autonomous driving applications. The lack of generalized datasets for diverse outdoor and indoor environments has left a large research gap, especially for in-thewild monocular scene depth estimation.
In this work we propose OmniHorizon, a synthetic omnidirectional dataset for jointly estimating the scene depth and normal information in diverse indoor and outdoor environments. Specifically, our dataset addresses several as- [30] Real Indoors 130 Static Stanford 2D-3D [2] Real Indoors 1,413 Static PanoSUNCG [41] Synthetic Indoors 25,000 Static Zillow [9] Real Indoors 71,474 Static Fukuoka [24] Real Outdoors + Indoors 650 Static Vehicles and Pedestrians OmniHorizon Synthetic Outdoors + Indoors 24,335 Dynamic + Static Vehicles and Pedestrians pects that are typically found lacking or under-represented in prior synthetic indoor and outdoor datasets. For example, OmniHorizon dataset represents both urban scenarios as well naturally occuring vegetation and rocks, and incorporates dynamic scene participants such as pedestrians and vehicles. Furthermore, we also consider different time of day settings for scenarios in the dataset allowing us to robustly estimate the depth and normals of the scene in various lighting conditions. We show through several ablation experiments that considering dynamic lighting and scene participants improve the accuracy and robustness of monocular depth and normal estimation. We further identify drawbacks with existing U-Netbased architectures for normal estimation, and propose a novel neural network architecture based on U-Net and Bottleneck transformer, dubbed UBotNet, which we find to improve the depth and normal estimation for both synthetic and real-world scenes. Moreover, we also analyze the cross-domain inference performance of UBotNet network trained on our OmniHorizon dataset and the state-of-the-art Fukuoka dataset [24]. The proposed dataset and network demonstrates significant improvements in cross-domain inference, wherein we train the network on purely synthetic scenes and use it for in-the-wild real-world scene understanding (see Figure 1).
To summarise, our two core contributions in this work:
• OmniHorizon: A synthetic omnidirectional dataset with 24,335 images for diverse scene depth and normal estimation, and cross-domain inference. The dataset includes dynamic scene elements such as dynamic lighting, cloud formations, pedestrians and vehicles.
• UBotNet: U-Net and Bottleneck Transformer based neural network for estimating depth and scene consistent normals, and generalizable for cross-domain inference. We also discuss UBotNet Lite, a compact version of UBotNet with 71.2% less parameters than the original model.
Related Work
The broadly classify the datasets in the literature related to depth and normal estimation into two categories based on whether the data was curated from the real world (Realistic dataset) or generated using a 3D rendering engine (Synthetic dataset). An overview of the datasets are also presented in Table 1.
Realistic datasets Matterport 3D [4] is a real world indoors dataset with 10,800 panoramic views captured from 90 building-scale scenes.It provides access to depth, normal, surface reconstruction, camera poses and semantic segmentations extracted from the scenes. Matterport3D 360 • [30] is an extension with 9684 high resolution equirectan-gular 360 samples for monocular depth estimation. Gibson [43] is a real-world based virtual environment that provides photorealistic interior spaces from 572 full buildings which are composed of 1447 floors. The dataset contains the 3D reconstruction, RGB images, depth, surface normals along with semantic annotations for few spaces. Stanford2D3D [2] is a dataset collected from 6 large-scale indoor areas providing 70,496 regular RGB and 1,413 equirectangular RGB images, along with their corresponding depths, surface normals and other data. HM3D [28] is currently the largest dataset for 3D indoor spaces with 1.4 − 3.7× the navigable space compared to other datasets. Replica [38] is a dataset with 18 3D indoor scene reconstructions of rooms. Replica 360 • 2k/4k RGBD [30] extends the above Replica dataset and provides 130 RGB-D pairs rendered at 2048 × 1024 and 4096 × 2048 resolution. Zillow [9] is one of the largest indoors dataset with 71,474 panoramas along with 21,596 room layouts, and 2,564 floor plans captured from 1524 homes. Fukuoka [24] is an outdoors dataset proposed for the challenge of place categorization. It provides 650 panoramic RGB views along with 3D depth and reflectance maps. The views are recorded from various outdoor settings such as forests, urban areas, coastal areas, parking and residential areas. KITTI-360 [22] is a dataset designed for autonomous driving systems that contains 320k images, 100k laser scans, and semantic annotations across both 2D and 3D domains.
Synthetic datasets Structured 3D [45] is a synthetic indoors dataset with 3500 scenes providing multiple furniture configurations. They also provide warm and cold lighting conditions in the dataset. PanoSUNCG [41] provides 103 scenes with 25K omnidirectional images rendered using environments from SUNCG [35]. 360D which was introduced in [46] provides 360 color images along with corresponding depth rendered from two synthetic (SunCG, SceneNet [23]) and two realistic (Matterport 3D, Stanford2D3D) datasets.
As compared in Table 1, our OmniHorizon dataset contains both indoors and outdoors scenes with dynamic and static scene lighting as well as static scene participants. This bridges a significant gap in existing datasets which predominantly considers indoor environments with static scene components and lack context for outdoor spaces.
Dataset
We rendered the OmniHorizon dataset using Unreal Engine 4. As part of the dataset, we provide color images, scene depth and world normal in stereo (top-bottom) format. All data is rendered at 1024 × 512 resolution. All the scene assets for the dataset were acquired from the Unreal Marketplace. We designed training path which is an animated sequence of 1521 frames captured using a moving camera. For each scene, we rendered 2-6 training paths depending on the scale of the scene. As a result, the dataset contains 24,335 omnidirectional views for outdoors scene depth and normal estimation. We clamp the depth to 150m (Unreal units) and use world-space normals for the normal maps (see supplementary material for discussion). We used an image split of 85 : 15 for the training (19392) and the validation (3423) sets. Specifically, we reserved the entire training-path 4 ( Figure 3 and Figure 6) from City Park scene as test image data (1520 images). The training-path 4 is isolated from other paths in the scene and contains underpass, stairs, uneven terrain, building and pedestrians, thereby making it an ideal test image data. We discuss in detail several attributes of the dataset in the following subsections
Scene Attributes
We notice that the scene attributes and context plays a crucial role in the performance of the neural networks on real-world scene inferences [1]. Our dataset is designed to capture a wider variety of scene attributes including urban environments consisting of buildings and roads, as well as more naturally occuring uneven terrains and vegetation, and several other entities that generally make an outdoors environment. Figure 2 provide a snapshot overview of our dataset. For example, Downtown [27] and CityPark [34] scenes of our dataset represent urban areas covering buildings, houses, parks and other street props. While CityPark scenes contain wider roads and streets, Downtown scenes are composed of narrower streets and alley ways. Scene such as Desert [15] is composed of rocks, roads, uneven terrain and wild vegetation. While we predominantly focused on outdoors scenes in OmniHorizon dataset, we also include an indoor environment called Dungeons [37]. There are darker rooms with details like arches, stone walls, narrow passages, and stairs that give the scene more context and variety.
Dynamic Lighting
A typical outdoor environment undergoes a range of lighting variations depending on the time of the day and complex cloud patterns in the sky. Existing datasets do not model such dynamic changes in lighting, causing compromised performance of trained neural networks, especially on scene understanding tasks in outdoor environments. Note that, the scene depth and normals are independent of the scene brightness or color. In other words, during inference, ideally the network has to generate consistent scene depth and normal estimates irrespective of the scene lighting conditions. To cater to this, we generated the scene depth and normal data in a separate rendering pass isolated from the scene color, allowing us to prototype changes in the scene lighting, brightness and color yet rendering consistent depth and normal data. We modeled the dynamic changes in lighting according to varying time of the day by modulating a directional light source (sunlight) and secondary light source (diffuse light from the sky). Both the position of the light sources and the respective intensities are changed throughout in an animated sequence to model an entire day. Figure 3 shows an exemplar illustration of the changes in lighting conditions of the scene via modulating the light sources.
To further model more complex lighting changes due to a variety of cloud formations in the sky during different times of the day, we also rendered various sky-cloud settings using a sky plugin [12] in our rendering pass. Our sky settings included formations such as Stratus, Cumulus and Cirriform clouds. We further varied the cloud coverage between a range of very light to extremely heavy. The overall time of day settings in our dataset span early morning to late evening. We restrict the implementation of the dynamic lighting only to City Park scene. For other scenes, we used a stationary light source to model real world scenarios where the lighting changes are not abrupt.
Dynamic Scene Participants
Of the several entities that make outdoor spaces, vehicles and pedestrians form important scene components. To modeled these dynamic scene participants, we model multiple classes of vehicles including trucks, hatchbacks, SUVs, pickup and sports cars. We rendered both automatically and randomly placed vehicles in outdoor environments as well as manually placed vehicles in parking lots and roads. We further used 3D scanned avatars [29] and Metahumans [13] to increase the visual diversity of the pedestrians used in the dataset. Metahumans used in our dataset are high-fidelity realistic virtual avatars with diverse skin tones and detailed grooming. We have used highest LOD level (LOD 0) for Metahumans (see supplementary material). For rendering human scene participants, we used three settings: idle poses, sitting and walking. The walking behaviour and trajectories of the pedestrians are controlled using spline path and the blueprint. Figure 4 depicts several examples of the placement of vehicles and human avatars in the dataset. It can be seen that the avatars are placed throughout the scene in various realistic locations such as sitting outside a cafeteria, walking on the street and discussing in a group.
Neural Cross-domain Inference
In this section, we describe UBotNet architecture based on U-Net [32] and a Bottleneck transformer [36] for crossdomain inference, and our network training procedure. We conduct four different experiments for evaluating the proposed dataset and network architectures: a) Benchmark on OmniHorizon dataset, b) Ablation study of the dataset, c) Sim-to-Real domain transfer, and d) In-the-wild depth and normal estimation from real-world omnidirectional images images.
UBotNet Architecture
Compared to high capacity encoders which include ResNet and DenseNet architectures, UNet with skipconnections has demonstrated better performance in Pano3D benchmarks [1]. However, we observed that U-Net architecture predicted inconsistent normals that vary for a range of images for both synthetic and real-world scenes. To circumvent this shortcoming, we intended to improvise the network such that it can learn information in a broader context and model long range dependencies, a restriction evident in CNN-based architectures. Vision Transformers (ViT), on the other hand, have achieved state-of-the-art results by using pure transformer architecture for image clas- sification [10]. It has also been demonstrated that transformer architectures have wider receptive fields compared to CNNs, thereby giving them an ability to integrate information globally across an image. Recently, it has been shown that replacing final three layers of a ResNet with a bottleneck transformer with global self-attention attain improved performance for instance segmentation and object detection tasks [36]. The fusion of the U-Net with attention or transformer-based architecture has also been explored in the application of medical image segmentation [6,25].
Motivated by the aforementioned prior work, we propose an improvised architecture called UBotNet that efficiently learns local features using convolutional layers and employs self-attention to aggregate information in a global context. The proposed UBotNet is based on U-Net and Bottleneck transformer [36]. Specifically, we place the selfattention transformer block at the lowest resolution feature maps in the U-Net bottleneck since self-attention requires O(n 2 d) memory and computation [40]. Figure 5 depicts the overview of our proposed architecture. We also replace the max pooling layer with anti-aliased max pooling layer [44].
Additionally, we also show a compact version of UBot-Net, UBotNet Lite, where we used separable convolution [8] similar to SepUNet [19] to reduce the number of parameters in the architecture. UBotNet Lite (38.3M) has 71.2% less parameters compared to it's bigger sibling UBotNet (133M). We further place two branches of fully-connected layers towards the end with sigmoid activation to predict scene depth and consistent normals. The CNN blocks learn the local image features whereas the Multi-Head Self-Attention (MHSA) block from the bottleneck transformer learns the global contextual features. Learning both local and global scale features are essential for robust normal estimation, as we also validate and demonstrate later in the experiments in Section 5.
Network Training and Experiments
The following network training and experiment configurations are maintained for all the benchmark, evaluation and ablation experiments.
Training configuration. We used an Nvidia RTX 3090 with 24GB onboard memory for training all network models. The batch size is set to 4 and Adam optimizer [18] is used with a learning rate of 1 × 10 −4 and decay rate of 1 × 10 −5 . The images were rescaled to a resolution of 512 × 256 for training and evaluation due to memory constraints. All the networks were trained for 40 epochs.
Loss Functions. The networks were all trained for jointlearning of both depth and normal information from the input monocular omnidirectional images. We used L berHu (Reverse-Huber) function [20] as the loss objective for depth and L 1 penalty as the objective function for estimating scene normals .
The overal loss function for joint learning is therefore a sum of both Depth and Normal objectives:
L T otal = L Depth + L N ormal(1)
Data Augmentation. We adapt two techniques augment the color data (Channel Shuffle [31] and Color Jitter [31]) of the input images and a third to perform rotation-based augmentation [16,17].
Baseline architectures and Evaluation criteria. We evaluated our dataset on five different architectures -URes-Net [46], RectNet [46], UNet 128 and the proposed UBotNet and UBotNet Lite architecture. The final layers of the URe-sNet and RectNet were modified to support joint learning of depth and normals. The UNet 128 is similar to the vanilla architecture but uses a base of 128 feature channels and goes upto 2048 channels. We have increased the number of feature channels to make it identical to the UBotNet resulting in 124 M parameters. The networks were evaluated using
Discussion and Evaluation
Benchmark Results on OmniHorizon
Quantitative results. Table 2 shows the quantitative results for the depth and normal estimation for all the networks. The RectNet and UResNet architectures show suboptimal results on the dataset. RectNet fails to converge after early iterations. On the other hand, both UBotNet and UNet 128 demonstrate better outcomes in the benchmark. It is evident from the results that both architectures greatly benefit from the skip-connections. Moreover, UBot-Net is consistently better than other architectures including the UNet 128 on the account of all the metrics except for MRE. For normal metrics, we observe a performance uplift of 14.92% for normal error and 4.45% for normal accuracy between UNet 128 and UBotNet. We observe that UBotNet Lite (38.3 M) performs slightly lower than UNet 128 (124 M) for certain metrics but shows better results for normal metrics while having 70% less parameters when compared to the UNet 128 .
Qualitative results. Figure 6 shows the visual differences for the depth 1 and normals between all the architectures and their validation against Ground Truth (GT) data. First image shows the building and stairs in close proximity. UBotNet shows better depth estimation for structures on the windows of the building and stairs compared to UNet 128 . Moreover, for UBotNet the normal estimates for the stairs and the building structure are more closely aligned to the GT compared to UNet 128 . We observe similar results for the UBotNet lite version but with a small inaccuracy in the pillar region. The second and third image emphasis on the elements that are farther from the camera. The images also test the networks' ability to identify trees and underpass structure in the shadows. The UNet 128 fails to identify the distant part of the tunnel in the third image, whereas the UBotNet is able to detect the end of the tunnel. Also, UNet 128 completely fails to estimate the normals for the underpass structure in both second and third image whereas the UBotNet and UBotNet lite estimates are more closer to the GT. The
Ablation Study
We perform the ablation study to address an important question : Does context matter in outdoor scenarios? It evaluates the contribution of our dataset in terms of the dynamic components: vehicles and pedestrians (VP), and dynamic lighting (DL). Therefore, we create two additional versions of the dataset. First, we generate the static version of the dataset which includes only static meshes and lacks any dynamic components. The second version of the dataset includes pedestrians and vehicles but lacks dynamic lighting component. Note that the full OmniHorizon dataset consists of all dynamic components. Table 3 shows the comparison between the various versions of the dataset using the UBotNet network.We observed incremental gain in the performance, specifically for depth accuracy and normal metrics on addition of the dynamic components to the dataset. Figure 7 shows the visual results from the ablation study. For the first image, Static version struggle with the normal estimation for water surface while the Static+VP version has issues with lighting and normal estimation.
Sim-to-Real Transfer
We evaluate the simulation-to-real domain transfer performance of our method on a real-world dataset -Fukuoka [24]. To achieve this task, we pretrain the UBotNet on our dataset and fine-tune it on Fukuoka for the task of depth estimation. Note that Fukuoka dataset does not provide ground truth for normal data and therefore we only evaluate the depth estimates. Table 4 summarizes the performance comparison between the networks pre-trained on our dataset and that trained on Fukuoka from scratch. We noted better performance of the pretrained network specifically for depth accuracy, where we see a gain of 12.2%. We also observed more accurate depth maps estimated from the test images when compared to training from scratch on Fukuoka as shown in Figure 7. When trained from scratch, the network struggles notably with vegetation. On the other hand, it benefits from a better understanding of scenes with complex vegetation when it was pre-trained on OmniHorizon.
Testing on the Real-world Images In-the-wild
The real-world omnidirectional images have been curated from the Polyhaven website [26] for testing the trained network on the images in the wild. We selected images that represent diverse outdoor scenarios cluttered with various objects and captured during different time of day settings. Figure 8 shows depth and normals estimated by UBotNet from the images. The images illustrate the ability of the network to estimate depth at a large range in various settings. Our network learns high level details from the vegetation (images 1, 3 and 4). This is reflected in the image 1 where the network was able to recognize the large tree in the foreground along with the walking path. It also captures the details from the cars in image 3. The network was able to identify sky region in cases with full clouds (image 2) and clean sky with no clouds(image 3 and 5). This demonstrates the advantage of the including various cloud formations and time of day settings in the dataset. The final image which shows a skating area is a good example of the ability of UBotNet to estimate normals of two upright structures in front of the buildings with a texture similar to the concrete floor. It highlights the capacity of the network to learn information in a global context understand the orientation of normal surfaces. Overall, the network demonstrates promising results for the estimation of depth and normal on real-world images. We show additional results in the supplementary material.
Limitations
There are specific scenarios where sunlight may overexpose parts of a scene while underexposing others. In such instances, the network struggles to correctly estimate depth and normals for the overexposed parts of the scene. Additionally, the network occasionally misinterprets vertical elements like handrails and bridge supports. Figure 8 shows both such challenging scenarios where our method compromised. We discuss the assumptions of our dataset in supplementary material.
Conclusion
We presented a new dataset called OmniHorizon and a hybrid architecture called UBotNet for depth and normal estimation in diverse scenarios. Firstly, our dataset includes diverse scenes and also dynamic scene participants such as pedestrians and vehicles. We noticed that dynamic components in the dataset are critical and not including them results in sub-optimal inferences on in-the-wild real-world examples. Secondly, our UBotNet, based on U-Net and Bottleneck transformer, trained on the OmniHorizon dataset demonstrated significantly improved and scene-consistent normal estimation against the vanilla U-Net architecture. Furthermore, we presented UBotNet Lite, a smaller version of the network that retains respectable depth and normal accuracy while having only 30% of the network parameters. We outlined the benefits of pretraining network on Omni-Horizon and fine-tuning it on Fukuoka dataset. Finally, we demonstrated the application of our model trained on Omni-Horizon for estimating the depth and normals of real-world omnidirectional images in-the-wild. We are very excited to release our dataset and models to the research community.
Supplementary Material
In this supplementary material, we discuss our approach on generating the OmniHorizon dataset in Unreal Engine 4. We elaborate on the factors and certain assumptions that we made in order to render the dataset. Additionally, we discuss about training the UBotNet on indoor datasets and architecture choices. Finally, we demonstrate additional results for depth and normal estimation from real-world images in the wild.
Depth clamping
Rendering engines such as Unreal Engine 4 work with a larger depth range compared to that captured by physical sensors. However, we were interested in exploring the range of depth information that can be used for covering a wide range of objects in outdoor scenarios. This motivated us to simulate the limitations of the physical sensors and restrict the depth range to 150 m, similar to the Fukuoka dataset [24]. The engine places the far plane at infinity, which results in depth values being generated for extremely distant objects. To avoid this, we modify the depth material to visualise the impact of constraining the depth to a maximum specified value. We show the results for the clamping of depth at a range of 10m, 75m and 150m in Figure 11. At a depth of 10 m, only the truck is visible. When the depth range is raised to 75 m, cars and building start to appear in the background. At 150 m, the trees and most of the background are visible. By limiting the depth in outdoor environments, it is possible to focus solely on nearby items, or, depending on the application, on distant objects as well.
View-space vs world-space normals
The view space normals are calculated relative to the camera orientation, whereas the world space normals are calculated with respect to the global axes of the scene. The normals in view space are desired when using a perspective camera as they are tied to the camera pose (extrinsic parameters). However, the panoramic image is obtained by rotating the camera on both the horizontal and vertical axis in increments of fixed angle steps (5°), followed by merging the multiple views. Since the coordinate system is relative to the camera in view space, it also gets modified with the rotation. This results in a gradient of normals with no basis vectors. The normals obtained in world space are absolute and independent of camera pose. Figure 9 shows the difference between the view-space and world-space normals. Therefore, we captured the normals in world space as it was consistent for both within and between the scenes. We show the convention used for the world-space normals in Figure 10.
Virtual Avatars
As discussed in main paper, we utilised Metahumans [13] for the virtual avatars in the scene. We have used premade MetaHumans available in the Quixel bridge. It allowed us to bring in highly detailed characters and more diversity in the pedestrians. But there were certain challenges while using the Metahumans for the dataset. They are generated with multiple level of details (LODs) for perfomance optimisation. As a result, there would be sudden popups and other artifacts when the camera is approaching a character. Figure 12 illustrates how the character hair and details change when the camera is approaching the character. Lower LOD level (LOD 8) indicates lowest detailed polygon mesh with no advanced features such as detail normal maps or hairs. The higher LOD level (Level 0/1) has higher polygons with extra detail maps for the skin and hair grooming system. Additionally, we also observed artifacts in the normal maps for the characters with detailed grooming such as facial hair. Figure 13 shows the issues with the normal maps of a character in the region with facial hair. For such characters we used LOD 1 or LOD 2 to resolve the problems.
Assumptions in the Dataset
Our dataset renders several realistic outdoor and indoor environments with dynamic scene components. While curating this dataset, we made certain assumptions especially about the outdoor scenes which we list below:
1. The sky is assumed to be situated at infinitely large distance from the camera, and is represented as a spherical mesh of large radius encompassing the entire scene. Additionally, normals are not rendered for the sky region. It is represent using black which indicates invalid normal values. This allows us to distinguish sky from other regions in the scene. c) The modified LOD system is used to maintain LODs at a fixed LOD so that the avatar's appearance is unaffected by distance. d) The LOD of the character is locked to 1 using Forced LOD. Figure 13. Artefacts in normal maps for facial hairs. When the camera is very close to the characters, the engine uses additional detail meshes for characters with facial hair at the highest LOD level (LOD 0). As a result, artefacts appear in the normal maps.We use LOD 1 or 2 for such characters.
2. Transparent and transluscent materials such as water, windows of the buildings and windshields of vehicles are replaced with fully reflective materials. We observed that inferring depth of such materials from color images is challenging and this limitation, for exam-ple, also applies to real-world datasets captured using lidars [39]. Figure 14 depicts the limitation of using transparent and translucent materials in the dataset. The original water shader in the scene was designed in such a way that it acted as a see-through material in case of depth. As a result, the depth map captures the terrain hidden underneath the water surface. We modified the the water shader to a reflective surface and thus depth is correctly rendered as a planar surface. We observed a similar case for the glass shader used for windows in the vehicles. The vehicles indeed have detailed indoors but due to reflections on the glass, the inside is not clearly visible. However, the depth map has much cleaner view of the indoors. To avoid this conflict of information, we use fully opaque and reflective materials for the windows.
UBotNet
UBotNet for Indoor datasets. In the main paper, we discussed about the UBotNet architecture and the results from training on the OmniHorizon dataset. We additonally trained UBotNet on real-world indoor dataset Pano3D [1] to validate the performance of the network on other datasets. Pano3D is proposed as a modification of Matteport3D [4] and Gibson3D [43]. We used the official splits provided by the authors for Matterport3D for training and validation. For, Gibson, we used the GibsonV2 Full Low Resolution for training and validated on Matterport. All the images used for training were of 512 x 256 resolution. We used the loss function and training parameters outlined in our main paper. We trained UBotNet Lite on the both the datasets for 60 epochs. Table 5 shows the quantitative results for the task of depth estimation by UBotNet Lite evaluated on Matter-port3D. We also show the qualitative results for the validation task in Figure 15. We observed better performance in overall metrics and the visual results when the network is trained on the Gibson3D.
Absolute vs Relative positional encoding. We utilised relative positional encoding [36] for self-attention in our proposed UBotNet architecture. We compare it against the absolute positional embeddings and show the quantitative results in Table 6. The absolute positional embeddings perform inferior to the relative positional embeddings used for self-attention. Moreover, the differences are more prominent in case of normal estimation. This is reaffirmed by the visual differences shown in Figure 16. The network loses the context required for learning the consistent representation of the normals. It behaves similar to the UNet 128 network discussed in the main paper.
Addition Results
We show additional results on the real-world images in the Figure 17 and Figure 18. The networks used were trained purely on OmniHorizon.
Figure 1 .
1Synthetic to Real cross-domain inference. UBotNet trained on the OmniHorizon dataset is used to estimate depth and normals from the real-world image captured outdoors.
Figure 2 .
2Overview of the OmniHorizon dataset. The dataset consists of three outdoors scenarios: City Park, Downtown and Desert, and one interior scene: Dungeons. City Park and Downtown represent urban landscape and streets with pedestrians and vehicles. Desert covers landscape with emphasis on vegetation and terrain. Dungeons is a detailed indoors scene with stone walls, archs and narrow passages.
Figure 3 .
3Dynamic lighting and varying time of day settings. a) The lighting of the scene is varied by modulating the directional light (sun) and secondary light source (skylight). b) Changes in the scene lighting condition achieved using the modulation of the light sources.
Figure 4 .
4Examples of vehicles and pedestrians in OmniHorizon dataset. a) Vehicles in a parking lot, b) virtual avatars sitting in a cafeteria, c) pedestrian walking on the street (spline path is highlighted in pink) and d) casual group hangout.
Figure 5 .
5Proposed UBotNet architecture. UBotNet is a hybrid architecture based on UNet and Bottleneck Transformer (BoTNet). Antialiased max pooling is used for the pooling operation. The transformer block is placed in the middle of the encoder and decoder paths of the UNet. UBotNet Lite uses separable convolutions in place of standard convolution layers; otherwise, it is identical to UBotNet. A simplified illustration of BoTNet is also shown which contains Multi-Head Self-Attention (MHSA) for learning global context.
Figure 7 .
7a) Ablation study of the OmniHorizon dataset. Comparison for the depth and normal estimation between the various versions of the dataset: Static, Static + VP, and Static + VP + DL. b) Sim-to-Real performance on Fukuoka dataset. We compare the performance of depth estimation between the network pre-trained on OmniHorizon and fine-tuned on Fukuoka against the network trained from scratch. results show that the proposed architectures perform relatively well for normal estimation due to the global context extracted by the MHSA from the encoder features.
For the second image, differences can be observed in the vehicle on the left, and building between Static and other ver-sions. The inclusion of vehicles to the dataset reflects in better results in the real-world examples. Finally, the last image demonstrates the shadow artifacts present in normal maps in both Static and Static+VP version which are absent in the result from the full dataset.
Figure 8 .
8a) Predictions on the real-world images in the wild. Depth and normals estimated from real-world images representing the diverse outdoor scenarios. The images show different sky conditions including overexposed sky, cloudy and clear sky with no clouds. It also features man-made constructions, vehicles and vegetation. b) Failure cases. Network fails to estimate depth and normals in scenarios with overexposed regions. It also fails to recognize vertical upright structure such as the bridge railing.
Figure 9 .
9Comparison between view-space and world-space normals. The normals captured in view-space appear as gradient with lack of clear distinction between the basis vectors. Normal maps recorded in world-space follow a consistent coordinate system.
Figure 10 .
10Convention for the world-space normals.
Figure 11 .
11Depth clamping experiment. Comparison between various depth ranges after clamping to a specific range: 10 m, 75 m and 150 m. Inverted depth maps are shown for better visualization.
Figure 12 .
12Dynamic LODs vs Constant LOD. a) The Dynamic LOD system loads different meshes with various level of details based on the proximity to camera. This however results in sudden poping up of the meshes which generates artefacts in the data. b) Default LOD settings used by the engine.
Figure 14 .
14Assumptions for the dataset. a) Modification of water shader to achieve constant depth across the surface of the water. b) Modification of glass shader into opaque reflective surface which hides the interior parts of the vehicles.
Figure 15 .
15Qualitative results for UBotNet Lite trained on Indoor datasets -Matterport3D and Gibson3D.
Figure 16 .
16Comparison between Abs. and Rel. positional embedding. Absolute positional embedding loses the context required for learning the normals when used for self-attention.
Figure 17 .
17Depth and Normal estimation on real-world images in the wild. Comparison between all the networks discussed in main paper for depth and normal estimation on real world images.
Figure 18 .
18Examples of depth and normal estimation using UBotNet on real-world images in the wild.
Table 1 .
1Comparison between the various proposed omnidirectional datasets. While existing datasets are predominantly indoors, our proposed dataset includes both indoor and outdoor environments, dynamic lighting and scene participants.Dataset
Domain
Type
No. of panoramic views
Scene Lighting
Dynamic components
Matterport3D 360 • [30]
Real
Indoors
9,684
Static
Replica 360 • 2k/4k RGBD
Table 2 .
2Quantitative Results for the benchmark evaluated on the OmniHorizon dataset. Values in bold highlight best results.Figure 6. Qualitative Results from the benchmark on OmniHorizon dataset. Three different instances of varying depths and lighting conditions are compared between all networks. UBotNet performs consistently better than UNet128 and other architectures when estimating depth and normals. UBotNet Lite shows small artefact in depth estimates but still preserves the global context required to learn normals.Depth Error ↓
Depth Accuracy ↑
Normal Error ↓
Normal Accuracy ↑
Method
# parameters RMSE MRE
RMSE log δ1 < 1.25 δ2 < 1.25 2 δ3 < 1.25 3 Mean Median RMSE 5.0 •
7.5 •
11.25 •
RectNet [46]
8.9 M
0.646 23.786
1.213
0.247
0.265
0.283
9.84
5.49
14.53 48.84 56.06 65.85
UResNet [46]
50.8 M
0.097
0.487
0.260
0.424
0.614
0.768
11.50
7.18
16.32 44.50 49.01 55.73
UBotNet Lite (Ours)
38.3 M
0.063
0.403
0.181
0.657
0.844
0.896
8.00
4.19
12.57 54.86 64.51 75.36
UNet128
124 M
0.052
0.259
0.157
0.641
0.849
0.925
9.01
4.01
14.71 54.00 62.58 72.68
UBotNet (Ours)
133 M
0.054
0.271
0.151
0.712
0.874
0.929
7.44
3.61
12.12 56.80 67.29 78.52
Table 3 .
3Quantitative results for the ablation study on the OmniHorizon dataset. Various versions of the dataset are compared by removing the dynamic elements from the scene. VP -Vehicles & Pedestrians and DL -Dynamic LightingDepth Error ↓
Depth Accuracy ↑
Normal Error ↓
Normal Accuracy ↑
Method
RMSE MRE RMSE log δ1 < 1.25 δ2 < 1.25 2 δ3 < 1.25 3 Mean Median RMSE 5.0 •
7.5 •
11.25 •
Static
0.055 0.293
0.155
0.656
0.854
0.924
7.67
3.74
12.55 56.16 66.49 77.60
Static + VP
0.053 0.289
0.154
0.713
0.868
0.924
7.53
3.64
12.26 56.72 67.05 78.18
Static + VP + DL 0.054 0.271
0.151
0.712
0.875
0.926
7.44
3.61
12.12 56.80 67.28 78.52
Table 4 .
4Quantitative results for Sim-to-Real performance of our dataset on Fukuoka dataset.Depth Error ↓
Depth Accuracy ↑
Pre-train
RMSE MRE RMSE log
δ1
δ2
δ3
None
0.036 0.633
0.014
0.265 0.497 0.664
OmniHorizon 0.033 0.682
0.012
0.402 0.630 0.761
Table 5 .
5Quantitative results for depth estimation using UBotNet Lite validated on indoor dataset -Matterport3D.Depth Error ↓
Depth Accuracy ↑
Dataset
RMSE MRE RMSE log
δ1
δ2
δ3
Matterport3D 0.639 0.142
0.064
0.817 0.952 0.981
Gibson 3D
0.591 0.154
0.061
0.830 0.965 0.986
Table 6 .
6Quantitative results for the comparison between the positional embedding used in the UBotNet architecture for self-attention. The results for the Relative Positional Embedding are repeated from our main paper for the comparison. Depth Error ↓ Depth Accuracy ↑ Normal Error ↓ Normal Accuracy ↑ Method RMSE MRE RMSE log δ1 < 1.25 δ2 < 1.25 2 δ3 < 1.25 3 Mean Median RMSE 5.0 • 7.5 • 11.25 •Absolute Pos. Emb. 0.053 0.290
0.152
0.691
0.871
0.925
8.65
3.98
13.99 54.26 63.00 73.23
Relative Pos. Emb.
0.054 0.271
0.151
0.712
0.875
0.926
7.44
3.61
12.12 56.80 67.28 78.52
Depth maps have been normalised for visualisation purposes.
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|
[] |
[
".3 mm WAVELENGTH VLBI OF SAGITTARIUS A*: DETECTION OF TIME-VARIABLE EMISSION ON EVENT HORIZON SCALES",
".3 mm WAVELENGTH VLBI OF SAGITTARIUS A*: DETECTION OF TIME-VARIABLE EMISSION ON EVENT HORIZON SCALES"
] |
[
"Vincent L Fish ",
"Sheperd S Doeleman ",
"Christopher Beaudoin ",
"Ray Blundell ",
"David E Bolin ",
"Geoffrey C Bower ",
"Richard Chamberlin ",
"Robert Freund ",
"Mark A Gurwell ",
"Mareki Honma ",
"Makoto Inoue ",
"Thomas P Krichbaum ",
"James Lamb ",
"Daniel P Marrone ",
"James M Moran ",
"Tomoaki Oyama ",
"Richard Plambeck ",
"Rurik Primiani ",
"Alan E E Rogers ",
"Daniel L Smythe ",
"Jason Soohoo ",
"Peter Strittmatter ",
"Remo P J Tilanus ",
"Michael Titus ",
"Jonathan Weintroub ",
"Melvyn Wright ",
"David Woody ",
"Ken H Young ",
"Lucy M Ziurys "
] |
[] |
[] |
Sagittarius A*, the ∼ 4 × 10 6 M ⊙ black hole candidate at the Galactic Center, can be studied on Schwarzschild radius scales with (sub)millimeter wavelength Very Long Baseline Interferometry (VLBI). We report on 1.3 mm wavelength observations of Sgr A* using a VLBI array consisting of the JCMT on Mauna Kea, the ARO/SMT on Mt. Graham in Arizona, and two telescopes of the CARMA array at Cedar Flat in California. Both Sgr A* and the quasar calibrator 1924−292 were observed over three consecutive nights, and both sources were clearly detected on all baselines. For the first time, we are able to extract 1.3 mm VLBI interferometer phase information on Sgr A* through measurement of closure phase on the triangle of baselines. On the third night of observing, the correlated flux density of Sgr A* on all VLBI baselines increased relative to the first two nights, providing strong evidence for time-variable change on scales of a few Schwarzschild radii. These results suggest that future VLBI observations with greater sensitivity and additional baselines will play a valuable role in determining the structure of emission near the event horizon of Sgr A*.
|
10.1088/2041-8205/727/2/l36
|
[
"https://arxiv.org/pdf/1011.2472v1.pdf"
] | 5,687,089 |
1011.2472
|
9a2a9c005204ebee74a1156670cb8f80860053a4
|
.3 mm WAVELENGTH VLBI OF SAGITTARIUS A*: DETECTION OF TIME-VARIABLE EMISSION ON EVENT HORIZON SCALES
10 Nov 2010 November 11, 2010
Vincent L Fish
Sheperd S Doeleman
Christopher Beaudoin
Ray Blundell
David E Bolin
Geoffrey C Bower
Richard Chamberlin
Robert Freund
Mark A Gurwell
Mareki Honma
Makoto Inoue
Thomas P Krichbaum
James Lamb
Daniel P Marrone
James M Moran
Tomoaki Oyama
Richard Plambeck
Rurik Primiani
Alan E E Rogers
Daniel L Smythe
Jason Soohoo
Peter Strittmatter
Remo P J Tilanus
Michael Titus
Jonathan Weintroub
Melvyn Wright
David Woody
Ken H Young
Lucy M Ziurys
.3 mm WAVELENGTH VLBI OF SAGITTARIUS A*: DETECTION OF TIME-VARIABLE EMISSION ON EVENT HORIZON SCALES
10 Nov 2010 November 11, 2010Preprint typeset using L A T E X style emulateapj v. 11/10/09 Draft versionSubject headings: Galaxy: center -submillimeter: general -techniques: high angular resolution -tech- niques: interferometric
Sagittarius A*, the ∼ 4 × 10 6 M ⊙ black hole candidate at the Galactic Center, can be studied on Schwarzschild radius scales with (sub)millimeter wavelength Very Long Baseline Interferometry (VLBI). We report on 1.3 mm wavelength observations of Sgr A* using a VLBI array consisting of the JCMT on Mauna Kea, the ARO/SMT on Mt. Graham in Arizona, and two telescopes of the CARMA array at Cedar Flat in California. Both Sgr A* and the quasar calibrator 1924−292 were observed over three consecutive nights, and both sources were clearly detected on all baselines. For the first time, we are able to extract 1.3 mm VLBI interferometer phase information on Sgr A* through measurement of closure phase on the triangle of baselines. On the third night of observing, the correlated flux density of Sgr A* on all VLBI baselines increased relative to the first two nights, providing strong evidence for time-variable change on scales of a few Schwarzschild radii. These results suggest that future VLBI observations with greater sensitivity and additional baselines will play a valuable role in determining the structure of emission near the event horizon of Sgr A*.
INTRODUCTION
The case for linking Sgr A*, the radio source at the center of the Milky Way, with a supermassive black hole is very strong. Mass estimates inferred from stellar orbits, proper motion studies that indicate Sgr A* is nearly motionless, VLBI observations that reveal it is ultracompact, and short-timescale variability from radio to X-rays all point towards Sgr A*'s association with a ∼ 4 × 10 6 M ⊙ black hole (Reid 2009, and references therein). At a distance of ∼ 8 kpc, the Schwarzschild radius of this black hole subtends R Sch ∼ 10 µas, making the apparent size of its event horizon the largest that we know of. VLBI at (sub)millimeter wavelengths is ideally suited to observing Sgr A* on these angular scales. Previous 1.3 mm VLBI detections of Sgr A* on a Hawaii-Arizona baseline established the existence of coherent structures on scales of a few R Sch (Doeleman et al. 2008). Current 1.3 mm VLBI observations can be used to address two fundamental questions concerning the nature of Sgr A*. The first is whether the accretion flow surrounding Sgr A* exhibits an expected "shadow" feature that occurs due to the strong gravitational lensing near the black hole. Emission from the accretion flow is preferentially lensed onto the last photon orbit, resulting in a relatively dim central region encircled by a brighter annulus (Falcke et al. 2000). A second question is whether the flaring behavior exhibited by Sgr A* has its origins in compact structures that arise near the black hole event horizon. Broadband flares on timescales ranging from minutes to hours are well-documented (Marrone et al. 2008;Yusef-Zadeh et al. 2009;Dodds-Eden et al. 2009) and imply time-variable structures in the innermost accretion region. If small-scale variable structures are present, 1.3 mm VLBI can sensitively monitor the changing morphology of Sgr A* using non-imaging techniques with time resolutions of tens of seconds Fish et al. 2009b).
We report on new 1.3 mm VLBI observations of Sgr A* using a four-telescope array. These observations confirm event horizon scale structure within Sgr A*, impose new constraints on accretion models for Sgr A*, and reveal time-dependent variability on scales of a few R Sch .
2. OBSERVATIONS Sgr A* and several calibrator sources were observed with four telescopes at three observatories: the James Clerk Maxwell Telescope (JCMT; henceforth also J) on Mauna Kea in Hawaii, the Arizona Radio Observatory's Submillimeter Telescope (ARO/SMT; S) in Arizona, and two telescopes of the Combined Array for Research in Millimeter-wave Astronomy (CARMA; C and D, located ∼ 60 m apart) in California. On Mauna Kea, the Submillimeter Array (SMA) housed the VLBI recording system and synthesized the hydrogen maser based VLBI reference used at the JCMT. Masers at all sites were checked against ultra-stable crystals; combined losses due to maser instabilities and local oscillator decoherence are estimated to be 5%. Observations occurred over three nights: 2009 April 5-7 (days 95-97). Sources were observed in left circular polarization in two 480 MHz bandwidths centered at 229.089 and 229.601 GHz (low and high bands). Data recorded at all sites was shipped to MIT Haystack Observatory in Westford, Massachusetts for processing on the Mark4 VLBI correlator. Once correlated, data for each scan (typically 10-15 minutes) were corrected for coherence losses due to atmospheric turbulence and searched for detections using methods detailed in Doeleman et al. (2001Doeleman et al. ( , 2008. Atmospheric coherence times ranged from a few to ∼ 20 s, depending on weather conditions at each telescope.
CALIBRATION
The VLBI correlation coefficient for each baseline was multiplied by the geometric mean of the System Equivalent Flux Density (SEFD) of both antennas. The SEFD is a product of antenna gain (Jy/K) and the opacity-corrected system temperature, which was measured just prior to each VLBI scan using a vane calibration technique that corrects for the atmosphere. For the JCMT and ARO/SMT, antenna gains were determined from observations of planets at several points during the multiple day campaign, and the gains were observed to be stable. Relative gains for the two CARMA dishes were estimated using observations taken by CARMA in interferometric array mode before each VLBI scan, and the gains were then set to a common flux scale using planet scans at the end of each night.
The flux densities of all VLBI targets (Sgr A*, 1924−292, M87, 3C273, 3C345, 1733−130, 3C279, 0854+201) were measured with CARMA. For Sgr A*, data with baselines shorter than 20 kλ were discarded to filter out extended emission in the Galactic center. The measured flux densities of all sources increased from day 95 to day 96 and from day 96 to day 97. We attribute this systematic trend to errors in the planet calibrations made shortly after sunrise, when antenna focus, pointing offsets, and atmospheric coherence typically change. The flux density measured for the calibrator 1924−294, observed over the same time and elevation ranges as Sgr A*, was 9.95, 10.21, and 10.75 Jy on days 95, 96, and 97. We normalized CARMA gains to a constant flux density of 10.25 Jy on all three days. The resulting measured flux densities for Sgr A* are 3.03, 3.16, and 3.61 Jy on days 95, 96, and 97, respectively. We adopt these fluxes for all subsequent analysis.
As shown in the upper panels of Figures 1 and 2, there are still noticeable variations in the correlated flux densities even after renormalizing the day-to-day flux scales. These residual calibration errors and amplitude variation can be corrected for by making three simplifying assumptions that allow us to use standard "self-calibration" techniques. First, the flux densities of detections in the low and high bands, which differ by only 0.2% in frequency, are assumed to be equal in each scan. Second, flux densities on the SC and SD baselines are assumed to be equal. While one could in principle require that JC and JD flux densities be equal as well, the signal-to-noise ratio (SNR) is generally much lower on the JC and JD baselines than on the shorter VLBI baselines (SC and SD), since both 1924−292 and Sgr A* are more resolved on longer baselines. Third, CARMA antenna gains are adjusted to make the correlated flux density on the CD baseline (with a fringe spacing measured in arcseconds) equal to the total flux density measured each night by CARMA. This final constraint enforces a constant source flux density over the duration of each night of observation. While some of the observed variation in Sgr A* over the course of a night may be due to intrinsic variability, the 1924−292 data exhibit similar scatter, suggesting that calibration errors may dominate over source variability. Combined, these assumptions result in a closed-form solution for gain-correction coefficients for telescopes C, D, and S in each band. Henceforth, we will use the term "gain-corrected" to refer to flux densities that have been multiplied by these gaincorrection coefficients. We note that if the total flux density (CD) is varied, the SJ flux densities are unchanged while other flux densities vary as the square root of the factor.
The quasar 1924−292 was easily detected on all baselines (Table 1). On each scan, low-band and high-band fluxes after a-priori calibration track each other consistently (Figure 1). After gain correction assuming a total flux density of 10.25 Jy, the data from all three days are highly consistent with one another. The SC and SD baselines show consistent variation in the correlated flux density that is repeated each day. The long-baseline detections (SJ, JC, and JD) also show day-to-day repeatable behavior, indicating detection of stable source structure presumably associated with a jet (Shen 1997). The consistency of these data demonstrates the validity of the gain-correction technique. Based on the statistics of the data on scans of 1924−292, systematic errors are estimated to be ∼ 5%.
RESULTS
We report the first 1.3 mm VLBI detections of Sgr A* on Hawaii-California baselines with correlated flux densities for several scans of 400 mJy on the JC and JD baselines during day 96 ( Figure 2). Nondetections on the JC and JD baselines on day 95 are attributable to the higher opacity at the JCMT on that day. The robust detections on the long (Hawaii-Arizona and Hawaii-California) baselines confirm the detection of event horizon scale structure reported in Doeleman et al. (2008).
Because Sgr A* was detected on baselines between all three sites, we are able to measure closure phase: the sum of interferometric phase around a closed triangle of baselines. This quantity is relatively immune to calibration errors and in general provides important constraints on source structure. On the CARMA-ARO/SMT-JCMT triangle, the phase closures on Sgr A* for day 96 (5 independent measurements) were computed using 10-second coherent integrations, which were averaged over full 10-minute VLBI scans to increase SNR (Rogers et al. 1995). Because of the relatively low SNR on the long baselines to the JCMT, it is only possible to say that the closure phases are consistent with a value of zero with a variation of ±40 • .
Variability
The arcsecond-scale flux density of Sgr A* on days 95 and 96 is the same to within uncertainties in the flux scale, but on day 97 the flux density of Sgr A* was ∼ 17% higher. This Detections are color-coded by baseline. Squares indicate low-band data and circles indicate high-band data. The day 97 a-priori calibration is noticeably poorer at CARMA. Bottom: Same, after gain correction. Errorbars include a 5% systematic component. The gain-corrected data exhibit much higher day-to-day repeatability. The red dashed line shows the assumed CD flux (the associated flux scale uncertainty is ∼ 5%, modulo uncertainties in planet fluxes). Gain-corrected SC and SD data are equal by definition.
brightening is accompanied by changes on VLBI scales as well. First, the SC/CD and SD/CD flux density ratios are higher on day 97 than on days 95 and 96 ( Figure 2). Second, there are more Hawaii-California detections during day 97, and the apparent flux densities on these baselines are on average higher than on day 96. Third, the flux densities on the SJ baseline are also larger on day 97. These differences are consistent with an episode of variability in Sgr A* during which the small-scale structure increased in flux density between days 96 and 97.
Geometrical Models of the Structure of Sgr A*
Though our data are better calibrated than in the previous epoch (Doeleman et al. 2008), the structure of Sgr A* is poorly constrained because millimeter VLBI detections of Sgr A* remain limited in terms of baseline length and orientation. As a result, many models can be made to fit the data: extended double sources, large rings, combinations of largeand small-scale components, etc.
Nevertheless, with the caveat that this small dataset should not be overinterpreted, it is instructive to investigate the two classes of models originally considered by Doeleman et al. (2008) to fit the 1.3 mm VLBI data obtained in 2007: circular Gaussians and rings. All of the 2007 data points could be fitted by a single Gaussian component. In contrast, we note a loss of ∼ 1 Jy of correlated flux density between the connected-element (CD) and SC/SD baselines (Figure 3). In the context of Gaussian models of emission on R Sch scales, this suggests the existence of additional variable structure on scales between those probed by the SC/SD (a few hundred microarcseconds) and the CD (a few arcseconds) baselines.
We adopt this assumption to estimate the size of the inner accretion flow in Sgr A*. Effectively, this reduces to fitting all of the VLBI data excluding the CD data points. For the Gaussian model, the best fits imply a flux density of 2.07 +0.14 −0 Ring models with three parameters (inner radius, outer radius, and flux density) can also fit the VLBI data ( Figure 3 and Table 2). However, no single set of ring model parameters consistently fits the data on all three days, which would suggest that the size and structure of Sgr A* are variable within the context of ring models. This stands in contrast with the Gaussian model, for which all epochs of data are consistent with a uniform size despite differences in the flux density. Longer-wavelength VLBI observations are inconclusive as to whether a significant correlation exists between the flux density and size of the emission in Sgr A* (Bower et al. 2004;Lu et al. 2010). However, the size of the emission at these wavelengths is dominated by interstellar scattering effects. Future 1.3 mm VLBI observations with higher sensitivity, sufficient to robustly measure the closure phase, will be an important discriminant between these and other models. For example, an elliptical Gaussian distribution will result in zero closure phase on any triangle of baselines, while a ring model can result in closure phases of 180 degrees depending on the orientation and length of the array baselines. The ring models shown in Figure 3 all have closure phases of zero on the CARMA-ARO/SMT-JCMT triangle of baselines, consistent with the measured closure phases (Section 4). However, a ring model with a null near 3.4 Gλ between the CARMA-JCMT and ARO/SMT-JCMT baselines, would produce a closure phase of 180 degrees, which is strongly ruled out by the April 2009 data. Measurement on an intermediate baseline in the 1 − 2 Gλ range would provide a powerful discriminant between large classes of geometrical models.
DISCUSSION
Implications for Accretion and Flare Models
The flux density of Sgr A* on VLBI scales is seen to increase from day 95/96 to day 97. During the first ∼ 1 hour on day 97, when the atmosphere at CARMA was relatively stable, the data are consistent with a constant flux density, suggesting that the flux density increased before observations on day 97 but held steady at a higher level than on the previous nights. This behavior is consistent with other (sub)millimeter observations, which show variability punctuated with periods when the flux density is stable (e.g., Marrone et al. 2006;Yusef-Zadeh et al. 2009;Kunneriath et al. 2010).
The flux density increase appears to be due to an event that establishes a new steady-state in Sgr A*. If instead the flux density increase is due to a short-duration event that concluded before the start of observations on day 97, the unchanging size of the compact region (as implied by Gaussian models in Section 4.2) and the timescale over which the compact flux density is seen to be constant limits the expansion speed of the region to be highly nonrelativistic (v 0.05c, consistent with Lu et al. 2010) and much lower than the sound speed (c/ √ 3) (Marrone et al. 2008), in contrast with relativistic jet models (e.g., Falcke et al. 2009). While a low expansion speed is predicted by adiabatically-expanding plasmon models (Yusef-Zadeh et al. 2009), these models also predict an increase in source size. Our observations detect Sgr A* after the increase in flux density has occurred, but we do not find evidence of an increase in source size as predicted by adiabatic expansion. Future, more sensitive observations of Sgr A* before, during, and after a flare event will be necessary to more fully understand the mechanism responsible for variability in Sgr A*.
Physical Models of the Structure of Sgr A*
Numerous physically-motivated models have been advanced to explain the multiwavelength emission from Sgr A* (e.g., Falcke et al. 2000;Yuan et al. 2003;Markoff et al. 2007;Noble et al. 2007). In recent years, several efforts to constrain disk models using 1.3 mm VLBI observations have placed limits on model parameters such as the incli- nation of the disk and the spin of the black hole. Radiatively inefficient accretion flow models and some general relativistic magnetohydrodynamic simulations found that the Doeleman et al. (2008) data were already sufficient to rule out a low-inclination (i.e., nearly face-on) disk Mościbrodzka et al. 2009). Low-inclination disk models are qualitatively similar to the ring model in Section 4.2, with a "null" in the correlated flux density as a function of baseline length whose depth decreases with increasing disk inclination (Dexter et al. 2009;Fish et al. 2009a). The location of this null as a function of baseline length is set by the ratio of the mass of the black hole and the distance to Sgr A*, which is determined to within a few percent (Ghez et al. 2008;Gillessen et al. 2009).
Emission in high-inclination models is dominated by the highly Doppler-boosted approaching side of the disk. Highinclination disk models exhibit a monotonic decrease in flux density as baselines approach the length of the SJ baseline, with long-baseline correlated flux densities highly dependent on the orientation of the disk in the plane of the sky. The decline of correlated flux density with baseline length from Hawaii-California to Hawaii-Arizona baselines requires that the inclination be 40 • in several disk models that were consistent with the 2007 data (Dexter et al. 2009;Broderick et al. 2009). The impact of the 2009 Hawaii-California detections on the 2007 fits is to strengthen the constraint on inclination in these models (e.g., Broderick et al., private communication). 6. SUMMARY VLBI observations of Sgr A* at 1.3mm wavelength in 2009 have robustly confirmed the detection of Schwarzschild ra-dius scale structures reported in Doeleman et al. (2008). On the third of three days of observations, the total flux density of Sgr A* was observed to have increased by ∼ 17%, indicating an episode of variability similar to those described in multi-wavelength monitoring campaigns (e.g., Marrone et al. 2008;Yusef-Zadeh et al. 2009). The VLBI data support a picture in which this flux density increase is due to a brightening of structure on scales of only a few R Sch . Having achieved the dual technical milestones of successfully detecting closure phase and developing robust methods of amplitude calibration, it is clear that future (sub)millimeter VLBI observations with higher sensitivity and many more baselines will be able to powerfully constrain models of Sgr A* on event horizon scales.
High-frequency VLBI work at MIT Haystack Observatory is supported by grants from the National Science Foundation (NSF). ARO receives partial support from the NSF ATI program. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. Funding for ongoing CARMA development and operations is supported by the NSF and the CARMA partner universities. DEB acknowledges support from the NSF Research Experiences for Undergraduates program.
Facilities: CARMA (), HHT (), JCMT (), SMA (), CSO ()
FIG. 1 .
1-Top: Observations of 1924 − 292.
.15 Jy and a size of 41.3 +5.4 −4.3 µas (FWHM; errors are 3σ) on day 95 and 2.07 +0.19 −0.19 Jy and 44.4 +3.0 −3.0 µas on day 96 (Figure 3and Table 2). These values are consistent with the single compact component Gaussian fit ofDoeleman et al. (2008), who estimated a flux density of 2.4 ± 0.5 Jy and a size of 43 +14 −8 µas (before deconvolution of the expected interstellar scattering; 37 µas unscattered) for the 230 GHz emission. On day 97, the best fit model has a much higher flux density of 2.85 +0.29 −0.28 Jy but a similar FWHM of 42.6 +3.1 −2.9 µas. Despite the increase in flux density observed on day 97, the diameter of the fitted compact component in Sgr A* on that day is identical (to within the errors) to the values for the size obtained on days 95 and 96.
FIG. 2 .
2-Same asFigure 1but for Sgr A*. Only scans for which detections are seen in both bands on the CD, SC, and SD baselines are shown.
FIG. 3 .
3-Correlated gain-corrected flux density plots. Solid lines show best-fit circular Gaussian models of the compact emission in Sgr A* and are consistent with a source size of approximately 43 µas on all days. The dotted lines show the best ring model fits to Sgr A* data.
TABLE 1
1GAIN-CORRECTED DETECTIONS OF 1924−292 AND SGR A* GAIN-CORRECTED DETECTIONS OF 1924−292 AND SGR A* GAIN-CORRECTED DETECTIONS OF 1924−292 AND SGR A* MODEL FITS TO SGR A* DATADay
UT Time
u
v
Flux Density
σ
Source
Number (hh mm) Baseline Band (Mλ) (Mλ)
(Jy)
(Jy)
1924−292
95
13 10
SC
both
-509
105
4.96 0.26
SJ
low -3103 -1561
1.47 0.11
SJ
high -3103 -1561
1.40 0.12
JC
low
2594
1667
1.63 0.19
JC
high
2594
1667
1.95 0.22
JD
low
2594
1667
1.72 0.20
JD
high
2594
1667
1.73 0.22
13 55
SC
both
-569
157
5.60 0.29
SJ
low -3325 -1251
1.24 0.09
SJ
high -3325 -1251
1.13 0.09
JC
low
2756
1408
1.33 0.14
JC
high
2756
1408
1.17 0.14
JD
low
2756
1408
1.28 0.14
JD
high
2756
1408
1.15 0.14
15 25
SC
both
-621
273
4.30 0.22
SJ
low -3380
-598
1.02 0.07
SJ
high -3380
-598
1.02 0.07
JC
low
2759
871
2.11 0.13
JC
high
2759
871
2.16 0.13
JD
low
2759
871
2.13 0.13
JD
high
2759
871
2.04 0.12
15 45
SC
both
-619
300
3.66 0.19
SJ
low -3320
-454
1.15 0.07
SJ
high -3320
-454
1.08 0.07
JC
low
2701
754
2.31 0.14
JC
high
2701
754
2.13 0.13
JD
low
2701
754
2.38 0.15
JD
high
2701
754
2.23 0.14
16 15
SC
both
-608
339
3.75 0.20
SJ
low -3184
-246
1.23 0.08
SJ
high -3184
-246
1.12 0.07
JC
low
2575
585
2.56 0.16
JC
high
2575
585
2.27 0.15
JD
low
2576
585
2.35 0.15
JD
high
2576
585
2.21 0.15
96
13 10
SC
both
-515
110
5.03 0.27
SJ
low -3127 -1535
1.67 0.12
SJ
high -3127 -1535
1.64 0.12
JC
low
2612
1645
1.96 0.16
JC
high
2612
1645
1.77 0.16
JD
low
2612
1645
2.04 0.17
JD
high
2612
1645
1.96 0.17
14 00
SC
both
-578
168
5.78 0.30
SJ
low -3354 -1187
1.48 0.10
SJ
high -3354 -1187
1.40 0.10
JC
low
2776
1356
1.48 0.13
JC
high
2776
1356
1.38 0.12
JD
low
2776
1356
1.47 0.13
JD
high
2776
1356
1.47 0.13
15 30
SC
both
-621
285
4.21 0.22
15 50
SC
both
-617
312
3.28 0.18
SJ
low -3286
-391
1.04 0.07
SJ
high -3286
-391
0.90 0.07
JC
low
2668
703
2.21 0.15
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|
[] |
[
"On new infinite families of completely regular and completely transitive codes",
"On new infinite families of completely regular and completely transitive codes"
] |
[
"J Borges ",
"J Rifà ",
"V A Zinoviev "
] |
[] |
[] |
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming un upper bound on the order of the monomial automorphism groups, according to computational results.
|
10.48550/arxiv.2303.11190
|
[
"https://export.arxiv.org/pdf/2303.11190v1.pdf"
] | 257,632,450 |
2303.11190
|
751e6135861dd6b3f7734248a473c9a751d2317e
|
On new infinite families of completely regular and completely transitive codes
20 Mar 2023 March 21, 2023
J Borges
J Rifà
V A Zinoviev
On new infinite families of completely regular and completely transitive codes
20 Mar 2023 March 21, 2023
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming un upper bound on the order of the monomial automorphism groups, according to computational results.
Introduction
Let F q be the finite field of order q, where q is a prime power. For a qary code C ⊂ F n q of length n, denote by d its minimum (Hamming) distance between any pair of distinct codewords. The packing radius of C is e = ⌊(d − 1)/2⌋. Given any vector v ∈ F n q , its distance to the code C is d(v, C) = min x∈C {d(v, x)} and the covering radius of the code C is ρ = max v∈F n 2 {d(v, C)}. Note that e ≤ ρ. A linear [n, k, d; ρ] q -code is a kdimensional subspace of F n q , with minimum distance d and covering radius ρ. We denote by D = C + x (x ∈ F n q ) a coset of C, where + means the component-wise addition in F q .
For a given code C of length n and covering radius ρ, define C(i) = {x ∈ F n q : d(x, C) = i}, i = 0, 1, . . . , ρ.
The sets C(0) = C, C(1), . . . , C(ρ) are called the subconstituents of C.
Say that two vectors x and y are neighbors if d(x, y) = 1. Denote by 0 the all-zero vector or the all-zero matrix, depending on the context. Let M be a monomial matrix, i.e. a matrix with exactly one nonzero entry in each row and column. If q is prime, then the automorphism group of C, Aut(C), consists of all monomial (n × n)-matrices M over F q such that cM ∈ C for all c ∈ C. If q is a power of a prime number, then Aut(C) also contains any field automorphism of F q which preserves C. The group Aut(C) acts on the set of cosets of C in the following way: for all π ∈ Aut(C) and
for every vector v ∈ F n q we have π(v + C) = π(v) + C. Fix the following notation for groups, which we need: let C t denote the cyclic group of order t, let S t denote the symmetric group of order t!, and let GL(m, q) be the q-ary general linear group (formed by all nonsingular q-ary (n × n)-matrices).
In this paper, we consider only monomial automorphisms. Thus, when q is not a prime, but a prime power, we omit the field automorphisms that fix the code. We denote by MAut(C) the monomial automorphism group of a code C. Definition 1.2 Let C be a q-ary linear code with covering radius ρ. Then C is completely transitive if MAut(C) has ρ + 1 orbits when acts on the cosets of C.
Note that this definition generalizes to the q-ary case the definition given in [14]. However, it is a particular case of the definition of coset-completely transitive code given in [9], where not only monomial automorphisms are considered, but also the field automorphisms are taken into account.
Since two cosets in the same orbit have the same weight distribution, it is clear that any completely transitive code is completely regular.
It is well known, e.g. see [11], that the monomial automorphism group of a Hamming code H m of length (q m − 1)/(q − 1) is isomorphic to the general linear group GL(m, q), which acts transitively on the set of vectors of weight one. As a consequence, all cosets of minimum weight one are in the same orbit and thus Hamming codes are completely transitive. In the binary case, the action of GL(m, 2) on the set of coordinate positions is even doubly transitive (but only in the binary case). Later, we will use this fact.
Completely regular and completely transitive codes are classical subjects in algebraic coding theory, which are closely connected with graph theory, combinatorial designs and algebraic combinatorics. Existence, construction and enumeration of all such codes are open hard problems (see [7,15,10,12,4] and references there).
It is well known that new completely regular codes can be obtained by the direct sum of perfect codes or, more general, by the direct sum of completely regular codes with covering radius 1 [1,14]. In the previous papers [3,5], we extend these constructions, giving several explicit constructions of new completely regular codes, based on concatenation methods. Here, we find the monomial automorphism groups for that families in some cases. This gives mutually nonequivalent binary linear completely regular codes with the same intersection arrays and isomorphic monomial automorphism groups.
We show cases in which the constructed families of codes are also completely transitive.
In Section 2 we give some preliminary results, mostly coming from the previous papers [2,3,5] (where we introduced two infinite families of CR codes A (r) and B (r) ), that help to us place ourselves in the problem we want to address. In Section 3 we study the completely transitive character of codes B (r) and calculate MAut(B (r) ) for some cases. Finally, we give a result that gives us the character of completely transitive for these B (r) codes with an extra assumption about the order of MAut(B (r) ). This assumption is given by computational considerations on the order of such monomial automorphism groups.
Preliminary results
In this section we recall the results of [2,3,5]. Combining such results, in the next section, we specify cases in which the constructed infinite families of completely regular codes are also completely transitive.
For any vector x = (x 1 , . . . , x n ) ∈ F n q , denote by σ(x) the right cyclic shift of x, i.e. σ(x) = (x n , x 1 , . . . , x n−1 ). Define recursively σ i (x) = σ(σ i−1 (x)), for i = 2, 3, . . . and σ 1 (x) = σ(x). For j < 0, we define σ j (x) = σ ℓ (x), where ℓ = j mod n. The next constructions are described in [3], although the dual codes of the resulting family of q-ary completely regular codes are well known as the family SU2 in [8].
Let H be the parity check matrix of a q-ary cyclic Hamming code H m of length n = (q m − 1)/(q − 1) (recall that a cyclic version of H m exists exactly when n and q − 1 are coprime numbers). Clearly the simplex code generated by H is also a cyclic code. For any r ∈ {1, 2, . . . , n}, consider the code B (r) of length nr with the following parity check matrix H b (r) (we call it Construction I in [3]):
H b (r) = H H · · · H H 1 H 2 · · · H r , r = 1, . . . , n;(1)
where H i , for 1 ≤ i ≤ n, is the matrix Hξ i (here ξ ∈ F q m is a primitive nth root of unity), hence H i is obtained from H by cyclically shifting i times its columns to the right (in these terms H = H n ). In [3] we also presented Construction II. In this case, the corresponding code of length n(r+3), which we denote A (r) , has parity check matrix H a (r) of the form:
H a (r) = H O H H H · · · H O H H H 1 H 2 · · · H r .(2)
For the codes A (r) we fix the following interval for r: r ∈ {−2, −1, 0, 1, . . . , n− 1} (for r = n we would have pairs of linear dependent columns), where A (0) , A (−1) and A (−2) are defined by the parity check matrices, respectively
H a (0) = H O H O H H , H a (−1) = H O O H , H a (−2) = H O .(3)
Note that the codes B (1) and A (−2) are the Hamming code of length (q m − 1)/(q − 1).
Lemma 2.1 Let i and j be positive integers such that i + j ≤ n. The code D with parity check matrix H H · · · H H i H i+1 · · · H i+j−1 (4) is equivalent to the code B (j) .
Proof. The statement is evident, since the parity check matrices for the codes D and B (j) differ from each other by the multiplier ξ i−1 , applied to the second bottom part of rows.
In [3] we proved the following theorem.
IA = {(q − 1)nr, ((q − 1)n − r + 2)(r − 1); 1, r(r − 1)}. (ii) For any r, 0 ≤ r ≤ n − 1, the code A (r) with parity check matrix H a (r)
given in (2) is a completely regular [(r + 3)n, (r + 3)n − 2m, 3; 2] q -code with intersection array
IA = {(q − 1)n(r + 3), ((q − 1)n − 1 − r)(r + 2); 1, (r + 2)(r + 3)}.
In [5] we defined the concept of supplementary codes. Let H m be the parity check matrix of a q-ary Hamming code of length n = (q m − 1)/(q − 1). In all cases, the parameters and intersection arrays are computed in [5].
Now, consider the parity check matrix H 2m of a q-ary Hamming code H 2m of length (q 2m − 1)/(q − 1), such that the q-ary Hamming code H m is cyclic. Equivalently, n = (q m − 1)/(q − 1) and q − 1 are coprime numbers.
We assume that m ≥ 3 to avoid trivial cases. In fact, for m = 2, H m is cyclic only if q is a power of 2. Present H 2m as follows:
H 2m = H b (r) H c (r) , r ∈ {1, . . . , q m },(5)
where
H b (r) is the matrix (1) with rn = r(q m − 1)/(q − 1) columns, hence it is a parity check matrix for B (r) . Call C (r) the supplementary code of B (r) ,
i.e. C (r) has parity check matrix H c (r). By combining the results of [3] and [5], we obtain the following theorem. Remark 1 In the binary case, q = 2, the matrix H 2m can be written as
H 2m = H H · · · H H · · · H H 0 H H 1 H 2 · · · H r H r+1 · · · H n−1 0 H H ,(6)
where H is a parity check matrix of a binary cyclic Hamming code H m .
Indeed, H 2m has n(n + 2) = 2 2m − 1 columns and there are no repeated columns. Therefore, in the binary case and by Lemma 2.1, the code C (r) is equivalent to the code A (n−r−1) (Construction II in [3]).
Completely transitive families
For the rest of the paper, let B (r) be the code with parity check matrix H b (r)
given in (1). Let T 1 , . . . , T r be the n-sets, which we call blocks, of coordinate positions corresponding to the columns of H 1 , . . . , H r . That is,
T j = {(j − 1)n + 1, (j − 1)n + 2, .
. . , jn}, ∀j = 1, . . . , r.
Consider also the vectors of weight one indexed as follows: where the superindices indicate the corresponding block.
In order to establish in which cases the codes of Theorem 2.4 are completely transitive, we need several previous results. Proof. It is well known that the order of GL(m, q) is m−1 k=0 (q m − q k ), which is even. Hence, if GL(m, q) is 2-homogeneous, it is also 2-transitive by [ Proof. Since B (r) has minimum distance 3, we have that the number of cosets of weight 1 is rn(q − 1) = r(q m − 1). The total number of cosets is q rn /(q rn−2m ) = q 2m . Hence, the number of cosets with minimum weight greater than 1 is q 2m − 1 − r(q m − 1). All such cosets have minimum weight 2, because the covering radius of B (r) is ρ = 2. Proof. The statement (i) is trivial, since B (1) is a Hamming code of length (q m − 1)/(q − 1). It is well known that its automorphism group is GL(m, q) [11].
For (ii), note that, after linear operations on the rows, the parity check matrix H b (2) of the code B (2) can be presented in the form
H b (2) = H O O H .
Since the Hamming code of length (q m − 1)/(q − 1) (defined by the parity check matrix H) is stabilized by the group GL(m, q), we have that the code B (2) is invariant under the action of any matrix G of type
G = G 1 O O G 2 ,(7)
where both matrices G 1 and G 2 are arbitrary matrices from the group GL(m, q).
We conclude that the monomial automorphism group MAut(B (2) ) contains as a subgroup the group GL(m, q) × GL(m, q). From the other side, we can clearly interchange the blocks corresponding to the columns of G 1 and G 2 , respectively, without changing the code B (2) . Hence the group MAut(B (2) ) contains as a subgroup the group of order 2. It is easy to see that any monomial automorphism of B (2) contains only automorphisms from these groups above. Indeed, assume that the matrix G belongs to GL(2m, q), but not to GL(m, q) × GL(m, q). Then we can see (by solving the corresponding system of linear equations) that any column of H b (2) can be moved to any other column. But this leads to a contradiction to the shape of H b (2). Hence, we conclude that MAut(B (2) ) contains as subgroups only GL(m, q) × GL(m, q) and C 2 . Since the two cosets of GL(m, q) × GL(m, q) induced by C 2 coincide, we deduce that MAut(B (2) ) is a direct product of all three subgroups, i.e.
MAut(B (2) ) = GL(m, q) × GL(m, q) × C 2 .
For (iii), the parity check matrix H b (3) (see (1)) of B (3) can be transformed, by linear operations on the rows and column permutations, to the following form:
H b (3) ≃ H O H O H H .
The exclusive property of this matrix is that any of all three blocks is stabilized by all matrices G of the type
G = G 1 O O G 1 ,
where G 1 is an arbitrary matrix from the group GL(m, q). We conclude that the group MAut(B (3) ) contains as a subgroup the group GL(m, q). Now we can see that any codeword z = (x, y, u) of B (3) can be presented in the form Similarly to the previous case, we can see that these are the only monomial automorphisms of the code B (3) . Indeed, if for example G 1 = G 2 , where G is of the form (7) (see case (ii)), we change the third block of H b (3) and obtain the column which does not belong to the third block of H b (3).
z = (u + x 1 , u + x 2 , u + x 3 ), where x i is
Since any element of GL(m, q) × GL(m, q) cannot interchange the blocks and since they intersect only in the identity element, we conclude that the group MAut(B (3) ) is the direct product of groups GL(m, q) and S 3 , i.e.
MAut(B (3) ) = GL(m, q) × S 3 .
Evidently, the actions of both GL(m, q) and S 3 do not depend on each
other.
The next statement shows cases in which the code B (r) is completely transitive (hence, so is C (r) ).
Proposition 1
The codes B (1) and B (2) are completely transitive. Furthermore, for q = 2, the codes B (3) , B (n−1) and B (n) are also completely transitive (here n = (q m − 1)/(q − 1)).
Proof. The code B (1) is a Hamming code. Its automorphism group is GL(m, q) [11], that acts transitively over the set of vectors of weight one.
Hence all cosets with minimum weight one are in the same orbit and thus B (1) is completely transitive.
For B (2) , we have that its automorphism group is GL(m, q) × GL(m, q) × C 2 (Lemma 3.3) which acts transitively over the set of vectors of weight one in each block and also both blocks can be interchanged. Therefore all cosets with minimum weight one are in the same orbit. By Theorem 2.2, the covering radius of B (2) is 2. Hence, we only need to prove that the cosets with minimum weight two are also in the same orbit. Note that any vector at distance 2 from the code has its nonzero coordinates in different blocks of coordinate positions. Since we can act to these blocks independently, we have that the cosets with minimum weight two of B (2) are in the same orbit. otherwise both vectors x and y would not be at distance 2 from the code.
Since GL(m, 2) is doubly transitive [11], there exists φ ∈ MAut(B (3) ) such that φ(e a i + e b j ) = e a k + e b ℓ . Now, consider θ ∈ S 3 such that θ(T a ) = T c and θ(T b ) = T d . We obtain that θ(φ(x)) = y. Therefore all cosets with minimum weight 2 are in the same orbit. Thus, B (3) is completely transitive since its covering radius is 2.
For q = 2, the supplementary codes of B (n−1) and B (n) have, respectively, parity check matrices H a (0) and H a (−1) defined in (3). We have seen that these codes are equivalent to B (3) and B (2) , respectively. From Theorem 2.4, we deduce that B (n−1) and B (n) are completely transitive codes. Proof. By Proposition 1 we know that B (3) is completely transitive for q = 2. We have to prove that it is not for q > 2.
The code B (3) is equivalent to the code C that has parity check matrix:
H b (3) = H O H O H H .
Consider the set of vectors
S = {αe 1 i + e 2 j | (i, α) = (j, 1), α ∈ F q \ {0}, i, j ∈ {1, . . . , n}}.
Clearly, any vector in S is at distance two from C. Indeed, if x = αe 1 i +e 2 j ∈ S, then H b (3)x T is a nonzero vector which is not a column of H b (3). Moreover, given two such vectors x = αe 1 i + e 2 j and y = α ′ e 1
i ′ + e 2 j ′ , where i = i ′ or j = j ′ or α = α ′ , we have that H b (3)(x − y) T is a nonzero vector, thus x − y / ∈ C.
Therefore, all vectors in S are in different cosets of C.
Compute the cardinality of S:
(i) Vectors of the form αe 1 i + e 2 i , (α = 1). There are n(q − 2) such vectors.
(ii) Vectors of the form αe 1 i + e 2 j , where i = j. There are n(q − 1)(n − 1) such vectors.
Hence,
|S| = n(q − 2) + n(q − 1)(n − 1) = n(q m − 2) = (q m − 1)(q m − 2) q − 1 .
As a consequence, we can find (q m − 1)(q m − 2)/(q − 1) vectors which are all in different cosets of minimum weight 2. Thus, if all such cosets are in the same orbit, we have that the monomial automorphisms in GL(q, m) of the form
g = g 1 O O g 1
acts 2-homogeneously on the first two blocks (indeed, for any vector αe 1 i +βe 2 j , we can consider a multiple α ′ e 1 i + e 2 j in the same coset). But this implies that the action of GL(q, m) in only one block is 2-homogeneous, since any automorphism like g acts identically in both blocks. Thus, we have q = 2, by Lemma 3.1. To determine the monomial automorphism group of B (r) in the general case seems to be a challenging problem. However, many computational results using magma [6], suggest that the order of the monomial automorphism group is not greater than 8n(q − 1), for r > 3 except when q = 2 and r ∈ {n − 1, n}. In these last two cases we have MAut(B (n−1) ) = MAut(B (3) ) = GL(m, 2) × S 3 and MAut(B (n) ) = MAut(B (2) ) = GL(m, 2) × GL(m, 2) × C 2 , respectively, and it is clear that in these two cases the order of MAut(B (r) ) is greater than 8n(q − 1). Assuming such upper bound we have that the codes not mentioned in Proposition 1 are not completely transitive. Proof. If B (r) is completely transitive, then MAut(B (r) ) must be transitive over the set of vectors of weight one since all cosets of minimum weight one must be in the same orbit. Hence | MAut(B (r) )| is a multiple of rn(q −1), say cn(q − 1). Moreover, cn(q − 1) ≥ rn(q − 1) since B (r) has rn(q − 1) cosets of minimum weight one. By Lemma 3.2, the number of cosets of minimum weight two is n(q − 1)(n(q − 1) + 2 − r). All such cosets must be in the same orbit and thus c ≥ n(q − 1) + 2 − r.
Corollary 1 Assume that | MAut(B (r) )| ≤ 8n(q − 1), for r > 3 except when q = 2 and r ∈ {n − 1, n}.
(i) If q = 2 and 4 ≤ r < n − 1, then B (r) is not completely transitive.
(ii) If q > 2 and 4 ≤ r ≤ n, then B (r) is not completely transitive.
Proof. From Proposition 2, we have r ≤ 8 and n(q − 1) + 2 − r ≤ 8.
Combining both inequalities, we obtain n(q − 1) ≤ 14. Thus, (ii) For q > 2, the codes B (r) and C (r) are completely transitive if r ∈ {1, 2}.
(iii) For q > 2, the codes B (3) and C (3) are not completely transitive.
Definition 1.1 ([12]) A code C of length n and covering radius ρ is completely regular, if for all l ≥ 0 every vector x ∈ C(l) has the same number c l of neighbors in C(l − 1) and the same number b l of neighbors in C(l + 1). Define a l = (q − 1)·n − b l − c l and set c 0 = b ρ = 0. The parameters a l , b l and c l (0 ≤ l ≤ ρ) are called intersection numbers and the sequence {b 0 , . . . , b ρ−1 ; c 1 , . . . , c ρ } is called the intersection array (shortly IA) of C.
) For any r, 2 ≤ r ≤ n, the code B (r) with parity check matrix H b (r) given in (1) is a completely regular [nr, nr − 2m, 3; 2] q -code with intersection array
Theorem 2. 4
4Let n = (q m − 1)/(q − 1) and let B (r) and C (r) be the supplementary codes as defined above.(i) B (1) is a Hamming [n, n − m, 3; 1] q -code and C (1) is a [q m n, q m n − 2m, 3, 2] q -code. Both codes are completely regular and completely transitive with intersection arraysIA(B (1) ) = {q m − 1; 1}; IA(C (1) ) = {q m (q m − 1), q m − 1; 1, q m (q m − 1)}. (ii) For 2 ≤ r ≤ n, B (r) is a [rn,rn − 2m, 3; 2] q -code and C (r) is a [(q m + 1 −r)n, (q m + 1 −r)n−2m, 3; 2] q -code. Both codes are completely regular with intersection arrays IA(B (r) ) = {r(q m − 1), (r − 1)(q m + 1 − r); 1, r(r − 1)}; IA(C (r) ) = {(q m + 1 − r)(q m − 1), r(q m − r); 1, (q m + 1 − r)(q m − r)}. Furthermore, B (r) is completely transitive if and only if C (r) is completely transitive.
. . . , e 2 n , . . . , . . . , e r 1 , . . . , e r n ;
Lemma 3. 1
1The general linear group GL(m, q) (m ≥ 3) is not 2-homogeneous, unless q = 2.
Lemma 3. 2
2Let n = (q m − 1)/(q − 1). For 1 < r ≤ n, let B (r) be the [nr, nr − 2m, 3; 2] q -code obtained by Construction I. Then, the number of cosets of B (r) with minimum weight 2 is q 2m − 1 − r(q m − 1).
Let B (r) be the [nr, nr−2m, 3; 2] q -code obtained by Construction I, where n = (q m − 1)/(q − 1). Then, (i) MAut(B (1) ) = GL(m, q).
( ii )
iiMAut(B (2) ) = GL(m, q) × GL(m, q) × C 2 . (iii) MAut(B (3) ) = GL(m, q) × S 3 .
a codeword of the Hamming code (of length n = (q m − 1)/(q − 1), i.e. it has the support only in the block T i , i = 1, 2, 3, and u is an arbitrary binary vector of length n). We deduce that all three blocks can be arbitrary permuted without changing of the code B(3) and, therefore, MAut(B (3) ) contains as a subgroup the group S 3 .
For q = 2
2, by Lemma 3.3, the automorphism group of B (3) is GL(m, 2) × S 3 , that clearly acts transitively over the coordinate positions. Hence, the cosets of B (3) with minimum weight one are in the same orbit. Consider two cosets of B (3) with minimum weight two, B (3) + x and B (3) + y, where x = e a i + e b j and y = e c k + e d ℓ . Note that i = j, k = ℓ, a = b, c = d;
Lemma 3. 4
4The code B (3) is completely transitive if and only if q = 2.
Remark 2
2According toLemma 3.2, the total number of cosets of minimum weight 2 is (q m − 1)(q m − 2). Note that for q = 2, the number of cosets induced by the set S coincides and since the action of GL(q, m) is doubly transitive, all the cosets of minimum weight 2 are in the same orbit. This is an alternative argument to see that B(3) is completely transitive in the binary case (Proposition 1).
Proposition 2
2If B (r) is completely transitive, then | MAut(B (r) )| = cn(q − 1), where c ≥ max{r, n(q − 1) + 2 − r}.
( i )
iIf q = 2, then m ≤ 3 (and n ≤ 7). Hence, the only possible cases are m = 3, r ∈ {4, 5}. (ii) If q > 2, then the only possible cases are q = 3, m = 2 (hence n = 4), r ∈ {3, 4}. For these remaining cases, (q, m, r) verified with magma that the corresponding codes are not completely transitive. From Proposition 1, Lemma 3.4, and Corollary 1 we can state the main result of this paper.
Theorem 3. 5
5Let B (r) be the code over F q of length rn = r(q m − 1)/(q − 1) (1 ≤ r ≤ n), obtained by Construction I. Let C (r) be the supplementary code of B (r) , such that the concatenation of the parity check matrices of B (r) and C (r) is H 2m , the parity check matrix of a Hamming code of length (q 2m − 1)/(q − 1). Then, (i) For q = 2, the codes B (r) and C (r) are completely transitive if r ∈ {1, 2, 3, n − 1, n}.
Consider a non-empty subset of n A < n columns of H m as a parity check matrix of a code A. Call B the code that has as parity check matrix the remaining n B = n − n A columns of H m . The code A is the supplementary code of B, and vice versa. We summarize the results of[5] in the following theorem. Let A and B be supplementary codes as defined above.(i) If A has covering radius 1, then it is a Hamming code. Both codes A and B are completely regular and completely transitive.(ii) If A has covering radius 2, dimension n A − m, and A is completely regular, then B is completely regular with covering radius at most 2.(iii) If A has covering radius 2, dimension n A − m, and A is completely transitive, then B is completely transitive.Theorem 2.3 ([5])
13, Lemma 2.1], implying q = 2.
For the remaining cases (not in (i), (ii) nor (iii)) the codes B (r) and. For the remaining cases (not in (i), (ii) nor (iii)) the codes B (r) and
C (r) are not completely transitive, assuming that in these cases the order of MAut(B (r) ). C (r) are not completely transitive, assuming that in these cases the order of MAut(B (r) )
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. J Koolen, D Krotov, B Martin, Completely regular codes. J. Koolen, D. Krotov and B. Martin, Completely regular codes, https://sites.google.com/site/completelyregularcodes.
The theory of error-correcting codes. F J Macwilliams, N J A Sloane, ElsevierF. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, Elsevier, 1977.
A Neumaier, Completely regular codes, Discrete mathematics. 106A. Neumaier, Completely regular codes, Discrete mathematics, 106 (1992), 353-360.
Algorithmic recognition of actions of 2-homogeneous groups on pairs. G R Sharp, LMS Journal of Computation and Mathematics. 1G. R. Sharp, Algorithmic recognition of actions of 2-homogeneous groups on pairs, LMS Journal of Computation and Mathematics, 1 (1998), 109--147.
P Solé, Completely regular codes and completely transitive codes. P. Solé, Completely regular codes and completely transitive codes, Discrete Mathematics, 81 (1990), 193-201.
Distance-regular graphs. E R Van Dam, J H Koolen, H Tanaka, The Journal of Combinatorics. 22E. R. van Dam, J. H. Koolen and H. Tanaka, Distance-regular graphs, The Journal of Combinatorics, (2016), DS22.
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"ON PROPERTY-pP 1 q AND SEMI-CONTINUITY PROPERTIES OF RESTRICTED CHEBYSHEV-CENTER MAPS IN ℓ 8 -DIRECT SUMS",
"ON PROPERTY-pP 1 q AND SEMI-CONTINUITY PROPERTIES OF RESTRICTED CHEBYSHEV-CENTER MAPS IN ℓ 8 -DIRECT SUMS"
] |
[
"Teena Thomas "
] |
[] |
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For a compact Hausdorff space S, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S, denoted by CpSq, satisfies property-pP1q (the setvalued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of CpSq. Various stability results related to property-pP1q and semi-continuity properties of restricted Chebyshevcenter maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c0 then the closed unit ball of Y satisfies property-pP1q for the non-empty closed bounded subsets of ℓ8 and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ8. We also investigate a variant of the transitivity property, similar to the one discussed in [C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl., 426(2): 1217-1231, 2015], for property-pP1q.2010 Mathematics Subject Classification. Primary 41A65; Secondary 46E15, 46A55, 46B20.Key words and phrases. property-pP1q, subalgebra, polyhedral Banach space, restricted Chebyshev-center map, lower and upper Hausdorff semi-continuity. 1 2 T. THOMAS to the set of restricted Chebyshev centers of F in V , is upper semi-continuous on F with respect to the Hausdorff metric.Throughout this paper, X denotes a real Banach space. We assume only norm closed and linear subspaces. For x P X and r ą 0, let B X rx, rs " ty P X : }x´y} ď ru and B X px, rq " ty P X : }x´y} ă ru. For simplicity, we denote B X r0, 1s by B X . The closed unit ball of a subspace Y of X is the set B Y :" B X X Y . Let CVpXq be the class of all non-empty closed convex subsets of X. For V P CVpXq, we denote the classes of all nonempty closed bounded, compact and finite subsets of V by CBpV q, KpV q and FpV q respectively. For x P X and a bounded subset A of X, we define dpx, Aq " inft}x´a} : a P Au and rpx, Aq " supt}x´a} : a P Au. For a set A Ď X and λ ą 0, let λA " tλa : a P Au.Let V P CVpXq and B P CBpXq. The restricted Chebyshev radius of B in V is defined as the number rad V pBq :" inf vPV rpv, Bq. The elements of the set cent V pBq :" tv P V : rad V pBq " rpv, Bqu are called the restricted Chebyshev centers of B in V . For each δ ą 0, we define cent V pB, δq " tv P V : rpv, F q ď rad V pF q`δu.Let F Ď CBpXq. The pair pV, F q satisfies the restricted center property pr.c.p.q if for each F P F , cent V pF q ‰ H. The set-valued map, cent V p.q, which maps each set F P F to the set cent V pF q, is called the restricted Chebyshev-center map of V on F . We refer to[15]for the terminologies defined above.In particular, let F be the class of all singleton subsets of X. If pV, F q has r.c.p. then V is said to be proximinal in X and the restricted Chebyshevcenter map of V on F , denoted by P V , is called the the metric projection from X onto V .We discuss the continuity properties of the metric projections and restricted Chebyshev-center maps with respect to the Hausdorff metric. We recall that for a Banach space X, the Hausdorff metric, denoted by d H , is defined as follows : for each A, B P CBpXq, d H pA, Bq " inftt ą 0 : A Ď B`tB X and B Ď A`tB X u.Let T be a topological space. Let Φ be a set-valued map from T into CBpXq. We say Φ is lower Hausdorff semi-continuous (l.H.s.c.) on T if for each t P T and ε ą 0, there exists a neighbourhood N pt, εq of t such that for each s P N pt, εq and z P Φptq, B X pz, εq X Φpsq ‰ H. The map Φ is upper Hausdorff semi-continuous (u.H.s.c.) on T if for each t P T
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2303.10676
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713f2da577eb7502ec21dfce3568caf7204e1d3f
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ON PROPERTY-pP 1 q AND SEMI-CONTINUITY PROPERTIES OF RESTRICTED CHEBYSHEV-CENTER MAPS IN ℓ 8 -DIRECT SUMS
19 Mar 2023
Teena Thomas
ON PROPERTY-pP 1 q AND SEMI-CONTINUITY PROPERTIES OF RESTRICTED CHEBYSHEV-CENTER MAPS IN ℓ 8 -DIRECT SUMS
19 Mar 2023
For a compact Hausdorff space S, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S, denoted by CpSq, satisfies property-pP1q (the setvalued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of CpSq. Various stability results related to property-pP1q and semi-continuity properties of restricted Chebyshevcenter maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c0 then the closed unit ball of Y satisfies property-pP1q for the non-empty closed bounded subsets of ℓ8 and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ8. We also investigate a variant of the transitivity property, similar to the one discussed in [C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl., 426(2): 1217-1231, 2015], for property-pP1q.2010 Mathematics Subject Classification. Primary 41A65; Secondary 46E15, 46A55, 46B20.Key words and phrases. property-pP1q, subalgebra, polyhedral Banach space, restricted Chebyshev-center map, lower and upper Hausdorff semi-continuity. 1 2 T. THOMAS to the set of restricted Chebyshev centers of F in V , is upper semi-continuous on F with respect to the Hausdorff metric.Throughout this paper, X denotes a real Banach space. We assume only norm closed and linear subspaces. For x P X and r ą 0, let B X rx, rs " ty P X : }x´y} ď ru and B X px, rq " ty P X : }x´y} ă ru. For simplicity, we denote B X r0, 1s by B X . The closed unit ball of a subspace Y of X is the set B Y :" B X X Y . Let CVpXq be the class of all non-empty closed convex subsets of X. For V P CVpXq, we denote the classes of all nonempty closed bounded, compact and finite subsets of V by CBpV q, KpV q and FpV q respectively. For x P X and a bounded subset A of X, we define dpx, Aq " inft}x´a} : a P Au and rpx, Aq " supt}x´a} : a P Au. For a set A Ď X and λ ą 0, let λA " tλa : a P Au.Let V P CVpXq and B P CBpXq. The restricted Chebyshev radius of B in V is defined as the number rad V pBq :" inf vPV rpv, Bq. The elements of the set cent V pBq :" tv P V : rad V pBq " rpv, Bqu are called the restricted Chebyshev centers of B in V . For each δ ą 0, we define cent V pB, δq " tv P V : rpv, F q ď rad V pF q`δu.Let F Ď CBpXq. The pair pV, F q satisfies the restricted center property pr.c.p.q if for each F P F , cent V pF q ‰ H. The set-valued map, cent V p.q, which maps each set F P F to the set cent V pF q, is called the restricted Chebyshev-center map of V on F . We refer to[15]for the terminologies defined above.In particular, let F be the class of all singleton subsets of X. If pV, F q has r.c.p. then V is said to be proximinal in X and the restricted Chebyshevcenter map of V on F , denoted by P V , is called the the metric projection from X onto V .We discuss the continuity properties of the metric projections and restricted Chebyshev-center maps with respect to the Hausdorff metric. We recall that for a Banach space X, the Hausdorff metric, denoted by d H , is defined as follows : for each A, B P CBpXq, d H pA, Bq " inftt ą 0 : A Ď B`tB X and B Ď A`tB X u.Let T be a topological space. Let Φ be a set-valued map from T into CBpXq. We say Φ is lower Hausdorff semi-continuous (l.H.s.c.) on T if for each t P T and ε ą 0, there exists a neighbourhood N pt, εq of t such that for each s P N pt, εq and z P Φptq, B X pz, εq X Φpsq ‰ H. The map Φ is upper Hausdorff semi-continuous (u.H.s.c.) on T if for each t P T
Introduction
In this article, we explore the (semi-)continuity properties of the restricted Chebyshev-center maps in certain classes of Banach spaces through the notion of property-pP 1 q, which is the set-valued analogue of strong proximinality. For a real Banach space X, a non-empty closed convex subset V of X and a family F of non-empty closed bounded subsets of X, property-pP 1 q is defined for a triplet pX, V, F q. The main motivation to study this property is Theorem 1.1 ([14, Theorem 5]). If the triplet pX, V, F q has property-pP 1 q then the restricted Chebyshev-center map of V , which maps every set F P F and ε ą 0, there exists a neighbourhood N pt, εq of t such that for each s P N pt, εq, Φpsq Ď Φptq`εB X . The map Φ is Hausdorff metric continuous on T if for each t P T , the single-valued map Φ from T into the metric space pCBpXq, d H q is continuous at t P T . By [8,Remark 2.8], Φ is Hausdorff metric continuous on T if and only if Φ is both l.H.s.c. and u.H.s.c. on T .
In this article, we consider the following definition of property-pP 1 q which is equivalent to that introduced by J. Mach in [14]. . Let X be a Banach space, V P CVpXq and F Ď CBpXq such that pV, F q has r.c.p.. Then the triplet pX, V, F q has property-pP 1 q if for each ε ą 0 and F P F , there exists δpε, F q ą 0 such that cent V pF, δq Ď cent V pF q`εB X . Remark 1.3. With the notations as above, let pV, F q have r.c.p..
piq If F is the class of all singleton subsets of X then we say V is strongly proximinal in X (see [6]). piiq It is easy to verify that pX, V, F q has property-pP 1 q if and only if for each ǫ ą 0 and F P F , there exists δ ą 0 such that SpF, δq :" suptdpv, cent V pF qq : v P V and rpv, F q ď rad V pF q`δu ă ε.
We now recall a notion in Banach spaces which is stronger than strong proximinality. A subspace J of a Banach space X is an M -ideal in X if there exists a linear projection P : X˚Ñ X˚such that for each x˚P X˚, }x˚} " }P x˚}`}x˚´P x˚} and the range of P is the annihilator of J in X˚. A particular case of M -ideals are M -summands. A subspace J of a Banach space X is an M -summand in X if there exists a linear projection P : X Ñ X such that for each x P X, }x} " maxt}P x}, }x´P x}u and the range of P is J. We refer to [7] for a detailed study on M -ideals. Mideals are strongly proximinal; see [7] and [3,Proposition 3.3]. However, the example provided by Veselý in [20] shows that in general, an M -ideal in a Banach space may not admit Chebyshev centers for its closed bounded subsets.
For a topological space T and a Banach space X, C b pT, Xq denote the Banach space of X-valued bounded continuous functions on T . If T is a compact Hausdorff space and X " R then C b pT, Xq is simply denoted as CpT q.
In [5], Godefroy and Indumathi proved that for a subspace X of c 0 and a finite co-dimensional linear subspace Y of X, if every hyperplane of X containing Y is proximinal in X then Y is proximinal in X. Later, Indumathi proved the following result in [8,Theorem 4.1].
Theorem 1.4. Let Y be a proximinal finite co-dimensional linear subspace of c 0 . Then Y is proximinal in ℓ 8 and the metric projection P Y from ℓ 8 onto Y is Hausdorff metric continuous.
Jayanarayanan and Lalithambigai improved Theorem 1.4 in [9, Corollary 3.7]. They proved that if Y is a proximinal finite co-dimensional subspace of c 0 then B Y is strongly proximinal in ℓ 8 .
The present paper generalizes Theorem 1.4 and proves that the triplet pℓ 8 , B Y , CBpℓ 8 qq satisfies property-pP 1 q and the map cent B Y p.q Hausdorff metric continuous on CBpℓ 8 q whenever Y is a proximinal finite codimensional subspace of c 0 in Sections 3 and 4 respectively. To this end, we recall Theorem 1.5 ([2, Theorem 2]). Let T be a topological space and X be a uniformly convex Banach space. Then for each B P CBpC b pT, Xqq, cent C b pT,Xq pBq ‰ H and the map cent C b pT,Xq p.q is uniformly Hausdorff continuous on CBpC b pT, Xqq.
In Section 2, we improve Theorem 1.5 and obtain a few consequences which are of interest. We prove that for a compact Hausdorff space S and a uniformly convex Banach space X, pC b pT, Xq, B C b pT,Xq , CBpC b pT, Xqqq has property-pP 1 q and the map cent B C b pT,Xq p.q is uniformly Hausdorff metric continuous on CBpC b pT, Xqq. Consequently, we obtain that pℓ 8 , B c 0 , CBpℓ 8 qq has property-pP 1 q which generalizes [9, Theorem 3.1]. Moreover, we prove that for a compact Hausdorff space S and a closed linear subalgebra A of CpSq, the triplet pCpSq˚˚, B A , CBpCpSq˚˚qq satisfies property-pP 1 q.
In Section 3, apart from the results mentioned earlier, we also establish the stability of r.c.p. and property-pP 1 q in the ℓ 8 -direct sums. Given two subspaces Y 1 and Y 2 of Banach spaces X 1 and X 2 , let X be the ℓ 8 -direct sum of X 1 and X 2 and Y be that of Y 1 and Y 2 . We prove that pX, Y, CBpXqq has property-pP 1 q if and only if for each i " 1, 2, pX i , Y i , CBpX i qq has property-pP 1 q.
In Section 4, we derive the stability of the lower and upper semi-continuity properties of the restricted Chebyshev-center maps of ℓ 8 -direct sums under certain assumptions. It is proved in [18] that for a subspace Y of a Banach space X, if pX, B Y , CBpXqq has property-pP 1 q then so does pX, Y, CBpXqq. In this section, we establish that for a subspace Y of a Banach space X, if pB Y , CBpXqq has r.c.p. then the Hausdorff metric continuity of the map cent B Y p.q on CBpXq implies that of the map cent Y p.q on CBpXq.
In [10], the authors proved a variation of the transitivity problem for strong proximinality. Theorem 1.6 ([10, Theorem 3.13]). Let X be a Banach space, J be an M -ideal in X and Y be a finite co-dimensional subspace of X such that
Y Ď J Ď X. If Y is strongly proximinal in J, then Y is strongly proximinal in X.
We pose a similar problem for property-pP 1 q as follows : Question 1.7. Let X be a Banach space, J be an M -ideal in X and Y be a finite co-dimensional subspace of X such that Y Ď J. If the triplets pJ, Y, CBpJqq (or pJ, Y, KpJqq or pJ, Y, FpJqq respectively) and pX, J, CBpXqq (or pX, J, KpXqq or pX, J, FpXqq respectively) have property-pP 1 q, then does pX, Y, CBpXqq (or pX, Y, KpXqq or pX, Y, FpXqq respectively) have property-pP 1 q?
We do not know the answer to Question 1.7 in its entirety. Nevertheless, we provide positive results for the following two particular cases in Section 5 : We prove that for a Banach space X, an M -summand Y in X and a subspace Z of Y , if pY, Z, CBpY qq has property-pP 1 q then so does pX, Z, CBpXqq. Further, let us recall that a Banach space X is said to be an L 1 -predual space if X˚is isometric to an L 1 pµq space for some positive measure µ. We refer to [13] for a detailed study on L 1 -preduals. We prove that for an L 1 -predual space X, an M -ideal J in X and a finite co-dimensional subspace Y of X such that Y Ď J Ď X, if Y is strongly proximinal in J, then pX, Y, KpXqq has property-pP 1 q.
Property-pP 1 q in vector-valued continuous function spaces
In this section, we discuss property-pP 1 q of the closed unit ball of the space C b pT, Xq whenever T is a topological space and X is a uniformly convex Banach space. To this end, we first recall the following characterization of a uniformly convex Banach space. . Let X be a Banach space. Then X is uniformly convex if and only if for each ε ą 0, there exists δ 1 pεq ą 0 such that if x, y P X and Φ P X˚such that }x} " }y} " 1 " }Φ} " Φpyq and Φpxq ą 1´δ 1 pεq then }x´y} ă ε. We can choose δ 1 pεq ď ε 2 .
We also recall the following estimation. The following result is obtained through some modifications in the proof of [2,Theorem 2]. For the sake of thoroughness, we present the modifications in its proof here. Theorem 2.3. Let T be a topological space and X be a uniformly convex Banach space. Then the triplet pC b pT, Xq, B C b pT,Xq , CBpC b pT, Xqqq satisfies property-pP 1 q and the map cent B C b pT,Xq p.q is uniformly Hausdorff metric continuous on CBpC b pT, Xqq.
Proof. Let B P CBpC b pT, Xqq. We define R " rad B C b pT,Xq pBq. Let us assume R " 1 and fix ε ą 0. Then we obtain δ 1 pεq ą 0 satisfying the condition in Lemma 2.1. Now there exists f 0 P B C b pT,Xq such that rpf 0 , Bq ď 1`δ 1 pεq. We claim that there exists f 1 P B C b pT,Xq such that rpf 1 , Bq ď 1`δ 1 pε{2q and }f 0´f1 } ď 2ε. Indeed, there exists g P B C b pT,Xq such that rpg, Bq ď 1`δ 1 pε{2q. We now define α : T Ñ r0, 1s and f 1 : T Ñ X as follows : for each t P T ,
αptq " $ ' & ' % 1, if }gptq´f 0 ptq} ď 2ε; 2ε }gptq´f 0 ptq}
, if }gptq´f 0 ptq} ą 2ε. and f 1 ptq " f 0 ptq`αptqpgptq´f 0 ptqq.
Clearly, f 1 P B C b pT,Xq and }f 1´f0 } ď 2ε. For each b P B, we apply the same arguments as in the proof of [2, Theorem 2] to show that for each t P T ,
}f 1 ptq´bptq} ď 1`δ 1 pε{2q(1)
and hence rpf 1 , Bq ď 1`δ 1 pε{2q. We now proceed inductively to find a sequence tf n u Ď B C b pT,Xq such that for each n " 1, 2, . . ., }f n`1´fn } ď 2 ε 2 n and rpf n`1 , Bq ď 1δ 1 pε{2 n`1 q. Since tf n u is Cauchy, there exists f P B C b pT,Xq such that lim nÑ8 f n " f . Hence }f´f 0 } ď 4ε and rpf, Bq ď lim nÑ8 rpf n , Bq ď 1. Thus f P cent B C b pT,Xq pBq. It also follows that cent B C b pT,Xq pB, δ 1 pεqq Ď cent B C b pT,Xq pBq`4εB X . Hence pC b pT, Xq, B C b pT,Xq , tBuq has property-pP 1 q.
Now assume 0 ă R ‰ 1. Then inf f PB C b pT,Xq r´f R , 1 R B¯" 1. In the argument above, if f 0 , g are chosen in 1 R B C b pT,Xq then f 1 P 1 R B C b pT,Xq . Hence replacing B C b pT,Xq and B by 1 R B C b pT,Xq and 1 R B respectively in the argument above, we can conclude that pC b pT, Xq, 1 R B C b pT,Xq , t 1 R Buq has property-pP 1 q. Thus it follows from [18, Proposition 2.3 (ii)] that pC b pT, Xq, B C b pT,Xq , tBuq has property-pP 1 q.
In order to show that the map cent B C b pT,Xq p.q is uniformly Hausdorff continuous on CBpC b pT, Xqq, let us fix ε, R ą 0. We now obtain a δ 1 pεq ą 0 satisfying the condition in Lemma 2.
1. Choose 0 ă δ ă Rδ 1 pεq 2 . Let A, B P CBpC b pT, Xqq such that rad B C b pT,Xq pBq, rad B C b pT,Xq pAq ă R and d H pB, Aq ă δ. Then by Lemma 2.2, |rad B C b pT,Xq pBq´rad B C b pT,Xq pAq| ă δ. Let f P cent B C b pT,Xq pAq. Thus rpf, Bq ă rad B C b pT,Xq pAq`δ ă rad B C b pT,Xq pBq`2δ ă p1`δ 1 pεqqR. Now using the arguments above, we obtain f 0 P cent B C b pT,Xq pBq such that }f´f 0 } ď 4εR. Similarly, cent B C b pT,Xq pBq Ď cent B C b pT,Xq pAq`4εRB C b pT,Xq . Thus d H pcent B C b pT,Xq pBqq, cent B C b pT,Xq pAqq ď 4εR.
Corollary 2.4. Let S be a compact Hausdorff space and A be a closed linear subspace of CpSq described as follows :
A " tf P CpSq : f pt i q " λ i f ps i q, for each i P Iu, for some index I and co-ordinates pt i , s i , λ i q P SˆSˆt´1, 0, 1u for each i P I. Then the triplet pCpSq, B A , CBpCpSqqq satisfies property-pP 1 q and the map cent B A p.q is uniformly Hausdorff metric continuous on CBpCpSqq.
Proof. In the proof of Theorem 2.3, if we choose f 0 and g in B A then clearly }f 1 } ď 1 and from the description of A, f 1 P B A . Hence the result follows.
By using the representation given in [17, Theorem 2.2] of closed linear subalgebras of CpSq and applying Corollary 2.4, we obtain the following result.
Corollary 2.5. Let S be a compact Hausdorff space and A be a closed linear subalgebra of CpSq. Then the triplet pCpSq, B A , CBpCpSqqq satisfies property-pP 1 q and the map cent B A p.q is uniformly Hausdorff metric continuous on CBpCpSqq.
For a compact Hausdorff space S, it is known that CpSq˚˚is a CpΩq space, for some compact Hausdorff space Ω (see [11]) and CpSq is a closed linear subalgebra of CpΩq under the canonical embedding (see [16]). Therefore, as a direct consequence of Corollary 2.5, we have Corollary 2.6. Let S be a compact Hausdorff space and A be a closed linear subalgebra of CpSq. Then the triplet pCpSq˚˚, B A , CBpCpSq˚˚qq satisfies property-pP 1 q and the map cent B A p.q is uniformly Hausdorff metric continuous on CBpCpSq˚˚q.
The triplet pℓ 8 , B c 0 , Kpℓ 8 qq has property-pP 1 q, and it follows from the well-known fact that c 0 is an M -ideal in ℓ 8 and [18, Theorem 3.2]. The next result follows from the fact that c 0 is a subalgebra in ℓ 8 -CpβNq (here βN is the Stone-Čech compactification of the natural numbers N) and Corollary 2.5.
Corollary 2.7. The triplet pℓ 8 , B c 0 , CBpℓ 8 qq has property-pP 1 q and the map cent Bc 0 p.q is uniformly Hausdorff metric continuous on CBpℓ 8 q.
3. Stability of property-pP 1 q in ℓ 8 -direct sums
We first establish some notations which are used in the present and subsequent sections. Let X 1 and X 2 be two Banach spaces. Then the ℓ 8 -direct sum of X 1 and X 2 ,
X :" X 1 ' 8 X 2 " tpx 1 , x 2 q P X 1ˆX2 : x 1 P X 1 and x 2 P X 2 u, is a Banach space equipped with the maximum norm. For i P t1, 2u, if V i P CVpX i q then V 1ˆV2 P CVpXq. For each B P CBpXq, let Bp1q " tb 1 P X 1 : there exists b P B and b 2 P X 2 such that b " pb 1 , b 2 qu and Bp2q " tb 2 P X 2 : there exists b P B and b 1 P X 1 such that b " pb 1 , b 2 qu.(2)
Clearly, for each i P t1, 2u, Bpiq P CBpX i q and B Ď Bp1qˆBp2q. Moreover, for each B P CBpXq and i P t1, 2u, let r i pBq " rad V i pBpiqq.
The following result gives a formula for the restricted Chebyshev radius of a closed bounded subset of an ℓ 8 -direct sum.
Proposition 3.1. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q. Let X " X 1 ' 8 X 2 , V " V 1ˆV2 and B P CBpXq.
Then rad V pBq " maxtr 1 pBq, r 2 pBqu and for each v " pv 1 , v 2 q P V , rpv, Bq " maxtrpv 1 , Bp1qq, rpv 2 , Bp2qqu.
Proof. Let v " pv 1 , v 2 q P V and for i P t1, 2u, b i P Bpiq. Then there exists b, b 1 P B and b 1 i P X i for i P t1, 2u such that b " pb 1 , b 1 2 q and b 1 " pb 1 1 , b 2 q.
Thus }b 1´v1 } ď }b´v} ď rpv, Bq and }b 2´v2 } ď }b 1´v } ď rpv, Bq. It follows that for i P t1, 2u, rpv i , Bpiqq ď rpv, Bq. It is now easy to conclude that maxtr 1 pBq, r 2 pBqu ď rad V pBq.
Conversely, for each ε ą 0 and i P t1, 2u, there
exists v i P V i such that rpv i , Bpiqq ă r i pBq`ε. Let v " pv 1 , v 2 q P V and b " pb 1 , b 2 q P B. Then for i P t1, 2u, b i P Bpiq. Hence }v´b} " maxt}v 1´b1 }, }v 2´b2 }u ă maxtr 1 pBq, r 2 pBqu`ε.
It follows that rad V pBq ď maxtr 1 pBq, r 2 pBqu`ε. Since ε ą 0 is arbitrary, we obtain the desired equality.
It follows from the arguments above that for each v " pv 1 , v 2 q P V , rpv, Bq " maxtrpv 1 , Bp1qq, rpv 2 , Bp2qqu.
We now establish the stability of r.c.p. under the ℓ 8 -direct sum.
Proposition 3.2. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 .
If for each i P t1, 2u, pV i , CBpX i qq has r.c.p. then pV, CBpXqq has r.c.p..
Proof. Let B P CBpXq. By our assumption, for i P t1, 2u, let v i P cent V i pBpiqq. Then it follows from Proposition 3.1 that rad V pBq " rppv 1 , v 2 q, Bq. Remark 3.3. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q. For each i P t1, 2u, suppose pV i , CBpX i qq has r.c.p.. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 . It is easy to verify the following facts.
piq For each B P CBpXq,
cent V pBq " $ ' ' ' ' ' ' & ' ' ' ' ' ' % cent V 1 pBp1qqˆcent V 2 pBp2qq, if r 1 pBq " r 2 pBq; č b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1ˆc ent V 2 pBp2qq, if r 1 pBq ă r 2 pBq; cent V 1 pBp1qqˆč b 2 PBp2q B X 2 rb 2 , r 1 pBqs X V 2 , if r 2 pBq ă r 1 pBq.
We note that in each case above,
cent V pBq Ě cent V 1 pBp1qqˆcent V 2 pBp2qq.
piiq For each B, A P CBpXq,
maxtd H pBp1q, Ap1qq, d H pBp2q, Ap2qqu ď d H pB, Aq.
Next, we prove a technical result. It is crucial to this study and is a generalization of [8,Fact 3.2].
Lemma 3.4. Let X be a Banach space, V P CVpXq and F Ď CBpXq such that pV, F q has r.c.p.. Let F P F and α ą rad V pF q. Then for each ε ą 0, there exists δ ą 0 such that for each F 1 P F with d H pF, F 1 q ă δ and scalar β such that |α´β| ă δ, we have
d H˜č zPF Brz, αs X V, č z 1 PF 1 Brz 1 , βs X V¸ă ε.
Proof. Let us define R F " rad V pF q and fix ε ą 0. Further, define 2γ " α´R F , L " α`R F`2 and δ " mint1, γ 2 , γε 2L u. Let F 1 P F be such that d H pF, F 1 q ă δ and β be a scalar such that |α´β| ă δ. For simplicity, let R F 1 " rad V pF 1 q. Then from Lemma 2.2,
|R F´RF 1 | ă δ. Moreover, β´R F 1 " β´α`R F´RF 1`α´R F ą 2γ´2δ ě 2γ´γ " γ. Now, let v P Ş zPF Brz, αs X V and z 1 P F 1 . Then there exists z P F such that }z´z 1 } ă δ. Thus }z 1´v } ď }z 1´z }`}z´v} ă δ`α ă β`2δ. Hence for each z 1 P F 1 , }z 1´v } ă β`2δ. Let v 0 P cent V pF 1 q. Define v 1 " λv`p1´λqv 0 where λ " β´R F 1 β´R F 1`2δ .
Then v 1 P V and for each z 1 P F 1 ,
}z 1´v1 } ď λ}z 1´v }`p1´λq}z 1´v 0 } ă λpβ`2δq`p1´λqR F 1 " λpβ´R F 1`2δq`R F 1 " β.
It follows that v 1 P Ş z 1 PF 1 Brz 1 , βs X V . Further, let z 1 P F 1 . Since d H pF, F 1 q ă δ, there exists z P F such that }z´z 1 } ă δ. Now,
}v´v 1 } " p1´λq}v´v 0 } " 2δ β´R F 1`2δ }v´v 0 } ă 2δ γ p}v´z}`}z´z 1 }`}z 1´v 0 }q ă 2δ γ pα`δ`rpv 0 , F 1 qq " 2δ γ pα`δ`R F 1 q ă 2δ γ pα`R F`2 δq ă 2δL γ ď ε.(3)
Therefore, v 1 P Ş z 1 PF 1 Brz 1 , βs X V such that }v´v 1 } ă ε. Similarly, for each w 1 P Ş z 1 PF 1 Brz 1 , βsXV , we obtain w P Ş zPF Brz, αsXV such that }w´w 1 } ă ε. This completes the proof.
The next result proves the stability of property-pP 1 q under ℓ 8 -direct sums.
Theorem 3.5. For i P t1, 2u, let X i be a Banach space and V i P CVpX i q. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 . If for i P t1, 2u, pX i , V i , CBpX i qq has property-pP 1 q then pX, V, CBpXqq has property-pP 1 q.
Proof. Assume that for i P t1, 2u, pX i , V i , CBpX i qq has property-pP 1 q. By Proposition 3.2, pV, CBpXqq has r.c.p.. Let B P CBpXq and ε ą 0. Then there exists δ ą 0 such that for i P t1, 2u,
cent V i pBpiq, δq Ď cent V i pBpiqq`εB X i .(4)
Case 1 : r 1 pBq " r 2 pBq By Remark 3.3 (i), cent V pBq " cent V 1 pBp1qqˆcent V 2 pBp2qq. Similarly, for each γ ą 0, we also have cent V pB, γq " cent V 1 pBp1q, γqˆcent V 2 pBp2q, γq. Therefore, it follows from p4q that cent V pB, δq Ď cent V pBq`εB X .
Case 2 : r 1 pBq ‰ r 2 pBq Without loss of generality, assume r 1 pBq ă r 2 pBq, since the same arguments work for the reverse inequality. By Remark 3.
3 (i), cent V pBq " Ş b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1ˆc ent V 2 pBp2qq. Similarly, for each γ ą 0, cent V pB, γq " č b 1 PBp1q B X 1 rb 1 , r 2 pBq`γs X V 1ˆc ent V 2 pBp2q, γq.
Let ε ą 0. Now replacing X, V and B by X 1 , V 1 and Bp1q respectively in Lemma 3.4, we obtain δ 1 ą 0 such that for each A P CBpXq with d H pB, Aq ă 2δ 1 and a scalar β with |α´β| ă 2δ 1 , we have
d H¨č b 1 PBp1q B X 1 rb 1 , αs X V 1 , č a 1 PAp1q B X 1 ra 1 , βs X V 1‚ ă ε.
Choose δ 0 " mintδ, δ 1 u. Let us take A " B, α " r 2 pBq and β " r 2 pBq`δ 0 . Then we have
d H¨č b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 , č b 1 PBp1q B X 1 rb 1 , r 2 pBq`δ 0 s X V 1‚ ă ε. (5)
Let v " pv 1 , v 2 q P cent V pB, δ 0 q such that v 1 P Ş b 1 PBp1q B X 1 rb 1 , r 2 pBqδ 0 s X V 1 and v 2 P cent V 2 pBp2q, δ 0 q. Then by p4q and p5q, there exists w 1 P Ş b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 and w 2 P cent V 2 pBp2qq such that for each i P t1, 2u, }w i´vi } ď ε. Hence pw 1 , w 2 q P cent V pBq and }v´pw 1 , w 2 q} ď ε.
One of the instances where the converses of Proposition 3.2 and Theorem 3.5 hold true is as follows: Proposition 3.6. For each i P t1, 2u, let X i be a Banach space and Y i be a closed non-trivial subspace of X i . Let X " X 1 ' 8 X 2 and Y " Y 1 ' 8 Y 2 . Then piq If pY, CBpXqq has r.c.p. then for each i P t1, 2u, pY i , CBpX i qq has r.c.p.. piiq If pX, Y, CBpXqq has property-pP 1 q then for each i P t1, 2u, pX i , Y i , CBpX i qq has property-pP 1 q Proof. piq : Without loss of generality, we only prove that pY 1 , pCBqpX 1 qq has r.c.p.. Let Let B 1 P CBpX 1 q. We can choose B 2 P CBpX 2 q such that rad Y 2 pB 2 q " rad Y 1 pB 1 q. Let B " B 1ˆB2 P CBpXq. Then Bp1q " B 1 and Bp2q " B 2 and hence r 1 pBq " r 2 pBq. By Remark 3.3 (i), cent Y pBq " cent Y 1 pB 1 qˆcent Y 2 pB 2 q. Thus by our assumption, it follows that cent Y 1 pB 1 q ‰ H.
piiq : Without loss of generality, we only prove that pX 1 , Y 1 , pCBqpX 1 qq has property-pP 1 q. By Proposition 3.2, pY 1 , CBpX 1 qq has r.c.p.. Let B 1 P CBpX 1 q. We choose B 2 P CBpX 2 q such that rad Y 2 pB 2 q " rad Y 1 pB 1 q. Define B " B 1ˆB2 P CBpXq. Then Bp1q " B 1 and Bp2q " B 2 and hence r 1 pBq " r 2 pBq. Let ε ą 0. Then there exists δ ą 0 such that cent Y pB, δq Ď cent Y pBq`εB X . By Remark 3.3 (i), cent Y pBq " cent Y 1 pB 1 qˆcent Y 2 pB 2 q. Similarly, cent Y pB, δq " cent Y 1 pB 1 , δqˆcent Y 2 pB 2 , δq. Let y 1 P cent Y 1 pB 1 , δq and y 2 P cent Y 2 pB 2 , δq. Hence py 1 , y 2 q P cent Y pB, δq. Then there exists y 1 1 P cent Y 1 pB 1 q and y 1 2 P cent Y 2 pB 2 q such that }py 1 , y 2 q´py 1 1 , y 1 2 q} ď ε. Thus }y 1´y
1 1 } ď ε.
An application of the above stability results is the following result. Proof. Let Y be a proximinal finite co-dimensional subspace of c 0 . The following decomposition is well-known: the spaces ℓ 8 and Y are isometrically isomorphic to X ' 8 ℓ 8 and Z ' 8 c 0 respectively for some finite dimensional spaces X and Z with Z Ď X Ď c 0 (see [8,Section 4] for a detailed explanation). Clearly, B Y " B ZˆBc 0 . Using a compactness argument, we easily observe that pX, B Z , CBpXqq has property-pP 1 q. Therefore by Corollary 2.7 and Theorem 3.5, we conclude that pℓ 8 , B Y , CBpℓ 8 qq has property-pP 1 q.
Semi-continuity of restricted Chebyshev-center maps of ℓ 8 -direct sums
In this section, we first derive few stability results concerning the semicontinuity properties of restricted Chebyshev-center maps of ℓ 8 -direct sums.
Proposition 4.1. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q such that pV i , CBpX i qq has r.c.p.. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 . If for each i P t1, 2u, cent V i p.q is l.H.s.c. on CBpX i q then the map cent V p.q is l.H.s.c. on CBpXq.
Proof. By Proposition 3.2, pV, CBpXqq has r.c.p.. Let B P CBpXq and ε ą 0. Using lower Hausdorff semi-continuity of the maps cent V 1 p.q and cent V 2 p.q at Bp1q and Bp2q respectively, there exists δ ą 0 such that for i P t1, 2u, if A P CBpXq with d H pB, Aq ă δ and ρ i P cent V i pBpiqq then
B X i pρ i , εq X cent V i pApiqq ‰ H.(6)
Case 1 : r 1 pBq " r 2 pBq By Remark 3.3 (i), cent V pBq " cent V 1 pBp1qqˆcent V 2 pBp2qq. Let i P t1, 2u. Let ρ i P cent V i pBpiqq and A P CBpXq such that d H pB, Aq ă δ. Then by Remark 3.3 (ii), d H pBpiq, Apiqq ă δ. Hence by p6q, there exists v i P B X i pρ i , εqX cent V i pApiqq. By Remark 3.3 (i), pρ 1 , ρ 2 q P cent V pBq and pv 1 , v 2 q P cent V pAq. Moreover, }pρ 1 , ρ 2 q´pv 1 , v 2 q} ă ε. Hence B X ppρ 1 , ρ 2 q, εq X cent V pAq ‰ H. Thus cent V p.q is l.H.s.c. at B.
Case 2 : r 1 pBq ‰ r 2 pBq Without loss of generality, assume r 1 pBq ă r 2 pBq, since the same arguments work for the reverse inequality. Let 2γ " r 2 pBq´r 1 pBq. Replacing X, V , B and α by X 1 , V 1 , Bp1q and r 2 pBq respectively in Lemma 3.4, we obtain 0 ă δ ă γ 2 such that whenever A P CBpXq with d H pB, Aq ă δ, we have r 2 pAq´r 1 pAq ą γ and
d H¨č b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 , č a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1‚ ă ε.(7)
Without loss of generality, assume that δ is so chosen that p6q is also satisfied. Let A P CBpXq with d H pB, Aq ă δ. Then r 2 pAq ą r 1 pAq and hence by Remark 3.3 (i), cent V pAq " Ş a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1ˆc ent V 2 pAp2qq. Let v 1 P Ş b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 and v 2 P cent V 2 pBp2qq and hence by Remark 3.3 (i), v " pv 1 , v 2 q P cent V pBq. Therefore, by (6) and (7), there exists w 1 P Ş a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1 and w 2 P cent V 2 pAp2qq such that }v 1´w1 } ă ε and }v 2´w2 } ă ε. Thus pw 1 , w 2 q P B X pv, εq X cent V pAq. Proposition 4.2. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 . If for each i P t1, 2u, pX i , V i , CBpX i qq has property-pP 1 q then the map cent V p.q is u.H.s.c. on CBpXq.
Proof. By Proposition 3.2, pV, CBpXqq has r.c.p.. By Theorem 1.1, for each i P t1, 2u, cent V i p.q is u.H.s.c. on CBpX i q. Let B P CBpXq and ε ą 0. Then there exists δ ą 0 such that for each i P t1, 2u, if A P CBpXq with
d H pB, Aq ă δ ñ cent V i pApiqq Ď cent V i pBpiqq`εB X i .(8)
Case 1 : r 1 pBq " r 2 pBq By Remark 3.3 (i), cent V pBq " cent V 1 pBp1qqˆcent V 2 pBp2qq. By our assumption and Remark 1.3 (ii), we can choose θ ą 0 such that for i P t1, 2u, SpBpiq, θq ă ε. We further choose 0 ă δ ă θ 4 such that p8q holds valid. Let A P CBpXq such that d H pB, Aq ă δ. If r 1 pAq " r 2 pAq then cent V pAq " cent V 1 pAp1qqˆcent V 2 pAp2qq. Thus by p8q, we obtain cent V pAq Ď cent V pBq`εB X . Now without loss of generality, we assume that r 1 pAq ă r 2 pAq. For i P t1, 2u, since
|r i pBq´r i pAq| ď d H pBpiq, Apiqq ď d H pB, Aq ă θ 4 ,
we have |r 1 pBq´r 2 pAq| ď |r 1 pBq´r 2 pBq|`|r 2 pBq´r 2 pAq| ă θ 4 .
Now, by Remark 3.3 (i),
cent V pAq " č a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1ˆc ent V 2 pAp2qq.
Let v 1 P Ş a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1 . Then using p9q and the assumption that d H pB, Aq ă δ, for each b 1 P Bp1q, there exists a 1 P Ap1q such that }a 1´b1 } ă θ 4 and hence }v 1´b1 } ď }v 1´a1 }`}a 1´b1 } ă r 2 pAq`θ 4 ă r 1 pBq`θ 2 .
It follows that rpv 1 , Bp1qq ď r 1 pBq`θ 2 . Now, since SpBp1q, θq ă ε, dpv 1 , cent V 1 pBp1qqq ă ε. Thus there exists w 1 P cent V 1 pBp1qq satisfying }v 1´w1 } ă ε. It follows that č a 1 PAp1q
B X 1 ra 1 , r 2 pAqs X V 1 Ď cent V 1 pBp1qq`εB X 1 . Moreover, cent V 2 pAp2qq Ď cent V 2 pBp2qq`εB X 2 . Hence, cent V pAq Ď cent V pBq`εB X .
Case 2 : r 1 pBq ‰ r 2 pBq We discuss only r 1 pBq ă r 2 pBq since the same arguments work for the reverse inequality. Let 2γ " r 2 pBq´r 1 pBq. Replacing X, V , B and α by X 1 , V 1 , Bp1q and r 2 pBq respectively in Lemma 3.4, we obtain 0 ă δ ă γ 2 such that whenever A P CBpXq with d H pB, Aq ă δ, we have r 2 pAq´r 1 pAq ą γ and
d H¨č b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 , č a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1‚ ă ε.(10)
We choose δ ą 0 such that p8q is also satisfied. If A P CBpXq such that d H pB, Aq ă δ then r 2 pAq ą r 1 pAq and hence by Remark 3.3 (i),
cent V pAq " č a 1 PAp1q B X 1 ra 1 , r 2 pAqs X V 1ˆc ent V 2 pAp2qq. Let A P CBpXq such that d H pB, Aq ă δ. If v 1 P Ş a 1
PAp1q B X 1 ra 1 , r 2 pAqs X V 1 and v 2 P cent V 2 pAp2qq then by p8q and p10q, we choose w 1 P Ş b 1 PBp1q B X 1 rb 1 , r 2 pBqs X V 1 and w 2 P cent V 2 pBp2qq satisfying }v i´wi } ă ε for i P t1, 2u. Thus pw 1 , w 2 q P cent V pBq and }pv 1 , v 2 q´pw 1 , w 2 q} ă ε. It follows that cent V pAq Ď cent V pBq`εB X .
The following result follows from Propositions 4.1 and 4.2 and [8, Remark 2.8].
Proposition 4.3. For each i P t1, 2u, let X i be a Banach space and V i P CVpX i q such that pX i , V i , CBpX i qq has property-pP 1 q. Let X " X 1 ' 8 X 2 and V " V 1ˆV2 . If for each i P t1, 2u, cent V i p.q is Hausdorff metric continuous on CBpX i q then cent V p.q is Hausdorff metric continuous on CBpXq.
We note here that the stability results proved in Sections 3 and 4 are similarly valid if we replace the class of all non-empty closed bounded subsets with the class of all non-empty compact or finite subsets of the respective spaces.
We recall that a finite dimensional Banach space X is called polyhedral if B X has only finitely many extreme points. An infinite dimensional Banach space X is called polyhedral if each of the finite dimensional subspace of X is polyhedral. The space c 0 is a well-known example of an infinite dimensional polyhedral space. We refer to [4] and the references therein for a study on polyhedral spaces. I. G. Tsar'kov recently proved Theorem 4.4 ([19,p. 243] and [1, Theorem 6.7, pg. 801]). Let V be a nonempty polyhedral subset of a finite dimensional polyhedral Banach space X. Then the map cent V p.q is globally Lipschitz Hausdorff metric continuous on CBpXq and admits a Lipschitz selection.
We are now equipped enough to prove the following result. Proof. Consider the decompositions of ℓ 8 and Y as X ' 8 ℓ 8 and Z ' 8 c 0 respectively where X and Z are defined as in the proof of Proposition 3.7. Clearly, B Y " B ZˆBc 0 . Now, X is a finite dimensional subspace of c 0 and hence is a polyhedral space. Thus by Theorem 4.4, cent B Z p.q is Hausdorff metric continuous on CBpXq. Therefore, we conclude from Corollary 2.7 and Theorems 3.5 and 4.3 that cent B Y p.q is Hausdorff metric continuous on CBpℓ 8 q.
Lastly, we discuss the interrelationship between the semi-continuity properties of maps cent B X p.q and cent X p.q in a Banach space X.
d H pB, Aq ă δ, y P cent λB Y pBq ñ B X py, λεq X cent λB Y pAq ‰ H.(11)
We now set γ " δ λ . Let A P CBpXq such that d H p 1 λ B, Aq ă γ and y P cent B Y p 1 λ Bq. This implies d H pB, λAq ă γλ " δ and from [18, Lemma 2.1], λy P cent λB Y pBq. Therefore, by (11), let z P B X pλy, λεq X cent λB Y pλAq. Thus by [18, Lemma 2.1], z λ P cent B Y pAq. It follows that } z λ´y } ă ε and hence B X py, εq X cent B Y pAq ‰ H.
Conversely, let ε ą 0. Then there exists δ ą 0 such that if A P CBpXq with
d Hˆ1 λ B, A˙ă δ, y P cent B Yˆ1 λ B˙ñ B X´y , ε λ¯X cent B Y pAq ‰ H.(12)
We now set γ " λδ. Let A P CBpXq such that d H pB, Aq ă γ and y P cent λB Y pBq. This implies d H p 1 λ B, 1 λ Aq ă γ λ " δ and from [18, Lemma 2.1], y λ P cent B Y p 1 λ Bq. Therefore, by (12), let z P B X p y λ , ε λ q X cent B Y p 1 λ Aq. Thus by [18, Lemma 2.1], λz P cent λB Y pAq. It follows that }λz´y} ă ε and hence B X py, εq X cent B Y pAq ‰ H.
piiq Let B P CBpXq and λ ą sup bPB }b}`rad Y pBq. From our assumption,
cent B Y p.q is l.H.s.c. at 1 λ B. Therefore from piq, cent λB Y p.q is l.H.s.c. at B. Let ε ą 0. Then there exists δ ą 0 such that if A P CBpXq with d H pB, Aq ă δ, y P cent λB Y pBq ñ B X py, εq X cent λB Y pAq ‰ H.(13)
Let us choose 0 ă γ ă mint2δ, λ´psup bPB }b}`rad Y pBqqu.
A variation of transitivity problem for property-pP 1 q
In this section, we provide answers to the transitivity type problem for property-pP 1 q, stated in Question 1.7, for certain cases. The following result generalizes the transitivity property proved in [10, Proposition 3.2].
Proposition 5.1. Let X be a Banach space, Y be an M -summand in X and Z be a subspace of Y . If pY, Z, CBpY qq satisfies property-pP 1 q then pX, Z, CBpXqq satisfies property-pP 1 q.
Proof. Let X " Y ' 8 W for some subspace W of X. Let B P CBpXq. We define Bp1q " ty P Y : there exists x P B and w P W such that x " y`wu and Bp2q " tw P W : there exists x P B and y P Y such that x " y`wu.
Clearly, Bp1q P CBpY q and Bp2q P CBpW q.
We first prove the following claim :
rad Z pBq " maxtrad Z pBp1qq, sup wPBp2q }w}u.
Let z P Z and y P Bp1q. Then there exists x P B and w P W such that x " y`w. Thus }z´y} ď maxt}z´y}, }w}u " }z´x} ď rpz, Bq. It follows that for each z P Z, rpz, Bp1qq ď rpz, Bq. Hence rad Z pBp1qq ď rad Z pBq. Now, let w P Bp2q and z P Z. Then there exists x P B and y P Y such that x " y`w. Thus }w} ď maxt}y´z}, }w}u " }x´z} ď rpz, Bq. It follows that sup wPBp2q }w} ď rad Z pBq. Thus max # rad Z pBp1qq, sup wPBp2q }w} + ď rad Z pBq.
Now, let ε ą 0 be such that max ! rad Z pBp1qq, sup wPBp2q }w} ) ă ε. Let us choose 0 ă δ ă ε such that rad Z pBp1qq ă δ and sup wPBp2q }w} ă δ. Thus rpz 0 , Bp1qq ă δ for some z 0 P Z. Let x " y`w P B such that y P Bp1q and w P Bp2q. Thus }x´z 0 } " maxt}y´z 0 }, }w}u ă δ. It follows that rpz 0 , Bq ď δ ă ε and hence rad Z pBq ă ε. Since ε ą 0 is arbitrary, it follows that rad Z pBq ď max # rad Z pBp1qq, sup wPBp2q }w} + .
This proves our claim. It is easy to verify that cent Z pBp1qq Ď cent Z pBq, using the claim above Case 1 : sup wPBp2q }w} ą rad Z pBp1qq Let us define R " rad Z pBq " sup wPBp2q }w}. It is easy to verify that cent Z pBq " Ş yPBp1q B X ry, Rs X Z and for each η ą 0, cent Z pB, ηq " Ş yPBp1q B X ry, R`ηs X Z. Let us fix ε ą 0. Then by Lemma 3.4, there exists δ ą 0 such that for each B 1 P CBpXq with d H pBp1q, B 1 q ă 2δ and a scalar β ą 0 with |β´R| ă 2δ, we have d H¨č yPBp1q B X ry, Rs X Z, č bPB 1 B X rb, βs X Z‚ă ε.
Let us choose B 1 " Bp1q and β " R`δ. Then d H¨č yPBp1q B X ry, Rs X Z, č yPBp1q B X ry, R`δs X Z‚ă ε.
Thus Ş yPBp1q B X ry, R`δs X Z Ď Ş yPBp1q B X ry, Rs X Z`εB X , that is, cent Z pB, δq Ď cent Z pBq`εB X . This proves that pX, Y, tBuq satsifies property-pP 1 q.
Case 2 : sup wPBp2q }w} ď rad Z pBp1qq Clearly, rad Z pBq " rad Z pBp1qq. Let z P cent Z pBq. Then rpz, Bq " rad Z pBp1qq. Let y P Bp1q. Then there exists x P B and w P W such that x " y`w. Hence }z´y} ď }z´x} ď rpz, Bq " rad Z pBp1qq. It follows that z P cent Z pBp1qq. Therefore, cent Z pBq " cent Z pBp1qq. Similarly, for each η ą 0, cent Z pB, ηq " cent Z pBp1q, ηq. Since pY, Z, tBp1quq satisfies property-pP 1 q, it follows that pX, Y, tBuq has property-pP 1 q.
Another instance where Question 1.7 is positively answered is as follows :
Proposition 5.2. Let X be an L 1 -predual space. Let Y be a finite codimensional subspace of X and J be an M -ideal in X such that Y Ď J. If Y is strongly proximinal in J, then the triplet pX, Y, KpXqq satisfies property-pP 1 q.
Proof. By Theorem 1.6, Y is strongly proximinal in X. Therefore, by [18,Theorem 4.7], pX, Y, KpXqq satisfies property-pP 1 q.
Declarations
(1) Funding: No funding was received to assist with the preparation of this manuscript. (2) Competing interests: The author has no competing interests to declare that are relevant to the content of this manuscript.
Lemma 2. 2 ([ 18 ,
218Lemma 4.1]). Let X be a Banach space and V P CVpXq. Then for each A, B P CBpXq and v P V , |rpv, Aq´rpv, Bq| ď d H pA, Bq and |rad V pAq´rad V pBq| ď d H pA, Bq.
Proposition 3. 7 .
7Let Y be a proximinal finite co-dimensional subspace of c 0 . Then the triplet pℓ 8 , B Y , CBpℓ 8 qq has property-pP 1 q.
Proposition 4 . 5 .
45Let Y be a proximinal finite co-dimensional subspace of c 0 . Then the map cent B Y p.q is Hausdorff metric continuous on CBpℓ 8 q.
Proposition
4.6. Let X be a Banach space and Y be a subspace of X. Let the pair pB Y , CBpXqq have r.c.p.. If cent B Y p.q is l.H.s.c. (resp. u.H.s.c.) on CBpXq then cent Y p.q is l.H.s.c. (resp. u.H.s.c.) on CBpXq. piiiq For each λ ą 0 and B P CBpXq, cent λB Y p.q is Hausdorff metric continuous at B if and only if cent B Y p.q is Hausdorff metric continuous at 1 λ B. pivq If cent B Y p.q is Hausdorff metric continuous on CBpXq then cent Y p.q is Hausdorff metric continuous on CBpXq. Proof. By [18, Proposition 2.2], pY, CBpXqq has r.c.p.. piq We first assume that cent λB Y p.q is l.H.s.c. at B. Let us fix ε ą 0. Then there exists δ ą 0 such that if A P CBpXq withpiq For each λ ą 0 and B P CBpXq, cent λB Y p.q is l.H.s.c. (resp.
u.H.s.c.) at B if and only if cent B Y p.q is l.H.s.c. (resp. u.H.s.c.) at
1
λ B.
piiq
Let A P CBpXq such that d H pB, Aq ă γ 2 and y P cent Y pBq. Now by Lemma 2.2, rad Y pAq`sup Hence by [18, Lemma 2.1 (iii)], cent Y pBq " cent λB Y pBq and cent Y pAq " cent λB Y pAq. Therefore, it follows from p13q that B X py, εq X cent B Y pAq ‰ H. Similar arguments as above apply for the u.H.s.c. part of the results in piq and piiq and hence we omit it. The result in piiiq follows from piq and [8, Remark 2.8] and the result in pivq follows from piiq and [8, Remark 2.8].aPA
}a} ď rad Y pBq`γ
2`γ 2`s
up
bPB
}b} ă λ.
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|
[] |
[
"Revisiting the Adversarial Robustness-Accuracy Tradeoff in Robot Learning",
"Revisiting the Adversarial Robustness-Accuracy Tradeoff in Robot Learning"
] |
[
"Mathias Lechner ",
"Alexander Amini ",
"Daniela Rus ",
"Thomas A Henzinger "
] |
[] |
[] |
Adversarial training (i.e., training on adversarially perturbed input data) is a well-studied method for making neural networks robust to potential adversarial attacks during inference. However, the improved robustness does not come for free but rather is accompanied by a decrease in overall model accuracy and performance. Recent work has shown that, in practical robot learning applications, the effects of adversarial training do not pose a fair trade-off but inflict a net loss when measured in holistic robot performance. This work revisits the robustness-accuracy trade-off in robot learning by systematically analyzing if recent advances in robust training methods and theory in conjunction with adversarial robot learning, are capable of making adversarial training suitable for real-world robot applications. We evaluate three different robot learning tasks ranging from autonomous driving in a high-fidelity environment amenable to sim-to-real deployment to mobile robot navigation and gesture recognition. Our results demonstrate that, while these techniques make incremental improvements on the trade-off on a relative scale, the negative impact on the nominal accuracy caused by adversarial training still outweighs the improved robustness by an order of magnitude. We conclude that although progress is happening, further advances in robust learning methods are necessary before they can benefit robot learning tasks in practice.
|
10.1109/lra.2023.3240930
|
[
"https://export.arxiv.org/pdf/2204.07373v2.pdf"
] | 248,218,519 |
2204.07373
|
faeb50ceb48dfa1590d92761c2830eeab4825e54
|
Revisiting the Adversarial Robustness-Accuracy Tradeoff in Robot Learning
Mathias Lechner
Alexander Amini
Daniela Rus
Thomas A Henzinger
Revisiting the Adversarial Robustness-Accuracy Tradeoff in Robot Learning
Adversarial training (i.e., training on adversarially perturbed input data) is a well-studied method for making neural networks robust to potential adversarial attacks during inference. However, the improved robustness does not come for free but rather is accompanied by a decrease in overall model accuracy and performance. Recent work has shown that, in practical robot learning applications, the effects of adversarial training do not pose a fair trade-off but inflict a net loss when measured in holistic robot performance. This work revisits the robustness-accuracy trade-off in robot learning by systematically analyzing if recent advances in robust training methods and theory in conjunction with adversarial robot learning, are capable of making adversarial training suitable for real-world robot applications. We evaluate three different robot learning tasks ranging from autonomous driving in a high-fidelity environment amenable to sim-to-real deployment to mobile robot navigation and gesture recognition. Our results demonstrate that, while these techniques make incremental improvements on the trade-off on a relative scale, the negative impact on the nominal accuracy caused by adversarial training still outweighs the improved robustness by an order of magnitude. We conclude that although progress is happening, further advances in robust learning methods are necessary before they can benefit robot learning tasks in practice.
I. INTRODUCTION
Adversarial attacks are well-studied vulnerabilities of deep neural networks [1], [2]. These norm-bounded input perturbations make the network change its decision compared to the unaltered input and can have catastrophic impact in practical robotics applications. Critically, the adversarially altered inputs are barely distinguishable from the original input by humans. Most realistic-sized computer vision networks can be fooled by perturbations that change each pixel by a maximum of 4% (i.e., a l ∞ -norm less or equal to 8) while being barely noticeable by humans.
Adversarial robustness is an important consideration in the development of robotic applications, as it ensures that the robot's behavior remains consistent and predictable in the presence of perturbations or attacks. In the real world, robots must be able to operate in a variety of environments and under a wide range of conditions, some of which may be outside of their training data or beyond their control. Adversarial robustness allows robots to continue functioning effectively even when faced with such challenges, improving their reliability and safety in real-world applications. Additionally, as robots become more integrated into society and are given greater autonomy, it becomes increasingly important to ensure that they are not susceptible to manipulation or exploitation by malicious actors. Adversarial robustness helps to protect against such threats and ensure that robots can be trusted to behave in a predictable and responsible manner.
Robust learning aims to tackle the problem by training networks that are immune to adversarial or other types of attacks [3], [4], [5], [6], [7], [8], [9], [10]. One of the most dominant approaches for training robust models is adversarial training which adds adversarial perturbations to the training data online during and throughout the learning procedure [3], [7]. Adversarial training methods improve the test-time robustness on adversarial examples at the critical cost of lower nominal accuracy [11], [12], [13]. For instance, the advanced adversarial training algorithm of [14], which won the NeurIPS 2018 Adversarial Vision Challenge, yielded a robust network with an accuracy of 89% on the CIFAR-10 dataset. In contrast, standard training algorithms can easily produce non-robust networks with an accuracy above 96% on this dataset [15]. This dilemma of choosing between an accurate but vulnerable and a robust but less accurate model is known as the robustness-accuracy trade-off [13], [14], [11].
Recent work [16] has investigated this trade-off specifically in the context of robot learning applications where both accuracy and robustness are critical as the system is ul-timately deployed into physical, safety-critical environments. The authors observed that this trade-off is not fair trade but poses a net loss when evaluating the robots' overall performance and concluded that adversarial training is not ready for robot learning. However, recent work has shown that multiple factors (e.g., model size, choice of the activation function, adversarial training procedure) contribute to the reduction in accuracy of robot learning methods [17], [18], [19], [20], [21], [22]. In particular, these works underline that larger models are necessary for robustly fitting the training data [23], [18]. Moreover, they emphasize that a more careful selection of the neural network architecture and hyperparameters is needed when replacing standard training with adversarial training methods [21], [19]. However, there remains a critically important and open question on if these recent advances are sufficient to quell the costs of adversarial training for robotics.
In this work, we assess whether the conclusion of [16] that adversarial training is not ready for robot learning remains true or is challenged by these recent advances in the field. In particular, we evaluate if overparametrized models [18], vision transformers [19], [24], smooth curvature activation functions [20], more careful hyperparameter selection [21], and advanced adversarial training methods [22] can provide acceptable accuracy and robustness on three robot learning and autonomous driving tasks.
Our results show that, although the techniques listed above pose a significant improvement in the robustnessaccuracy gap, the negative impact on the nominal accuracy from adversarial training still outweighs the benefits of the induced robustness. Specifically, while the methods from the literature make single digits improvements on the robustnessaccuracy Pareto front, i.e., improving both accuracy and robustness, the negative side-effects of adversarial training methods still outweigh these advances by an order of magnitude. Nonetheless, we observed the trend that combining multiple individually introduced robustness enhancement methods provided the most promising future path toward closing the robustness-accuracy gap, e.g., as outlined in Figure 1.
We summarize our contributions as: • We evaluate five advancements in robust learning methods in three different real-world robotic applications (456 models tested in total) for their suitability in closing the robustness-accuracy tradeoff gap in robot learning tasks in practice. • We provide strong empirical evidence that, while robustness can be improved by the methods from literature, the negative effect on the nominal accuracy of adversarial training still outweighs the improvements of these methods by an order of magnitude. • Our results show that adversarial training is most effective when multiple individual robust learning approaches are combined. This suggests that the most promising path to closing the robustness-accuracy gap entirely in the future is the integration of multiple independent approaches for enhancing robustness.
The remainder of this paper is structured as follows. In section II, we recapitulate robustness of neural networks, adversarial training, and the robustness-accuracy trade-off. In section III, we describe related work on improving the robustness of neural networks and avoiding the reduced clean accuracy of adversarial training. Finally, in section IV, we experimentally evaluate these improvements on three robot learning tasks.
II. BACKGROUND AND RELATED WORK
A neural network is a function f θ : X → Y parameterized by θ. In supervised learning, the training objective is to fit the function to a given dataset in the form of {(x 1 , y 1 ), . . . (x n , y n )} assumed to be i.i.d. sampled from a probability distribution over X × Y. This fitting process is done via empirical risk minimization (ERM) that minimizes
1 n n i=1 L(f θ (x i ), y i )(1)
via stochastic gradient descent. The differentiable loss function L : Y × Y → R characterizes how well the network's prediction f θ (x i ) matches the ground truth label y i . An adversarial attack is a sample (x i , y i ) from the data distribution and a corresponding attack vector µ with µ ≤ ε such that f (x i ) = f (x i + µ) with ε being a threshold. For image data, L ∞ thresholds δ ≤ 8 are usually not recognizable or appear as noise for human observers. It has been shown that most neural networks, irrespective of network types, input domains, or learning setting, are susceptible to adversarial attacks [2], [25], [26], [27], [28], [29], [30].
Typical norms used in adversarial attacks are the 1 , 2 , and the ∞ norm. In this work, we focus on the ∞ norm. A network is robust on a given sample if no such attack µ exists within a threshold ε. The robust accuracy is the standard metric for measuring the robustness of a network aggregated over an entire dataset {(x 1 , y 1 ), . . . (x n , y n )} by counting the ratio of correctly classified samples that are also robust.
In practice, deciding whether a network is robust for a sample is an NP-hard problem [31], [32], [33] and, therefore, cannot be computed for typically sized networks in a reasonable time. Instead, the robustness of networks is often studied with respect to empirical gradient and black-boxbased attack methods. The fast gradient sign method (FGSM) [2] computes an attack by
µ = ε sign ∂L(f θ (x i ), y i ) ∂x i .(2)
Despite its simplicity, adversarial training often uses the FGSM method due to its speed. The iterative fast gradient sign method (I-FGSM) [34] is a more sophisticated generalization of the FSGM. It computes an attack iteratively in k steps starting from µ 0 = 0 and updating it by
µ i = ε k sign ∂L(f θ (x i + µ i−1 ), y i ) ∂x i .(3)
DeepFool [35], the C&W method [36], and projected gradient descent [7] are other common iterative attack methods that are used for evaluating robustness but are too computationally expensive to incorporate in adversarial training. DeepFool [35] linearizes the network in each iteration of updating µ i . Projected gradient descent [7] applies unconstrained gradient descent but divides each µ i by its norm and multiplies the results with ε to project it back into the given threshold. The C&W method [36] avoids such projection by parametrizing the attack vector µ by another variable and a transformation that already normalizes the attack to stay within a given threshold. It has been experimentally shown that any network of non-trivial size is, at least in parts, vulnerable to such attacks [7].
Robust learning methods aim to train networks that are robust [3], [4], [5], [6], [7], [8], [9]. One of the most common robust learning methods is adversarial training which changes the standard ERM objective to the min-max objective
1 n n i=1 max µ: µ ≤ε L(f θ (x i + µ), y i ),(4)
where ε > 0 is some attack budget controlling how much each input can be perturbed. Due to the computation overhead by this training objective, fast attack-generating methods are typically used for computing the max in Equation 4, e.g., the FGSM or I-FGSM. Alternative approaches to adversarial training make minor modifications to the objective term in Equation 4. For instance, the TRADES algorithm [14] replaces the label y i in Equation 4 with the network's prediction of the original input, i.e., f θ (x i ), and optimizes a joint objective of the standard ERM term and the robustness term. The approach of [37] removes the overhead imposed by the maximization step in Equation 4 by pre-computing µ in the previous gradient descent step. Although such pre-computed µ can become inaccurate, i.e., stale, [37] showed that it improves robustness in practice. Adversarial weight perturbation (AWP) [22] improves the generalization of adversarially trained networks by injecting adversarial noise into the weights of the network and smoothing the loss surface. Data augmentation applied to adversarial training has also been shown to positively affect the robustness, and the generalization of neural networks [17]. The work of [38] has shown that the negative impact of adversarial training on the clean accuracy of a network can be further reduced by combining it with advanced data augmentation techniques such as MixUp [39].
The major limitation of adversarial training methods is that they negatively affect the network's standard accuracy (or other performance metrics). For example, medium-sized networks achieve an accuracy of 96% on the CIFAR-10 dataset when trained with standard ERM [15]. However, in [14] the best-performing network trained with the TRADES algorithm could only achieve a standard accuracy of 89% on this dataset. This phenomenon of an antagonistic relation between accuracy and robustness was first studied in [13] and is known as the accuracy-robustness trade-off. The tradeoff was studied in the context of robot learning in [16] by investigating whether the gained robustness is worth the reduction in nominal accuracy in real-world robotic tasks. The authors observed that the adversarially trained networks resulted in a worse robot performance than by using a network trained in the standard way.
The concept of adversarial training and the min-max objective of robust learning has been adopted for other taskspecific types of specifications, such as safety. For example, [16] has introduced safety-domain training by replacing the norm-bounded neighborhoods of labeled samples with arbitrary sets and corresponding labels, i.e., a min-max training objective over labeled sets. Some modifications of the minmax objective have been studied in feedback systems with closed-loop safety and stability specifications. For instance, [40], [41], [42] propose to learn a safety certificate via a learner-verifier framework where the maximization step is replaced by a verification module that provides formal guarantees on the certificate.
Adversarial training has also been studied as a regularizer for improving the generalization of neural networks. In particular, [43] used mild adversarial attacks based on a hierarchical structure to improve the clean accuracy of vision transformer models [44]. The work of [45] studied human adversaries to improve the performance in robotic object manipulation tasks.
III. METHODS
In this section we describe three directions from the literature that point to paths of how to improve robustness without sacrificing standard accuracy.
A. Smooth activations and bag of tricks
Recent work suggests that the common ReLU activation function, i.e., max{0, x}, is not well suited for adversarial training methods [20]. Instead, the authors observed that activation functions with smooth curvatures provide better robustness at roughly the same standard accuracy. Specifically, the sigmoid-weighted linear unit (SiLU) activation function [46], i.e., x · 1 1+exp(−x) , was highlighted as having a smooth second derivative and observed to improve robustness compared to alternative activations. We note that the SiLU activation was concurrently proposed as swish activation function in [47].
The work of [21] investigated how hyperparameters of the learning process affect adversarial training compared to standard ERM. For example, the authors experiment with learning rate schedules, early stopping, and batch size, among other settings. The authors observed that adversarial training benefits from a higher weight decay factor than standard training. Moreover, the authors confirmed that a smooth activation function improves robustness over the ReLU activation.
B. Robustness requires overparametrization
Theoretical contributions to the robustness-accuracy tradeoff recently discovered that overparametrization is necessary for smoothly fitting the training data [23]. While empirical
Summer (in-distribution) Winter (in-distribution) Autumn (out-of-distribution) Night (out-of-distribution) Fig. 2.
Test conditions of our closed-loop driving experiment using a data-driven simulation environment [48]. The training data are collected in summer and winter conditions (separated from the testing data). results already suggested that the accuracy of larger models suffers less from adversarial training than for small models, the critical insight is that such large models are necessary. In particular, the authors proved that for a dataset of n samples with d-dimensional features, a model with n parameters can fit the training samples but cannot smoothly interpolate between them. Moreover, the authors show that a model needs at least nd parameters to fit the training data and interpolate them smoothly. The authors also demonstrated that contemporary models for standard datasets do not contain enough parameters with respect to their proven results.
C. Vision transformers are more robust than CNNs
The vision transformer (ViT) [44] is a powerful machine learning architecture that represents an image as a sequence of patches and processes this sequence using a self-attention mechanism [49]. Detailed experimental comparisons between vision transformer and convolutional neural networks suggest that ViTs are naturally more robust with respect to object occlusions and distributions shifts [50]. Concurrent work on comparing ViTs to CNNs with respect to adversarial attacks has found that vision transformers seem to be naturally more robust to adversarial attacks as well.
All advances on the robustness-accuracy tradeoff discussed above are either theoretical or were evaluated on static image classification tasks. Moreover, the methods are typically evaluated on research datasets such as CIFAR and Ima-geNet. While these datasets allow studying machine learning models' general performance, they significantly differ from real-world robot learning tasks. For example, the CIFAR datasets consist of very low-resolution images, i.e., 32-by-32 pixel, whereas robotic vision processing systems handle images with much higher resolution, e.g., 256-by-256 pixels in [16]. Although the samples of the ImageNet dataset have a realistic image resolution, typical robot learning datasets consist of multiple orders of magnitude fewer samples than the ImageNet dataset. Moreover, experiments on research datasets often report static test metrics, whereas learned robotic controllers are deployed in a closed-loop on a robot.
The next section evaluates the methods described above on multiple real-world robot learning tasks, including open- loop training and closed-loop evaluation on an autonomous driving task.
IV. EXPERIMENTS
In this section we study the advances in adversarial training methods on three robot learning tasks.
A. End-to-end driving
Our first experiment considers an autonomous driving task. In particular, a network is trained to predict the curvature of the road ahead of a car from images received at a camera that is mounted on top of the vehicle. The training data is collected by a human driver who maneuvers the car around a test track. The networks are then trained on collected data using supervised learning. Finally, we deploy the networks in a closed-loop autonomous driving simulator. We use the VISTA simulation environment [48] for this purpose.
We compare the performance of a baseline CNN with four variations. First, we compare with an enlarged variant of the baseline CNN to validate the necessity of overparametrization for robustness empirically. Next, we equip the baseline with the smoother SiLU activation and increase the weight decay (wd+). We also test the CNN trained with adversarial weight perturbation (AWP) [22] instead of training via the objective in Eq. (4). Finally, we test a vision transformer model. The baseline model (CNN) consists of 440k, the enlarged model (CNN-large) of 7.7M, and the tested vision transformer (ViT) of 2.0M trainable parameters. The inputs of all architectures are 160-by-48 RGB images that are normalized per-image to have zero mean and unit standard deviation. The architecture details of the two convolutional networks are listed in the Appendix. Our vision transforms splits the input image into non-overlapping patches of 16-by-12 pixels, uses a latent dimension of 256, with 4 attention heads, 384 feed-forward dimensions, and 4 layers in total. For the training, we use the Adam optimizer [51] with a learning rate of 0.0003 and a batch size of 64. The weight decay is set to 10 −5 , except for the wd+ variant, which is trained with a decay factor of 5 · 10 −5 . We train all networks for a total of 900,000 steps. We train all models with standard and adversarial training with increasing attack budget (ε = 0, 1, . . . 8) and I-FGSM as attack methods.
For each model and attack budget pair, we run a total of 400 simulations, split into 200 in-training distribution, and 200 out-of-training distribution condition runs. The intraining data were collected in summer and winter and were separated from the training data, i.e., there is no overlap between the training data and the evaluation data. The out-of-training data were collected in autumn and during the night, with no such condition present in the training data. The four conditions are visualized in Figure 2. As an evaluation metric, we report the number of crashes during the simulation, i.e., when the vehicle leaves the road.
The top row in Figure 3 shows the crashes during the summer and winter simulations. The results show that the overparametrized model and the vision transformer indeed provide better performance at a larger adversarial training budget than the baseline. An increased weight decay improved the performance only at lower attack budget training, while the networks with SiLU activation performed worse in the closed-loop tests. At larger attack budgets, no model could drive the car safely, while most models learned by standard ERM could drive all 200 runs flawlessly.
The out-of-training distribution simulation results for autumn and night conditions are shown in the middle and bottom row in Figure 3. We observe that adversarial training significantly hurt the out-of-distribution performance of all models, i.e., especially in the autumn data. A video demonstration of the simulated runs is available at https:// youtu.be/TQKP7l9PfNo. In summary, the best driving performance across all four tested conditions was observed with networks trained with standard ERM.
B. Visual gesture recognition
Our second experiment concerns training an image classifier that controls the operating modes of a mobile robot as reported in [16]. The dataset consists of 2029 sample 256-by-256 pixel images corresponding to three classes, i.e., idle (905 samples), enable (552 samples), and disable (572 samples), which are split into a training and a validation set with a 90%:10% ratio. The experiments on the physical robot in [16] suggest that a validation accuracy of above 90% is necessary for acceptable robot performance. Due to the small size of the dataset, we resort to transfer learning of a pre-trained classifier using the big-transfer (BiT) finetuning protocol of initializing the output layer with all zeros 59.0% ± 9.5 84.6% ± 1.9 ε = 8 (+AWP) 76.0% ± 11.0 80.0% ± 9.5
ViT-Large/16 ε = 0 20.8% ± 11.0 85.9% ± 2.5 ε = 1 35.3% ± 1.2 89.9% ± 3.6 ε = 2 67.1% ± 9.9 89.6% ± 3.2 ε = 4
77.4% ± 6.6 71.0% ± 15.9 ε = 8 58.0% ± 17.5 47.2% ± 9.8 ε = 4 (+AWP)
73.9% ± 7.3 86.3% ± 7.4 ε = 8 (+AWP)
89.7% ± 0.6 88.9% ± 3.1 and training all layers [52].
In this experiment, we test the theoretical necessity of overparametrization in practice. We train networks of different sizes using adversarial training with increasing attack budget (ε ∈ {0, 1, 2, 4, 8}) and report the robust validation accuracy under I-FGSM attacks with a radius of ε = 8. We also evaluate models trained with adversarial weight perturbation (AWP) [22] and ε ∈ {4, 8}.
As a proxy for real-world test accuracy, we collect a new dataset comprising 190 idle samples, 129 enable samples, and 140 disable samples. Particularly, the test set resembles a real-world deployment of the model on the robot and ensures that there is no spurious temporal or spatial correlation with the original data source. We use the clean accuracy of the new set as our test metric to estimate real-world performance.
For increasing the size of the model, we test a ResNet50 (24M), ResNet101 (43M), and ResNet152 (58M) with the number of trainable parameters reported in parenthesis [53]. We also evaluate the vision transformer models ViT-Small (22M), ViT-Base (86M), and ViT-Large (304M) that process the images in the form of 16-by-16 pixel patches [44]. For the training, we use the Adam optimizer [51] with a learning rate of 0.00005 and a batch size of 64, except for the ResNet152 where a batch size of 32 is used due to out-of-memory errors. We repeat each training run with 5 random seeds and report the mean and standard deviation.
The results in Table I and Table II show that the overall best test accuracy could be achieved with standard empirical risk minimization and a ResNet50 or ViT-Small model. As expected, however, these models provide no robustness to adversarial attacks. Still, acceptable test performance (≥80%) at non-trivial robustness was realized by models trained with a small attack budget, e.g. ε = {1, 2}. Nonetheless, the gap between the overall best test accuracy and the top-scoring adversarially trained models is significant, i.e., over one and two standard deviations of the standard trained ResNet50 and ViT-Small model, respectively.
The results in Table I and Table II show the trend that with an increase in model size, the models become more accurate under adversarial training. This effect is even more amplified when considering the more advanced adversarial weight perturbation training (+AWP). Specifically, the most robust ResNet and vision transformer are both their largest variant trained with AWP. Moreover, we observe an advantage of the ViT architecture over the ResNets in terms of robustness, which has been studied in more detail in [24], [19]. This result suggests that even larger ViT-based models combined with even more advanced adversarial training schemes may be able to close the robustness-accuracy gap entirely.
C. Certified safety-domain training
Adversarial training methods do not ensure robustness but provide only empirical improvements over common attack methods. Certified training methods such as the interval bound propagation [54] can learn networks with formal robustness or safety guarantees. In this experiment, we study the safety-domain training of LiDAR-based mobile robot navigation controller [16]. The objective of the learned controller is to map 541-dimensional laser range scans to 7 possible categories, i.e., stay, straight forward, left forward, right forward, straight backward, left backward, and right backward. The dataset consists of 2705 training and 570 validation samples uniformly distributed across the seven classes. Using safety-domain training, we want to ensure that the robot never crashes into an object in front of it. This is achieved by training an abstract interpretation representation of the network to never output a forward locomotion class in case the LiDAR input indicates an obstacle. There are four safety levels with different strictness of what accounts for an obstacle, e.g., several consecutive rays or just a single ray, defined in [16]. Safety level 0 corresponds to standard training, while safety level 3 represents the strictest level.
We test the overparametrization, increased weight decay (from 0 to 10 −5 ), and smooth activation function methods on this task. As a baseline, we use the 1D-CNN from [16], which is comprised of 360k parameters. Our overparametrized models increase the width of the network to obtain CNNs with 1.4M, 3.2M, and 5.7M parameters respectively. We use the exponential linear unit (ELU) activation function [55] to represent a smooth activation due to the non-monotonicity of SiLU being incompatible with the used abstract interpretation domains. We train all models with the Adam optimizer [51] with a learning rate of 0.0001 and a batch size of 64. The safety level 0 models are trained for 20 epochs, while the networks trained using safetydomain training for 2000 epochs. The network architectures are shown in the Appendix.
We report the validation accuracy as an evaluation metric. The experiments on the physical robot in [16] suggest that a validation accuracy above 80% is necessary to achieve an acceptable real-world performance. Note that all models, except those trained with safety level 0, provide some form of formal safety guarantees. Therefore, this experiment studies how much validation accuracy is traded for the ensured safety. We repeat each training run with 5 random seeds and report the mean and standard deviation.
The result in Table III shows that safety-domain training benefits from an increased number of parameters (width). However, the improvement over the baseline is rather incremental and accounts only for a few percent. In contrast, the accuracy reduction caused by the safety-domain training is several times more significant, e.g., around 10%, and no network trained with safety-domain training exceeds the threshold of 80% accuracy. The networks with smooth activation function and increased weight decay performed worse than the baseline when using safety-domain training. This suggests that certified training methods such as safetydomain training may require different hyperparameters and learning settings than adversarial training.
V. DISCUSSION AND CONCLUSION
Adversarial training (i.e., training on adversarially perturbed input data) is a well-studied method for making neural networks robust to potential adversarial attacks during inference. However, the improved robustness does not come for free but rather is accompanied by a decrease in nominal model accuracy and performance [14]. Recent work [16] has shown that, in practical robot learning applications, the effects of adversarial training do not pose a fair tradeoff but inflict a net loss when measured in holistic robot performance. This work revisited the robustness-accuracy trade-off in robot learning by systematically analyzing if recent advances in robust training methods and theory in conjunction with adversarial robot learning can make adversarial training suitable for real-world robot applications.
We evaluated a total of five robust training methods on three different robot learning tasks ranging from autonomous driving in a high-fidelity environment amenable to simto-real deployment to mobile robot navigation and gesture recognition. Our results indicate that the negative impact on the nominal accuracy from adversarial training still outweighs the induced robustness. In other words, while adversarial training can improve the model's ability to withstand attacks, it does not justify the reduced accuracy on clean, non-adversarial data.
Nonetheless, our results suggest that, in aggregate, when combining these methods, a significant improvement in the robustness-accuracy gap is made. For instance, the combination of overparametrization, a vision transformer, and a more advanced training scheme (adversarial weight perturbation) performs much better under adversarial training than the models tested in [16]. This suggests that future research directions that can be further combined, e.g., data augmentation or other training schemes, may be able to close the robustness-accuracy entirely.
ACKNOWLEDGMENT
This work was supported in parts by the AI2050 program at Schmidt Futures (Grant G-22-63172), Capgemini SE, ERC-2020-AdG 101020093, National Science Foundation (NSF), and JP Morgan Graduate Fellowships. We thank Christoph Lampert for inspiring this work. Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
APPENDIX
Here we describe the details of the neural network architectures used in our experiments.
1
Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory, Cambridge, MA, 02139 USA {mlechner, amini, rus}@mit.edu) Fig. 1. High-level summary of our results. Adversarial training improves robustness at the cost of significantly reduced accuracy. We show that methods to counteract this decrease in accuracy are most effective when multiple approaches are combined, i.e., an overparametrized network, a vision transformer neural architecture, and advanced adversarial training procedures.
Fig. 3 .
3Number of crashes out of 100 simulation runs in each data setting (summer, winter, fall, night) with respect to varying the adversarial training budget. All models were trained in summer and winter conditions (on a different data split than the evaluations). The large CNN and the ViT model perform best under heavy adversarial training, but no adversarially trained model could handle distribution shifts, i.e., fall and night conditions.
TEST ACCURACY ON THE VISUAL GESTURE RECOGNITION DATASET OF VARIOUS ADVERSARIALLY FINE-TUNED MODELS. BEST ROBUST VALIDATION ACCURACY AND TEST ACCURACIES GREATER THAN 80% ARE HIGHLIGHTED IN BOLD. BEST VALUES ARE UNDERLINED.Model
Adversarial
Robust
Test
training budget
validation
accuracy
accuracy
ResNet50
ε = 0
2.3% ± 1.9
93.5% ± 3.8
ε = 1
18.9% ± 3.2
86.1% ± 3.1
ε = 2
55.3% ± 3.1
76.0% ± 2.5
ε = 4
77.0% ± 3.8
68.6% ± 3.2
ε = 8
60.7% ± 2.7
51.8% ± 6.6
ε = 4 (+AWP)
75.0% ± 1.0
67.7% ± 8.6
ε = 8 (+AWP)
47.6% ± 5.5
41.4% ± 0.4
ResNet101
ε = 0
3.8% ± 1.3
85.9% ± 7.3
ε = 1
20.7% ± 2.2
82.9% ± 4.4
ε = 2
54.8% ± 1.4
76.1% ± 2.6
ε = 4
44.4% ± 0.2
41.5% ± 1.2
ε = 8
43.9% ± 0.5
41.7% ± 0.0
ε = 4 (+AWP)
43.8% ± 0.4
41.7% ± 0.0
ε = 8 (+AWP)
44.2% ± 0.4
41.7% ± 0.0
ResNet152
ε = 0
4.4% ± 4.4
87.2% ± 4.2
ε = 1
29.6% ± 5.7
82.1% ± 7.0
ε = 2
66.7% ± 4.2
75.8% ± 4.4
ε = 4
52.6% ± 9.8
50.8% ± 12.4
ε = 8
44.5% ± 0.4
41.7% ± 0.0
ε = 4 (+AWP)
78.0% ± 3.5
67.3% ± 1.6
ε = 8 (+AWP)
44.2% ± 0.4
41.7% ± 0.0
TABLE I
ROBUST VALIDATION ACCURACY (UNDER I-FGSM WITH ε = 8) AND
Model
Adversarial
Robust
Test
training budget
validation
accuracy
accuracy
ViT-Small/16
ε = 0
0.8% ± 1.5
94.1% ± 2.1
ε = 1
21.3% ± 5.6
84.6% ± 6.0
ε = 2
48.0% ± 4.8
83.3% ± 4.7
ε = 4
72.3% ± 5.0
75.2% ± 3.1
ε = 8
43.9% ± 0.4
41.1% ± 1.9
ε = 4 (+AWP)
58.3% ± 2.0
85.5% ± 0.4
ε = 8 (+AWP)
61.4% ± 1.8
66.6% ± 3.7
ViT-Base/16
ε = 0
11.7% ± 4.5
82.3% ± 5.7
ε = 1
37.9% ± 8.4
79.4% ± 1.7
ε = 2
61.4% ± 3.8
84.5% ± 4.6
ε = 4
67.1% ± 11.7
76.0% ± 1.9
ε = 8
48.0% ± 3.7
54.6% ± 10.2
ε = 4 (+AWP)
TABLE II ROBUST
IIVALIDATION ACCURACY (UNDER I-FGSM WITH ε = 8) AND TEST ACCURACY ON THE VISUAL GESTURE RECOGNITION DATASET OF VARIOUS ADVERSARIALLY FINE-TUNED MODELS. BEST ROBUST VALIDATION ACCURACY AND TEST ACCURACIES GREATER THAN 80% ARE HIGHLIGHTED IN BOLD. BEST VALUES ARE UNDERLINED.
TABLE III VALIDATION ACCURACY ON THE ROBOT FOLLOW DATASET [16] OF 1D-CONVOLUTIONAL NNS WITH VARIOUS HYPERPARAMETERS AND TRAINED WITH STANDARD AND SAFETY-DOMAIN TRAINING. VALUES GREATER THAN 80% ARE HIGHLIGHTED IN BOLD. SAFETY LEVEL 0 CORRESPONDS TO STANDARD TRAINING, WHILE THE NETWORK TRAINED WITH SAFETY LEVEL 1 AND ABOVE PROVIDE FORMAL SAFETY GUARANTEES OF NEVER CRASHING THE ROBOT INTO AN OBSTACLE. THE COLUMNS SHOW NETWORKS WITH DIFFERENT WIDENING FACTOR. THE NUMBER OF LEARNABLE PARAMETERS ARE SHOWN IN PARENTHESIS.Safety
Validation accuracy
level
Width 1 (360k) Width 2 (1.4M) Width 3 (3.2M) Width 4 (5.7M)
0
Baseline
83.2% ± 0.8
84.7% ± 1.6
83.9% ± 1.9
85.2% ± 0.8
ELU
73.3% ± 1.5
72.5% ± 3.3
73.3% ± 0.8
71.3% ± 1.3
wd+
82.5% ± 2.0
84.0% ± 2.1
85.7% ± 1.3
85.7% ± 1.2
1
Baseline
75.1% ± 2.6
78.6% ± 3.7
77.4% ± 2.1
78.7% ± 3.4
ELU
53.1% ± 0.6
53.5% ± 0.4
52.9% ± 0.6
52.3% ± 0.8
wd+
74.2% ± 3.4
75.0% ± 1.8
65.9% ± 10.7
67.4% ± 12.0
2
Baseline
76.3% ± 3.1
76.8% ± 4.9
76.1% ± 2.8
78.5% ± 3.2
ELU
53.6% ± 0.3
53.1% ± 0.3
53.2% ± 0.4
52.9% ± 0.6
wd+
72.9% ± 3.3
75.5% ± 2.1
68.4% ± 8.6
70.7% ± 10.0
3
Baseline
51.8% ± 0.9
52.8% ± 0.5
53.3% ± 0.1
53.9% ± 0.3
ELU
53.2% ± 0.8
53.8% ± 0.5
53.1% ± 0.1
53.2% ± 0.4
wd+
51.4% ± 1.1
52.8% ± 0.7
52.8% ± 0.6
53.4% ± 0.4
TABLE IV CONVOLUTIONAL NEURAL NETWORK BASELINE ARCHITECTURE FOR OUR AUTONOMOUS DRIVING EXPERIMENT (440K PARAMETERS). F REFERS TO THE NUMBER OF FILTERS, K TO THE KERNEL SIZE, AND S TO THE STRIDE. ENLARGED NEURAL NETWORK ARCHITECTURE FOR OUR AUTONOMOUS DRIVING EXPERIMENT (7.7M PARAMETERS). F REFERS TO THE NUMBER OF FILTERS, K TO THE KERNEL SIZE, AND S TO THE STRIDE.TABLE VI NETWORK ARCHITECTURE OF THE 1D-CNN TRAINED WITH SAFETY-DOMAIN TRAINING (F= NUMBER OF FILTERS, K = KERNEL SIZE, S = STRIDE). W IS THE WIDENING FACTOR.Layer
Parameter
Conv2D
F=32, K=5, S=2, ReLU
Conv2D
F=64, K=5, S=1, ReLU
Conv2D
F=96, K=3, S=2, ReLU
Conv2D
F=128, K=3, S=1, ReLU
GlobalAveragePool2D
Fully-connected
1000 units, ReLU
Fully-connected
100 units, ReLU
Fully-connected
1 unit
Layer
Parameter
Conv2D
F=32, K=5, S=2
BatchNorm2D
ReLU (post BN)
Conv2D
F=128, K=5, S=1
BatchNorm2D
ReLU (post BN)
Conv2D
F=256, K=3, S=2
BatchNorm2D
ReLU (post BN)
Conv2D
F=512, K=3, S=1
BatchNorm2D
ReLU (post BN)
Conv2D
F=1024, K=3, S=1
GlobalAveragePool2D
Fully-connected
1024 units, ReLU
Fully-connected
256 units, ReLU
Fully-connected
1 unit
TABLE V
Layer
Parameter
Conv1D
F=w*32, K=5, S=1, ReLU
Conv1D
F=w*96, K=5, S=2, ReLU
Conv1D
F=w*96, K=5, S=2, ReLU
Conv1D
F=w*96, K=5, S=2, ReLU
Conv1D
F=w*96, K=5, S=2, ReLU
Conv1D
F=w*96, K=5, S=2, ReLU
Flatten
Fully-connected
w*128 units, ReLU
Fully-connected 7
softmax
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|
[] |
[
"A Systematic Literature Review on relationship between agile methods and Open Source Software Development methodology",
"A Systematic Literature Review on relationship between agile methods and Open Source Software Development methodology"
] |
[
"Taghi Javdani Gandomani ",
"Hazura Zulzalil ",
"Abul Azim ",
"Abd Ghani ",
"MDAbu Bakar "
] |
[] |
[] |
Agile software development methods (ASD) and open source software development methods (OSSD) are two different approaches which were introduced in last decade and both of them have their fanatical advocators. Yet, it seems that relation and interface between ASD and OSSD is a fertile area and few rigorous studies have been done in this matter. Major goal of this study was assessment of the relation and integration of ASD and OSSD. Analyzing of collected data shows that ASD and OSSD are able to support each other. Some practices in one of them are useful in the other. Another finding is that however there are some case studies using ASD and OSSD simultaneously, but there is not enough evidence about comprehensive integration of them.
| null |
[
"https://arxiv.org/pdf/1302.2748v1.pdf"
] | 15,022,665 |
1302.2748
|
fbc083d1f0938825fcd6a176d40b5482f489fc09
|
A Systematic Literature Review on relationship between agile methods and Open Source Software Development methodology
Taghi Javdani Gandomani
Hazura Zulzalil
Abul Azim
Abd Ghani
MDAbu Bakar
A Systematic Literature Review on relationship between agile methods and Open Source Software Development methodology
Agile Software Development methods (ASD)Open Source Software Development methods (OSSD)Systematic literature reviewSoftware engineering
Agile software development methods (ASD) and open source software development methods (OSSD) are two different approaches which were introduced in last decade and both of them have their fanatical advocators. Yet, it seems that relation and interface between ASD and OSSD is a fertile area and few rigorous studies have been done in this matter. Major goal of this study was assessment of the relation and integration of ASD and OSSD. Analyzing of collected data shows that ASD and OSSD are able to support each other. Some practices in one of them are useful in the other. Another finding is that however there are some case studies using ASD and OSSD simultaneously, but there is not enough evidence about comprehensive integration of them.
I. Introduction
Agile software development methodology (ASD) which was introduced formally within last decades is a reaction to traditional methods (e.g. waterfall) which also are recognized as heavy-weight methods [1]. Since technology and industry grow too fast, requirements changes rapidly and so, new innovation of software development was introduced [2]. Four agile principles introduced as agile manifesto [3], have attracted many software producers and software engineers to migrate from plan-based software development methodology to agile methods [4]- [5]- [6]- [7]- [8] . Also, focusing on user requirements and embracing of changes during development cycle causing customers to accept these methods as well as welcoming. Open Source Software Development (OSSD) is another famous approach which has been used increasingly in last decade. OSSD relies on skilled volunteers and their experiences in distributed teams [9]. However there is no specific definition, OSSD is defined as a revolutionary software process allowing source codes to be redistributed and modified freely and encourage developing software in a collaborating environment. This approach is also increasingly used in many projects and organizations [10]. Both ASD and OSSD provide many benefits and facilities that could be effective in software production. From the early years, the relationship between these approaches was studied [11]- [12]. Aim of this study is assessing of the relationship between ASD and OSSD, and if so, extent of the relation and probably integration of them by doing a systematic Literature Review (SLR) on simple and clear question in this regard.
In the next section we describe our method for conducting review, in section III results will be shown, in section IV we answer our questions and finally in section V, conclusion will be presented.
II. Method
The goal of this study is conducting SLR to assess the relationship between ASD and OSSD. We used guidelines proposed by Kichenham [13] for performing our study. The main steps are explained in the next parts of this section.
II.1. Research Questions
The addressed questions in this study are: RQ1: Could ASD and OSSD have any relationship? RQ2: Are practices of one of them applicable in the second? RQ3: Can they integrate with each other?
As regards to RQ1, it was important for us to know any possible relation between these approaches. Our aim was not to address adequately relation of them; we were only looking for finding any relation. If so, then we could focus on next questions.
As regards to RQ2, we were looking for finding any practices or rules in one them which could be applicable by the second. For this question, we focused on software engineering concepts in either method.
As regards to RQ3, We were looking for researches and case studies that had reported in combining, integrating or collaborating of both methods in software projects or organizations. We were looking for feasibility of application of both methods simultaneously, even in many specific practices.
II.2. Research Process
Our search process for review was based on online searching in famous online databases which are addressed as table I. Since these databases cover almost all major journals and conference proceedings, manually review of journal was not required. Review has been carried on by mean of search facilities in these databases and using appropriate logical expressions. In first stage, our focus was on title and abstract of articles found in search process and select appropriate and relevant studies. If there was any doubt, our decision was based on reviewing it at one glance. If not sure for choosing it, in next step we asked an expert to help us in decision making. Our final step was direct contact to authors, which was not used in selection process.
II.3. Inclusion and exclusion criteria
Papers which only had focused on one approach and did not discuss the other were ignored. There were some papers which were relevant to our study indirectly, but, in our defined process, we could not find them. This is not a threat for our review, because all appropriate papers were included and our review covered enough direct studies in this research.
II.4. Quality Assessment
For assessing studies we defined the following questions: QA1. Does study agree with existence of any relation between ASD and OSSD?
QA2. Does study report any similar practice in both methods?
QA3. Does study report successful use of both methods simultaneously?
QA4. Is there any practice in one of them which is useful and helpful in the second?
QA5. Is there any successful case study in integrating them?
QA6. Does study agree with feasibility of integration or collaboration of ASD and OSSD?
We scored questions as bellow:
QA1. Y (Yes) study explicitly agrees with existence of any relationship; P (Partially) study implicitly agrees and N (No) study disagrees with existence of any relation. QA2. Y, the authors address one or more similar practices; P, some of the ones practices could be tailored and customized in the second and N, there is no similar and adaptable practices in them.
QA3. Y, Authors address successful case study using both ASD and OSSD simultaneously; P, Authors address some case studies which use some of the ASD practices and some of the OSSD practices simultaneously and N, there is no case study of application of ASD and OSSD simultaneously.
QA4. Y, the authors report any useful practices of one that is applicable and helpful in the second; P, customized and tailored practices of each one, could be benefit in the second and N, there is no practice of each one, usable in the second.
QA5. Y, study addresses successful case study of integration of ASD and OSSD; P, study addresses case study of integration ASD and OSSD which is partly successful and N, there is no successful case study of integration of ASD and OSSD.
QA6. Y, study agrees with feasibility of integration of ASD and OSSD; P, study partly agrees (or implicitly) with integration ASD and OSSD and N, study rejects feasibility of integration between ASD and OSSD or has no idea about it.
We defined Y=1, P=0.5 and N=0 or Unknown where information is not clearly specified. All authors assessed every article and if there is no agreement in scoring, we discussed enough to reach agreement. For unknown questions we should ask study's authors via email and rescore question based on received answers.
II.5. Data Collection
These data were extracted from each article:
• The full source and references
• The author(s) information and details
• Research issues
• Main ideas and our questions related information and candidate answers All articles were reviewed and data was extracted by one person and checked by another. This idea was chosen for better consistency in reviewing all papers and improving quality of review. In any disagreement, authors discussed to reach to an agreement.
II.6. Data Analysis
Our collect data was organized to address:
• Whether study agrees with existence of any relationship between ASD and OSSD or not? (Addressing RQ1)
• Whether study mentions similar practice/concept in either methods or no? (Addressing RQ2)
• Whether study reports simultaneously application of two methods or no? (Addressing RQ2, RQ3)
• Whether study addresses any useful practice of one method that is applicable in the other or not? (Addressing RQ2 and RQ3)
• Whether study reports any successful integration of two methods or not? (Addressing RQ3)
• Whether authors believe that ASD and OSSD integration or collaboration is feasible or no? (Addressing QR3)
III. Results
In this section we explain results of our review. Table II shows the results of our selection procedure. In this table, results of searching in all databases are provided, but, some of the studies were repeated in more than one online database, so, final number of unique studies selected for our review was distinguished after elimination of repeated articles. Final selected studies are listed in table III.
III.1. Search Results
III.2. Quality evaluation of studies
During this phase, we found out that some of the selected articles however claimed to be related to both ASD and OSSD, but, they do not provide any valuable information to our research, so, we decided to delete them from scope of our study. List of this study and main reason for eliminating of them is provided in table IV.
Assessment of each study was done by means of criteria explained in section 2.4 and the scores for each of them are shown in table V.
III.3. Quality factors
For assessing results of our quality questions, we use average of total scores. This average is useful for some questions, but it is not useful for some other. For instance, we cannot answer the question about possibility of integration with average of scores because of the nature of the question; instead, we use negative ideas for rejecting possibility.
IV. Discussion
In this part, the answers to our study questions will be discussed.
IV.1. Relation between ASD and OSSD: yes or no?
Most of the articles agree that there are relationship between ASD and OSSD. By reviewing them, it seems that this relation is mostly in how to manage ASD and OSSD project. Also in some studies authors claim that OSSD is one type of ASD [11]- [14]. Hence, our research results support our first question strongly. It seems that most of the authors agree with the perception of relationship between two approaches. This relation sometimes could be found in comparing their features [15]- [16].
IV.2. Support Practices
17 studies strongly believe that ASD and OSSD have similar practices. These similar practices mainly return to principles of ASD and OSSD; e.g. both of them rely on self-organized teams and shared goals in team inputs [17]. Of course they have different management [15], but, teams are based on creativity of individuals. Another main common issue is incremental development. However there is different point of view about software development in these methods, but, multi releases is a common concept in both of them.
Nevertheless, only one half of studies present case studies that have used both methodologies simultaneously. In two studies, using TDD in an open source project was not only successful but also caused better code quality [18]- [19]. Three other studies were reported about one European project, 'PyPy'. However they had focused on different view, but, all had talked about successfully using ASD and OSSD simultaneously [20]- [21]- [22]. Eclipse was another project which has used ASD and OSSD simultaneously [23]. Interestingly, all of the papers agree that ASD and OSSD are able to support each other, even at least in some specific practices or areas [9] . One study explained how they used agile practices in a safety-critical open source project [9]. Authors had claimed that using these approaches together afford benefits to both of them. Another author had claimed that agile method helped his team to track the progress in open source project [24]. In some other studies, authors had tried to use concepts of one methodology to another, which were out of our scope [25]- [26].
In sum up, our research shows that ASD and OSSD can help each other and collaborate in some practices.
IV.3. Integration of ASD and OSSD
One critical issue in our study was integration of ASD and OSSD. We do not see any claim on comprehensive integration of these methods. Only about a quarter of studies have presented evidence about integration. We were looking for case studies which claim on integration of these methods directly, but we only found that there is collaboration between them and nothing more. It seems that authors were cautious about this matter in their case studies. Most of them implicitly believe that integration of ASD and OSSD is possible, at least in some specific practices or projects [27].Meanwhile some studies did not disagree with possibility of ASD and OSSD integration [9]- [15]- [27]- [28].Also some of them mentioned that adoption is a necessary activity for using ASD and OSSD together [29]. Three studies [23]- [30]- [31], by presenting successful case studies and another one [32], without any case study agreed with possibility of successful integration. So, we have not found any clear case study on successful integration of both methodologies, but it seems that integration in some practices is feasible by doing appropriate adoption. It discusses on SD methodologies in point of view of knowledge without concern about their relation to each other.
Detecting agility of open source projects through developer engagement Adams et al. 2008 It focuses on determination of agility in open source projects. Finding agility of OSS is its main aim and not relation or even combination of ASD and OSSD.
Detecting agility of open source projects through developer engagement Dimitropoulos 2009 This paper cannot answer none of our questions, it only explain that open source can play a role as agility enabler in projects.
V. Conclusion
Software engineers in last decade have been interested in agile methodology and open source software development. Both of them present some new features and they seem beneficial for better and faster software development. By doing an SLR we were looking for relationship between ASD and OSSD. Fortunately our study shows that both ASD and OSSD can help each other and collaborate in doing software projects by sharing their practices. There are enough evidences that agile and open source practices can support each other, mainly because of some of their common concepts and principles. Also, however, there are a few successful experiences on integration of ASD and OSSD, but, most of the studies are optimistic in possibility of their integration, but there is no empirical successful case study for supporting this idea in software producing industry.
TABLE I
ISTUDIES RESOURCE
Source
Address
Scopus
www.scopus.com
IEEE Xplore
ieeexplore.ieee.org
ACM Digital Library
Portal.acm.org
Springer Link
www.springerlink.com
Tailor and Francis
www.tandfonline.com
Science Direct
www.sciencedirect.com
TABLE II RESULTS
IIOF STUDY SELECTION PROCEDURESource
Search Results
Selected Studies
Scopus
113
18
IEEE Xplore
70
4
ACM Digital Library
18
4
Springer Link
33
9
Science Direct
4
3
Total
-
38
Repeated articles
11
Finally selected articles
27
TABLE III SELECTED
IIISTUDIES FOR CONDUCTING REVIEW S20 Open Source and Agile Methods: Two Worlds Closer than It Seems K. Gray et al. A case study in using agile methods in OSSD 2011ID
Title
Author(s)
Main Topic
Year
S1
Open source development and Agile methods
Simmons and Dillon
Relation between ASD and OSSD
2003
S2
Is Open Source Software Development Essentially an
Agile Method?
Warsta and
Abrahamsson
Relation between ASD and OSSD
2003
S3
Introducing TDD on a free libre open source software
project: a simulation experiment
Turnu et al.
The effects of adopting TDD on our open source
2004
S4
Agile Principles and Open Source Software
Development: A Theoretical and Empirical Discussion
Koch
Investigation on accordance of ASD and OSSD
2004
S4
Agile, open source, distributed, and on-time -Inside
the eclipse development process
Gamma
Discuss on Eclipse project
2005
S5
In search of the sweet spot: agile open collaborative
corporate software development
Theunissen et al.
Combining ASD and OSSD
2005
S6
Open source development and Agile methods
Fraser et al.
Strategies, tools, and communities focused on
OSSD
2006
S7
Open source software in an agile world
B. Düring
A case study in combining ASD and OSSD(PyPy)
2006
S8
Sprint driven development: Agile methodologies in a
distributed open source project (PyPy)
B. Düring
A case study in combining ASD and
OSSD(PyPy)
2006
S9
Trouble in paradise: the open source project PyPy,
EU-funding and agile practices
Turnu et al.
Study the effects of the adoption of agile
practices on OSSD
2006
S10
Modeling and simulation of open source development
using an agile practice
Porruvecchio et al.
Relation between ASD and OSSD
2007
S11
An agile approach for integration of an open source
health information system
Avotins et al.
How agile can help OSSD
2007
S12
The case for innovative open source development and
agile methods
Goth
A case study in combining ASD and
OSSD(PyPy)
2007
S13
Sprinting toward open source development
Deshpande and Riehle
Investigation on impact of continuous integration
of ASD on OSSD
2008
S14
Continuous integration in open source software
development
Theunissen et al.
Relation between ASD and OSSD
2008
S15
Corporate, Agile and Open Source Software
development: A witch's brew or an elixir of life?
Adams and Capiluppi
evaluating the impact of sprinting on a Free
Software project
2009
S16
Bridging the gap between agile and free software
approaches: The impact of sprinting
Tsirakidis et al.
Similarity and differences of ASD and OSSD
2009
S17
Identification of success and failure factors of two
agile software development teams in an open source
organization
Wusteman
Adoption of ASD and OSSD
2009
S18
OJAX: A case study in agile Web 2.0 open source
development
Lavazza et al.
A case study in using SCRUM for the
development of an OSSD
2010
S19
Applying SCRUM in an OSS Development Process: An
Empirical Evaluation
Corbucci and
Goldman
Identify communication issues encountered in
ASD and OSSD
2010
S21
Agile methods for open source safety-critical software
Okoli and Carillo
Comparing OSSD with ASD and Disciplined
methods
2011
S22
The best of adaptive and predictive methodologies:
Open source software development, a balance between
agility and discipline
Magdaleno et al.
A review study on relation of famous software
development methods.
2012
S23
Reconciling software development models: A quasi-
systematic review
Simmons and Dillon
Relation between ASD and OSSD
TABLE IV REJECTED
IVSTUDIES DURING QUALITY ASSESSMENTAOSTA: Agile Open Source Tools AcademyWild 2006 It is a workshop report about a tool and cannot be valid for our study.Title
Author(s)
Year
Reason for Rejection
System development methodologies: A
knowledge perspective
Kerley et al.
2006
TABLE V QUALITY
VEVALUATION
Source
QA1
QA2
QA3
QA4
QA5
QA6
1
Y
N
P
Y
N
P
2
y
y
n
y
n
p
3
Y
N
Y
Y
p
p
4
Y
P
N
Y
n
P
5
Y
y
y
y
p
y
6
y
y
n
y
n
y
7
y
y
n
y
n
p
8
Y
y
y
y
p
P
9
y
y
y
y
p
p
10
y
y
y
y
p
p
11
y
y
p
y
n
p
12
y
y
n
y
n
p
13
y
y
y
y
p
p
14
y
y
n
y
n
n
15
y
y
y
y
n
p
16
y
n
n
y
n
n
17
y
y
p
y
p
p
18
y
y
y
y
y
y
19
y
y
y
y
y
y
20
y
y
n
y
n
p
21
p
p
y
y
p
n
22
p
p
n
y
n
n
23
y
y
n
y
n
y
Average 0.96
0.80
0.50
1.00
26.1
0.52
Abdul Azim Abd Ghani obtained his Ph.D. from University of Strathclyde. Currently, he is a Professor at the Faculty of Computer Science and Information Technology, University Putra Malaysia. His research interests are software engineering, software measurement, software quality, and security in computing.Authors' information1,2,3,4Information System Deptartment Computer Science and Information Technology University Putra Malaysia Taghi Javdani Gandomani with 10 years experience in both industry and academic research in software methodologies and project management and with the other authors is working in Software Engineering Laboratory in University Putra Malaysia (UPM). They work on different research in SE, such as software metrics, software quality, software modeling and software methodologies. Mr. T. Javdani works on agile software methods and he is PhD student in software engineering in University Putra Malaysia.Hazura Zulzalil holds a Ph.D. from University Putra Malaysia. Currently, she is a senior lecturer at the Faculty of Computer Science and Information Technology, University Putra Malaysia. Her research interests are software metrics, software quality and software product evaluation.
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|
[] |
[
"online Gambling of pure chance: Wager Distribution, Risk Attitude, and Anomalous Diffusion",
"online Gambling of pure chance: Wager Distribution, Risk Attitude, and Anomalous Diffusion"
] |
[
"Xiangwen Wang \nDepartment of Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA\n\nCenter for Soft Matter and Biological Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA\n",
"Michel Pleimling \nDepartment of Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA\n\nCenter for Soft Matter and Biological Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA\n\nAcademy of Integrated Science\nVirginia Tech\n24061-0563BlacksburgVAUSA\n"
] |
[
"Department of Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA",
"Center for Soft Matter and Biological Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA",
"Department of Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA",
"Center for Soft Matter and Biological Physics\nVirginia Tech\n24061-0435BlacksburgVAUSA",
"Academy of Integrated Science\nVirginia Tech\n24061-0563BlacksburgVAUSA"
] |
[] |
Online gambling sites offer many different gambling games. In this work we analyse the gambling logs of numerous solely probability-based gambling games and extract the wager and odds distributions. We find that the log-normal distribution describes the wager distribution at the aggregate level. Viewing the gamblers' net incomes as random walks, we study the mean-squared displacement of net income and related quantities and find different diffusive behaviors for different games. We discuss possible origins for the observed anomalous diffusion.Today, gambling is a huge industry with a huge social impact. According to a report by the American Gaming Association 1 , commercial casinos in the United States alone made total revenue of over 40 billion US dollars in 2017. On the other hand, different studies reported that 0.12%-5.8% of the adults and 0.2-12.3% of the adolescents across different countries in the world are experiencing problematic gambling 2,3 . Studying the gamblers' behavior patterns not only contributes to the prevention of problematic gambling and adolescent gambling, but also helps to better understand human decision-making processes. Researchers have put a lot of attention on studying gambling-related activities. Economists have proposed many theories about how humans make decisions under different risk conditions. Several of them can also be applied to model gambling behaviors. For example, the prospect theory introduced by Kahneman and Tversky 4 and its variant cumulative prospect theory 5 have been adopted in modeling casino gambling 6 . In parallel to the theoretical approach, numerous studies focus on the empirical analysis of gambling behaviors, aiming at explaining the motivations behind problematic gambling behaviors. However, parametric models that quantitatively describe empirical gambling behaviors are still missing. Such models can contribute to evaluating gambling theories proposed by economists, as well as yield a better understanding of the gamblers' behaviors. Our goal is to provide such a parametric model for describing human wagering activities and risk attitude during gambling from empirical gambling logs. However, it is very difficult to obtain gambling logs from traditional casinos, and it is hard to collect large amounts of behavior data in a lab-controlled environment. Therefore in this paper we will focus on analyzing online gambling logs collected from online casinos.Whereas historically the development of probability theory, which then became the foundation of statistics, was tied to chance games, today we use statistical tools to analyze gamblers' behaviors.Recent years have seen an increasing trend of online gambling due to its low barriers to entry, high anonymity and instant payout. For researchers of gambling behaviors, online gambling games present two advantages: simple rules and the availability of large amounts of gambling logs. In addition to the usual forms of gambling games that can be found in traditional casinos, many online casinos also offer games that follow very simple rules, which makes analyzing the gambling behavior much easier as there are much fewer degrees of freedom required to be considered. On the other hand, many online casinos have made gambling logs publicly available on their websites, mainly for verification purposes, which provides researchers with abundant data to work on. Due to the high popularity of online gambling, in a dataset provided by an online casino there are often thousands or even hundreds of thousands of gamblers listed. Such a large scale of data can hardly be obtained in a lab environment. Prior research has begun to make use of online gambling logs. For example, Meng's thesis 7 presented a pattern analysis of typical gamblers in Bitcoin gambling. It is worth arguing that although our work only focuses on the behaviors of online gamblers, there is no reason to think that our conclusions cannot be extended to traditional gamblers.
|
10.1038/s41598-019-50168-2
| null | 202,541,318 |
1909.02343
|
b825cff3848b4cef3bc556eeb00a1e54d37cd4ee
|
online Gambling of pure chance: Wager Distribution, Risk Attitude, and Anomalous Diffusion
Xiangwen Wang
Department of Physics
Virginia Tech
24061-0435BlacksburgVAUSA
Center for Soft Matter and Biological Physics
Virginia Tech
24061-0435BlacksburgVAUSA
Michel Pleimling
Department of Physics
Virginia Tech
24061-0435BlacksburgVAUSA
Center for Soft Matter and Biological Physics
Virginia Tech
24061-0435BlacksburgVAUSA
Academy of Integrated Science
Virginia Tech
24061-0563BlacksburgVAUSA
online Gambling of pure chance: Wager Distribution, Risk Attitude, and Anomalous Diffusion
10.1038/s41598-019-50168-2Received: 1 July 2019 Accepted: 5 September 20191 Scientific RepoRtS | (2019) 9:14712 | https://. Correspondence and requests for materials should be addressed to M.P. (email: [email protected])
Online gambling sites offer many different gambling games. In this work we analyse the gambling logs of numerous solely probability-based gambling games and extract the wager and odds distributions. We find that the log-normal distribution describes the wager distribution at the aggregate level. Viewing the gamblers' net incomes as random walks, we study the mean-squared displacement of net income and related quantities and find different diffusive behaviors for different games. We discuss possible origins for the observed anomalous diffusion.Today, gambling is a huge industry with a huge social impact. According to a report by the American Gaming Association 1 , commercial casinos in the United States alone made total revenue of over 40 billion US dollars in 2017. On the other hand, different studies reported that 0.12%-5.8% of the adults and 0.2-12.3% of the adolescents across different countries in the world are experiencing problematic gambling 2,3 . Studying the gamblers' behavior patterns not only contributes to the prevention of problematic gambling and adolescent gambling, but also helps to better understand human decision-making processes. Researchers have put a lot of attention on studying gambling-related activities. Economists have proposed many theories about how humans make decisions under different risk conditions. Several of them can also be applied to model gambling behaviors. For example, the prospect theory introduced by Kahneman and Tversky 4 and its variant cumulative prospect theory 5 have been adopted in modeling casino gambling 6 . In parallel to the theoretical approach, numerous studies focus on the empirical analysis of gambling behaviors, aiming at explaining the motivations behind problematic gambling behaviors. However, parametric models that quantitatively describe empirical gambling behaviors are still missing. Such models can contribute to evaluating gambling theories proposed by economists, as well as yield a better understanding of the gamblers' behaviors. Our goal is to provide such a parametric model for describing human wagering activities and risk attitude during gambling from empirical gambling logs. However, it is very difficult to obtain gambling logs from traditional casinos, and it is hard to collect large amounts of behavior data in a lab-controlled environment. Therefore in this paper we will focus on analyzing online gambling logs collected from online casinos.Whereas historically the development of probability theory, which then became the foundation of statistics, was tied to chance games, today we use statistical tools to analyze gamblers' behaviors.Recent years have seen an increasing trend of online gambling due to its low barriers to entry, high anonymity and instant payout. For researchers of gambling behaviors, online gambling games present two advantages: simple rules and the availability of large amounts of gambling logs. In addition to the usual forms of gambling games that can be found in traditional casinos, many online casinos also offer games that follow very simple rules, which makes analyzing the gambling behavior much easier as there are much fewer degrees of freedom required to be considered. On the other hand, many online casinos have made gambling logs publicly available on their websites, mainly for verification purposes, which provides researchers with abundant data to work on. Due to the high popularity of online gambling, in a dataset provided by an online casino there are often thousands or even hundreds of thousands of gamblers listed. Such a large scale of data can hardly be obtained in a lab environment. Prior research has begun to make use of online gambling logs. For example, Meng's thesis 7 presented a pattern analysis of typical gamblers in Bitcoin gambling. It is worth arguing that although our work only focuses on the behaviors of online gamblers, there is no reason to think that our conclusions cannot be extended to traditional gamblers.
Naturally, we can treat the changing cumulative net income of a player during their gambling activities as a random walk process 8 . We are particularly interested in the diffusive characteristics of the gambler's net income. This is another reason why we want to analyze the wager distribution and risk attitude of gamblers, since both distributions are closely related to the displacement distribution for the gambler's random walks. Within this paper, we will mainly focus on the analysis at the population level. Physicists have long been studying diffusion processes in different systems, and recently anomalous diffusive properties have been reported in many human activities, including human spatial movement [9][10][11] , and information foraging 12 . In a previous study of skin gambling 8 , we have shown that in a parimutuel betting game (where players gamble against each other), a gambler's net income displays a crossover from superdiffusion to normal diffusion. We have reproduced this crossover in simulations by introducing finite and overall conserved gamblers' wealth (see 13 for a different way of modeling this using kinetic equations of Boltzmann and Fokker-Planck type). However, this explanation cannot be used in other types of gambling games where there is no interaction among gamblers (e.g., fixed-odds betting games, which will be introduced below), as they violate the conservation of gamblers' overall wealth. In this paper, we want to expand the scope of our study to more general gambling games, check the corresponding diffusive properties, and propose some explanations for the observed behaviors.
One of our goals is to uncover the commonalities behind the behavior of online gamblers. To implement this, we analyze the data from different online gambling systems. The first one is skin gambling, where the bettors are mostly video game players and where cosmetic skins from online video games are used as virtual currency for wagering 8,14 . The other system is crypto-currency gambling, where the bettors are mostly crypto-currency users. Different types of crypto-currencies are used for wagering. Commonly used crypto-currencies include Bitcoin, Ethereum, and Bitcoin Cash, whose basic units are BTC, ETH and BCH, respectively. As the overlap of these two communities, video game players and crypto-currency users, is relatively small for now, features of gambling patterns common between these two gambling systems are possibly features common among all online gamblers.
Not only do we consider different gambling systems, but we also discuss different types of gambling games. In this paper, we discuss four types of solely probability-based gambling games (Roulette, Crash, Satoshi Dice and Jackpot), whose outcomes in theory will not benefit from the gamblers' skill or experience when the in-game random number generators are well designed. In general, there are two frameworks of betting in gambling: fixed-odds betting, where the odds is fixed and known before players wager in one round; and parimutuel betting, where the odds can still change after players place the bets until all players finish wagering. In fixed-odds betting, usually players bet against the house/website, and there is no direct interaction among players; and in parimutuel betting, usually players bet against each other. The four types of games we discuss in this paper will cover both betting frameworks (see the Methods section).
When a player attends one round in any of those games, there are only two possible outcomes: either win or lose. When losing, the player will lose the wager they placed during that round; whereas when winning, the prize winner receives equals their original wager multiplied by a coefficient. This coefficient is generally larger than 1, and in gambling terminology, it is called odds in decimal format 15,16 . Here we will simply refer to it as odds. Note that the definition of odds in gambling is different than the definition of odds in statistics, and in this paper we follow the former one. When a player attends one round, their chance of winning is usually close to, but less than the inverse of the odds. The difference is caused by the players' statistical disadvantage in winning compared to the house due to the design of the game rules. In addition, the website usually charges the winner with a site cut (commission fee), which is a fixed percentage of the prize.
We further define the payoff, o p , to be the net change of one player's wealth after they attend one round. Although the four types of games are based on different rules, the payoffs all follow the same expression which is always negative since either η or f m are non-zero. In gambling terminology, ξ is called the house edge, from which the websites make profits. The house edge represents the proportion the website will benefit on average when players wager. In the four types of games we discuss, the house edge ξ ranges from 1% to 8%. If there is no house edge ξ = 0, that means it is a fair game. In a fair game or when we ignore the house edge, the expected payoff would be 0.
= − − + + − − − = − − + − + ≡ − ( ) E o m b m f m m f b mf m f b b , ( (1 1/ ) (1 )( 1)(1/ ) ) ((1 ) ( 1 1/ ) ) ,(2)
In the Results section, we begin with an analysis of wager distribution and log-ratios between successive wagers, which helps us to understand the gamblers' wagering strategy. We then focus on an analysis of risk attitude by studying the distribution of the odds players choose to wager with. We conclude by extending our discussion to the analysis of net incomes of gamblers viewed as random walks. This allows us to gain insights into the
Results
Wager distribution. From the viewpoint of the interaction among players, the games discussed in this paper can be grouped into two classes: in Roulette, Crash, and Satoshi Dice games, there is little or no interaction among players, whereas in Jackpot games, players need to gamble against each other. At the same time, from the viewpoint of wager itself, the games can also be grouped into two classes: In games (A-G), the wagers can be an arbitrary amount of virtual currencies, such as virtual skin tickets or crypto-currency units, whereas in game (H), the wagers are placed in the form of in-game skins, which means the wager distribution further involves the distributions of the market price and availability of the skins.
Furthermore, from the viewpoint of the odds, considering the empirical datasets we have, when analyzing the wager distribution, there are three situations: i) For Roulette and Satoshi Dice games, the odds are fixed constants, and wagers placed with the same odds are analyzed to find the distribution. ii) For Crash games, the odds are selected by the players, and wagers placed with different odds are mixed together during distribution analysis. iii) For the Jackpot game, the odds are not fixed at the moment when the player wagers.
In Table 1 we categorize the 8 datasets based on the above information. At the same time, for each dataset we perform a distribution analysis of wagers at the aggregate level. Within the same dataset wagers placed under different maximum allowed bet values are discussed separately. We plot the complementary cumulative distribution function (CCDF) of the empirical data and the fitted distribution to check the goodness-of-fit, see Fig. 1. CCDF, sometimes also referred to as the survival function, is given by
= > = − ≤ F x P X x PX x ( ) ( ) 1 (
). It turns out that when players are allowed to place arbitrary wagers (games A-G in Table 1), the wager distributions can in general be best-fitted by log-normal distributions. In particular, in games (A, B, C, E, F, G), the wager distribution can be approximated by the following expression
= Φ − Φ − Φ μ σ μ σ μ σ + − − − ( ) ( ) ( ) P x ( ) 1 ,(3)
x x
x ln( 1) l n( ) ln( ) min with ≤ x x min and σ > 0. Φ ⋅ ( ) is the cumulative distribution function of the standard normal distribution. Meanwhile in game (D), the fitted log-normal distribution is truncated at an upper boundary x max , which might result from the maximum allowed small bet value and the huge variation of the market price of crypto-currencies.
During model selection, we notice that when we select different x min , occasionally a power-law distribution with exponential cutoff is reported to be a better fit, but often it does not provide a decent absolute fit on the tail, and overall the log-normal distribution provides smaller Kolmogorov-Smirnov distances, see the Methods section. Table 1. The best-fitted distribution and estimated parameters of wagers. For games (A, B, C, E, F, G) the best-fitted model is a log-normal distribution, and for game (D) the log-normal distribution is truncated at a maximum value. For game (H) the wager distribution follows a power law -exponential -power law pattern. In the rightmost column, μ (respectively σ 2 ) represents the mean (respectively variance) of the logarithms of bet values.
On the other hand, as we have pointed out in the previous study 8 , when players are restricted to use in-game skins as wagers for gambling, the wager distribution can be best fitted by a shifted power law with exponential cutoff. Now, with a similar situation in game (H), where wagers can only be in-game skins, we find that the early part of the curve can be again fitted by a power law with exponential cutoff, as shown in Fig. 1(H). However, this time it does not maintain the exponential decay of its tail; instead, it changes back to a power-law decay. The overall distribution contains six parameters, given by the expression x trans ( trans ) . We believe that when players are restricted to use in-game skins as wagers, the decision to include one particular skin in their wager is further influenced by the price and availability of that skin. These factors make the wager distribution deviate from the log-normal distribution, which is observed in games (A-G). This is very clear when comparing the wager distributions of games (G) and (H) as both games are jackpot games of skin gambling, and the only difference is whether players are directly using skins as wagers or are using virtual skin tickets obtained from depositing skins. The power-law tail, which was not observed in the previous study 8 , might result from the increment of the maximum allowed skin price (from $400 to $1800).
δ = + − + ≤ + > α λ β η − − − P x c c c x e x x c c c c x x x ( ) 1 ( ) 1 , for , , f or ,(4)
The above discussions, including the results for games (A-G) in Table 1, show that the wager distributions in pure probability-based gambling games, no matter whether the game follows parimutuel betting or fixed-odds (preset/player-selected) betting, stay log-normal as long as the players are allowed to place arbitrary amounts of wagers. This commonality of log-normal distribution no longer holds when this arbitrariness of wager value is violated, e.g., in the scenario where the player can only wager items (in-game skins).
Log-normal distribution has been reported in a wide range of economic, biological, and sociological systems 17 , including income, species abundance, family size, etc. Economists have proposed different kinds of www.nature.com/scientificreports www.nature.com/scientificreports/ generative mechanisms for log-normal distributions (and power-law distributions as well). One particular interest for us is the multiplicative process 18,19 . Starting from an initial value X 0 , random variables in a multiplicative process follow an iterative formula ν =
+ X X exp( ) i i i 1 or ν = + + X X ln l n i i i 1 .
If the v i has finite mean and variance, and is independent and identically distributed, then according to the central limit theorem, for large i, ln X i will follow a normal distribution, which means X i will follow a log-normal distribution.
If we want to check whether gamblers follow multiplicative processes when they wager, we can first check the correlation between consecutive bets
+ b b ( , ) i i 1 .
Due to the large variances of the wager distributions, Pearson's correlation coefficient may perform poorly. Instead, we adopt two rank-based correlation coefficients, Kendall's Tau 20 τ K and Spearman's Rho 21 ρ S . At the same time, we also check the mean and variance of the log-ratios
+ b b ln( / ) i i 1
between consecutive bets. These statistics can be found in Table 2. The results reveal that the values of consecutive bets exhibit a strong positive correlation, with all the correlation coefficients larger than 0.5. It shows that players' next bet values are largely dependent on their previous bet values. At the same time, the bet values are following gradual changes, rather than rapid changes. These conclusions can be confirmed by the small mean values and small variances of log-ratios between consecutive bets.
Further analysis of the distribution of ν shows an exponential decay on both of its tails, see Fig. 2. This means that ν approximately follows a Laplace distribution. However, compared to a Laplace distribution, the empirical log-ratio distribution shows a much higher probability at ν = 0, whose value can be found in the last column of Table 2. We also observe that ν presents higher probability densities around small integers/half-integers and their inverses. Due to the existence of these differences, we will skip the parameter fitting for the distribution of ν. The high probability of staying on the same wager indicates that betting with fixed wager is one of the common strategies adopted by gamblers.
Meanwhile, the high positive auto-correlations, along with the higher probability densities at small integers/ half-integers and their inverses, provide evidence that gamblers often follow a multiplicative process when wagering. The multiplication process can be explained by the wide adoption of multiplicative betting systems. "Betting system" here refers to the strategy of wagering where the next bet value depends on both the previous bet value and the previous outcome 22,23 . Although betting systems will not provide a long-term benefit, as the expected payoff will always be 0 in a fair game, still they are widely adopted among gamblers. A well-known multiplicative betting system is the Martingale (sometimes called geometric progression) 23 . In Martingale betting, starting with an initial wager, the gambler will double their wager each time they lose one round, and return to the initial wager once they win. Martingale is a negative-progression betting system where the gambler will increase their wager when they lose and/or decrease their wager when they win.
Apart from multiplicative betting, there are many other types of betting systems, such as additive betting and linear betting 23 . The reasons why multiplicative betting systems are dominant in our datasets are: 1) Martingale is a well-known betting system among gamblers; 2) Many online gambling websites provide a service for changing the bet value in a multiplicative way. For example, for the Crash games csgofast-Crash (C) and ethCrash (D), both websites provide a simple program for automatically wagering in a multiplicative way. For the Roulette games and Coinroll (F), the websites provide an interface with which the gambler can quickly double or half their wager. However, for Satoshi Dice (E) and csgospeed-Jackpot (G), no such function is provided, yet we still observe similar results, indicating that gamblers will follow a multiplicative betting themselves. Figure 2 provides us with the distribution of ν, however, it will not tell us whether the gamblers adopt the negative/positive-progression betting systems. Therefore we further analyze the effect on the bet values of winning/losing a round. How the gamblers adjust their wager after winning/losing rounds is shown in Table 3. We can see that although there is a high probability for sticking to the same bet values, the most likely outcome after losing a round is that the gambler increases their wager. When winning one round, gamblers are more likely to decrease their wager. This means that negative-progression strategies are more common among gamblers than positive-progression strategies.
Risk attitude.
We now turn to the following question: When a player is allowed to choose the odds themselves in a near-fair game, how would they balance the risk and potential return? Higher odds means a lower chance of winning and higher potential return, for example, setting odds of 10 means that the winning chance is only 1/10, but the potential winning payoff equals 9 times the original wager. In our analysis, we can examine such www.nature.com/scientificreports www.nature.com/scientificreports/ behaviors based on the gambling logs from Crash and Satoshi Dice games. For the Crash game only CSGOFAST. COM provides the player-selected odds even when players lose that round, whereas for the Satoshi Dice game only Coinroll accepts player-selected odds. We will therefore focus on the data collected on these two websites. For the Crash game on CSGOFAST.COM, the odds can only be set as multiples of 0.01, whereas for the Satoshi Dice game on Coinroll the odds can be set to 0.99·65536/i where i is a positive integer less than 64000. To simplify our modeling work, we will convert the odds on Coinroll to be multiples of 0.01 (same as for the Crash game).
Dataset τ K (b i , b i+1 ) ρ s (b i , b i+1 ) 〈log 10 (b i+1 /b i )〉 var(log 10 (b i+1 /b i )) P(b i = b i+1 )
It turns out that in both cases the odds can be modeled with a truncated shifted power-law distribution,
δ ζ α δ ζ α δ ζ α δ = − − ≤ < − − = α − P m m m m m m m m m m ( ) ( ) ( , ) , for ,( , ) ( , )
, for , where ζ ⋅ ⋅ ( , ) is the incomplete Zeta function, and m max is the upper truncation. Note that there is a jump at m max , meaning that the players are more likely to place bets on the maximum allowed odds than on a slightly smaller Figure 2. The distribution of the logarithmic of the ratio (log-ratio) between consecutive bet values. For games (A-C), the log-ratio can be described by a Laplace distribution. For games (D,F-H), the log-ratio presents bellshaped distribution. In general, the distributions are symmetric with respect to the y-axis, except in games (D,F). The x-coordinate
+ b b log ( / ) i i 10 1
is proportional to the parameter ν. Table 3. Statistics about how gamblers change their bet values after winning/losing rounds. Apart from fixedwagering betting, a comparison between the probabilities suggests gamblers prefer negative-progression betting rather than positive-progression betting. See the caption of Table 2 www.nature.com/scientificreports www.nature.com/scientificreports/ odds. The estimated parameters α = .
Dataset
After Losing After Winning
P(b i+1 > b i ) P(b i+1 = b i ) P(b i+1 < b i ) P(b i+1 > b i ) P(b i+1 = b i ) P(b i+1 < b i )
1 881, δ = . 0 849, and = . m 1 15 min for csgofast-Jackpot on CSGOFAST. COM, whereas for Coinroll the parameters are found to be α = .
1 423, δ = .
2 217, and = . m 2 58 min . From the comparison between the CCDFs of empirical data and fitting curves, as shown in Fig. 3, we can see that the truncated shifted power law can capture the overall decaying trends of odds distribution. The stepped behavior results from the gamblers' preference of simple numbers.
A distribution that is close to a power law indicates that a gambler's free choice of odds displays scaling characteristics (within the allowed range) in near-fair games. It also means that when gamblers are free to determine the risks of their games, although in most times they will stick to low risks, showing a risk-aversion attitude, they still present a non-negligible probability of accepting high risks in exchange for high potential returns. The scaling properties of risk attitude might not be unique to gamblers, but also may help to explain some of the risk-seeking behaviors in stock markets or financial trading.
We now re-examine the distributions from the point of view of estimating the crash point m C (Satoshi Dice games can be explained with the same mechanism). The true distribution of m C generated by the websites follow a power-law decay with an exponent of 2 (with some small deviation due to the house edge). Meanwhile, a closer look at the fitted exponents listed above gives us two empirical exponents of 1.423 and 1.881, both of which are smaller than 2. The smaller exponents reveal that gamblers believe that they have a larger chance to win a high-odds game than they actually do. Or equivalently, it means the gamblers over-weight the winning chance of low-probability games. At the same time, the "shifted" characteristics here lead to more bets on small odds, which also indicates that the gamblers over-estimate the winning chance of high-probability games. As a result, they under-weight the winning chances of mild-probability games. These are clear empirical evidence of probability weighting among gamblers, which is believed to be one of the fundamental mechanisms in economics 6 .
Wealth distribution. In the previous study of skin gambling 8 , we pointed out that the wealth distribution of skin gamblers shows a pairwise power-law tail. This time, by considering the players' deposits to their wallets on a gambling site as the wealth data, we find that the pairwise power-law tails are also observed for bitcoin gambling. We find that on the gambling website Coinroll, starting from 5660 cents, the players' wealth distribution follows a pairwise power-law distribution, with the power of the first regime to be 1.585, and the power of the second regime to be 3.258, see Fig. 4. The crossover happens at 1.221 × 10 5 cents. As both wealth distributions of skin gambling and bitcoin gambling can be approximated by a pairwise power distribution, we believe that it is a good option for modeling the tails of gambler wealth distribution in different scenarios.
Removing effects due to inequality in the number of bets. In the above sections, we have analyzed the distributions of several quantities at the population level. However, there is a huge inequality of the number of placed bets among gamblers. We therefore wonder whether those distributions we obtain result from the inequality of number of bets among individuals. To remove the effects of this inequality, we randomly sample in each dataset the same number of bets from heavy gamblers. We re-analyze the wager distribution and odds distribution with the sample data to see if we obtain the same distribution as before. In each dataset we randomly sample 500 bets from each of those gamblers who placed at least 500 bets above b min given in Table 1. Some datasets are excluded here as either they do not have enough data or we cannot identify individual gamblers. When re-analyzing the odds distribution, to ensure we have enough data, we respectively sample 100 and 2000 bets from each of those gamblers in games (C) and (F) who have at least 100 and 2000 valid player-selected odds above m min . According to the results in Fig. 5, after removing the inequality the wager distributions can still be approximated by log-normal distributions, but some deviation can be observed. Similarly, the odds distributions again follow truncated shifted power-law distributions after removing the inequality. These results demonstrate that the shape of the distributions we obtained in the above sections is not a result of the inequality of the number of bets. Now our question becomes whether the conclusion regarding the distribution at the population level can be extended to the individual level. Here due to the limitation of data, we will only discuss the wager distribution. Analyzing the individual distribution of top gamblers, we find that although heavy-tailed properties can be www.nature.com/scientificreports www.nature.com/scientificreports/ widely observed at the individual level, only a small proportion of top gamblers presents log-normal distributed wagers. Other distributions encountered include log-normal distributions, power-law distributions, power-law distributions with exponential cutoff, pair-wise power-law distributions, irregular heavy-tailed distributions, as well as distributions that only have a few values. The diversity of the wager distributions at the individual level suggests a diversity of individual betting strategies. Also, it indicates that a gambler may not stick to only one betting strategy. It follows that the log-normal wager distribution observed at the population level is very likely an aggregate result.
Diffusive process. For an individual player's gambling sequence we define "time" t as the number of bets one player has placed so far, and define as net income the sum of the payoffs of those bets. In all the games we analyze, there are only two possible outcomes: a win or a loss. The player's net income will change each time they place a bet in a round, with the step length to be the payoff from that bet. We can treat the change of one player's net income as a random walk in a one-dimensional space (see Fig. 1 in ref. 8 for an example of such a trajectory). The time t will increase by 1 when the player places a new bet, therefore the process is a discrete-time random walk. Now, let us focus on the analysis of the diffusive process of the gamblers' net incomes, starting with the analysis of the change of the mean net income with the number of rounds played (time),
∆ = − = ∑ = w t w t w o i ( ) ( ) ( ) i t p 0 1
, where w 0 is the player's initial wealth, w(t) is the player's wealth after attending t rounds, and o i ( ) p is the payoff from the i th round the player attended. 〈⋅〉 represents an ensemble average over a population of players placing bets. In the rest of this paper, 〈⋅〉 will always be used for representing an ensemble average. In Fig. 6 we show the change of ∆w t ( ) over time. In most of the datasets, players' mean net www.nature.com/scientificreports www.nature.com/scientificreports/ income decreases over time, which suggests that in general players will lose more as they gamble more. At the same time, in some datasets such as Ethcrash (D) and Coinroll (F), large fluctuations can be observed.
An useful tool for studying the diffusive process is the ensemble-averaged mean-squared displacement (MSD), defined as For a normal diffusive process, ∆ ∼ w t t ( ) 2 , otherwise an anomalous diffusive behavior prevails. More specifically, when the MSD growth is faster (respectively, slower) than linear, superdiffusion (respectively, subdiffusion) is observed.
∑ ∆ = − = = w t w t w o i ( ) ( ( ) ) ( ) ,(6)
In Fig. 7, we present the growth of the ensemble-averaged MSD against time for each of the datasets. To reduce the coarseness, MSD curves are smoothed with log-binning technique. The error bars in Fig. 7 represent 95% confidence intervals computed with bootstrapping using 2000 independent re-sampling runs. It is interesting to see that for different datasets we observe different diffusive behaviors. For games csgofast-Crash (C) we observe that the MSD grows faster than a linear function, suggesting superdiffusive behavior. Meanwhile, for games csgofast-Double (A), ethCrash (D), csgospeed (G), and csgofast-Jackpot (H), the MSD first presents a superdiffusive regime, followed by a crossover to a normal diffusive regime. For games csgofast-X50 (B) and Coinroll (F), although the ensemble-averaged MSD roughly presents a linear/sublinear growth, a careful inspection shows that both curves consist of several convex-shaped regimes, indicating a more complex behavior. Convex-shaped regimes can also be observed in csgofast-Crash games (C).
In ref. 8 we argued that the crossover from a superdiffusive regime to a normal diffusive regime in a parimutuel game is due to the limitation of individuals' wealth and the conservation of total wealth. Similar crossovers are observed in games (G) and (H), two parimutuel betting games, where the same explanation can be applied. On the other hand, this crossover is also found in a Roulette game and in a Crash game, where there is no interaction among gamblers. The limitation of an individual's wealth can still be a partial explanation, but the conservation of total wealth no longer holds. A different explanation needs to be proposed to model this crossover.
In the following we briefly discuss how we can obtain from gambling models the different diffusive processes observed in the data. We will not attempt to reproduce the parameters we obtained from the gambling logs, but rather try to explore the possible reasons for the anomalous diffusion we reported.
For a gambling process, if the gambler's behavior is independent among different rounds, i.e., the wager and odds are respectively independent and identically distributed (IID), with no influence from the previous outcomes, and if the wager b has finite variance and the odds m has finite mean, then MSD's growth will be a linear function of time t: www.nature.com/scientificreports www.nature.com/scientificreports/ where 〈 〉 m is the mean value of odds distribution and b 2 is the second moment of the wager distribution. But normal diffusion is only found in few datasets, the remaining datasets presenting anomalous diffusion which conflicts with the IID assumption.
∆ = − = 〈 〉 − w t w t w m b t ( ) ( ( ) ) ( 1 ) ,(7)
Having shown the popularity of betting systems among gamblers, we would like to check how different betting systems affect diffusive behaviors. First, we simulate gamblers that follow Martingale strategies in a Crash game. We assume that the selection of odds follows a power-law distribution with an exponent α, with a minimum odds of 1 and a maximum odds of 50, where the maximum odds is set to ensure a finite mean of the odds distribution. Starting from a minimum bet of 1, we multiply wagers by a ratio γ each time the gamblers lose one round and return to the minimum bet each time they win. Once the wager reaches a preset maximum bet value 10000, we reset the gambler with a minimum bet. MSD obtained from 10 billion individual simulations is shown in Fig. 8. Different curves correspond to different exponents in odds distribution. We can see that the MSD initially presents an exponential-like growth, before the growths reduce to a linear function. It is easy to explain the exponential growth since many gamblers lose the rounds and therefore increase their wager by the factor γ, which leads to an increase in the average bet value. The superdiffusion here suggests that Martingale strategy increases gamblers' risks of huge losses. Considering the wide adoption of Martingale among gamblers, this could be a reason for the superdiffusion as well as the crossover to normal diffusion we found in several datasets. Comparison of the MSD curves of different α suggests that a more aggressive risk attitude leads to a higher risk of huge losses (as well as higher potential winnings).
Next we examine the ergodicity of the random walk process of net income by computing the time-averaged mean-squared displacement and the ergodicity breaking parameter. The time-averaged MSD is defined as
∑ δ = − + − = − t T t w k t w k ( ) 1 ( ( ) ( )) ,(8)k T t 2 1 2
where T is the length of the player's betting history, i.e. total number of rounds they attend, and is used for representing a time average. To calculate the time-averaged MSD, we need to make sure the player has played enough rounds so that we have a long enough series of net income data, therefore in each dataset we filter out the players who played less than T = 1000 rounds. As shown in Fig. 9 For an ergodic process, the parameter EB should be close to 0. However, as shown in Fig. 10, in most datasets, with the exception of csgospeed (G) and csgofast-Jackpot (H), EB is large. It follows that non-ergodicity is observed in most games and that gambling processes indeed often deviate from normal diffusion, which further highlights the complexity of human gambling behavior.
Another way to examine the diffusive behavior of a process is through the analysis of the first-passage time distribution. The first-passage time t FP is the time required for a random walker at location w to leave the region
− + w V w V [ , ]
FP FP for the first time, where V FP is the target value or first-passage value. The first-passage time distribution P t ( ) where Θ ⋅ ( ) is the Heaviside step function. We use = V 200 FP (US cents), with the exception of csgofast-Jackpot (H) for which V FP is chosen to be 5000. For a normal diffusive process, the tail of P t ( ) FP should decay with an exponent of 3/2. In Fig. 11 we plot the first-passage time distribution for each dataset, where again diverse diffusive behaviors are observed. In the games csgofast-Double (A) and csgofast-Jackpot (H), the tails of P t ( ) FP approximately decay with an exponent of 3/2 (see the thin green lines), indicating normal diffusive processes. For the game csgospeed (G), the exponent is found to be larger than 3/2, indicating a superdiffusive process. And in games csgofast-X50 (B), csgofast-Crash (C), ethCrash (D), and Coinroll (F), the exponents are clearly smaller than 3/2, indicating a subdiffusive behavior. We note that the results obtained from ensemble-averaged MSD sometimes differ from the results obtained from the first-passage time distributions. Nonetheless, anomalous diffusive behavior is widely observed.
∑ ∑ = Θ + − − − Θ + − − − →∞ = →∞ = P t T w k t wk V T w k t wk V ( ) lim 1 ( ( ) ( ) ) lim 1 ( ( 1) ( ) ),(10)
To confirm our conclusion about the wide existence of anomalous diffusive behavior in gambling activities, we further calculate the non-Gaussian parameter (NGP) 26,28,29
= ∆ ∆ − . NGP t w t w t () ( ) 3 ( ) 1 (11) 4 2 2
For a Gaussian process, the NGP should approach 0 when t gets large. In Fig. 12 we show the NGP as a function of time. In most of the games, except Coinroll (F), NGP shows a clear decreasing trend as t increases. In the game Coinroll (F), a decrease is not apparent, and most likely this game does not follow a Gaussian process. In the other games, although the NGP is still decreasing, we can not discriminate whether for large t this quantity will tend to 0 or instead reach a plateau value larger than zero. For example, for the game csgospeed (G) the NGP seems to reach a plateau ≈ . NGP t ( ) 1 5 instead of continuing to decrease, but this could also be the consequence of insufficient data. Still, our analysis does not provide clear evidence for the presence of Gaussianity in gambling behaviors.
Figure 8.
A betting system similar to Martingale will lead to a crossover from superdiffusion to normal diffusion according to the growth of mean-squared displacement. Comparison between curves of different parameters shows that higher γ and lower α both will lead to a higher chance of huge losses/winnings. www.nature.com/scientificreports www.nature.com/scientificreports/ To sum up our analysis of the players' net incomes viewed as random walks, the diverse diffusive behaviors found in the datasets indicate that human gambling behavior is more complex than random betting and simple betting systems. Further studies are required in order to fully understand the observed differences. At the individual level, as has been pointed out by Meng 7 , gamblers show a huge diversity of betting strategies, and even individual gamblers constantly change their betting strategy. Differences in the fractions of gamblers playing specific betting strategies could be a reason why we see a variety of diffusive behaviors in the datasets.
Discussion
The quick development of the video gaming industry has also resulted in an explosive growth of other online entertainment. This is especially true for online gambling that has evolved quickly into a booming industry with multi-billion levels. Every day million of bets are placed on websites all around the globe as many different gambling games are available online for gamblers.
Analysing different types of gambling games (ranging from Roulette to Jackpot games), we have shown that log-normal distributions can be widely used to describe the wager distributions of online gamblers at the aggregate level. The risk attitude of online gamblers shows scaling properties too, which indicates that although most gamblers are risk-averse, they sometime will take large risks in exchange for high potential gains.
Viewing the gamblers' net income as a random walk in time (where for each gambler time is increased by one unit every time they play a game), we can analyze the mean-squared displacement of net income and related quantities like the ergodicity breaking parameter or the non-Gaussian parameter with the goal to gain an understanding of the gamblers' betting strategies through the diffusive behaviors emerging from the datasets. For some games the mean-squared displacement and the first-passage time distribution reveal a transition from superdiffusion to normal diffusion as time increases. For all games the ergodicity breaking parameter and the non-Gaussian parameter reveal deviations from normal diffusion. All this indicates that gamblers' behaviors are very diverse and more complex than what would be expected from simple betting systems. We speculate that one of the reasons for the observed diverse diffusive behaviors at the aggregate level can be found in the differences in the fractions of gamblers playing specific betting strategies, but more work is required to fully understand the gamblers' complex behaviors. www.nature.com/scientificreports www.nature.com/scientificreports/
Methods
Detailed rules of the different games. Roulette. We focus on a simplified version of Roulette games that appears in online casinos, where a wheel with multiple slots painted with different colors will be spun, after which a winning slot will be selected. The Roulette table of a traditional Roulette game is composed of 38 slots, among which 18 slots are painted in black, 18 slots are painted in red, and two slots ("0" and "00") are painted in green. The online Roulette games are similar to the traditional ones, except that the number of colors and the number of slots for each color might be different. Each slot has the same probability to be chosen as the winning slot. Players will guess the color of the winning slot before the game starts. The players have a certain time for wagering, after which the game ends and a winning slot is selected by the website. Those players who successfully wagered on the correct color win, the others lose. As the chance of winning and odds for each color are directly provided by the website, roulette is a fixed-odds betting game.
Crash. "Crash" describes a type of gambling games mainly hosted in online casinos. Before the game starts, the site will generate a crash point m C , which is initially hidden to the players. With a lower boundary of 1, the crash point is distributed approximately in an inverse square law. The players need to place their wager in order to enter one round. After the game starts, on the player's user interface a number, called multiplier, will show up and gradually increase from 1 to the predetermined crash point m C , after which the game ends. During this process, if the player "cash-outs" at a certain multiplier m, before the game ends, they win the round; otherwise they lose. This multiplier m they cashed out at is the odds, which means when winning, the player will receive a prize that equals his wager multiplied by m. When m C is generated with a strict inverse-square-law distribution, the winning chance exactly equals the inverse of the player-selected odds m. The player can also set up the cash-out multipliers automatically before the game starts, to avoid the possible time delay of manual cash-out. Since in a manual cash-out scenario, after the game starts, the multiplier will show up on the screen, at a given moment the decision of the cash-out multiplier is based on the player's satisfaction with the current multiplier, and involves more complicated dynamics of decision-making processes. Meanwhile, in an auto cash-out scenario, the multiplier m is chosen before the game starts, which means the decision making is more "static. " Crash is also a fixed-odds betting game where the odds are player-selected.
Satoshi dice. Satoshi Dice is one of the most popular games in crptocurrency gambling. In 2013, the transactions resulting from playing Satoshi Dice games accounted for about 60% of overall Bitcoin transactions 30 . When playing Satoshi Dice, the player needs to pick a number A within a range U (0, ) provided by the website. The odds can be calculated with the expression = m U A / . Once the player finishes wagering, the website will pick another www.nature.com/scientificreports www.nature.com/scientificreports/ number B which is uniformly distributed on U (0, ). If B is less than A, then the player wins the round, otherwise they lose. Satoshi Dice is a fixed-odds betting game. In some online casinos, players cannot choose A arbitrarily, but instead, they have to select A from a preset list provided by the gambling website. Since the odds m is determined from A, we are more interested in the case where the players can choose A arbitrarily, from which we can obtain a more detailed distribution of the odds m, which helps us to understand the players' risk attitude. According to the rules of Satoshi Dice games, the maximum allowed bet is proportional to the inverse of A, which means the accepted range of wager is directly related to the odds.
Jackpot. Unlike the games discussed above, Jackpot is a parimutuel betting game, where players gamble against each other. During the game, each player attending the same round will deposit their wager to a pool. The game-ending condition varies across different websites, it could be a certain pool size, a certain amount of players, or a preset time span. When the game ending condition is reached, each player's winning chance will be determined by the fraction of their wager in the wager pool, based on which one player will be chosen as the winner by the website. The winner will obtain the whole wager pool as the prize, after excluding the site cut. The odds can be calculated by the pool size divided by the player's wager, but it is unknown to the players at the moment they wager. In the previous study 8 , we have already discussed the player's behavior in Jackpot games of skin gambling where in-game skins are directly used as wagers. In this paper, we extend the analysis to a case where wagers can be arbitrary amounts of virtual skin tickets (players need to first exchange in-game skins into virtual skin tickets).
Data summary. For each type of game, we collect two datasets. In total, we analyze 8 datasets collected from 4 different online gambling websites, and the number of bet logs contained in each dataset ranges from 0.3 million to 19.2 million. Due to the high variation of market prices of crypto-currencies and in-game skins, the wager and deposits are first converted into US cents based on their daily market prices.
CSGOFAST. From the skin gambling website CSGOFAST 31 we collected four datasets on the Roulette, Crash and Jackpot games (csgofast-Double, csgofast-X50, csgofast-Crash, csgofast-Jackpot) it provides.
csgofast-Double (A) is a Roulette game in which players can bet on 3 different colors (Red, Black, Green), which respectively provide odds of (2,2,14). The data were collected in two different time periods, and the only difference between them is a change of the maximum allowed bet values. csgofast-X50 (B) is also a Roulette game in which players can bet on 4 different colors (Blue, Red, Green, Gold) with odds (2, 3, 5, 50).
csgofast-Crash (C) is a Crash game. As we mentioned earlier, when analyzing the risk attitude of gamblers in Crash game, we are more interested in how players set up the odds (multiplier) with the automatically cash-out option. On CSGOFAST, under the automatically cash-out option, players can only setup odds ranging from 1.10 to 50. The interesting point about this dataset is that even if the player loses the round, if they used the automatically cash-out option, it still displays the player-selected odds (which is set before the game starts); meanwhile if they used the manually cash-out option, no odds is displayed. Therefore in early-crashed games ( < . m 1 10 C ), all the displayed odds that are larger than 1.10 were placed with automatically cash-out option. These displayed odds will be used in odds distribution analysis. The data are also collected in two different periods, where the only difference is still a change of the maximum allowed bet value. Roulette and Crash games on CSGOFAST all use virtual skin tickets for wagering.
csgofast-Jackpot (H) is a Jackpot game, where in-game skins are directly placed as wagers. Each skin has a market value that ranges from 3 to 180000 US cents. A player can place at most 10 skins in one round.
CSGOSpeed. From the skin gambling website CSGOSpeed 32 we collected one dataset from its Jackpot game csgospeed-Jackpot (G), in which arbitrary amounts of virtual skin tickets can be used as wagers. The difference between datasets (H) and (G) focuses on whether the wagers are in-game skins or virtual skin tickets.
ethCrash. ethCrash 33 is a cryto-currency gambling website providing a Crash game ethCrash (D). Players need to place wagers in Ethereum (ETH), one type of crypto-currency.
SatoshiDice. SatoshiDice 34 is a cryto-currency gambling website which accepts Bitcoin Cash (BCH) as wagers. It provides a Satoshi Dice game satoshidice (E), where only 11 preset odds can be wagered on, ranging from 1.05 to 1013.74. Among the preset odds, we find that more than 30% of the bets are placed under the odds 1.98, and we will analyze those bets for wager distribution.
Coinroll. Coinroll 35 is a cryto-currency gambling website which accepts Bitcoin (BTC) as wagers. It provides a Satoshi Dice game Coinroll (F), where players can either wager on the 8 preset odds listed by the website, or choose an odds of their own. When further analyzing the data, we find that a few players placed an unusual large amount of bets, where the top player placed more than 11 million bets. Although these large number of bets prove the heavy-tailed distribution of the number of bets of individuals, we have doubts that these players are playing for the purpose of gambling. As we have pointed out, all the games discussed in this paper have negative expected payoffs. Indeed, prior studies have raised suspicion about the use of crypo-currency gambling websites as a way for money laundering 36 . We will therefore exclude from our analysis gamblers who placed more than half a million bets. For bets wagered on the preset odds, we find that more than 57% are placed under the odds 1.98, and we use these bets to analyze the wager distribution. On the other hand, since player-selected odds show a broader spectrum regarding the risk attitude of gamblers, we focus on the odds distribution of the player-selected odds. www.nature.com/scientificreports www.nature.com/scientificreports/ As already mentioned, we will exclude the bets from those players who placed at least half a million bets from our odds distribution analysis.
Although crypto-currency has gained decent popularity in the financial and technological world, in this paper we still measure the wager/wealth deposited in forms of crypto-currencies in US dollars, since the wagers in skin gambling are measured in US dollars. The historical daily price data of crypto-currencies (Bitcion, Ethereum, Bitcoin Cash) are obtained from CoinDesk 37 (for Bitcoin) and CoinMetrics 38 (for Ethereum and Bitcoin Cash).
Ethics for data analysis. The data collected and analyzed in this paper are all publicly accessible on the internet, and we collect the data either with the consent of the website administrators or without violating the terms of service or acceptance usage listed on the hosting website. The data we use do not include any personally identifiable information (PII), and we further anonymize account-related information before storing them into our databases to preserve players' privacy. In addition, our data collection and analysis procedures are performed solely passively, with absolutely no interaction with any human subject. To avoid abusing the hosting websites (i.e., the gambling websites), the request rates of data-collecting are limited to 1 request per second. Considering the legal concerns and potential negative effects of online gambling [39][40][41][42][43][44][45][46] , our analysis aims only to help better prevent adolescent gambling and problem gambling. Parameter estimation and model selection. In our analysis, the parameters of different distribution models are obtained by applying Maximum Likelihood Estimation (MLE) 47 . To select the best-fit distribution, we compare the models' Akaike weights 48 derived from Akaike Information Criterion (AIC). Note that analyzing the fitting results, we constantly found that players show a tendency of using simple numbers when allowed to place wagers with arbitrary amounts of virtual currency. As a result, the curves of probability distribution functions appear to peak at simple numbers, and the corresponding cumulative distribution function shows a stepped behavior. This makes the fitting more difficult, especially for the determination of the start of the tail. To address this issue, we choose the start of the tail x min such that we obtain a small Kolmogorov-Smirnov (K-S) distance between the empirical distribution and the fitting distribution, while maintaining a good absolute fit between the complementary cumulative distribution functions (CCDF) of the empirical distribution and the best-fitted distribution. Candidate models for model selection in this paper include exponential distribution, power-law distribution, log-normal distribution, power-law distribution with sharp truncation, power-law distribution with exponential cutoff, and pairwise power-law distribution. More details about parameter fitting and model selection can be found in the article by Clauset et al. article 49 as well as in the previous paper by the authors 8 .
Data Availability
The datasets generated and/or analysed during the current study are available from the authors on reasonable request.
b 0 is the wager the player places, > m 1 is the odds, η > ≥ 1 0 corresponds to the site cut, and f m is a non-negative value based on the odds representing the players' statistical disadvantage in winning, as mentioned earlier. At least either η or f m are non-zero. From Eq. (1), we can obtain the expected payoff of attending one round
Figure 1 .
1In games (A-G), where players are allowed to choose arbitrary bet values, the wager distribution can be best fitted by log-normal distributions(3). In game (D), the log-normal distribution is truncated at its maximum bet value, indicated by *. The fitting lines represent the log-normal fittings. Wagers placed under the different maximum allowed bet values are discussed separately, e.g., in game (A), (A 1 ) has a maximum bet value of 500,000, and (A 2 ) has a maximum bet value of 50,000. On the other hand, in game (H) where wagers can only be in-game skins, the wager distribution is best described by a pairwise power law with an exponential transition, see Eq. (4). The red dotted line represents the log-normal fitting and the blue solid line represents the fitting of a pairwise power law with an exponential transition.Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Figure 3 .
3Odds distributions can be well-fitted by truncated shifted power-law distributions. Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Figure 4 .
4The tail of the wealth distribution of Bitcoin gamblers follows a pairwise power-law distribution.
Figure 5 .
5The wagers obtained from random sampling of top gamblers' bets still present log-normal distributions, although there are some observable deviations. Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Figure 6 .
6Change of the mean net income with time for the different datasets. Most of the datasets present a decreasing net income as time t increases. Each point is obtained from an average over at least 200 players. Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Figure 7 .
7the time-averaged MSD shows huge deviations from player to player, suggesting diverse betting behaviors at the individual level. At the same time, comparison between the ensemble-averaged time-averaged MSD δ t The growth of ensemble-averaged mean-squared displacement in different datasets presents different diffusive behaviors. In the figures, the error bars represent 95% confidence intervals, blue dashed lines follow linear functions (slope = 1), and green dotted lines follow quadratic functions (slope = 2). Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-in most datasets, except in the Coinroll (F), csgospeed (G) and csgofast-Jackpot (H) games. To further examine breaking of ergodicity, we have calculated the ergodicity breaking parameter EB 24-26 defined as
FP 8 ,
827 , defined as the survival probability that the random walker, who is located at w at time t
Figure 9 .
9The growth of the time-averaged MSD for individual gamblers, presented as thin lines, suggests diverse betting behaviors at the individual level. The comparison between δ t ( )2 (thick dashed black lines) and ∆w t ( ( )) 2 (thick full red lines) reveals that these quantities are different for most games, with the exception of the Coinroll (F), csgospeed (G) and csgofast-Jackpot (H) games. Players who played less than 1000 rounds are filtered out in each dataset.
Figure 10 .
10The change of the ergodicity breaking parameter with time. For all games, with the exception of the games csgospeed (G) and csgofast-Jackpot (H), EB is found to be much larger than 0, suggesting non-ergodic behavior.
Figure 11 .
11The tails of first-passage time distributions for the different datasets indicate different diffusive behaviors. The green lines represent a power-law decay with an exponent 3/2. The blue error bars indicate 95% confidence intervals. Only gamblers who attended more than 1000 rounds of games have been included in these calculations.
Figure 12 .
12In most datasets, except Coinroll (F), the non-Gaussian parameter shows a decreasing trend as t increases. However, in none of the studied cases does the non-Gaussian parameter fall below the value 1. (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Table 2. Correlation analysis shows that there is a strong positive correlation between consecutive bets, along with the small mean values and variances of log-ratio between consecutive bets. Satoshi Dice (E) is excluded here as individual gamblers in the dataset are not distinguishable. csgofast-Jackpot (H) is excluded in the calculation of P(b i = b i+1 ) due to the low precision of bet values in this dataset. (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2csgofast-Double (A)
0.596
0.737
0.010
0.183
0.342
csgofast-X50 (B)
0.692
0.803
0.007
0.102
0.512
csgofast-Crash (C)
0.858
0.909
0.004
0.038
0.802
ethCrash (D)
0.866
0.949
0.000
0.147
0.549
Coinroll (F)
0.826
0.925
0.000
0.282
0.497
csgospeed-Jackpot (G)
0.522
0.675
0.002
0.288
0.136
csgofast-Jackpot (H)
0.591
0.759
0.002
0.206
-
for some additional details. (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
Scientific RepoRtS | (2019) 9:14712 | https://doi.org/10.1038/s41598-019-50168-2
© The Author(s) 2019
AcknowledgementsThis work is supported by the US National Science Foundation through grant DMR-1606814.Author contributionsX.W. and M.P. conceived the study, X.W. wrote the computer codes and conducted the data analysis, X.W. and M.P. discussed the results and wrote the manuscript.Additional informationCompeting Interests: The authors declare no competing interests.Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
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|
[] |
[
"N-best Response-based Analysis of Contradiction-awareness in Neural Response Generation Models",
"N-best Response-based Analysis of Contradiction-awareness in Neural Response Generation Models"
] |
[
"Shiki Sato [email protected] \nTohoku University\n\n",
"Reina Akama [email protected] \nTohoku University\n\n",
"Hiroki Ouchi [email protected] \nNara Institute of Science and Technology 3 RIKEN\n\n",
"Ryoko Tokuhisa [email protected] \nTohoku University\n\n",
"Jun Suzuki [email protected] \nTohoku University\n\n",
"Kentaro Inui [email protected] \nTohoku University\n\n"
] |
[
"Tohoku University\n",
"Tohoku University\n",
"Nara Institute of Science and Technology 3 RIKEN\n",
"Tohoku University\n",
"Tohoku University\n",
"Tohoku University\n"
] |
[
"Proceedings of the SIGdial 2022 Conference"
] |
Avoiding the generation of responses that contradict the preceding context is a significant challenge in dialogue response generation. One feasible method is post-processing, such as filtering out contradicting responses from a resulting n-best response list. In this scenario, the quality of the n-best list considerably affects the occurrence of contradictions because the final response is chosen from this n-best list. This study quantitatively analyzes the contextual contradiction-awareness of neural response generation models using the consistency of the n-best lists. Particularly, we used polar questions as stimulus inputs for concise and quantitative analyses. Our tests illustrate the contradiction-awareness of recent neural response generation models and methodologies, followed by a discussion of their properties and limitations.
|
10.48550/arxiv.2208.02578
|
[
"https://www.aclanthology.org/2022.sigdial-1.60.pdf"
] | 251,320,564 |
2208.02578
|
07d67175550a6cf28cb7a7273ab5a2b1e03160e9
|
N-best Response-based Analysis of Contradiction-awareness in Neural Response Generation Models
September, 2022
Shiki Sato [email protected]
Tohoku University
Reina Akama [email protected]
Tohoku University
Hiroki Ouchi [email protected]
Nara Institute of Science and Technology 3 RIKEN
Ryoko Tokuhisa [email protected]
Tohoku University
Jun Suzuki [email protected]
Tohoku University
Kentaro Inui [email protected]
Tohoku University
N-best Response-based Analysis of Contradiction-awareness in Neural Response Generation Models
Proceedings of the SIGdial 2022 Conference
the SIGdial 2022 ConferenceEdinburgh, UKSeptember, 2022637
Avoiding the generation of responses that contradict the preceding context is a significant challenge in dialogue response generation. One feasible method is post-processing, such as filtering out contradicting responses from a resulting n-best response list. In this scenario, the quality of the n-best list considerably affects the occurrence of contradictions because the final response is chosen from this n-best list. This study quantitatively analyzes the contextual contradiction-awareness of neural response generation models using the consistency of the n-best lists. Particularly, we used polar questions as stimulus inputs for concise and quantitative analyses. Our tests illustrate the contradiction-awareness of recent neural response generation models and methodologies, followed by a discussion of their properties and limitations.
Introduction
Recent advanced response generation models (Zhang et al., 2020;Adiwardana et al., 2020;Roller et al., 2021) can generate relevant and meaningful responses, which can resolve dull response problems (Vinyals and Le, 2015;Sordoni et al., 2015;Serban et al., 2016). This advancement reveals additional flaws in the quality of neural model responses, such as contradiction. Contradiction is a critical error in dialogue because a single contradictory response can disrupt the flow of the dialogue (Higashinaka et al., 2015).
A generation model outputs a response by selecting the candidate with the highest likelihood (1best) from an n-best candidate list. Prior work has demonstrated that generating the n-best lists with noncontradictory 1-bests is an open challenge (Nie et al., 2020;Kim et al., 2020;Li et al., 2021). Thus, one practical technique for avoiding contradiction is to have an accurate contradiction detector that eliminates all contradictory candidates from the nbest list (Nie et al., 2020). In this scenario, the con- sistency of all candidates in the n-best list, not just the 1-best, substantially impacts whether the final output is contradictory because the final response is chosen from the n-best list. Nonetheless, earlier quantitative investigations of contradiction relied solely on 1-bests from models (Li et al., 2021).
In this study, we analyze the n-best lists generated by the models to explore methods for enhancing neural response generation to avoid contradiction. Specifically, we first consider how analyzing an n-best list should be approached. Then, we propose a method for statistically analyzing the n-best lists (Figure 1). Since it is impractical to study all conceivable contradictions in a dialogue, we first focus on contradictions in response to polar questions. 1 We use our method to highlight the contradiction-awareness of recent highperformance neural response generation models and methodologies. Our results show that beam search has limitations in terms of avoiding contradiction and that the newer techniques, such as unlikelihood training (Welleck et al., 2020), can help overcome these limitations.
NLI data
Dialogue context for our test
Analysis perspectives
First, n-best lists must be generated to prevent contradiction, assuming the filters can remove contradictory responses. An ideal model produces output that is noncontradictory and outperforms in many other criteria, such as relevance or informativeness. A model must generate at least one noncontradictory candidate to deliver a noncontradictory output. Furthermore, even noncontradictory candidates could be eliminated based on other criteria (e.g., relevance, informativeness). Therefore, it can be hypothesized that having more noncontradictory responses in an n-best list would enhance the final output quality across various criteria. Taking the above into account, we examine n-best lists based on the certainty of the existence of noncontradictory responses (Certainty), and the variety of noncontradictory responses (Variety):
• Certainty: The proportion of the n-best lists that have at least one noncontradictory response.
• Variety: The proportion of noncontradictory responses in each n-best list when only the n-best lists with at least one noncontradictory response are collected.
Given a set of inputs Q, we calculate them as follows:
Certainty = |Q ′ | |Q| , Variety = 1 |Q ′ | q∈Q ′ cnt(f (q)) |f (q)| Q ′ = {q | cnt(f (q)) > 0, q ∈ Q}
where f (·) is an n-best list generation function and cnt(·) is a function that returns the number of noncontradictory responses from a given n-best list. For example, the Certainty of a model that generates n-best lists with a combination of noncontradictory and contradictory responses is high, but its Variety is low. However, a model that always generates n-best lists with only noncontradictory or contradictory responses has a high Variety but a low Certainty. We anticipate that n-best lists must include noncontradictory responses (Certainty= 1.0), with a high proportion (high Variety).
Analytical inputs and evaluation
To analyze a model from the aforementioned viewpoints, we consider how to prepare the analytical inputs and evaluate the generated responses in this section.
Inputs for highlighting contradictions
Polar echo question. An echo question (Noh, 1998) confirms or clarifies the context information by repeating the utterance of another speaker. It is commonly used when the speaker did not hear or understand what was said correctly, or when the speaker wishes to express incredulity. Based on Li et al. (2021)'s discovery, contradictions emerge mostly when speakers refer to earlier information communicated in dialogue; we use echo questions as stimulus input in our analysis to elicit contradictory responses. We use polar-typed echo questions to make our analysis more succinct and quantitative. Since polar questions allow for basically only two responses, yes or no, we can clearly determine whether the generated response is contradictory or not. Furthermore, by analyzing the produced responses as a yes/no binary classification issue, it allows for quantitative discussion of experimental outcomes based on the probability level.
Input preparation. We use the dataset from the natural language inference (NLI) task to effectively obtain the analytical inputs described in the preceding paragraph. This dataset specifies the logical relationship (i.e., entailment, neutrality, or contradiction) between a premise and its associated hypothesis. We transform the NLI dataset into dialogue data using a set of basic rewriting rules. 2 Our test involves two types of inputs, which can be classified as follows:
• ENTQ: generating a confirmation response.
• CNTQ: generating a refutation response. Table 1 displays the input samples and how they are transformed from the initial NLI data. Each input is made up of the following two utterances: the history and message. In our analysis, the model generates responses to a given message, assuming the model has generated the history in the preceding turn.
Contradiction detection for output
To compute the Certainty and Variety, we must first determine whether each generated response in the n-bests compared to the inputs is contradictory. The simplest method for detecting the contradictions is to check whether the response begins with yes or no. However, in the event of an indirect expression (e.g., Why not?), this method cannot detect the contradictions. Therefore, we use an automated yes-no classifier to categorize the n-best responses to ENTQ/CNTQ. We train the classifier by fine-tuning RoBERTa (Liu et al., 2019) using the Circa dataset (Louis et al., 2020), which comprises pairs of polar questions and indirect responses, as well as annotations for the answer's interpretation, to categorize utterances as affirmations or refutations. 3
Experiments
We demonstrate how our framework shows the properties of n-best lists, which could be quite influential in terms of avoiding contradiction. We demonstrate this by comparing the n-bests generated by conventional beam search (BS) versus recently proposed techniques.
Experimental settings
Inputs preparation. We used the Multi-Genre NLI Corpus (Williams et al., 2018) to obtain analytical inputs, which is a large scale and is consistent in good quality NLI data. We created 2,000 ENTQ/CNTQ inputs by extracting 2,000 samples labeled with entailment or contradiction. 4
Response generation models. We used the following two recently developed high-performance models: DialoGPT (Zhang et al., 2020) and
Blender (Roller et al., 2021). 5
Analysis of n-best using beam search
Let B denote the beam size during generation. It has been empirically found that using beam search with B = 10 to generate a response yields excellent quality results and has a frequently used value (Zhang et al., 2020;Roller et al., 2021). Table 2 displays the Certainty and Variety of 10-best lists generated using these methods. Figure 2 also depicts the Certainty and Variety of n-best lists generated using different beam sizes.
Certainty. Table 2 illustrates that in approximately 10% of CNTQ-type inputs, even the highest scoring model generates 10-best lists full of contradictory responses. Even with a perfect response filter, the models are unable to provide noncontradictory answers to these questions. It should be emphasized that the error rate is not low, given that the inputs are polar questions with highly restricted viable responses. Expanding the beam size can increase the number of n-best lists with at least one noncontradictory response. Indeed, increas-ing the beam size enhances the Certainty ((a) and (b) in Figure 2). By increasing B to 40, the Certainty of using DialoGPT 345M for both ENTQand CNTQ-type inputs achieve 1.0.
Variety. With B = 10, all the models' Variety are more than 0.5 (chance rate) ( Table 2). Therefore, rather than being fully random, the models generate n-best lists with a degree of directionality toward avoiding contradictions. However, increasing the size of beam reduces the Variety ((c) and (d) in Figure 2), resulting in lower output quality. For example, the Variety of DialoGPT 345M with B = 40 for CNTQ-type inputs (a model with Certainty of 1.0 for both ENTQ-and CNTQ-type inputs) decreases to 0.58.
Overall. In terms of avoiding contradiction, our analytical framework demonstrated the features of the n-best lists of the beam search. The Certainty did not achieve 1.0 in the commonly used configuration (B = 10). When the beam size is increased, the Certainty increases to 1.0, whereas the Variety reduces dramatically. These results show the tradeoff between Certainty and Variety as a function of beam size; in this example, we found constraints in obtaining high Certainty and Variety with beam search. Furthermore, it is found that the Certainty obtained using DialoGPT is greater than that obtained using Blender, whereas the opposite is true for Variety, suggesting that various models behave differently in terms of Certainty and Variety. This study emphasizes the significance of examining the Certainty and Variety of each model.
Analysis of n-best by various techniques
How to achieve high Certainty and Variety? One method to increase Certainty is to generate n-best lists with a wider range of responses, such that each n-best list is guaranteed to contain a specific number of noncontradictory responses. The diverse beam search (DBS) (Vijayakumar et al., 2016) and nucleus sampling (NS) (Holtzman et al., 2020) methods are used to construct such n-best lists. Furthermore, Li et al. (2020) recently proposed models that use unlikelihood (UL) training to assign low probabilities to contradict responses. Using these models to generate n-best lists will almost certainly enhance both Certainty and Variety. We assess the n-best lists generated using these three strategies to see how much these techniques enhance Certainty and Variety (n-best lists generated using DBS and NS, and n-best lists generated using beam search together with the UL training). Appendix C contains a description of the techniques used for this analysis.
Result. Table 3 displays the Certainty and Variety of the 10-best lists generated using BS, DBS, NS, and UL. 6 The values of α show the degree of UL loss during fine-tuning. Here UL with α = 0 used the response generation model fine-tuned with maximum likelihood in the same training settings as those used for UL with α > 0. Thus, note that comparing UL with α = 0 and α > 0 allows a fair comparison between likelihood and unlikelihood training. The results reveal the properties of the nbest lists obtained for the three techniques, as well as the extent to which the techniques increase Certainty and Variety. The Certainty obtained using the DBS and NS method reach 1.0 for significantly lower search sizes than that for the BS to attain a Certainty of 1.0; the Variety for CNTQ-type inputs are less than 0.5 (chance rate). Thus, using the DBS and NS methods efficiently improves Certainty compared with the results obtained using the beam search; nevertheless, the methods do not simultaneously attain high Certainty and Variety. However, the Certainty obtained using UL with α > 0 are greater than those obtained using the BS, and this was accomplished while maintaining higher Variety than those obtained using the BS and UL with α = 0 (likelihood training). Our findings show that generation models are advancing toward high Certainty and Variety, which is particularly true for the recently proposed UL loss method. Despite the highly restricted viable responses, i.e., yes or no, the Certainty obtained using UL with α > 0 does not reach 1.0. Thus, we conclude that there is still room for improvement in n-best list generation in terms of avoiding contradiction.
Conclusion
Based on the recent development of contradiction detectors, removing contradictory candidates from models' n-best lists is a practical method for avoiding contradiction. In this method, the consistency of all candidates in the n-best lists substantially affects whether the final outputs are contradictory. We quantitatively examined the properties of the n-best lists in terms of avoiding contradiction, using polar-typed questions as analytical inputs. We demonstrated that the proposed framework exhibits the properties of n-best lists based on Certainty and Variety. Certainty determines whether an nbest list has at least one noncontradictory response, whereas Variety evaluates how many noncontradictory responses each n-best list has. The results, particularly, demonstrated the present limitations on achieving high Certainty and Variety when using the well-established beam search method. In addition, our method emphasizes the improvements in Certainty and Variety achieved by recently proposed response generation strategies.
Our approach, which analyzes models' n-best lists based on Certainty and Variety, can be applied to any response generation problem, not just polartyped response generation, which will be future work. detected the contradiction in the model response using an indirect expression, in Example 1. However, in Example 2, the classifier failed to detect the contradiction of the model response, having both a noncontradictory direct expression ("No") and a contradictory indirect expression (the part of the response after "No"). We found that the classifier tended to misclassify model responses containing the contradictions with themselves, such as Example 2.
C Details of experiments
Number of analyzed stimulus inputs. To simplify the analysis, we omitted from Section 4 and
Certainty: 3 /Figure 1 :
313=1.00, Variety: (0.8+1+1)/3=0.93 Certainty: 1/3=0.33, Variety: 0.2/1=0.20 (ii) Determine whether responses in n-best list are non-contradictory (i) Synthesize inputs (iii) Compute Certainty and Variety Overview of our analysis framework. The framework analyzes n-best lists by (i) synthesizing a stimulus input that induces contradictions, (ii) automatically determining whether responses in the n-best lists are contradictory, and (iii) computing Certainty and Variety.
Figure 2 :
2Certainty and Variety of n-best lists using beam search with various beam sizes.
Entailment Premise: yeah i'm in North Carolina −→ ENTQ History: Yeah I'm in North Carolina. Hypothesis: I'm in North Carolina. Message: Are you in North Carolina?Contradiction
Premise: yeah i'm in North Carolina −→
CNTQ
History: Yeah I'm in North Carolina.
Hypothesis: I'm in South Carolina.
Message: Aren't you in South Carolina?
Table 1 :
1Acquiring dialogue context by transforming the Natural Language Inference (NLI) data.
Table 2 :
2Certainty and Variety of 10-best lists using beam search with beam size B = 10.1
10
20
30
40
50
Beam size
0.4
0.6
0.8
1.0
Certainty
(a) Certainty for EntQ
Blender 400M
Blender 1B
Blender 3B
DialoGPT 345M
DialoGPT 762M
1
10
20
30
40
50
Beam size
Table 3 :
3Certainty and Variety of 10-best lists using various techniques with Blender 3B.
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Table 5 :
5Number of stimulus inputs analyzed to calculate the Certainty and Variety described inTable 2.Model
ENTQ
CNTQ
BS
1566 / 2000 1403 / 2000
DBS
991 / 2000
882 / 2000
NS
818 / 2000
684 / 2000
UL (α = 0)
1914 / 2000 1871 / 2000
UL (α = 1)
1806 / 2000 1887 / 2000
UL (α = 10)
1654 / 2000 1811 / 2000
Table 6 :
6Number of stimulus inputs analyzed to calculate the Certainty and Variety described inTable 3.
Codes and test set are available at https://github.com/shiki-sato/ nbest-contradiction-analysis
The details are described in Appendix A.
The details are described in Appendix B.4 We used the samples in the TELEPHONE domain; this domain covers open-domain conversations.5 The details of the settings are described in Appendix C.
For the BS, DBS, and UL, we obtained the 10-best lists setting beam size to 10. For the NS, we got the 10-best lists by performing nucleus sampling ten times.
AcknowledgmentsWe would like to thank all anonymous reviewers for their insightful comments. We also thank Ana Brassard and Yosuke Kishinami for their valuable feedback and support. This work was partly supported by JSPS KAKENHI Grant Numbers JP21J22383, JP22K17943, JST Moonshot R&D Grant Number JPMJMS2011, and a Bilateral Joint Research Program between RIKEN AIP Center and Tohoku University.As described in Section 3.1, we obtain an analytical input from the NLI dataset. Specifically, we convert the hypothesis sentence of an NLI sample into a yes-no question. We describe the procedure as follows:1. Detect the first verb of a sentence.2. Move the verb to the beginning of the sentence, or put one of {Do, Does, Did} at the front of the sentence, changing the verb back to its base (e.g., made → make).3. Change first-person pronouns to secondperson pronouns and second-person pronouns to first-person pronouns (e.g., my → your).4. Change the punctuation mark at the end of the sentence to a question mark.We used spaCy (en_core_web_sm)(Honnibal and Montani, 2017)to detect the verbs of hypothesis sentences. We did not use NLI samples with syntactically complex hypothesis sentences, such as those containing coordinating conjunctions, to avoid obtaining ungrammatical inputs. Further details are provided in our source codes. 7B Details of yes-no classifierTraining settings. On the Circa dataset, we fine-tuned the pretrained RoBERTa (roberta-large) implemented by Hugging Face(Wolf et al., 2020). We divided the dataset at random into train:valid = 8 : 2. The other training parameters were identical to those used byLouis et al. (2020).Performance of classifier. To investigate the performance of the classifier, we measured the classification accuracy. First, we manually labeled the top-1 responses in the 10-best lists generated by the analysis presented in Section 4.2 with one of the two following labels: Contradictory or Noncontradictory. The accuracy with which the automated evaluation categorized the labeled responses was then evaluated. We selected 500 responses 8 from 50 ENTQ/CNTQ inputs drawn at random from our test for the evaluation. The classifier classified 433/500 responses (see Appendix C), and the accuracy was 0.921. Some examples of the classification are shown inTable 4. The classifier correctly 7 https://github.com/shiki-sato/ nbest-contradiction-analysis 8 100 responses generated by each of 5 generation models. Appendix B the analytical inputs with one or more ambiguous responses in the n-best lists. We defined ambiguous responses as those that were not identified by the classifier as either affirmations or refutations. 9Table 5andTable 6display the number of analytical inputs from the total of 2, 000 ENTQ/CNTQ used for the two analyses in Section 4.Generation model settings. In Section 4 experiments, we used DialoGPT(Zhang et al., 2020)and Blender(Roller et al., 2021)as response generation models. We used the codes of ParlAI(Miller et al., 2017)with its default settings, except for beam_length_penalty= 0 to generate responses.Unlikelihood training settings. We used unlikelihood training with Blender 3B for the study of Section 4.3. To use the unlikelihood training proposed by Li et al.(2020), we require training data that includes the following three elements: input (here, history, and message), gold response, and negative response. These training samples were created by altering the NLI data with entailing and contradicting hypotheses. 10Table 7displays the original NLI data and the transformed training samples. One NLI data set yields four types of questions (PositiveQ1, PositiveQ2, NegativeQ1, and NegativeQ2). We synthesized 8,000 samples from 2,000 NLI data and randomly divided them into train : valid = 9 : 1. We tuned the learning rate {7.0 × 10 −4 , 7.0 × 10 −5 , 7.0 × 10 −6 , 7.0 × 10 −7 , 7.0 × 10 −8 } and the number of warmup updates {50, 100} for each α = {0, 1, 10} for training. The rest of the training parameters are identical to those used byRoller et al. (2021). It is worth noting that we only trained the models marked as UL in Section 4.3 on these transformed data.9Circa dataset has seven different labels such as "Yes" and "Probably/sometimes yes." We regard the responses classified into "In the middle" or "I am not sure" as ambiguous ones.10Note that we did not use the identical NLI samples to synthesize ENTQ/CNTQ.
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. 1b Blender, Blender 1B 1413 / 2000 1316 / 2000
. 3b Blender, Blender 3B 1566 / 2000 1403 / 2000
. 345m Dialogpt, DialoGPT 345M 1126 / 2000 924 / 2000
. 762m Dialogpt, DialoGPT 762M 1044 / 2000 956 / 2000
Negative: No, I'm in South Carolina. PositiveQ2 History: Yeah I'm in North Carolina. Message: Are you in South Carolina? Gold: No, I'm in North Carolina. Negative: Yes, I'm in South Carolina. NegativeQ1 History: Yeah I'm in North Carolina. Message: Aren't you in North Carolina? Gold: Yes, I'm in North Carolina. Negative: No, I'm in South Carolina. NegativeQ2 History: Yeah I'm in North Carolina. Message: Aren't you in South Carolina? Gold: No, I'm in North Carolina. Negative: Yes, I'm in South Carolina. (b) Training samples for UL. PositiveQ1 History: Yeah I'm in North Carolina. Message: Are you in North Carolina? Gold: Yes, I'm in North CarolinaPositiveQ1 History: Yeah I'm in North Carolina. Message: Are you in North Carolina? Gold: Yes, I'm in North Carolina. Negative: No, I'm in South Carolina. PositiveQ2 History: Yeah I'm in North Carolina. Message: Are you in South Carolina? Gold: No, I'm in North Carolina. Negative: Yes, I'm in South Carolina. NegativeQ1 History: Yeah I'm in North Carolina. Message: Aren't you in North Carolina? Gold: Yes, I'm in North Carolina. Negative: No, I'm in South Carolina. NegativeQ2 History: Yeah I'm in North Carolina. Message: Aren't you in South Carolina? Gold: No, I'm in North Carolina. Negative: Yes, I'm in South Carolina. (b) Training samples for UL
Example of transforming (a) original NLI data to (b) training sample for UL. We synthesized four questions. 7i.e., PositiveQ1, PositiveQ2, NegativeQ1, and NegativeQ2, from each NLI sampleTable 7: Example of transforming (a) original NLI data to (b) training sample for UL. We synthesized four ques- tions, i.e., PositiveQ1, PositiveQ2, NegativeQ1, and NegativeQ2, from each NLI sample.
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"GROUP-ALGEBRAIC CHARACTERIZATION OF SPIN PARTICLES: SEMI-SIMPLICITY, SO(2N) STRUCTURE AND IWASAWA DECOMPOSITION",
"GROUP-ALGEBRAIC CHARACTERIZATION OF SPIN PARTICLES: SEMI-SIMPLICITY, SO(2N) STRUCTURE AND IWASAWA DECOMPOSITION"
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[
"Mahouton Norbert Hounkonnou ",
"Francis Atta Howard [email protected]:[email protected] ",
"Kangni Kinvi \nUniversité de Cocody\nUFR-Mathematiques et Informatique\nAbidjan 2222 BP. 1214Côte d'Ivoire\n",
"\nUniversity of Abomey-Calavi\nInternational Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair)\n072 B.P. 50CotonouBenin Republic\n"
] |
[
"Université de Cocody\nUFR-Mathematiques et Informatique\nAbidjan 2222 BP. 1214Côte d'Ivoire",
"University of Abomey-Calavi\nInternational Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair)\n072 B.P. 50CotonouBenin Republic"
] |
[] |
In this paper, we focus on the characterization of Lie algebras of fermionic, bosonic and parastatistic operators of spin particles. We provide a method to construct a Lie group structure for the quantum spin particles. We show the semi-simplicity of a quantum spin particle Lie algebra, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition of spin particles at both the Lie algebra and Lie group levels. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.
| null | null | 218,889,322 |
2005.12274
|
3491ba520dcfab85f8e74ec31cbe4169034c0ca4
|
GROUP-ALGEBRAIC CHARACTERIZATION OF SPIN PARTICLES: SEMI-SIMPLICITY, SO(2N) STRUCTURE AND IWASAWA DECOMPOSITION
25 May 2020
Mahouton Norbert Hounkonnou
Francis Atta Howard [email protected]:[email protected]
Kangni Kinvi
Université de Cocody
UFR-Mathematiques et Informatique
Abidjan 2222 BP. 1214Côte d'Ivoire
University of Abomey-Calavi
International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair)
072 B.P. 50CotonouBenin Republic
GROUP-ALGEBRAIC CHARACTERIZATION OF SPIN PARTICLES: SEMI-SIMPLICITY, SO(2N) STRUCTURE AND IWASAWA DECOMPOSITION
25 May 2020(Dated: 27 May 2020)Angular momentum coupling · Clifford Algebra · Connectedness · Spin Lie group · Iwasawa decomposition · Spin particles· Semi-simplicity
In this paper, we focus on the characterization of Lie algebras of fermionic, bosonic and parastatistic operators of spin particles. We provide a method to construct a Lie group structure for the quantum spin particles. We show the semi-simplicity of a quantum spin particle Lie algebra, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition of spin particles at both the Lie algebra and Lie group levels. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.
I. INTRODUCTION
A. motivations
Doran's work 2 showed that every linear transformation can be represented as a monomial of vectors in geometric algebra, every Lie algebra as a bivector algebra, and every Lie group as a spin group. Schwinger's 12 realization of su(1, 1) Lie algebra with creation and annihilation operators was defined with spatial reference in the Pauli matrix representation 16 .
Several relations as well as connections were observed in spin particles such as fermionic, bosonic, parastatistic Lie algebras, and in geometric algebras such as the Clifford algebra, Grassmannian algebra and so on 13 . Sobczyk 13 also proved that the spin half particles can be represented by geometric algebras. Palev 11 highlighted that a semi-simple Lie algebra can be generated by the creation and annihilation operators. In all the above mentioned works, the classical groups such as B n and D n play a crucial role in the spin particle Lie algebra.
Moreover, several evidences from particle and theoretical physics showed the connection between quantum spin particle Lie algebra and Clifford algebra 2 . The spin of elementary particles obeying the Fermi-Dirac statistics, the Bose-Einstein statistics, the quantization of parastatistics such as parafermions and parabosons also gained much attention in the literature 1 , 2 , 12 , 13 , 15 , 16 .
In the opposite, exhaustive investigations on spin particle creation and annihilation and their angular momentum in connection with Lie groups, Lie algebras, Clifford algebras, and their representations are still lacking. This study aims at fulfilling this gap. The Iwasawa decomposition, introduced by the Japanese mathematician Kenkichi Iwasawa, generalizes the Gram-Schmidt orthogonality process in linear algebra 7 .
Motivated by all the above mentioned works, we prove, in this paper, that the spin particles admit a Lie group structure, show its connectedness and semi-simplicity, and construct the Iwasawa decomposition at both the Lie algebra and Lie group levels of the spin particles.
But before dealing with the main results, and as a matter of clarity in the development, let us briefly recall the main definitions, the known results, and the appropriate notations useful in the sequel.
B. Para-fermionic algebra
Let a ± 1 , ..., a ± n be the creation and annihilation operators for a system consisting of nfermions with commutator relations 4 :
[a − i , a + j ] = δ ij (1) [a − i , a − j ] = [a + i , a + j ] = 0,(2)
or, of n-parafermions with
[[a − i , a + i ], a ± j ] = ±2δ ij a + j ,(3)
where [X, Y ] := XY − Y X.
Let T be the associative free algebra of a i , a j ; i, j ∈ N = {1, 2, ...n}, and I be the two sided ideal in T generated by the relation (3). The Quotient (factor algebra)
Q = T I(5)
is called para-Fermi algebra, for all X, Y ∈ Q. This is an infinite dimensional Lie algebra with respect to the bracket defined by the equation (4).
C. Semi-simple Lie algebra generated by creation and annihilation operators
In this subsection, we quickly review main Lie algebraic properties retrieved from the work by Palev 11 , which are useful for our construction performed by sticking to the same notations.
Let g be a semi-simple Lie algebra generated by n pairs a ± 1 , ..., a ± n of creation and annihilation operators. The elements
h i = 1 2 [a − i , a + i ], i = 1, ...n
are contained in a Cartan subalgebra H of g. The rank of g ≥ n. If the semi-simple Lie algebra g of rank n is generated by n pairs of creation and annihilation operators, then, with respect to the basis of the Cartan subalgebra, the creation (resp. annihilation) operators are negative (resp. positive) root vectors. The correspondence with their roots is:
a ± i ←→ ±h * i .
where ±h * i is a basis in the space dual to the Cartan subalgebra 11 .
The semi-simple Lie algebra g of rank n is generated by n pairs of creation and annihilation operators if and only if it contains a complete system Φ of roots orthogonal with respect to the Cartan-Killing form. The semi-simple Lie algebra g of rank n is generated by n pairs of creation and annihilation operators if and only if it is a direct sum of classical Lie algebras
g = B m 1 ⊕ ... ⊕ B m k where m 1 + ... + m k = n.
The simple Lie algebra g of rank n is generated by n pairs of creation and annihilation operators if and only if it is isomorphic to the classical Lie algebra B n . 11
To construct an example of a semi-simple Lie algebra, we adapt Schwinger notation for the su(1, 1) Lie algebra: Let a + r = (a + + , a + − ) and a r = (a + , a − ) be the spin creation and annihilation operators, which obey the following commutation relations:
[a r , a r ′ ] = 0; [a r , a + r ′ ] = δ rr ′ ; [a + r , a + r ′ ] = 0 and the number of spins and the resultant angular momentum(j = 1 2 ) are respectively given by; n = r a + r a r , j = rr ′ a + r r | 1 2 σ | r ′ a r , where the creation and annihilation operators a + and a may be defined as:
a + = 1 √ 2 x − ∂ ∂x ; a = 1 √ 2 x + ∂ ∂x ,
which satisfy the Schwringer 12 su(1, 1) Lie algebra for the one-dimensional harmonic oscillator characterized by:
K + = 1 2 a + a + , K − = 1 2 aa, K z = 1 2 (a + a + 1), K ± = K x ± iK y , and
the Casimir operator C = −K 2 x − K 2 y + K 2 z , where the operators K + , K − and K z obey the commutation relation:
[K z , K ± ] = ±K z ; [K + , K − ] = −2K z . Remark I.1
The above su(1, 1) quasi-boson Lie algebra is a semi-simple Lie algebra.
D. Lie algebra of spin group
Let now m be an n-dimensional oriented real vector space with an inner product <, >.
We define the Clifford algebra 5 Cl(m) over m by the quotient T (m)/I, where T (m) is a tensor algebra over m and I is the ideal generated by all elements
v ⊗ v+ < v, v > 1, v ∈ m.
The multiplication of Cl(m) will be denoted by x · y. Let p : T (m) −→ Cl(m) be the canonical projection. Then, Cl(m) is decomposed into the direct sum Cl + (m) ⊕ Cl − (m) of the p-images of the elements of even and odd degrees of T (m), and m is identified with the subspace of Cl(m) through the projection p. Let e 1 , e 2 , · · · , e n be an oriented orthonormal basis of m. The map: e i2 · e i2 · · · · · e ip → (−1) p e ip · · · · e i2 · e i1 defines a linear map of Cl(m) and the image of x ∈ Cl(m) by this linear map is denoted byx. The spin group is defined by:
Spin (m) = {x ∈ Cl + (m) : x is invertible, x · m · x −1 ⊂ m and x ·x = 1}.(6)
Moreover, the subspace spin (m) of Cl(m) spanned by {e i ·e j } i<j is a Lie algebra of Spin (m)
in such a way that exp : spin (m) → Spin (m) is just the restriction of the exponential map of the algebra Cl(m) into Cl(m). The differentialπ of π is given by: with n = 2m(n = 2m + 1), (B n ) = {±e p , ±e p ± e q } n p,q = 1(p = q; e 1 , · · · , e 2n+1 is an orthonormal basis of R 2n+1 ),
π(x)v = x · v − v · x,
and D n is the group of orthogonal transfromations of a (2n)-dimensional complex vector space 3 L 2n : D n , n ≥ 4; g = SO(2n, C), K = SO(n, C) × SO(n, C)
with n = 2m(n = 2m + 1), (D n ) = {±e p ± e q } n p,q = 1(p = q; e 1 , · · · , e 2n is an orthonormal basis of R 2n ),
B n and D n are of great importance in Particle Physics 3 .
Let G be a connected Lie group. A semi-simple Lie group G is completely determined by the system Π(G) of its simple roots.
F. Spin Lie group and its Lie algebra
Suppose now M is an oriented Riemannian manifold. Let ξ denote a principal fibre bundle of an almost complex manifold with structural group SO(2n); n is a positive integer.
Let E(ξ) be the total space and B be the base space.
Definition I.1 9 A spin structure on ξ is a pair (ψ, f ) consisting of (i) a principal bundle ψ over B with the spin group (Spin (m)) as structural group; and
(ii) a map f : E(ψ) → E(ξ) such that the following diagram is commutative:
E(ψ) × Spin (m) −→ E(ψ) ց B ր E(ξ) × SO(m) −→ E(ξ)
Definition I.2 9 A spin manifold is an orientable manifold M together with a spin structure on the tangent bundle of M.
Definition I.3 2 A spin group is a compact dimensional Lie group.
Lie algebras are vector spaces that are convex and thus connected. The Lie algebra of spin particles spin (j) can be represented by classical matrices, which make it easier to see their algebraic nature 11 , 13 , 16 :
spin (j) = higgs j = 0; f ermions j = Z 1 2 ; Z is odd integer; bosons j = Z; Z is an integer.
The Lie algebra sl(2n, C) can represent the fermion spin Lie algebra of elementary particles in quantum physics 15 . As indicated in the mapping below, Z is an odd integer with natural numbers 4 , 15 , 16 n = 1, 2, 3, · · · , :
sl(2n, C) / / spin Z 1 2 / / f ermions .
The Lie algebra sl(2n+1, C) can represent the boson spin Lie algebra of elementary particles.
The map below gives a clear view with Z as an integer and natural numbers 4 , 15 , 16 n = 1, 2, 3, · · · .
sl(2n The groups SL(n, C) and SU(n) are simply connected. The groups GL(n, R) and SO(p, q) (0 < p < p + q) have two connected components. The group SO(2n + 1, C) is doubly connected and SO(2n, C) is fourfold connected 6 . We denote by Spin (J) the spin Lie group of a quantum spin particle as follows:
+ 1, C) / / spin (Z) / / bosons .Spin (J) = Higgs J = 0; F ermions J = Z 1 2 ; Z is odd integer; Bosons J = Z; Z is an integer.
The Lie group SL(2n, C) structure can represent the fermion Spin Lie group analog 4 , 15 , 16 :
SL(2n, C) / / Spin Z 2 / / F ermions,
while the Lie group SL(2n + 1, C) represents the boson Spin Lie group analog 4 , 15 , 16 :
SL(2n + 1, C) / / Spin (Z) / / Bosons .
Using the angular momentum coupling of spin particles, one can easily observe the following 4 , 13 , 15 , 16 : For integer n = 1, 2, 3 · · · Spin 1 2 = {particles spanned by 2 states with 2 × 2 matrix basis},
Spin 2n − 1 2 = {particles spanned by 2n states with (2n) × (2n) matrix basis},Spin ( 1) = {particles spanned by 3 states with 3 × 3 matrix basis}, Spin ( n) = {particles spanned by 2n + 1 states with (2n + 1) × (2n + 1) matrix basis}.
Spin half odd integer particles are fermions described by Fermi-Dirac statistics and have quantum numbers described by the Pauli exclusion principle 16 . They include the electron, proton, neutron, quarks and leptons. In particle physics, all these particles have symmetry and matrix representations.
G. SU(2) and Wigner coefficients
We now review some classical groups and the group theoretical approach. We consider the same notations as in 7 . SU (2) is the group of transformations in 2-dimensional unitary space, that is, the group of transformation leaving the form |
x 1 | 2 + | x 2 | 2 invariant.
This group is simply compact and possesses the Lie algebra composed of the three generators J 1 , J 2 , J 3 , obeying the commutation rule 7 :
[J i , J j ] = ie ijk J k .(9)
Next, we define a set of creation and annihilation operators a * 1 , a * 2 , a 1 , a 1 , which obey the commutation rule
[a i , a * j ] = δ ij , i, j = 1, 2,
while all other commutators vanish. The vacuum is defined by
a i | 0 = 0, and the states | j, m = (a * 1 ) j+m (a * 2 ) j−m [(j + m)!(j − m)!] 1 2 | 0 ,
where j is defined as the negative of the minimum of m or positive of the maximum of m.
Since SU(2) is a compact group, the representation must be finite dimensional 7 :
J + = a * 1 a 2 ; , J − = a * 2 a 1 , J z = 1 2 (a * 1 a 1 − a * 2 a 2 ).
We can find familiar results 7
J z |JM = M |JM , J ± |JM = J(J + 1) − M(M ± 1)|J, M ± 1 .
Furthermore, the operators e i obey the commutation relations (9), where
e 1 = J + + J − 2 , e 2 = J + − J − 2i , e 3 = J z .
This gives a Lie algebra realization of SU (2), since equation (9) constitutes a necessary and sufficient condition; thus we have established a mapping from the generators of the group onto the operators 1 , 7 e i :
J i → e i , i = 1, 2, 3,
which provides a representation states for the group algebra on the states | j, m . We now derive the Wigner coefficients of the group, that is, the coefficients coupling two states
| JM = m 1 m 2 C j 1 j 2 J m 1 m 2 M | j 1 , m 1 | j 2 , m 2 δ m 1 +m 2 ,M .
To do this, we consider the coupling of two spin half. Denoting by S 1 and S 2 two spins ( 1 2 ) angular momenta, we define the total spin
S = S 1 + S 2 .
Let S 2 1 , S 1z , S 2 2 and S 2z be the individual eigenstates of the eigenstates of S 2 and S z satisfying:
S 2 1 |s 1 m 1 = S 1 (S 1 + 1) 2 |s 1 m 1 ; S 2 2 |s 2 m 2 = S 2 (S 2 + 1) 2 |s 2 m 2 ; S 1z |s 1 m 1 = m |s 1 m 1 ; S 2z |s 2 m 2 = m |s 2 m 2 .
We define the raising and lowering operators as follows 7 , 15 :
S 1± |s 1 m 1 = s 1 (s 1 + 1) − m 1 (m 1 + 1)|s 1 , m 1 ± 1 , S 2± |s 2 m 2 = s 2 (s 2 + 1) − m 2 (m 2 + 1)|s 2 , m 2 ± 1 ,
and the tensor product basis vectors as:
|s 1 m 1 ; s 2 m 2 = |s 1 m 1 ⊗ |s 2 m 2 , where m 1 = −s 1 ...s 1 and m 2 = −s 2 ...s 2 .
We seek a transformation to a set basis denoted |SM , which obeys 1 , 7 , 15 :
S 2 |SM = S(S + 1) 2 |SM ,(10)S z |SM = M |SM ,(11)S ± |SM = S(S + 1) − M(M ± 1)|S, M ± 1 .(12)
In relation to unitary transformation
|SM = m 1 m 2 U s 1 s 2 m 1 m 2 ;sm |m 1 m 2 ,(13)
where
U s 1 s 2 i,
and comparing equation (13) and (14) lead to:
U s 1 s 2 m 1 m 2 ;sm ≡ s 1 s 2 m 1 m 2 |sm .
The Clebsch-Gordan coefficients (C.G) are obtained as:
U s 1 s 2 m 1 m 2 ;sm := C s 1 s 2 s m 1 m 2 m .
The Wigner coefficients 17 of the SU(2) group are then derived as:
C s 1 s 2 S m 1 m 2 M = [2S + 1] 1 2 (−1) s 2 +m 2 (S + s 1 − s 2 )!(S − s 1 + s 2 )!(s 1 + s 2 − S)! (S + s 1 + s 2 + 1)!(s 1 − m 1 )!(s 1 + m 1 )! × (S + M)!(S − M)! (s 2 + m 2 )!(s 2 − m 2 )! 1 2 k (−1) k × (S + s 2 + m 1 − k)!(s 1 − m 1 + k)! k!(S − s 1 + s 2 − k)!(M + S + k)!(s 1 − s 2 − M + K)
.
Details can be found in 1 , 7 , 15 .
The paper is organized as follows. Section II deals with the semi-simplicity of spin particle Lie structure. In section III, we develop the real Lie algebra of a spin particle. We construct the Iwasawa decomposition in section IV. Finally, we end with some concluding remarks in section V.
II. SPIN SEMI-SIMPLICITY
While investigating the general semi-simple Lie group structure, one can examine a similar structure in its Lie algebra. In this section, we prove some lemmas, which help us lift the notion of spin particle Lie algebra to the Lie group level, and prove a statement giving a clear picture of spin particles as Lie groups. Finally, we prove a theorem on its semi-simplicity 2 .
Lemma II.1 Any spin particle Lie algebra admits a Clifford algebra and a spin group structure.
Proof Consider any spin (j) , with j = 0, 1 2 , 1, · · · , satisfying the spin particle commutator and anticommutator relations equations (1) (2) (3) as well as the spin Lie algebra commutation bracket rule. From the equations (5) and (6) it is obvious that the Lie algebra of spin (j) is a Clifford algebra. Thus, the spin (j) Lie algebra is connected and its exponential is just
exp : spin (j) → Spin ( J) ,
where Spin (J) is the spin group. Hence, any spin particle admits a spin group.
Lemma II.2 Any spin group of a spin particle admits an almost complex spin manifold and a spin Lie group structure.
Proof From Lemma II.1, any spin particle admits a spin group. Also, from Definition I.1, the spin group, say Spin (J) , has a group structure with an almost complex manifold.
Thus, from Definition I.2, the spin particle, say Spin (J) , with J = 0, 1 2 , · · · , admits a spin manifold. Next, we see that any spin particle has a spin group, say Spin (J) . Since any spin particle has a spin manifold, we observe that Spin (J) is a spin group and, hence, a spin Lie group.
Proposition II.1 Any spin half odd integer, (resp. integer spin) Lie group, is a fourfold cover of the compact Lie group SO(2n), (resp. a double cover SO(2n + 1)).
Proof The fermion quantum structure can be given as:
Spin (J) / / Spin ( Z 2 ),
where Z is odd integer. The map
SL(2n, C) / / Spin ( Z 2 ) / / SO(2n) ,
where SO(2n) is the group of all matrices, conserves the quadratic form in C 2n . The compact simple Lie group SO(2n) is fourfold connected, and has a center, Z(G), Z 4 if n is odd and Z 2 × Z 2 if n is even. Since Spin ( Z 2 ) is a fermion with Z as odd integer, then, the diagram
SL(2n, C) −→ Spin ( Z 2 ) ց ↑ ↓ D n → f ermions ր SU * (2n) −→ SO(2n)
must commute. Thus, a fermion is a fourfold cover of SO(2n). Similarly, the boson quantum structure can be given as:
Spin (J) / / Spin (Z ),
where Z is an integer. The map
SL(2n + 1, C) / / Spin (Z) / / SO(2n + 1) ,
where SO(2n + 1) is the group of all matrices, conserves the quadratic form in C 2n+1 . The compact simple Lie group SO(2n + 1) is doubly connected and has center, Z(G), Z 2 . Since
Spin (Z) is a boson, where Z is an integer, then the diagram
SL(2n + 1, C) / / ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗
Spin (Z) ( ( P P P P P P P P P P P P SO(2n + 1) / / B n → boson must commute. Thus, a boson is a double cover of SO(2n + 1). See 6 for more details.
Theorem II.1 Any spin Lie group, Spin (J) of a spin particle is:
(i) connected;
(ii) semi-simple if and only if its simple roots are one of the Dynkin's root system Π(B n )
or Π(D n ).
Proof For (i) we let Spin (J) be a spin Lie group with J = 0, 1 2 , 1 · · · . For Spin (0), Spin 1 2 , Spin (1), we have, respectively, the diagrams: where m 1 + ... + m k = n. Therefore, the creation and annihilation operators of spin particles generate simple Lie algebra g of rank n isomorphic to the classical algebra B n , which contains a complete system Φ of roots orthogonal with respect to the Killing form 11 . Also, from equations (1) and (2), when we compare the bracket relation to that of the Dynkin's root D n , (see equation (8)), we observe that there is a correspondence. From Lemma II.2, we showed that every spin group of a spin particle is a spin Lie group. We can determine the system of simple roots for the groups Π(B n ) and Π(D n ). A semi-simple Lie group G is completely determined by the system Π(G) of its simple roots 3 . Thus, the spin Lie group of a spin particle is completely determined by the Π(G) of its simple roots. The converse is trivial since the groups B n and D n , which are Π(B n ) and Π(D n ) (Dynkin's root system), are the operators of the quantum spin particles generated by the creation and annihilation operators of rank n, since the spin Lie group is connected and its Lie algebra is also semisimple. Thus, the spin Lie group is semi-simple.
SL(1, C) / / & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Spin (0) ' ' P P P P P P P P P P P P SO(1) / / boson(Higgs), SL(2, C) / / & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Spin ( 1 2 ) & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ SO(2) / / f ermion, SL(3, C) / / & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Spin (1) % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ SO(3) / / boson.
III. REAL LIE ALGEBRA OF SPIN PARTICLE
The sl(2, C) Lie algebra can be decomposed into the compact real su(2) and imaginary isu(2) forms, or sl(2, R) and isl(2, R). It is only natural to seek the real form of the spin half particle Lie algebra in terms of Pauli matrices 16 , which are sl(2, C) matrix basis elements.
S x = S + + S − 2 and S y = S + − S − 2i .
From the above equations (10) and (11), we can write S 2 and S z in terms of spinors. Indeed,
S 2 χ1 2 = 2 1 2 1 2 + 1 χ1 2 = 3 4 2 χ1 2 , S 2 χ − 1 2 = 2 3 4 χ − 1 2 .(15)
From equations (15), we can deduce
S 2 = 3 4 2 1 0 0 1 = 3 4 2 I, where I = 1 0 0 1 is the identity matrix. Similarly, S z χ1 2 = 2 χ1 2 and S z χ − 1 2 = − 2 χ − 1 2 .
Therefore,
S z = 2 1 0 0 −1 = 2 σ z .
By analogous computations, we get:
S + = 0 1 0 0 = σ + and S − = 0 0 1 0 = σ − .
Similarly,
S x = S + + S − 2 = 2 0 1 1 0 = 2 σ x(16)
and
S y = S + − S − 2i = −iS k = −i 2 0 1 −1 0 = −i 2 σ k = 2 0 −i i 0 = 2 σ y .(17)
Defining the bracket [X, Y ] of X, Y by
[X, Y ] := XY − Y X,(18)
we observe that:
(i) [X, Y ] ∈ spin ( 1 2 ) if X, Y ∈ spin ( 1 2 ); (ii) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 f or X, Y, Z ∈ spin ( 1 2 ); (iii) [X, Y ] = −[Y, X] f or X, Y ∈ spin ( 1 2 ).
Thus, the spin ( 1 2 ) has a real Lie algebra structure on R. One can easily check that
[S k , S z ] = − S x , [S k , S + ] = S z and [S z , S + ] = S + .
Lemma III.1 For any Spin 1 2 , there exists an orthogonal (skew symmetric) basis element S k , which can be transformed into SO(2), a compact and rotational matrix.
Proof From equation (17), we have:
S y = S + − S − 2i = −iS k = −i 2 0 1 −1 0 = −i 2 σ k = 2 0 −i i 0 = 2 σ y ,
where S k is a basis of the spin half particle from Proposition III.1. For skew symmetric matrix,
S T k = −S k .
Similarly, for orthogonal matrix, we have:
S −1 k = S T k .
Also, we see that for any t ∈ R,
exp(tS k ) = cos t 2 sin t 2 − sin t 2 cos t 2 = k t .
When =1, det k t = 1. Thus, k t is compact. Next, we show that St(i) = {g ∈ G|g · i = i} is the stabilizer of i. Indeed,
g · i = i ⇐⇒ ai + b ci + d = i ⇐⇒ ai + b = i(ci + d) = id − c ⇐⇒ a = d and c = −b.
Then,
St(i) = {g = a b −b a ∈ G|a 2 + b 2 = 1}.
The relation a 2 + b 2 = 1 implies there exists θ ∈ [0, 4π] such that: a = cos θ 2 and b = sin θ 2 , where
St(i) = {g = cos θ 2 sin θ 2 − sin θ 2 cos θ 2 |0 ≤ θ ≤ 4π} = K.
Thus, K = SO(2) is the rotational matrix which is compact. Therefore, S k is compact for =1. Hence, the proof is completed. where sl(2, R) and su(2) are the real forms of the complex group sl(2, C) and su(2, C) 10 .
Similarly,
spin 1 2 = spin 1 2 ⊕ i spin 1 2 ,
where spin 1 2 is the real form of the spin half Lie algebra. The spin 1 2 ∈ sl(2, C). Thus, it is complex, and for good notation, we write spin 1 2 , C ∈ sl(2, C). For the real form, we write spin 1 2 , R ∈ sl(2, R).Finally,
spin 1 2 , C = spin 1 2 , R ⊕ i spin 1 2 , R .
For simplicity, in the next section, we shall use the usual notation spin( 1 2 ) to be the real form spin 1 2 , R of the spin half particle.
IV. IWASAWA DECOMPOSITION ON LIE ALGEBRA AND LIE GROUP LEVELS
Following the Iwasawa decompostion, we can uniquely decompose any semi-simple spin 1 2 particle Lie algebra as follows:
g = k ⊕ d 1 2 t ⊕ n ξ ,
where g is the Lie algebra of Spin 1 2 with k θ = {skew symmetric 2 × 2 matrices}, d (ii) Any spin 1 2 particle is uniquely decomposable in the form:
spin 1 2 = 3 k θ d 1 2 t n ξ = exp (θ s m |S k | s m ) · exp (t s m |S z | s m ) · exp (ξ s m |S + | s m ) . (19) If spin ( 1 2 ) = a b
c d ∈ Spin 1 2 , then, θ, t, ξ in Theorem IV.1(i) are given by the relations:
exp i θ 2 = a − ic 3 √ a 2 + c 2 ,(20)exp(t) = a 2 + c 2 6 ,(21)
and ξ = 6 (ab + cd) a 2 + c 2 .
Proof Since
k θ = exp(θS k ) = exp (θ s m |S k | s m ) = exp θ 2 0 1 −1 0 = exp θ 2 0 1 −1 0 (23) and K θ = ∞ n=0 (−1) n (2n)! θ 2 2n · I + ∞ n=0 (−1) n (2n + 1)! θ 2 2n+1 · σ k = cos θ 2 sin θ 2 − sin θ 2 cos θ 2 .
By isomorphism θ −→ K θ , we obtain:
K ∼ = R 4πZ ∼ = T. Moreover, d 1 2 t = exp (t(S z )) = exp (t s m |S z | s m ) = ∞ n=0 1 n! (t S z ) n = ∞ n=0 1 n! (t σ z ) n = e t 2 0 0 e − t 2 .
By isomorphism t −→ d 1 2 t , we also have: D ∼ = R. Now, since (S + ) 2 = 0,
n ξ = exp (ξS + ) = exp (ξ σ + ) = exp (ξ s m |S + | s m ) = 1 ξ 0 1 .
By matrix multiplication, we have:
a b c d = 3 k θ d 1 2 t n ξ = exp (θ s m |S k | s m ) · exp (t s m |S z | s m ) · exp (ξ s m |S + | s m ) = 3 exp t 2 cos θ 2 3 cos θ 2 exp t 2 ξ + 3 sin θ 2 exp − t 2 − 3 exp t 2 sin θ 2 − 3 sin θ 2 exp t 2 ξ + 3 cos θ 2 exp − t 2 . yielding a = 3 exp t 2 cos θ 2 , c = − 3 exp t 2 sin θ 2 , and a − ic = 3 exp t 2 + i θ 2 .
Hence, |a − ic| = 3 exp t 2 , we easily get equations (20) and (21), and
ab + cd = exp(t)ξ(24)
from which we can clearly obtain equation (22). Now we know that Spin 1 2 is spanned by two states: {| 1 2 1 2 , | 1 2 , − 1 2 }. From equations (10), (11) and (12), we can calculate the angular momentum for spin half integers such as of fermions, where n = 1, 2, 3 · · · , we have:
(i) S 2 |SM n = 4n 2 − 1 4 2 |SM . (ii) S z |SM n = ± 2n − k 2 |SM ,
where k ≤ 2n, and n = 1, 2, 3, · · · with k = 1, 3, 5, · · · .
(iii) The n th possible states of a spin half particle is given by:
M sn = 2S n + 1 = 2n,
where n = 1, 2, 3, · · · The particle is spanned by 2n states as below:
2n − 1 2 ± 2n − 1 2 , · · · , 2n − 1 2 ± ( 2n − k 2 ,
where k = 1, 3, 5, · · · , with k ≤ 2n.
(iv) The ladder operators act as follows:
S +n 2n − 1 2 2n − k 2 = (k − 1)n − (k − 1)(k − 1) 4 S, M + 1 , S +n 2n − 1 2 , − 2n − k 2 = (k + 1)n − (k + 1)(k + 1) 4 S, M + 1 , S −n 2n − 1 2 2n − k 2 = (k + 1)n − (k + 1)(k + 1) 4 S, M − 1 , S −n 2n − 1 2 k − 2n 2 = (k − 1)n − (k − 1)(k − 1) 4 S, M − 1 .
(v) The ladder operators can be splitted as:
S ±n = S xn ± S kn .
Proof For the spin half integer, we have the sequence:
1 2 , 3 2 , 5 2 · · · , 2n − 1 2 ,(25)
where n = 1, 2, 3, · · · . Similarly, we can use the sequence
1 2 , 3 2 , 5 2 · · · , 2n + 1 2 ,
where n = 0, 1, 2, 3, · · · . However, we will stick to that of equation (25). From equation (10), we have: S 2 |SM = S(S + 1) 2 |SM . For the spin half, S = 2n−1 2 , we have:
S 2 |SM = 2n − 1 2 2n − 1 2 + 1 2 |SM .
By simple computations, we arrive at:
S 2 |SM n = 4n 2 − 1 4 2 |SM .
This proves Theorem IV.1(i). Next, for (ii), we check the n th term for S zn :
S zn |SM = M |SM , S z |SM n = ± 2n − k 2 |SM ,
where k ≤ 2n, and n = 1, 2, 3, · · · with k = 1, 3, 5, · · · , as required. The n th possible states of a spin half particle are given by:
M sn = 2S n + 1 = 2 2n − 1 2 + 1 = 2n,
where n = 1, 2, 3, · · · For spin ( 1 2 ), we have ;
M s = 2n = 2(1) = 2 states, i.e. particles, the spin is spanned by 2n states. It is easy to check that:
2n − 1 2 ± 2n − 1 2 , · · · , 2n − 1 2 ± 2n − k 2 ,
where k = 1, 3, 5, · · · with k ≤ 2n as required by Theorem IV.2(ii).
For the (iv), we define the ladder operators for the Spin ( 2n−1 2 ):
S +n 2n − 1 2 ± 2n − k 2 .
From equation (12), we have:
S ± |SM = (S ∓ M)(S ± M + 1)|S, M ± 1(26)
and the computation provides the actions of the raising operator as given by the relations:
S +n 2n − 1 2 2n − k 2 = 2n − 1 2 − 2n − k 2 2n − 1 2 + 2n − k 2 + 1 S, M + 1 = k − 1 2 4n − k + 1 2 S, M + 1 = (k − 1)n − (k − 1)(k − 1) 4 S, M + 1 , and S +n 2n − 1 2 , − 2n − k 2 = S +n 2n − 1 2 k − 2n 2 = 2n − 1 2 − k − 2n 2 2n − 1 2 + k − 2n 2 + 1 S, M + 1 = 4n − k − 1 2 k + 1 2 S, M + 1 = (k + 1)n − (k + 1)(k + 1) 4 S, M + 1 .
Similarly, it is easy to check for the lowering operator S −n to obtain:
S −n 2n − 1 2 2n − k 2 = (k + 1)n − (k + 1)(k + 1) 4 S, M − 1 and S −n 2n − 1 2 k − 2n 2 = (k − 1)n − (k − 1)(k − 1) 4 S, M − 1 .
For (v), we observe the equations (16) and (17), and apply these operators to the spin ( 2n−1 2 ) to obtain:
S ±n = S xn ± iS yn = S xn ± i(−iS kn ) = S xn ± S kn .
Hence, the proof is completed.
Remark IV.1 Note that S T +n = S −n . From equation (16), we have:
S xn = S +n + S −n 2 .
Similarly, from equation (17), we obtain:
S yn = S +n − S −n 2i = σ kn 2i = −iS kn .(27)
For k ≤ 2n, the above ladder operators S +n and S −n act as:
S +n 2n − 1 2 ± 2n − k 2 and S −n 2n − 1 2 ± 2n − k 2 .
We observe that with natural numbers n = 1, 2, 3, · · · .
Proof From Lemma III.1, we observe that this is true for n = 1. For Spin 2n−1 2 particle spanned by 2n states, we consider similar arguments as for Theorem IV.3, replacing the S k matrix by the n th matrix S kn and deducing in same manner as in Lemma III.1 to get the above Theorem IV.3. Specifically, from Theorem IV.2, there exists S kn in the operator S yn from equation (27). One can check that these matrices are orthogonal and generate SO(2n) with n = 1, 2, 3 · · · . For n = 1 we have the compact and rotational matrix SO(2) as in the above Lemma III.1. This achieves the proof.
V. CONCLUDING REMARKS
In this paper, we provided an extension of semi-simplicity of spin particle Lie algebra to the Lie group level. We showed that a spin particle Lie algebra admits a Clifford algebra, an almost complex manifold and a spin Lie group structure. We demonstrated that any spin half particle, (resp. integer spin), spin Lie group is a fourfold, (resp. double), cover of the SO(2n), (resp. SO(2n + 1)). We also proved that any spin Lie group of a spin particle is connected and semi-simple. We constructed the real Lie algebra of the Spin 1 2 particle. We also performed the Iwasawa decomposition of the spin half into KDN. Finally, we applied the angular momentum coupling to the Spin 2n−1 2 particle and demonstrated that the orthogonal basis transforms into the SO(2n) one, which is nothing but the Dynkin's root D n .
DATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
for x ∈ spin (m) and v ∈ m. E. Root system for semi-simple Lie group Eugene Dynkin, based on a geometric method of classifying all simple Lie groups, proved that the semi-simple Lie group is determined by its system of simple roots. The root system of the group B n is the group of orthogonal transfromations of a (2n+1)-dimensional complex vector space 3 L 2n+1 : B n , n ≥ 2; g = SO(2n + 1, C), K = SO(n, C) × SO(n + 1, C)
Parafermions and parabosons have creation and annihilation operators that correspond to the Dynkin's root B n . We can lift the results from the Lie algebra level to the Lie group level. The classical Lie algebra matrices have corresponding Lie group analogues. The groups GL(n, C), SL(n, C), SL(n, C), SU(p, q), SU * (2n), SU(n), U(n), SO(n, C), SO(n), SO * (2n), Sp(n, C), Sp(n), Sp(2, R), Sp(p, q) are all connected. For more details, see 6 .
|s 1 s 2 m 1 m 2 s 1 s 2 m 1 m 2
2j is the ij th element of the unitary matrix U s 1 s 2 that transforms the basis |m 1 m 2 to the basis |SM 15 . Using the closure property of the basis |m 1 m 2 , |SM ,
From section II, the Lie groups SO(1), SO(2) and SO(3) are connected 6 . The results can be extended to all spin Lie groups of elementary particles as shown in Proposition II.1. Fermions and bosons are fourfold connected and double connected, respectively. More specifically, the spin Lie groups such as the Spin ( 1 2 ) fourfold covers the SO(2) compact Lie group, while the Spin (1) double covers the SO(3). Next, for (ii), we know that the creation and annihilation operators generate the semi-simple Lie algebra g of rank n, which is a direct sum of classical Lie algebras g = B m 1 ⊕ ... ⊕ B m k ,
Proposition III. 1
1The real Lie algebra spin 1 2 of spin half particles (Spin ( 1 2 )) is given by spin 1 2 = {S ∈ M 2 (R)|T rS = 0}. The commutation relations are given by :[S k , S z ] = − S x , [S k , S + ] = S z , [S z , S + ] = S + .Proof Take an arbitrary angular momentum spin ( 1 2 ) with spinors χ =
Remark III. 1
1The Pauli matrices 16 , as seen in Proposition III.1, are just the basis of sl(2, C). Moreover, sl(2, C) = sl(2, R) ⊕ i sl(2, R) = su(2) ⊕ i su(2),
∼
trace zero matrices} and n ξ ={ upper triangular 2 × 2 matrices with zeros on the diagonal}. It is just like the Iwasawa decomposition of the sl(2, r) Lie algebra when the value of = 1 8 , 10 . Theorem IV.1 Iwasawa Decomposition of Spin ( 1 2 ) particle 14 (i) Let θ, t, ξ be arbitrary real numbers, and put k θ = exp(θS k ), d 1 2 t = exp(tS z ), and n ξ = exp(ξS + ). Then, the subgroups 3 KDN of Spin 1 2 are defined by: K θ = { k θ |θ ∈ R}, D = { d 1 2 t |t ∈ R} and N = { n ξ |ξ ∈ R}. We have: = T, D ∼ = R , N ∼ = R.
so on· · · 15 . A question arises: what can be the n th term of spin half integer? From theoretical point of view, this can be useful in the study of particles rotational forms. We have the following statement: Theorem IV.2 For any Spin 2n−1 2
by 2n states, there exists an orthogonal basis S kn in the S yn operator which can be transformed into the group D n of SO(2n)
ACKNOWLEDGMENTS
This work is supported by the NLAGA-SIMONS grant and the TWAS Research Grant RGRA No. 17 − 1542RG/MAT HS/AF/AC − G − F R32400147. The ICMPA-UNESCO chair is in partnership with Daniel lagolnitzer Foundation(DIF), France, and the Association pour la Promotion Scientifique de l'Afrique (APSA), supporting the development of Mathematical Physics in Africa.
The case n = 1 corresponds to spin ( 1 2 ). Thus, for Spin 2n−11
2
1
2
and
1
2
−
1
2
,
since 2n−1
2
= 2(1)−1
2
= 1
2 . 2
From equation(28), we arrive at:Differentiating both sides of equation(28)with respect to θ, we have:By equating the imaginary parts of both sides of this relation, we get:(g, θ)) .Similarly, from direct computation as in Theorem IV.1, we have:Remark IV.2 The Iwasawa decomposition of the spin half particle into compact, rotational (Abelian), and nilpotent functions (subgroups) can also be performed for integer spin particles as well as for isospins.
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The Clifford algebra and the spinor group of a Hilbert space. P De , La Harpe, Compositio Mathematica. 253P. de. La Harpe, "The Clifford algebra and the spinor group of a Hilbert space". Compositio Mathematica, tome 25, No. 3 p. 245-261 (1972).
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Complex angular momenta and the groups SU(1, 1) and SU(2). W J Holman, L C BiedenharnJr, Ann. Phys. 47W. J. Holman III and L. C. Biedenharn Jr., Complex angular momenta and the groups SU(1, 1) and SU(2), Ann. Phys. 47 (1966).
On some types of topological groups. K Iwasawa, Ann. Math. 503K. Iwasawa, "On some types of topological groups", Ann. Math., Vol. 50, No. 3 (1949), pp. 507-558.
Remarks concerning spin manifolds. J Milnor, Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse. Princeton, NJPrinceton Univ. PressJ. Milnor, "Remarks concerning spin manifolds", In Differential and Combinatorial Topol- ogy, a Symposium in Honor of Marston Morse, Princeton Univ. Press, Princeton, NJ. (1965).
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Habilitation thesis, Inst. Nucl. Research and Nucl. Energy, Sofia. T D Palev, BulgarianLie algebraical aspects of the quantum statisticsT. D. Palev, "Lie algebraical aspects of the quantum statistics", Habilitation thesis, Inst. Nucl. Research and Nucl. Energy, Sofia, in Bulgarian (1976).
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|
[] |
[
"APPROACHES TOWARD PHYSICAL AND GENERAL VIDEO ANOMALY DETECTION",
"APPROACHES TOWARD PHYSICAL AND GENERAL VIDEO ANOMALY DETECTION"
] |
[
"Laura Kart \nSchool of Computer Science and Engineering\nThe Hebrew University of Jerusalem\nIsrael\n",
"Niv Cohen \nSchool of Computer Science and Engineering\nThe Hebrew University of Jerusalem\nIsrael\n"
] |
[
"School of Computer Science and Engineering\nThe Hebrew University of Jerusalem\nIsrael",
"School of Computer Science and Engineering\nThe Hebrew University of Jerusalem\nIsrael"
] |
[] |
In recent years, many works have addressed the problem of finding never-seen-before anomalies in videos. Yet, most work has been focused on detecting anomalous frames in surveillance videos taken from security cameras. Meanwhile, the task of anomaly detection (AD) in videos exhibiting anomalous mechanical behavior, has been mostly overlooked. Anomaly detection in such videos is both of academic and practical interest, as they may enable automatic detection of malfunctions in many manufacturing, maintenance, and real-life settings. To assess the potential of the different approaches to detect such anomalies, we evaluate two simple baseline approaches: (i) Temporal-pooled image AD techniques. (ii) Density estimation of videos represented with features pretrained for video-classification.Development of such methods calls for new benchmarks to allow evaluation of different possible approaches. We introduce the Physical Anomalous Trajectory or Motion (PHANTOM) dataset 1 , which contains six different video classes. Each class consists of normal and anomalous videos. The classes differ in the presented phenomena, the normal class variability, and the kind of anomalies in the videos. We also suggest an even harder benchmark where anomalous activities should be spotted on highly variable scenes.
|
10.1109/icassp43922.2022.9747367
|
[
"https://arxiv.org/pdf/2112.07661v1.pdf"
] | 245,131,228 |
2112.07661
|
8d6d799d6723a751b8808ab783d223671f89e552
|
APPROACHES TOWARD PHYSICAL AND GENERAL VIDEO ANOMALY DETECTION
Laura Kart
School of Computer Science and Engineering
The Hebrew University of Jerusalem
Israel
Niv Cohen
School of Computer Science and Engineering
The Hebrew University of Jerusalem
Israel
APPROACHES TOWARD PHYSICAL AND GENERAL VIDEO ANOMALY DETECTION
Index Terms-Anomaly Detection, Video Anomaly Detection
In recent years, many works have addressed the problem of finding never-seen-before anomalies in videos. Yet, most work has been focused on detecting anomalous frames in surveillance videos taken from security cameras. Meanwhile, the task of anomaly detection (AD) in videos exhibiting anomalous mechanical behavior, has been mostly overlooked. Anomaly detection in such videos is both of academic and practical interest, as they may enable automatic detection of malfunctions in many manufacturing, maintenance, and real-life settings. To assess the potential of the different approaches to detect such anomalies, we evaluate two simple baseline approaches: (i) Temporal-pooled image AD techniques. (ii) Density estimation of videos represented with features pretrained for video-classification.Development of such methods calls for new benchmarks to allow evaluation of different possible approaches. We introduce the Physical Anomalous Trajectory or Motion (PHANTOM) dataset 1 , which contains six different video classes. Each class consists of normal and anomalous videos. The classes differ in the presented phenomena, the normal class variability, and the kind of anomalies in the videos. We also suggest an even harder benchmark where anomalous activities should be spotted on highly variable scenes.
INTRODUCTION
Detecting never-seen-before novelties, or anomalies, is a key ability humans use to raise awareness of new dangers and opportunities. Examples of such include spotting new behaviors in natural systems, detecting security threats, or spotting equipment malfunctions. While anomaly detection (AD) methods aimed at images [1,2,3] usually address a large variety of data domains, video AD has been mostly focused on surveillance videos. In such videos [4,5], typically taken from security cameras, normal walking behavior is usually defined as the normal class, and other behaviors (riding a bike, crowd gathering) are defined as anomalies. In recent years, many new techniques have been suggested, pushing forward performance on such benchmarks [6,7]. Nevertheless, other kinds of video anomalies remain mostly out of the scope of contemporary research.
In this work we explore different kinds of anomalies, examining different types of methods to address them. Our work addresses physical abnormalities in the motion or trajectory of an object that have been largely overlooked and presents a new benchmarks for video anomaly detection.
RELATED WORK
Video Anomaly Detection As most work on video anomaly detection has focused on surveillance videos, various methods have been tried to tackle this problem. Deep learning methods have been able to make significant progress toward solving this problem, using autoencoders [8,9], generative models [10], or prediction [11]. More recent methods have been able to outperform using pretraining and self-supervised learning [12,6,7].
Image Anomaly Detection Image anomaly detection is a fastgrowing field, often using similar techniques to those used in video AD. Deep learning models have been able to outperform classical methods using auto-encoders and later RotNet-type self-supervised methods [2,13]. Lately, contrastive learning methods have been used to further improve performance [3,14]. A promising line of work suggests to detect anomalies using pretrained features [14,1]. Pretrained features robustly outperform self-supervised methods, especially on small datasets or when dealing with subtle anomalies. We use similar approaches in this paper.
Existing Datasets Challenging datasets and benchmarks have been key in advancing the field of computer vision and machine learning. In image anomaly detection, most work has been focused on utilizing standard image classification datasets, while specialized benchmarks focusing on surveillance have been suggested for video anomaly detection [15,16,17,18]. So far, video AD has so far been heavily reliant on these datasets. On one hand, this ensured that this rapidly-developing field continued to stay relevant to the evaluated task. On the other hand, this may have limited the generality of the methods. Table 2. Anomaly Detection Performance on the PHANTOM dataset (ROCAUC). The results of the image features were obtained using ViT with average pooling and the video features using TimeSformer. These are compared to MNAD surveillance method.
EVALUATED METHODS
In our work, we use feature extractors from pretrained image and video networks to represent the videos. As the train set contains normal-only videos, we assume that this set makes up a single highdensity region in the feature space. With this assumption, for new unseen videos we elect to use the k-nearest neighbors (kNN) distance from the train set as the anomaly score. This simple baseline approach of feature extraction combined with kNN outperforms all previous state-of-the-art methods on small datasets [1].
Image features-based AD
In the first approach, we work with networks pretrained for singleimage classification. As these networks have been shown to be very useful on image anomaly detection, the first approach we try is to adapt them to represent entire videos. The train set Vtrain = V1, . . . , Vn is comprised of normal videos only. Each video is separated into frames Vi = vi,1, . . . , vi,t i . The frames are evenly sampled and a feature extractor F is used to extract frame features.
F (vi,j) = fi,j(1)
In order to return a single feature vector for each video, we use maximum or average temporal pooling. We refer to each of the above as a time series operation T , taking the vector representation of each frame fi,1, . . . , fi,t i and returning a single vector representation for the entire video fi.
F (Vi) = T (fi,1, . . . , fi,t i ) = fi(2)
We now have a set of embeddings representing the train set, Ftrain = fi, . . . , fn. Given a new video sample Vy in the test set, we score its abnormality by extracting its features fy and then by computing its kNN distance from Ftrain.
d(Vy) = 1 k f ∈N k (fy ) f − fy 2 (3)
Here, N k (fy) represents the embeddings of the k-nearest neighbors to fy in Ftrain. After obtaining this distance, we determine whether the video is normal or not by verifying that the distance is greater than some threshold. This approach can be viewed as a simplified version of [19,20]. As official implementations of these methods were not available during this study, we are unable to provide a comparison.
Video features-based AD
Aiming not only to capture the semantics of each image, but also the dynamics of the video, we also utilize pretrained video representations. We use networks pretrained for classification on large video datasets, as such networks are likely to represent a video's content and dynamics in a meaningful way to humans.
Here too, the train set Vtrain = V1, . . . , Vn is normal. This time, the feature vector can be extracted directly from the video.
F (Vi) = fi(4)
We use feature extractors to obtain the train set embeddings Ftrain = f1, . . . , fn. Given a new test video Vy, we proceed as discussed above to compute the kNN distance from Ftrain and use it as the anomaly score.
Surveillance method
To compare our methods to the rapidly developing field of algorithms for AD in surveillance videos, we choose to run the Memoryguided Normality for Anomaly Detection (MNAD [7]) method on our dataset. MNAD is a state-of-the-art method for video AD for which we were able to find an official implementation. This method returns a score for each video frame. To adapt it such that it returns a single score per video, we take the average of the frame scores. Table 3. Anomaly Detection Performance on the SSv2 dataset (ROCAUC)
EXPERIMENTS
Our experimental results detail when our method performs well and establish a baseline for future work. We compare our method to MNAD on our dataset as well as on Something-Something-V2 [21]. Lastly, we run our method on the UCSD Pedestrian 1 and 2 [5] datasets commonly used in anomaly detection for surveillance.
Datasets
Physical Anomalous Trajectory or Motion (PHANTOM) Dataset
To evaluate the presented approaches, we created the Physical Anomalous Trajectory or Motion (PHANTOM) dataset consisting of six classes featuring everyday objects or physical setups, and showing nine different kinds of anomalies. We designed our classes to evaluate detection of various modes of video abnormalities that are generally excluded in video AD settings. The train and test sets of each class contain approximately 30 videos of varying lengths. The train set contains only normal videos, while the test set is evenly balanced between normal and anomalous videos. The classes were designed to be of varying difficulties and to feature different types of anomalies. For example, the window class was filmed in multiple lighting scenarios to increase variance.
The normal videos include motion that follows an expected trajectory (pendulum, keyboard) or an expected movement (window). The sushi class features procedural motion, while candle and magnets feature more subtle motion that only appears locally. The anomalous videos can feature an interference of the regular motion (window, candle, magnets), an added or removed step in the usual procedure (sushi), motion that follows a different trajectory (pendulum, keyboard), or contains a different object (pendulum). The pendulum and magnets classes contain more than one type of anomaly. An overview of these classes is displayed in Tab.2.
General Activity Dataset To examine a harder, even more general video AD setting, we work with the Something-Something-V2 (SSv2) dataset. It features 174 classes of various activities such as Throwing something in the air and catching it where something is not limited to a single object type. We adapt this dataset for use for general AD as described in Sec. 4.3 and it can be thought of as suitable crossover between physical and surveillance videos.
Physical AD on PHANTOM dataset
To determine the ability of our proposed method at finding anomalies in videos, we test it on our dataset. We explore the different modes Tearing Closing Fig. 2. Examples of frames from two of the ten video types in the SSv2 classes that we used. For each class, two frames are shown to demonstrate the high class variability.
of the method and uncover the strengths and weaknesses of each setting. We measure our results using the area under the curve of the receiver operating characteristic (ROCAUC) as is common in previous works [1,2]. Image Features: Here we run the videos in our dataset framewise through different pretrained image networks. We choose a Residual Network (ResNet [22]) and the Vision Transformer (ViT [23]) as previously they have been tested extensively on numerous image datasets and have been shown to achieve excellent transfer learning results. Each network comes pretrained on ImageNet and ImageNet-21K. In order to summarize the frame features into a single feature vector we use temporal pooling. After obtaining the video features, we give each video an anomaly score. The results of the best performing image-based method on the PHANTOM dataset appear in Tab.2.
Video Features: Video networks return a single feature vector per video, therefore it is not necessary to pool the frame-level features. We use the following pretrained video networks: TimeSformer [24] pretrained on Kinetics-600 [25] and SSv2, and the Big-Little-Video-Net Temporal Aggregation Module (bLVNet-TAM [26]) architecture pretrained on SSv2. A summary of the best-performing method is given in Tab
Comparison to MNAD on SSv2
Next, we choose to contrast our simple method of feature extraction and kNN to MNAD on the SSv2 dataset. We choose 10 classes and from among these classes, we determine the largest subclass and label it normal. We run this subclass against the videos in the largest subclass of the other 9 classes which are labeled anomalous. This evaluation is similar to that of the PHANTOM dataset. The names of the chosen classes and subclasses appear in Tab.3 together with the results.This task proved challenging for all methods and the results show that while our method works well on our PHANTOM dataset, it is not well-adapted for general videos. This highlights general and physical videos as different modalities which may call for different AD solutions.
Comparison on surveillance dataset
In order to examine the performance of our proposed benchmarks against surveillance videos, we test it on other types of data. To this end, we select the UCSD Pedestrian 1 and 2 datasets. These datasets contain surveillance footage of two pedestrian scenes and are commonly used for video AD evaluation. Surveillance AD methods differ from ours in that each frame is given an anomaly score. To overcome this, we adapt our procedure in the following way. We break the video into overlapping windows of constant length such that each frame appears in several windows. Thus, each video in the train set contributes several feature vectors equal to the number of overlapping windows in that video. Given an unseen video, we again divide the video into windows and compute the window features. We compute the kNN distance between each window and the windows in the train set. As each frame can appear in several windows, the anomaly score that is given is the average of the distances computed for the windows in which that frame ap-pears. A summary of the results appears in Tab.5 and are discussed in Sec. 5. We find that the better performing method on physical and general benchmarks under-performs on surveillance videos. This emphasizes the different nature of these tasks.
DISCUSSION
Video features vs. image features: Video features often outperformed the image features with the exception of the features obtained by ViT pretrained on ImageNet-1K. Still, we note that among the classes that favored the image features were those that either had subtle differences in the motion of the normal and anomalous classes (candle, magnets) or had similar motion but feature an entirely different object (pendulum). Conversely, the classes that favored the video features were those that feature an interruption in the motion (window) or motion that follows a different trajectory (keyboard, sushi). Intuitively, this is in line with what one may expect as the image networks are pretrained to classify objects of different classes while the video networks are pretrained to classify different types of motion. This may serve as a general guideline in order to determine when one technique may outperform the other.
The need for a general video AD method: While our method works well at finding dynamics-related anomalies like those seen in our dataset, the results in Tab.5 show that our pretrained methods are not well-adapted to surveillance datasets. On the other hand, while MNAD attains state-of-the-art results on UCSD, it is not able to solve our dataset as easily. When running these methods on the SSv2 dataset, Tab.3 shows that they both struggle with this task. The videos in the train and test sets that we chose contain high variability both in the video background and the object in question. For example, the Tearing paper subclass is filmed in numerous scenes with the paper being either blank or with words printed on it, and being of different sizes and colors. This variability made solving these classes difficult for both our method and MNAD. These results stress the difference between these video AD settings and the need to develop a technique that will be able to solve general video AD.
CONCLUSION
In our work, we focus on detecting anomalies in physical and general videos. We introduce new such benchmarks and evaluate baseline approaches on them. We find that the simple pretrained approach that struggles to outperform on surveillance data outperforms on our own suggested benchmarks. Taken together, our work highlights physical and general video anomaly detection as new tasks that call for the development of new approaches.
Fig. 1 .
1Examples of frames from each of the six video types in the PHANTOM dataset.
arXiv:2112.07661v1 [cs.CV] 14 Dec 2021Classname
normal class description
anomaly description
candle
burning candle flame
flame made to flicker
keyboard
sentence being typed
different sentence being typed
magnets: knocks
graphite oscillating between two magnets
tabletop is striked
magnets: starting still
. . .
graphite is still before oscillating
magnets: wind
. . .
wind blows on the graphite
pendulum: motion
ball swinging freely
swinging is manipulated
pendulum: object
. . .
ball replaced with other objects
sushi
ginger placed atop sushi, sushi dipped into sauce procedure that deviates from normal
window
window shutters being opened at constant speed
shutters opened with interruptions
Table 1. PHANTOM Dataset Class Descriptions
Classname
VIT TimeSformer MNAD
candle
0.76
0.71
0.49
keyboard
0.94
0.95
0.89
magnets: knocks
0.52
0.41
0.18
magnets: starting still 0.79
0.64
0.42
magnets: wind
0.25
0.45
0.34
pendulum: motion
0.87
1.00
0.89
pendulum: object
1.00
0.83
0.86
sushi
1.00
1.00
0.61
window
0.86
0.90
0.24
Average
0.78
0.76
0.55
ACKNOWLEDGEMENTSThis work was partly supported by the Federmann Cyber Security Research Center in conjunction with the Israel National Cyber Directorate.
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|
[] |
[
"A µ-mode integrator for solving evolution equations in Kronecker form",
"A µ-mode integrator for solving evolution equations in Kronecker form"
] |
[
"Marco Caliari \nDepartment of Computer Science\nUniversity of Verona\nItaly\n",
"Fabio Cassini \nDepartment of Mathematics\nUniversity of Trento\nItaly\n",
"Lukas Einkemmer \nDepartment of Mathematics\nUniversity of Innsbruck\nAustria\n",
"Alexander Ostermann \nDepartment of Mathematics\nUniversity of Innsbruck\nAustria\n",
"Franco Zivcovich \nLaboratoire Jacques-Louis Lions\nSorbonne University\nFrance\n"
] |
[
"Department of Computer Science\nUniversity of Verona\nItaly",
"Department of Mathematics\nUniversity of Trento\nItaly",
"Department of Mathematics\nUniversity of Innsbruck\nAustria",
"Department of Mathematics\nUniversity of Innsbruck\nAustria",
"Laboratoire Jacques-Louis Lions\nSorbonne University\nFrance"
] |
[] |
In this paper, we propose a µ-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how µ-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schrödinger equations, and we show that the µ-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem.
|
10.1016/j.jcp.2022.110989
|
[
"https://arxiv.org/pdf/2103.01691v2.pdf"
] | 232,092,892 |
2103.01691
|
79c3889540b925c8358b70bf394d6218fff3dd50
|
A µ-mode integrator for solving evolution equations in Kronecker form
Sep 2021
Marco Caliari
Department of Computer Science
University of Verona
Italy
Fabio Cassini
Department of Mathematics
University of Trento
Italy
Lukas Einkemmer
Department of Mathematics
University of Innsbruck
Austria
Alexander Ostermann
Department of Mathematics
University of Innsbruck
Austria
Franco Zivcovich
Laboratoire Jacques-Louis Lions
Sorbonne University
France
A µ-mode integrator for solving evolution equations in Kronecker form
Sep 2021numerical solution of evolution equationsµ-mode productdimension splittingspectral transformSchrödinger equationGraphic Processing Unit (GPU)
In this paper, we propose a µ-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how µ-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schrödinger equations, and we show that the µ-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem.
Introduction
Due to the importance of simulation in various fields of science and engineering, devising efficient numerical methods for solving evolutionary partial differential equations has received considerable interest in the literature. For linear problems with time-invariant coefficients, after discretizing in space, the task of solving the partial differential equation is equivalent to computing the action of a matrix exponential to a given initial value. Computing the action of matrix exponentials is also a crucial ingredient to devise efficient numerical methods for nonlinear partial differential equations; for example, in the context of exponential integrators [28] or splitting methods [35].
Despite the significant advances made in constructing more efficient numerical algorithms, efficiently computing the action of large matrix functions remains a significant challenge. In this paper, we propose a µ-mode integrator that performs this computation for matrices in Kronecker form by computing the action of one-dimensional matrix exponentials only. In d dimensions and with n grid points per dimension the number of arithmetic operations required scales as O(n d+1 ). Nevertheless, such an approach would not have been viable in the past. With the increasing gap between the amount of floating point operations compared to the amount of memory transactions modern computer systems can perform, however, this is no longer a consequential drawback. In fact, (batched) matrix-matrix multiplications, as are required for this algorithm, can achieve performance close to the theoretical limit of the hardware, and they do not suffer from the irregular memory accesses that plague implementations based on sparse matrix formats. This is particularly true on accelerators, such as Graphic Processing Units (GPUs). Thus, on modern computer hardware, the proposed method is extremely effective. In this paper, we will show that for a range of problems the proposed µ-mode integrator can outperform well established integrators that are commonly used in the field. We investigate the performances of the method for a two-dimensional pipe flow example. Then, we consider three-dimensional linear Schrödinger equations with time dependent and time independent potentials, in combination with Hermite spectral discretization, as well as a cubic nonlinear Schrödinger equation (Gross-Pitaevskii equation) in three space dimensions. In this context, we will also provide a discussion on the implementation of the method for multi-core CPUs and GPUs.
The µ-mode integrator is exact for linear problems in Kronecker form (see section 2 for more details). The discretization of many differential operators with constant coefficients fits into this class (e.g., the Laplacian operator ∆ and the i∆ operator that is commonly needed in quantum mechanics), as well as some more complicated problems (e.g. the Hamiltonian for a particle in a harmonic potential). For nonlinear partial differential equations, the approach can be used to solve the part of the problem that is in Kronecker form: for example, in the framework of a splitting method.
The µ-mode integrator is related to dimension splitting schemes such as alternating direction implicit (ADI) schemes (see, e.g., [24,27,37,41]). However, while the main motivation for the dimension splitting in ADI is to obtain one-dimensional matrix equations, for which efficient solvers such as the Thomas algorithm are known, for the µ-mode integrator the main utility of the dimension splitting is the reduction to one-dimensional problems for which matrix exponentials can be computed efficiently. Because of the exactness property described above, for many problems the µ-mode integrator can be employed with a much larger step size compared to implicit methods such as ADI. This is particularly true for highly oscillatory problems, where both implicit and explicit integrators do suffer from small time steps (see, e.g., [4]).
In the context of spectral decompositions, commonly employed for pseudospectral meth-ods, the structure of the problem also allows us to use µ-mode products to efficiently compute spectral transforms from the space of values to the space of coefficients (and vice versa) even if no d-dimensional fast transform is available. The outline of the paper is as follows. In section 2 we describe the proposed µ-mode integrator and explain in detail what it means for a differential equation to be in Kronecker form. We also discuss for which class of problems the integrator is particularly efficient. We then show, in section 3, how µ-mode products can be used to efficiently compute arbitrary spectral transforms. Numerical results that highlight the efficiency of the approach will be presented in section 4. The implementation on modern computer architectures, which includes performance results for multi-core CPU and GPU based systems, will be discussed in section 5. Finally, in section 6 we draw some conclusions.
2. The µ-mode integrator for differential equations in Kronecker form As a simple example that introduces the main idea, we consider the two-dimensional heat equation
∂ t u(t, x) = ∆u(t, x) = ∂ 2 1 + ∂ 2 2 u(t, x), x ∈ Ω ⊂ R 2 , t > 0, u(0, x) = u 0 (x)(1)
on a rectangle, subject to appropriate boundary conditions (e.g. periodic, homogeneous Dirichlet or homogeneous Neumann). Its analytic solution is given by
u(t, ·) = e t∆ u 0 = e t∂ 2 1 e t∂ 2 2 u 0 = e t∂ 2 2 e t∂ 2 1 u 0 ,(2)
where the last two equalities result from the fact that the partial differential operators ∂ 2 1 and ∂ 2 2 commute. Discretizing (1) by finite differences on a Cartesian grid with n 1 × n 2 grid points results in the linear differential equation
u ′ (t) = (I 2 ⊗ A 1 + A 2 ⊗ I 1 ) u(t), u(0) = u 0(3)
for the unknown vector u(t). Here, A 1 is a (one-dimensional) stencil matrix for ∂ 2 1 on the grid points x i 1 1 , 1 ≤ i 1 ≤ n 1 , and A 2 is a (one-dimensional) stencil matrix for ∂ 2 2 on the grid points x i 2 2 , 1 ≤ i 2 ≤ n 2 . The symbol ⊗ denotes the standard Kronecker product between two matrices. Since the matrices I 2 ⊗ A 1 and A 2 ⊗ I 1 trivially commute, the solution of (3) is given by
u(t) = e t(I 2 ⊗A 1 +A 2 ⊗I 1 ) u 0 = e tI 2 ⊗A 1 e tA 2 ⊗I 1 u 0 = e tA 2 ⊗I 1 e tI 2 ⊗A 1 u 0 ,
which is the discrete analog of (2).
Using the tensor structure of the problem, the required actions of the large matrices e tI 2 ⊗A 1 and e tA 2 ⊗I 1 on a vector can easily be reformulated. Let U(t) be the order two tensor of size n 1 × n 2 (in fact, a matrix) whose stacked columns form the vector u(t). The indices of this matrix reflect the structure of the grid. In particular
U(t)(i 1 , i 2 ) = u(t, x i 1 1 , x i 2 2 ), i 1 = 1, .
. . , n 1 , i 2 = 1, . . . , n 2 .
Using this tensor notation, problem (3) takes the form
U ′ (t) = A 1 U(t) + U(t)A T 2 , U(0) = U 0 ,
and its solution can be expressed as
U(t) = e tA 1 U 0 e tA T 2 ,(4)
see [38]. From this representation, it is clear that U(t) can be computed as the action of the small matrices e tA 1 and e tA 2 on the tensor U 0 . More precisely, the matrices e tA 1 and e tA 2 act on the first and second indices of U, respectively. The computation of (4) can thus be performed by the simple algorithm
U (0) = U 0 , U (1) (·, i 2 ) = e tA 1 U (0) (·, i 2 ), i 2 = 1, . . . , n 2 ,U (2) (i 1 , ·) = e tA 2 U (1) (i 1 , ·), i 1 = 1, . . . , n 1 ,U(t) = U (2) .
It should be duly noted that the µ-mode integrator is not restricted to the simple example considered until now. Indeed, let us consider the differential equation
u ′ (t) = Mu(t), u(0) = u 0 ,(5)
where
M = d µ=1 A ⊗µ and A ⊗µ = I d ⊗ · · · ⊗ I µ+1 ⊗ A µ ⊗ I µ−1 ⊗ · · · ⊗ I 1 .(6)
Here, A µ denotes an arbitrary n µ × n µ matrix while I µ is the identity matrix of size n µ , 1 ≤ µ ≤ d. The matrix M is also known in the literature as the Kronecker sum of the matrices A µ and is denoted by
M = A d ⊕ A d−1 ⊕ · · · ⊕ A 2 ⊕ A 1 .
Condition (6) holds true for a range of equations with linear and constant coefficient differential operators on tensor product domains. Examples in this class include, after space discretization, the diffusion-advection-absorption equation
∂ t u(t, x) = α∆u(t, x) + β · ∇u(t, x) − γu(t, x)
or the Schrödinger equation with potential in Kronecker form
i∂ t ψ(t, x) = − 1 2 ∆ψ(t, x) + d µ=1 V (t, x µ ) ψ(t, x).
Condition (6) is fulfilled also for some problems with non-constant coefficient differential operators, see section 4.2 for an example. We will consider these and other equations later in the paper to perform numerical examples. Equation (5) is what we call a linear problem in Kronecker form, and its solution is obviously given by
u(t) = e tA ⊗1 · · · e tA ⊗d u 0 ,
where the single factors e tA ⊗µ mutually commute. Again, the computation of u(t) just requires the actions of the small matrices e tAµ . More precisely, consider the order d tensor U(t) of size n 1 × · · · × n d that collects the values of a function u on a Cartesian grid, i.e.
U(t)(i 1 , . . . , i d ) = u(t, x i 1 1 , . . . , x i d d ), 1 ≤ i µ ≤ n µ , 1 ≤ µ ≤ d.
Then, in the same way as in the two-dimensional heat equation case, the computation of u(t) can be performed by
U (0) = U 0 , U (1) (·, i 2 , . . . , i d ) = e tA 1 U (0) (·, i 2 , . . . , i d ), 1 ≤ i µ ≤ n µ , 2 ≤ µ ≤ d, · · · U (d) (i 1 , . . . , i d−1 , ·) = e tA d U (d−1) (i 1 , . . . , i d−1 , ·), 1 ≤ i µ ≤ n µ , 1 ≤ µ ≤ d − 1, U(t) = U (d) .(7)
We remark that scheme (7) can also be useful as a building block for solving nonlinear partial differential equations. In this case, an exponential or splitting scheme would be used to separate the linear part, which is treated exactly by the integrator (7), from the nonlinear part which is treated in a different fashion. This is useful for a number of problems. For example, when solving the drift-kinetic equations in plasma physics using an exponential integrator [15,16], Fourier spectral methods are commonly used. While such FFT based schemes are efficient, it is also well known that they can lead to numerical oscillations [22]. Using integrator (7) would allow us to choose a more appropriate space discretization while still retaining efficiency. Another example are diffusion-reaction equations with nonlinear reaction terms that are treated using splitting methods (see, e.g., [20,21,29]). In this case scheme (7) would be used to efficiently solve the subflow corresponding to the linear diffusion. We further note that a related approach was pursued by [39] in order to produce schemes that solve two-and three-dimensional biological models.
Implementing integrator (7) requires the computation of d small exponentials of sizes n 1 × n 1 , . . . , n d × n d , respectively. If a marching scheme with constant time step is applied to (5), then these matrices can be precomputed once and for all, and their storage cost is negligible compared to that required by the solution U(t). Otherwise, we need to compute at every time step new matrix exponentials, whose computational cost still represents only a small fraction of the entire algorithm (see section 4.1). Indeed, the main component of the final cost is represented by the computation of matrix-matrix products of size n µ × n µ times n µ × (n 1 · · · n µ−1 n µ+1 · · · n d ). Thus, the computational complexity of the algorithm is O(N max µ n µ ), where N = n 1 · · · n d is the total number of degrees of freedom.
Clearly, we can solve equation (5) also by directly computing the vector e tM u 0 . In fact M is an N × N sparse matrix and, when it is too large for the explicit computation of e tM , the action of the matrix exponential can be approximated by polynomial methods such as Krylov projection (see, for instance, [25,40]), Taylor series [3], or polynomial interpolation (see, for instance, [9,10,11]). All these iterative methods require one matrix-vector product per iteration, which costs O(N) plus additional vector operations. The number of iterations, however, highly depends on the norm and some properties of the matrix, such as the normality, the condition number, and the stiffness, and it is not easy to predict it. Moreover, for Krylov methods, one has to take into account the storage of a full matrix with N rows and as many columns as the dimension of the Krylov subspace.
Also, an implicit scheme based on a Krylov solver could be applied to integrate equation (5). In particular, if we restrict our attention to the heat equation case and the conjugate gradient method, for example, O(max µ n µ ) iterations are needed for the solution (see the convergence analysis in [44,Chap. 6.11]), and each iteration requires a sparse matrix-vector product which is O(N). Hence, the resulting computational complexity is the same as for the proposed algorithm. However, on modern hardware architectures memory transactions are much more costly than performing floating point operations. A modern CPU or GPU can easily perform many tens of arithmetic operations in the same time it takes to read/write a single number from/to memory (see the discussion in section 5).
Summarizing, our scheme has the following advantages:
• For a heat equation the proposed integrator only requires O(N) memory operations, compared to an implicit scheme which requires O(N max µ n µ ) memory operations. This has huge performance implications on all modern computer architectures. For other classes of PDEs the analysis is more complicated. However, in many situations similar results can be obtained.
• Very efficient implementations of matrix-matrix products that operate close to the limit of the hardware are available. This is not the case for iterative schemes which are based on sparse matrix-vector products.
• The computation of pure matrix exponentials of small matrices is less prone to the problems that affect the approximation of the action of the (large) matrix exponential.
• The proposed integrator is often able to take much larger time step sizes than, for example, an ADI scheme, as it computes the exact result for equations in Kronecker form.
• Conserved quantities of the underlying system, such as mass, are preserved by the integrator.
We will in fact see that the proposed integrator can outperform algorithms with linear computational complexity (see sections 4.3 and 4.4). Equation (7) gives perhaps the most intuitive picture of the proposed approach. However, we can also formulate this problem in terms of µ-fibers. Indeed, let U ∈ C n 1 ×···×n d be an order d tensor. A µ-fiber of U is a vector in C nµ obtained by fixing every index of the tensor but the µth. In these terms, U (µ−1) (i 1 , . . . , i µ−1 , ·, i µ+1 , . . . , i d ) is a µ-fiber of the tensor U (µ−1) , and every line in formula (7) corresponds to the action of the matrix e tAµ on the µ-fibers of U (µ−1) . By means of µ-fibers, it is possible to define the following operation.
Definition 2.1. Let L ∈ C m×nµ be a matrix. Then the µ-mode product 1 of L with U, denoted by S = U × µ L, is the tensor S ∈ C n 1 ×...×n µ−1 ×m×n µ+1 ×...×n d obtained by multiplying the matrix L onto the µ-fibers of U, that is
S(i 1 , · · · , i µ−1 , i, i µ+1 , · · · , i d ) = nµ j=1 L ij U(i 1 , · · · , i µ−1 , j, i µ+1 , · · · , i d ), 1 ≤ i ≤ m.
According to this definition, it is clear that in formula (7) we are performing d consecutive µ-mode products with the matrices e tAµ , 1 ≤ µ ≤ d. We can therefore write scheme (7) as follows
U(t) = U 0 × 1 e tA 1 × 2 . . . × d e tA d .
This is the reason why we call the proposed method the µ-mode integrator. Notice that the concatenation of µ-mode products of d matrices with a tensor is also known as the Tucker operator (see [31]), and it can be performed using efficient level-3 BLAS operations. For more information on tensor algebra and the µ-mode product we refer the reader to [32].
Application of the µ-mode product to spectral decomposition and reconstruction
Problems of quantum mechanics with vanishing boundary conditions are often set in an unbounded spatial domain. In this case, the spectral decomposition in space by Hermite functions is appealing (see [7,48]), since it allows to treat boundary conditions in a natural way (without imposing artificial periodic boundary conditions as required by Fourier spectral methods, for example).
Consider the multi-index i = (i 1 , . . . , i d ) ∈ N d 0 and the coordinate vector
x = (x 1 , . . . , x d ) belonging to R d . We define the d-variate functions H i (x) as H i (x) = d µ=1 H iµ (x µ )e −x 2 µ /2 , where {H iµ (x µ )} iµ is the family of Hermite polynomials orthonormal with respect to the weight function e −x 2 µ on R, that is R d H i (x)H j (x)dx = δ ij .
1 Also known as mode-n product, n-mode product or mode-α multiplication, depending on the convention.
We recall that Hermite functions satisfy
− 1 2 d µ=1 (∂ 2 µ − x 2 µ ) H i (x) = λ i H i (x), where λ i = d µ=1 1 2 + i µ .
In general, we can consider a family of functions φ i :
R 1 × · · · × R d → C in tensor form φ i (x) = d µ=1 φ µ iµ (x µ )
which are orthonormal on the Cartesian product of intervals R 1 , . . . , R d of R.
If a function f can be expanded into a series
f (x) = i f i φ i (x), f i ∈ C, then its ith coefficient is f i = R 1 ×···×R d f (x)φ i (x)dx.
In order to approximate the integral on the right-hand side, we rely on a tensor-product quadrature formula. To do so, we consider for each direction µ a set of m µ uni-variate quadrature nodes X µ ℓµ and weights W µ ℓµ , 0 ≤ ℓ µ ≤ m µ , and fix to k µ the number of univariate functions φ µ iµ (x µ ) to be considered. We have then
f i = ℓ<m f (x ℓ )φ i (x ℓ )w ℓ , i < k,(8)where x ℓ = (X 1 ℓ 1 , · · · , X d ℓ d ) ∈ R d , w ℓ = d µ=1 W µ ℓµ
and k is the multi-index which collects the values {k µ } µ . We show now how µ-mode products can be employed to compute the coefficients of the spectral decomposition
f (x) = i<kf i φ i (x) ≈ f (x) (9)
of a d-variate function and its evaluation on a Cartesian grid. First of all, for each fixed µ,
1 ≤ µ ≤ d, we define the matrix Φ µ ∈ C kµ×mµ with components (Φ µ ) iℓ = φ µ i (X µ ℓ )
, and we denote by F W ∈ C m 1 ×···×m d the tensor with elements f (x ℓ )w ℓ and byF ∈ C k 1 ×···×k d the tensor with elementsf i . Then, in terms of the Tucker operator, we can write equation
(8) as followsF = F W × 1 Φ 1 × 2 · · · × d Φ d .(10)
It is then possible to evaluate the functionf (x) in (9) at a Cartesian grid
y p = (Y 1 p 1 , . . . , Y d p d ), that isf (y p ) = i<kf i φ i (y p ), p < q,(11)
by the Tucker operator, too. Here the component q µ of the multi-index q is the number of uni-variate evaluation points Y µ pµ . Indeed, if we collect the elementsf (y p ) in the tensor F ∈ C q 1 ×···×q d and, for fixed µ, we define the matrix Ψ µ ∈ C qµ×kµ with components (
Ψ µ ) pi = φ µ i (Y µ p ), thenF =F × 1 Ψ 1 × 2 · · · × d Ψ d(12)
is the tensor formulation of formula (11). Now, we restrict our attention to the common case where the quadrature nodes are chosen in such a way that
ℓ<m φ i (x ℓ )φ j (x ℓ )w ℓ = δ ij , i, j < k
with m = k, that is, the orthonormality relation among the φ i functions is true also at the discrete level. This is the case, for instance, when using Gauss-Hermite quadrature nodes for φ i (x) = H i (x). Then, the matrices Φ µ ∈ C mµ×mµ turn out to be square and formula (10) is the spectral transform from the space of values to the space of coefficients. Moreover, if the evaluation points coincide with the quadrature nodes, then we have Ψ µ = Φ * µ , where the symbol * denotes the conjugate transpose of the matrix, and formula (12) is the inverse spectral transform from the space of coefficients to the space of values.
As mentioned at the beginning of the section, we will employ the Hermite spectral decomposition in some of our experiments (see sections 4.3 and 4.4). Hence, we will use (10) and (12) for the required spectral transforms.
We also remark that a similar approach was pursued in [26] in the framework of threedimensional Chebyshev interpolation.
Numerical comparison
In this section, we will compare the proposed µ-mode integrator with some widely used techniques to solve partial differential equations. For that purpose a range of PDEs, mainly from quantum mechanics, is considered. Concerning the experiments in sections 4.1, 4.2 and 4.5, we will test the proposed method against the following iterative schemes commonly employed to compute the action of the matrix exponential e tM :
• expmv: a polynomial method described in [3] which is based on a Taylor expansion of the exponential;
• phipm: a full Krylov method presented in [40];
• kiops: a Krylov method based on an incomplete orthogonalization process, described in [25].
The MATLAB source code of these methods is publicly available. Although the underlying algorithms of these schemes only require the action of the matrix on a vector, only kiops is readily available to do that. Therefore, in order to ensure a fair comparison, we feed the functions with the matrix. Moreover, considering the action of the matrix on a vector (which in our case could be performed entirely in tensor formulation by means of sums of µ-mode products) instead of the matrix itself would not result in a speedup for the schemes (see section 4.1). The tolerance for all the algorithms considered has been set to 2 −53 , which corresponds to the machine epsilon for double precision computations. As a measure of cost, we consider the computational time (wall-clock time) needed to solve numerically the differential equation under consideration up to a fixed final time. As mentioned in section 2, the µ-mode integrator requires the explicit computation of small matrix exponentials. This is performed using the internal MATLAB ® function expm, which is based on the scaling and squaring rational Padé approximation described in [2]. In this context, another method which could be directly used in MATLAB is exptayotf from [12]. It is based on a backward stable Taylor approximation for the matrix exponential and is faster than expm. Moreover, as it works in single, double and variable precision arithmetic data types, it produces approximations with the desired accuracy. This is not possible for the iterative schemes which approximate the action of e tM , because the MATLAB ® sparse format is restricted to double precision. Another fast method using a similar technique and suited for double precision is expmpol from [46]. We will demonstrate that our MATLAB implementation of the proposed µ-mode integrator outperforms all the other schemes by at least a factor of 7.
Concerning the experiments in sections 4.3 and 4.4, we compare our µ-mode based approach with a splitting scheme/FFT based space discretization that is well established and efficient. In order to perform direct and inverse Fourier transforms, we employ the internal MATLAB ® functions fftn and ifftn respectively, which are in turn based on the very efficient FFTW library [23]. Care has been taken to ensure that comparisons conducted in MATLAB ® give a good indication of the performance that would be obtained in a compiled language. This is possible here as the majority part of the computational time is spent in the FFT routines. For these problems, we will show that the µ-mode integrator can reach a speedup of at least 5.
All the tests in this section have been conducted on an Intel Core i7-5500U CPU with 12GB of RAM using MATLAB ® R2020b.
Code validation
As an introductory test problem, in order to highlight some qualities of our µ-mode method, we consider the three-dimensional heat equation
∂ t u(t, x) = ∆u(t, x), x ∈ [0, 2π) 3 , t ∈ [0, T ], u(0, x) = cos x 1 + cos x 2 + cos x 3(13)
with periodic boundary conditions. The equation is discretized in space using centered finite differences with n µ grid points in the µth direction (the total number of degrees of freedom stored in computer memory is hence equal to N = n 1 n 2 n 3 ). By doing so we obtain the following ordinary differential
equation (ODE) u ′ (t) = Mu(t),(14)
where u denotes the vector in which the degrees of freedom are assembled. The exact solution of equation (14) is given by the action of the matrix exponential
u(t) = e tM u(0).(15)
The matrix M has the following Kronecker structure
M = I 3 ⊗ I 2 ⊗ A 1 + I 3 ⊗ A 2 ⊗ I 1 + A 3 ⊗ I 2 ⊗ I 1 ,
where A µ ∈ R nµ×nµ results from the one-dimensional discretization of the operator ∂ 2 µ , and I µ ∈ R nµ×nµ is the identity matrix. The quantity u(t) can be seen as vectorization of the tensor U(t), and we can write (15) in tensor form as
U(t) = U(0) × 1 e tA 1 × 2 e tA 2 × 3 e tA 3 , where U(t)(i 1 , i 2 , i 3 ) = u(t) i 1 +n 1 (i 2 −1)+n 1 n 2 (i 3 −1) .
We now present three numerical tests.
Test 1. We consider second order centered finite differences and compute the solution at time T = 1 for n µ = n, µ = 1, 2, 3 with various n. We investigate the wall-clock time as a function of the problem size.
Test 2. We fix the problem size (n µ = 40, µ = 1, 2, 3) and compute the solution at time T = 1 for different orders p of the finite difference scheme. We thereby investigate the wall-clock time as a function of the sparsity pattern of M.
Test 3. We consider second order centered finite differences and fix the problem size (n µ = 40, µ = 1, 2, 3). We then compute the solution at different final times T . By doing so we investigate the wall-clock time as a function of the norm of M.
The corresponding results are shown in Figure 1. We see that the proposed µ-mode integrator is always the fastest algorithm. The difference in computational time is at least a factor of 60.
Concerning the first test, we measure also the relative error between the analytical solution and the numerical one. As the dimensional splitting performed by the µ-mode integrator is exact, its errors are equal to the ones obtained by computing (15) using the other algorithms. Indeed, for the values of n under consideration, we obtain 2.06e-03, 1.09e-03, 6.71e-04, 4.55e-04 and 3.29e-04 for all the methods. We highlight also that the main cost of the µ-mode integrator is represented by the computation of the µ-mode products and not by the exponentiation of the matrices A µ (see Table 1). Lastly, notice that the iterative algorithms would not have taken advantage from the computation of the internal matrixvector products, which constitute their main cost, in tensor formulation (i.e. by means of sums of µ-mode products). Indeed, if we measure the wall-clock time for a single action of the matrix on a vector we observe, for the values of n under consideration, a speedup of averagely 1.5 times by using the standard sparse matrix-vector product as opposed to the tensor formulation. Figure 1).
The second test shows that the iterative schemes see a decrease in performance when decreasing the sparsity of the matrix (i.e. by increasing the order of the method p or by using a spectral approximation). This effect is particularly visible when performing a spectral discretization, which results in full matrices A µ . On the other hand, the µ-mode integrator is largely unaffected as it computes the exponential of the full matrices A µ , independently of the initial sparsity pattern, by using expm.
Similar observations can be made for the third test. While the iterative schemes suffer from increasing computational time as the norm of the matrix increases, for the µ-mode integrator this is not the case. The reason for this is that the scaling and squaring algorithm in expm scales very favorably as the norm of the matrix increases.
Pipe flow
To demonstrate that the µ-mode integrator can be used for some problems with nonconstant coefficients, we consider a model for a fluid flowing in a pipe. The main assumptions are that of radial symmetry (i.e. the solution does not depend on the angle variable in the circular cross section, see for example [47]) and a prescribed length-dependent flow velocity. In this case we obtain the following diffusion-advection equation for the concentration c ∂ t c(t, ρ, z) = α ∂ ρρ c(t, ρ, z)
+ 1 ρ ∂ ρ c(t, ρ, z) + ∂ zz c(t, ρ, z) − s(z)∂ z c(t, ρ, z),(16)
where t ∈ [0, T ], ρ ∈ [ρ min , ρ max ] and z ∈ [0, z max ]. Here α is the diffusivity and s(z) represents the advection velocity. After space discretization, which in our case is performed by means of second order centered finite differences with equal number of discretization points n µ in each direction (i.e. n µ = n, with µ = 1, 2), the resulting ODE is a linear problem in Kronecker form (6). The system can then be integrated exactly by the µ-mode integrator.
For the simulations conducted, we use the following initial and boundary conditions
c(0, ρ, z) = exp(−8(ρ − ρ 0 ) 2 − 8(z − z 0 ) 2 ), c(t, ρ, 0) = 0, ∂ z c(t, ρ, z max ) = 0, ∂ ρ c(t, ρ min , z) = 0, ∂ ρ c(t, ρ max , z) = 0,
while the flow velocity is set to s(z) = 2 + tanh(4(z − 5/2)) − tanh(4(z − 5)).
The parameters are chosen as ρ min = 0.1, ρ max = 5, z max = 8, α = 1/90, ρ 0 = (ρ min + ρ max )/2 and z 0 = 3/2. The structure of the problem does not allow an effective use of FFT based methods.
The results of the experiment are presented in Figure 2. The µ-mode integrator outperforms all the iterative methods by a consistent factor, with an average speedup of 45 times with respect to kiops, the fastest competitor in this simulation.
Schrödinger equation with time independent potential
In this section we solve the Schrödinger equation in three space dimensions with a time independent potential V (
i∂ t ψ(t, x) = − 1 2 ∆ψ(t, x) + V (x)ψ(t, x), x ∈ R 3 , t ∈ [0, 1] ψ(0, x) = ψ 0 (x)(17)x) = V 1 (x 1 ) + V 2 (x 2 ) + V 3 (x 3 ), where V 1 (x 1 ) = cos(2πx 1 ), V 2 (x 2 ) = x 2 2 /2, V 3 (x 3 ) = x 2 3 /2.
The initial condition is given by
ψ 0 (x) = 2 − 5 2 π − 3 4 (x 1 + ix 2 ) exp −x 2 1 /4 − x 2 2 /4 − x 2 3 /4 .
This equation could be integrated using any of the iterative methods considered in the previous section. However, for reasons of efficiency a time splitting approach is commonly employed. This treats the Laplacian and the potential part of the equations separately. For the former the fast Fourier transform (FFT) can be employed, while an analytic solution is available for the latter. The two partial flows are then combined by means of the Strang splitting scheme. For more details on this Time Splitting Fourier Pseudospectral method (TSFP) we refer the reader to [30].
Another approach is to use a Hermite pseudospectral space discretization. This has the advantage that harmonic potentials are treated exactly, which is desirable in many applications. However, for most of the other potentials, the resulting matrices are full which, for traditional integration schemes, means that using a Hermite pseudospectral discretization is not competitive with respect to TSFP. However, as long as the potential is in Kronecker form, we can employ the µ-mode integrator to perform computations very efficiently. Moreover, the resulting method based on the µ-mode integrator combined with a Hermite pseudospectral space discretization can take arbitrarily large time steps without incurring any time discretization error (as it is exact in time). We call this scheme the Hermite Kronecker Pseudospectral method (HKP).
Before proceeding, let us note that for the TSFP method it is necessary to truncate the unbounded domain. In order to relate the size of the truncated domain to the chosen degrees of freedom, we considered that, in practice, in the HKP method the domain is implicitly truncated. This truncation is given by the convex hull of the quadrature points necessary to compute the Hermite coefficients corresponding to the initial solution. For any choice of degrees of freedom of the TSFP method, we decided to truncate the unbounded domain to the corresponding convex hull of the quadrature points of the HKP method. In this way, for the same degrees of freedom, the two methods use the same amount of information coming from the same computational domain. The TSFP and the HKP methods are compared in Figure 3. In both cases, we consider a constant number of space discretization points n µ = n for every direction µ = 1, 2, 3 (total number of degrees of freedom N = n 3 ) and integrate the equation until final time T = 1 with constant time step size. We see that in terms of wall-clock time the HK method outperforms the TSFP scheme for all levels of accuracy considered here. Also note that the difference in performance increases as we move to more stringent tolerances. The reason for this is that the splitting error forces the TSFP scheme to take relatively small time steps.
Schrödinger equation with time dependent potential
Let us now consider the Schrödinger equation
∂ t ψ(t, x) = H(t, x)ψ(t, x), x ∈ R 3 , t ∈ [0, 1] ψ(0, x) = 2 − 5 2 π − 3 4 (x 1 + ix 2 ) exp −x 2 1 /4 − x 2 2 /4 − x 2 3 /4 ,(18)
where the Hamiltonian is given by
H(x, t) = i 2 ∆ − x 2 1 − x 2 2 − x 2 3 − 2x 3 sin 2 t .
Note that the potential is now time dependent, as opposed to the case presented in section 4.3. Such potentials commonly occur in applications, e.g. when studying laser-atom interactions (see, for example, [42]). Similarly to what we did in the time independent case, we can use a time splitting approach: the Laplacian part can still be computed efficiently in Fourier space, but now the potential part has no known analytical solution. Hence, for the numerical solution of the latter, we will employ an order two Magnus integrator, also known as the exponential midpoint rule. Let
u ′ (t) = A(t)u(t)
be the considered ODE with time dependent coefficients, and let u n be the numerical approximation to the solution at time t n . Then, the exponential midpoint rule provides the numerical solution u n+1 = exp τ n A(t n + τ n /2) u n (19) at time t n+1 = t n + τ n , where τ n denotes the chosen step size. The two partial flows are then combined together by means of the Strang splitting scheme. We call this scheme the Time Splitting Fourier Magnus Pseudospectral method (TSFMP). For the domain truncation needed in this approach, the same reasoning as in the time independent case applies. Another technique is to perform a Hermite pseudospectral space discretization. However, as opposed to the case in section 4.3, the resulting ODE cannot be integrated exactly in time. For the time discretization, we will then use the order two Magnus integrator (19). We call the resulting scheme Hermite Kronecker Magnus Pseudospectral method (HKMP).
The results of the experiments are depicted in Figure 4. In both cases, we consider a constant number of space discretization points n µ = n for every direction µ = 1, 2, 3 (total number of degrees of freedom N = n 3 ) and solve the equation until final time T = 1 with constant time step size. Moreover, concerning the TSFMP method, we integrate the subflow corresponding to the potential part with a single time step. Again, as we observed in the time independent case, the HKMP method outperforms the TSFMP scheme in any case. Notice in particular that, for the chosen degrees of freedom and time steps, the TSFMP method is not able to reach an accuracy of 1e-07, while the HKMP is.
Nonlinear Schrödinger/Gross-Pitaevskii equation
In this section we consider the nonlinear Schrödinger equation
∂ t ψ(t, x) = i 2 ∆ψ(t, x) + i 2 1 − |ψ(t, x)| 2 ψ(t, x),(20)
which is also known as Gross-Pitaevskii equation. The unknown ψ represents the wave function, x ∈ R 3 , t ∈ [0, 25], and the initial condition is constituted by the superimposition of two straight vortices in a background density |ψ ∞ | 2 = 1, in order to replicate the classical experiment of vortex reconnection (see [13] and the references therein for more details). The initial datum and the boundary conditions given by the background density make it quite difficult to use artificial periodic boundary conditions in a truncated domain, unless an expensive mirroring of the domain in the three dimensions is carried out. Therefore, in order to solve (20) numerically, we consider the Time Splitting Finite Difference method proposed in [13]. More specifically, we truncate the unbounded domain to x ∈ [−20, 20] 3 and discretize by non-uniform finite differences with homogeneous Neumann boundary conditions. The number n µ of discretization points is the same in each direction, i.e. n µ = n, with µ = 1, 2, 3. After a proper transformation of variables in order to recover symmetry, we end up with a system of ODEs of the form
ψ ′ (t) = i 2 M W ψ(t) + i 2 1 − W −1 |ψ(t)| 2 ψ(t),
where M W is a matrix in Kronecker form and W is a diagonal weight matrix. Then, we employ a Strang splitting scheme for the time integration, in which the linear part is solved either by means of the µ-mode integrator or by using the iterative methods indicated at the beginning of section 4. The nonlinear subflow is integrated exactly. The results of the experiment are presented in Figure 5. The µ-mode integrator outperforms expmv by approximately a factor of 7. The speedup compared to both phipm and kiops is even larger.
Implementation on multi-core CPUs and GPUs
It has increasingly been realized that in order to fully exploit present and future highperformance computing systems we require algorithms that parallelize well and which can be implemented efficiently on accelerators, such as GPUs [5]. In particular, for GPU computing much research effort has been undertaken to obtain efficient implementations (see, e.g., [6,8,17,18,19,33,36,43,45,49]). In this section we will consider an efficient implementation of the proposed µ-mode integrator on multi-core CPUs and GPUs. We note that all modern hardware platforms are much better at performing floating point operations (such as addition and multiplication) than they are at accessing data in memory. This favors algorithms with a high flop/byte ratio; that is, algorithms that perform many floating point operations for every byte that is loaded from or written to memory. The µ-mode product of a square matrix for an array of size n 1 ×· · ·×n µ−1 ×n µ ×n µ+1 ×· · ·×n d is computed using a matrix-matrix multiplication of size n µ × n µ times n µ × (n 1 · · · n µ−1 n µ+1 · · · n d ), see section 2 for more details. For moderate n µ the relatively small n µ × n µ matrix can be kept in cache and thus O(n µ N) arithmetic operations are performed compared to O(N) memory operations, where N = n 1 · · · n d is the total number of degrees of freedom. Thus, the flop/byte ratio of the algorithm is O(n µ ), which makes it ideally suited to modern computer hardware. This is particularly true when the µ-mode integrator is compared to an implicit scheme implemented with sparse matrixvector products. In this case the flop/byte ratio is only O(1) and modern CPU and GPUs will spend most of their time waiting for data that is fetched from memory.
To make this analysis more precise, we have to compare the flop/byte ratio of the algorithm to that of the hardware. For the benchmarks in this section we will use a multi-core CPU system based on a dual socket Intel Xeon Gold 5118 with 2 × 12 cores. The system has a peak floating point performance of 1.8 TFlops/s (double precision) and a theoretical peak memory bandwidth of 256 GB/s. Thus, during the time a double precision floating point number is fetched from memory approximately 56 arithmetic operations can be performed. In addition, we will use a NVIDIA V100 GPU with 7.5 TFlop/s double precision performance and 900 GB/s peak memory bandwidth (approximately 67 arithmetic operations can be performed for each number that is fetched from memory). Due to their large floating point performance we expect the algorithm to perform well on GPUs. A feature of the V100 GPU is that it contains so-called tensor cores that can dramatically accelerate half-precision computations (up to 125 Tflops/s). Tensor cores are primarily designed for machine learning tasks, but they can also be exploited for matrix-matrix products (see, e.g., [1,34]).
For reasonably large n µ the proposed µ-mode integrator is thus compute bound. However, since very efficient (close to the theoretical peak performance) matrix-matrix routines are available on both of these platforms, one can not be entirely indifferent towards memory operations. There are two basic ways to implement the algorithm. The first is to explicitly form the n µ × (n 1 · · · n µ−1 n µ+1 · · · n d ) matrix. This has the advantage that a single matrixmatrix multiplication (gemm) can be used to perform each µ-mode product and that the corresponding operands have the proper sequential memory layout. The disadvantage is that a permute operation has to be performed before each µ-mode product is computed. This is an extremely memory bound operation with strided access for which the floating point unit in the CPU or GPU lies entirely dormant. Thus, while this is clearly the favored approach in a MATLAB implementation, it does not achieve optimal performance. The approach we have chosen in this section is to directly perform the µ-mode products on the multi-dimensional array stored in memory (without altering the memory layout in between such operations).
Both Intel MKL and cuBLAS provide appropriate batched gemm routines (cblas gemm batch for Intel MKL and cublasGemmStridedBatched for cuBLAS) that are heavily optimized, and we will make use of those library functions in our implementation (for more details on these routines we refer to [14]). Our code is written in C++ and uses CUDA for the GPU implementation.
Before proceeding, let us briefly discuss how the µ-mode integrator would perform in a distributed memory setting (i.e. when parallelized using MPI). Since, in general, the matrix exponentials are full matrices, each degree of freedom along a coordinate axis couples with each other degree of freedom on that same axis. This data communication pattern is similar to computing a FFT. Thus, we would expect the µ-mode product to scale comparable to FFT on a distributed memory system. This would be worse than a stencil code. However, one should keep in mind that the µ-mode integrator can take much larger time steps. Thus, the overall communication overhead to compute the solution at a specified final time could still be larger for an explicit or an iterative method.
In the remainder of this section we will present benchmark results for our implementations. The speedups are always calculated as ratio between the wall-clock time needed by the CPU and the one needed by the GPU.
Heat equation
We consider the same problem as in section 4.1, Test 1. The wall-clock time for computing the matrix exponentials and a single time step of the proposed algorithm is listed in Table 2. We consider both a CPU implementation using MKL (double and single precision) and a GPU implementation based on cuBLAS (double, single, and half precision). The GPU implementation outperforms the CPU implementation by a factor of approximately 13. Using half-precision computations on the GPU results in another performance increase by approximately a factor of 2. The relative error with respect to the analytical solution reached by the double precision and single precision, for both CPU and GPU and the values of n under consideration, are 8.22e-05, 3.66e-05, 2.06e-05, 1.47e-05. Results in half precision are not reported as the accuracy of the method is lower than the precision itself. (13) discretized using second-order centered finite differences with n 3 degrees of freedom. The time for computing the matrix exponentials (exp) and for one step of the µ-mode integrator are listed (in ms). The speedup is the ratio between the single step performed in CPU and GPU, in double and single precision. The matrix exponential is always computed in double precision.
For a number of simulations conducted we observed a drastic reduction in performance for single precision computations when using Intel MKL. To illustrate this we consider the heat equation
∂ t u(t, x) = ∆u(t, x), x ∈ − 11 4 , 11 4 3 , t ∈ [0, 1], u(0, x) = x 4 1 + x 4 2 + x 4 3 exp −x 4 1 − x 4 2 − x 4 3(21)
with (artificial) Dirichlet boundary conditions, discretized in space as above. From Table 3 we see that the performance of single precision computations with Intel MKL can be worse by a factor of 3.5 compared to double precision, which obviously completely defeats the purpose of doing so. The reason for this performance degradation are so-called denormal numbers, i.e. floating point numbers with leading zeros in the mantissa. Since there is no reliable way to disable denormal numbers on modern x86-64 systems, we avoid them by scaling the initial value in an appropriate way. Since this is a linear problem, the scaling can easily be undone after the computation. The results with the scaling workaround, listed in Table 3, now show the expected behavior (that is, single precision computations are approximately twice as fast as double precision ones). We note that this is not an issue with our µ-mode integrator but rather an issue with Intel MKL. The cuBLAS implementation is free from this artifact and thus no normalization is necessary on the GPU.
Schrödinger equation with time independent potential
We consider the Schrödinger equation with time independent potential from section 4.3. The equation is integrated up to T = 1 in a single step, as for this problem no error is introduced by the µ-mode integrator. For the space discretization the Hermite pseudospectral discretization is used. The results for both the CPU and GPU implementation are listed in Table 4 (17) integrated with the HKP method (n 3 degrees of freedom). The time for computing the matrix exponential (exp) and for one step of the µ-mode integrator are listed (in ms). The speedup is the ratio between the single step performed in CPU and GPU, in double and single precision.
Schrödinger equation with time dependent potential
We consider once again the Schrödinger equation with the time dependent potential from section 4.4 solved with the HKMP method. The equation is integrated up to T = 1 with time step τ = 0.02. The results are given in Table 5. In this case, the matrix exponential changes as we evolve the system in time. Thus, the performance of computing the matrix exponential has to be considered alongside the µ-mode products. On the CPU this is not an issue as the time required for the matrix exponential is significantly smaller than the time required for the µ-mode products. However, for the GPU implementation and small problem sizes it is necessary to perform the matrix exponential on the GPU as well. To do this we have implemented an algorithm based on a Taylor backward stable approach. Overall, we observe a speedup of approximately 15 by going from the CPU to the GPU (for both single and double precision).
Conclusions
We have shown that with the proposed µ-mode integrator we can make use of modern computer hardware to efficiently solve a number of partial differential equations. In particular, we have demonstrated that for Schrödinger equations the approach can outperform well (18) integrated with the HKMP method (n 3 degrees of freedom). The time for computing the matrix exponentials and for one step of the µ-mode integrator is listed (in ms). The acronym exp (ext) refers to exponentiation of the time independent matrices, which are diagonal, while exp (int) refers to the time dependent ones that have to be computed at each time step. The speedup is the ratio between the single step performed in CPU and GPU, in double precision (top) and single precision (bottom).
established integrators in the literature by a significant margin. This was also possible thanks to the usage of the µ-mode product to efficiently compute spectral transforms, which can be beneficial even in applications that are not related to solving partial differential equations. The proposed integrator is particularly efficient on GPUs too, as we have demonstrated, which is a significant asset for running simulation on the current and next generation of supercomputers.
Figure 1 :
1The wall-clock time for solving the heat equation(13)is shown as a function of n (left), of the order of the finite difference scheme p (middle), and of the final time T (right). Note that p = ∞ corresponds to a spectral space discretization.
Figure 2 :
2Wall-clock time (in seconds) for the integration of (16) up to T = 4 as a function of n (total number of degrees of freedom N = n 2 ).
Figure 3 :
3Precision diagram for the integration of the Schrödinger equation with a time independent potential (17) up to T = 1. The number of degrees of freedom N and the number of time steps (ts) are varied in order to achieve a result which is accurate up to the given tolerance. The reference solution has been computed by the HKP method with N = 300 3 .
Figure 4 :
4Precision diagram for the integration of the Schrödinger equation with a time dependent potential (18) up to T = 1. The number of degrees of freedom N and the number of time steps (ts) are varied in order to achieve a result which is accurate up to the given tolerance. The reference solution has been computed by the HKMP method with N = 100 3 and ts = 2048.
Figure 5 :
5Wall-clock time (in seconds) for the integration of (20) up to T = 25 as a function of n (total number of degrees of freedom N = n 3 ). A constant time step size τ = 0.1 is employed.
Table 1 :
1Breakdown of wall-clock time (in ms) for the µ-mode integrator for different values of n (cf. left plot of
Table 2 :
2Wall-clock time for the heat equation
. The GPU implementation, for both single and double precision, shows a speedup of approximately 15 compared to the CPU implementation.n
exp
double
single
scaled single half
CPU GPU
CPU
GPU
CPU
GPU
200 2.92
38.80 2.64
92.19
1.34
19.98
0.38
300 6.01 157.41 8.87
385.84 5.22
71.24
2.71
400 13.40 314.96 29.85 1059.78 16.86
154.84
6.67
500 30.19 702.48 52.92 2567.56 30.42
367.34
13.44
Table 3 :
3Wall-clock time for the heat equation (21) discretized using second order centered finite differences with n 3 degrees of freedom. The performance degradation in CPU due to denormal numbers disappears when using the scaling workaround (scaled single). Speedups are not computed in this case.n
double
single
exp
CPU
GPU speedup
exp
CPU
GPU speedup
127 5.56
20.89
1.27
16.4x
4.71
13.71
0.64
21.4x
255 8.31 224.13 16.02
13.9x
5.16 134.21
8.11
16.5x
511 50.79 3121.42 219.13
14.2x
28.01 1824.93 119.46
15.2x
Table 4 :
4Wall-clock time for the linear Schrödinger equation with time independent potential
Table 5 :
5Wall-clock time for the Schrödinger equation with time dependent potential
AcknowledgmentsThe authors would like to thank the Italian Ministry of Instruction, University and Research (MIUR) for partially supporting this research with funds coming from PRIN Project 2017 (No. 2017KKJP4X entitled "Innovative numerical methods for evolutionary partial differential equations and applications").The authors acknowledge partial support from the Program Ricerca di Base 2019 of the University of Verona entitled "Geometric Evolution of Multi Agent Systems".FZ has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 850941).This work is supported by the Austrian Science Fund (FWF) -project id: P32143-N32.
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|
[] |
[
"Modular Functor and Representation Theory of sl 2 at a Rational Level",
"Modular Functor and Representation Theory of sl 2 at a Rational Level"
] |
[
"Boris Feigin \nLandau Institute for Theoretical Physics\nDepartment of Mathematics\nUniversity of Southern\nCalifornia\n",
"Feodor Malikov \nLandau Institute for Theoretical Physics\nDepartment of Mathematics\nUniversity of Southern\nCalifornia\n"
] |
[
"Landau Institute for Theoretical Physics\nDepartment of Mathematics\nUniversity of Southern\nCalifornia",
"Landau Institute for Theoretical Physics\nDepartment of Mathematics\nUniversity of Southern\nCalifornia"
] |
[] |
We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D−module on the moduli space of rank 2 vector bundles with the parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2), sl 2 " in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl 2 , sl 2 ".March, 1995, and at Service de Physique Theorique at Saclay, in November, 1994. We are grateful to J.-B. Zuber for invitation and warm hospitality. Considerable part of this work was done over the 2 years one of us spent at Yale. Inspiring and friendly atmosphere at the Department of Mathematics contributed a lot -and so did the discussions with Igor Frenkel, Ian Grojnowski, Gregg Zuckerman. We are grateful to Itzhak Bars for bringing to our attention the paper [37], to Sanjaye Ramgoolam for sending his work, and to David Kazhdan for an interesting conversation at Harvard.Notations and known results2.1Some notations from commutative algebra are as follows:] is its completion by positive powers of t; C[t, t −1 ] is a ring of Laurent polynomials and C((t)) is its completion by positive powers of t.By functions on the formal (punctured) neighborhood of a non-singular point on a curve we will mean a ring isomorphic to C[[t]] (C((t)) resp.); to specify such an isomorphism means to pick a local coordinate t. The analogous meaning will be given to the phrase " sections of a vector bundle on the formal (punctured) neighborhood of a non-singular point on a curve".
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Modular Functor and Representation Theory of sl 2 at a Rational Level
arXiv:q-alg/9511011v1 18 Nov 1995
Boris Feigin
Landau Institute for Theoretical Physics
Department of Mathematics
University of Southern
California
Feodor Malikov
Landau Institute for Theoretical Physics
Department of Mathematics
University of Southern
California
Modular Functor and Representation Theory of sl 2 at a Rational Level
arXiv:q-alg/9511011v1 18 Nov 1995
We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D−module on the moduli space of rank 2 vector bundles with the parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2), sl 2 " in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl 2 , sl 2 ".March, 1995, and at Service de Physique Theorique at Saclay, in November, 1994. We are grateful to J.-B. Zuber for invitation and warm hospitality. Considerable part of this work was done over the 2 years one of us spent at Yale. Inspiring and friendly atmosphere at the Department of Mathematics contributed a lot -and so did the discussions with Igor Frenkel, Ian Grojnowski, Gregg Zuckerman. We are grateful to Itzhak Bars for bringing to our attention the paper [37], to Sanjaye Ramgoolam for sending his work, and to David Kazhdan for an interesting conversation at Harvard.Notations and known results2.1Some notations from commutative algebra are as follows:] is its completion by positive powers of t; C[t, t −1 ] is a ring of Laurent polynomials and C((t)) is its completion by positive powers of t.By functions on the formal (punctured) neighborhood of a non-singular point on a curve we will mean a ring isomorphic to C[[t]] (C((t)) resp.); to specify such an isomorphism means to pick a local coordinate t. The analogous meaning will be given to the phrase " sections of a vector bundle on the formal (punctured) neighborhood of a non-singular point on a curve".
Introduction
In this paper we define a new modular functor based on Kac-Wakimoto admissible representations over sl 2 . The modular functor introduced by Segal [41] assigns a finite-dimensional vector space to the data consisting of a punctured curve, a rank 2 vector bundle and a collection of integral dominant highest weights attached to the punctures. Our modular functor does the same for the Segal's data (with integral dominant highest weights replaced with admissible highest weights) extended by the lines in the fibers over the punctures.
As the data " surface, vector bundle, punctures, lines in fibers over punctures" evolve, so does the corresponding finite dimensional vector space. This leads to a new D−module on the moduli space of rank 2 vector bundles with parabolic structure (fixed lines in certain fibers). The main feature of this D−module, as opposed to the standard one (see Tsuchiya-Ueno-Yamada [43], or Beilinson-Feigin-Mazur [4]), or Moore-Seiberg [36]) is that it is singular over a certain set of exceptional vector bundles. The latter is closely related to the Hitchin's global nilpotent cone.
We also prove that our D−module has (in a proper sense) regular singularities at infinity and that dimension of the generic fiber can be calculated by the usual combinatorial algorithm: by pinching the surface the problem is reduced to the case of a sphere with ≥ 3 punctures and further to a collection of spheres with 3 punctures. Dimension of the space attached to the datum "3 modules sitting at 3 points on a sphere" is calculated explicitly. It is a pure linear algebra calculation of dimension of the space of coinvariants of a certain infnite dimensional algebra with coefficients in a certain infinite dimensional representation. As the result is amusing we will record it here.
First of all, and it is important, in the genus zero case, one can work with modules at a generic level, as opposed to admissible representations which only exist when the level is rational. It is in complete analogy with the usual WZW model, where the famous theory of Knizhnik-Zamolodchikov equations arises from a collection of the so-called Weyl modules sitting on a sphere (terminology is borrowed from [29]). The family of Weyl modules is good for the purpose of studying integrable representations because each integrable representation is a quotient of some Weyl module. This is no longer the case as far as admissible representations are concerned. A family of modules suitable for our needs is that of what we call generalized Weyl modules; the latter is defined to be a Verma module quotiented out by a singular vector.
Generalized Weyl modules are naturally parametrized by the symbols (V ǫ r , V s ) r, s ≥ 0, ǫ ∈ Z/2Z. Here V r is to be thought of as the r + 1−dimensional irreducible sl 2 −module; meaning of V ǫ r will be explained soon. It is appropriate to keep in mind that the conventional Weyl module is defined to be the module induced from V r . Therefore usually Weyl modules are labelled by sl 2 −modules. In our situation Weyl modules are those related to symbols (V 0 0 , V s ). According to Verlinde, dimensions of the spaces associated to 3 modules on a sphere are structure constants of Verlinde algebra. Result of calculation of Verlinde algebra in our situation is as follows:
(V α r 1 , V s 1 ) • (V β r 2 , V s 2 ) =(1)
(V α+β r 1 +r 2 , V s 1 ⊗ V s 2 ) + (V α+β+1 r 1 +r 2 −1 , V s 1 ⊗ V s 2 ) + (V α+β r 1 +r 2 −2 , V s 1 ⊗ V s 2 ) + · · · + (V α+β |r 1 −r 2 | , V s 1 ⊗ V s 2 ) .
Recall that the usual Verlinde algebra built on Weyl modules is as follows:
V s 1 • V s 2 = V s 1 ⊗ V s 2 ,
i.e. it is the Grothendieck ring if the category of finite dimensional representations of sl 2 .
Observe that our formula agrees with the latter one on Weyl modules.
The first component of the right hand side of our formula is equally easy to interpret. It is known that the symbols V ǫ r naturally parametrize finite dimensional representations of the simplest rank 1 superalgebra osp(1|2). The category of finite dimensional osp(1|2)−modules is a tensor category and (1) reads as follows: Verlinde algebra is isomorphic to the product of Grothendieck rings of the categories of finite dimensional representations of osp(1|2) and sl 2 .
It is known in principle what to do when passing from modules to their quotients, in our case from generalized Weyl modules at a generic level to admissible representations at a rational level: one has to replace Lie algebras with quantized universal enveloping algebras at roots of unity and consider Grothendieck rings of the corresponding semisimple "quotient categories". Examples: Verlinde algebra built on integrable sl 2 −modules has to do with sl 2 in this way, and Verlinde algebra built on minimal representations of Virasoro algebra in this way has to do with 2 copies of sl 2 . It appears that Verlinde algebra built on admissible representations is related to the pair (osp(1|2), sl 2 ) in exactly the same way as V ir−Verlinde algebra is related to the pair of sl 2 's.
Interest in admissible representation originates in the fact that the characters of admissible representations representations at a fixed level give a representation of the modular group. However realization of this fact immediately gave rise to two puzzles:
(i) Given a representation of the modular group, Verlinde formula produces structure constants of Verlinde algebra; in the case of admissible representations some of the structure constants are negative. This does not make much sense as they are supposed to count dimensions.
(ii) Quantum Drinfeld-Sokolov reduction provides a functor from the category of sl 2 −modules to the category of V ir−modules, which sends admissible representations to minimal representations. It should give an epimorphism (or some weakened version of it) of a suitably defined Verlinde algebra for sl 2 on the well-known Verlinde algebra for V ir.
We are able to give an answer to (ii), and a partial answer to (i).
As far as (ii) is concerned, let us for simplicity step aside and consider V ir−modules at a generic (not necessarily rational) level. Then there is an analogue of a generalized Weyl module -Verma module quotiented out by a singular vector -and these are naturally parametrized by the symbols (V r , V s ). The desired epimorphism is given by:
(V ǫ r , V s ) → (V r , V s ) + (V r−1 , V s ).
This map is naturally related to the Drinfeld-Sokolov reduction in the following way.
As we have fixed the category of representations, we have triangular decomposition of sl 2 ;
in particular we have 2 opposite nilpotent subalgebras, Ce, Cf . Therefore there are in fact 2 Drinfeld-Sokolov functors, φ e , φ f . It happens that the map above is induced by the direct sum φ e ⊕ φ f .
As to (i), the situation is as follows. The structure constants naturally arrange in a tensor {c r ij }, the indices running through a set of representations in question. Let us compare the set {c r ij } of the structure coefficients of our algebra and the set {b r ij } of structure coefficients of the algebra calculated by Verlinde formula:
If our c r ij = 0, then b r ij = 0. If c r ij = 0, then b r ij is "most certainly" zero, however in some exceptional cases it is non-zero. The latter cases in our situation are interpreted in the following way. Recall that we have not only 3 modules, i, j, r, but also 3 Borel subalgebras, b i , b j , b r , which vary. Now as c r ij = 0, the fiber of our D−module is = 0 (in fact it is 1-dimensional), if the 3 Borel subalgebras are pairwise different. If however 2 of them meet, the fiber usually vanishes, but sometimes survives. It survives if and only if b r ij = 0. If non-zero, b r ij can be ±1. There is no doubt that b r ij is a result of some cohomological calculation related to the D−module. Unfortunately we cannot make it more precise at the moment.
Just as in the usual case Weyl modules on a sphere produce a trivial vector bundle with the flat (Knizhnik-Zamolodchikov)connection, in our case we get a bundle with a flat connection on a space of the 2 times greater dimension. The extra coordinates come from the flag manifold, recall that we are dealing with moduli of vector bundles with parabolic structure. Horizontal sections of this connection satisfy a system of differential equations;
we get twice as many equations as there are KZ equations: half of them are indeed KZ equations and the other half comes from singular vectors in Verma modules over sl 2 . The latter is but natural -it is exactly one of the lessons of the pioneering work [6]. This allows to put the integral formulas for solutions of Knizhnik-Zamolodchikov equations, which we wrote in [19], in a proper context: they give horizontal sections of this new connection.
We conjecture that our methods, in fact, provide all horizontal sections. The relation of our formulas to those in [42] is that the latter are necessarily polynomials as functions on the flag manifold while ours are not.
We wish to acknowledge that there has been a number of works approaching WZW model for admissible representation from different points of view, see for example [1,16,23,37,39]. It would be interesting to relate our integral formulas with those in [37] and the new Hopf algebra of [39] to the above mentioned "osp(1|2) × sl 2 " at roots of unity. To the best of our knowledge, Verlinde algebras proposed in these work do not solve (ii) abovethose algebras are rather trivial when compared to the V ir−analogue. Our starting point, see [18], was the work [1], where Verlinde algebra for admissible representations was first calculated (in the form equivalent but much less illuminating than the one described above), using the language which left completely open the problem of existence of a D−module, such that dimension of the fiber is calculated through this algebra.
2.2
Set g = sl 2 ,ĝ = sl 2 = sl 2 ⊗C[z, z −1 ]⊕Cc.g ≥ = Ce ⊕ Ch andĝ ≥ = g ⊗ zC[[z]] ⊕ b ⊕
Cc are standard Borel subalgebras of g and g resp;
g > = Ce andĝ > = g ⊗ zC[[z]
] ⊕ g > are standard "maximal nilpotent subalgebras" of g andĝ resp.;
Ch and Ch ⊕ Cc are standard Cartan subalgebras of g andĝ resp.
The Verma module M λ,k is a module induced from the character of g ⊗ zC[[z]] ⊕ b ⊕ Cc annihilating g ⊗ zC[z] ⊕ Ce and sending h and c to λ and k resp. k is often referred to as a level. Generator of M λ,k is usually denoted by v λ,k . A quotient of a Verma module is called highest weight module.
The algebraĝ is Z 2 + −graded by assigning f ⊗ z n → (1, −n), e ⊗ z n → (−1, −n) and so is a Verma module ( as well as its quotients):
M λ,k = ⊕ i,j M i,j λ,k .
There is a canonical antiinvolution ω :ĝ →ĝ interchangingĝ > andĝ < and constant on the Cartan subalgebra. For any highest weight module V denote by V c and call contragredient the module equal to the restricted dual V * as a vector space with the following action ofĝ: < gx, y >=< x, ω(g)y >, g ∈ĝ, x ∈ V * , y ∈ V.
If a highest weight module V is irreducible then it is isomorphic to V c . A morphism of highest weight modules V 1 → V 2 naturally induces the morphism of the corresponding
contragredient modules: V c 2 → V c 1 . A morphism of Verma modules M λ,k → M µ,k is determined by the image of v λ,k .
The image can be written as Sv µ,k for a uniquely determined element S of the universal enveloping algebra of g ⊗ z −1 C[z −1 ] ⊕ Cf . If non-zero, the vector Sv µ,k , or even S for this matter, is called singular. The singular vector can be equivalently defined as an eigenvector of the Cartan subalgebra ofĝ annihilated byĝ > . In this form definition applies to an arbitraryĝ−module.
Singular vector formula
It follows from Kac-Kazhdan determinant formula that a singular vector generically appears in the homogeneous components of degree either n(−1, m), m > 0, n > 0 or n(1, m), m ≥ 0, n > 0. Denote the corresponding singular vectors by S 1 n,m and S 0 n,m resp. Singular vectors S i nm were found in [32] in an unconventional form containing nonintegral powers of elements ofĝ ( see also [3] for another approach):
S 1 nm = (e ⊗ z −1 ) n+mt f n+(m−1)t (e ⊗ z −1 ) n+(m−2)t · · · (e ⊗ z −1 ) n−mt ,(2)S 0 nm = f n+mt (e ⊗ z −1 ) n+(m−1)t f n+(m−2)t · · · f n−mt ,(3)
where t = k + 2.
This form is not always convenient to calculate a singular vector. It is, however, a useful tool to derive properties of a singular vector. For example, denoting by π :ĝ → g, g ⊗ z n → g the evaluation map, one uses ( 2,3) to derive that (see [22], also [33] for the proof in a more general quantum case):
πS 1 nm = ( m i=1 N j=1 P (−it − j))e N (4) πS 0 nm = ( m i=1 N −1 j=0 P (it + j))f N ,(5)
where P (t) = ef − (t + 1)h − t(t + 1).
Generalized Weyl modules and admissible representations
The structure of Verma modules overĝ is known in full detail ( [34] (i) k is generic (not rational) and M λ,k contains only one singular vector;
(ii)k + 2 = p/q > 0 is a ratio of 2 positive integers and M λ,k contains infinitely many singular vectors.
It can of course happen that k + 2 = p/q < 0. We will not be interested in this case and confine to mentioning that here the situation is in a sense dual to (ii).
2.4.1
Case (i) M λ,k contains a unique proper submodule M generated by the singular vector. M is, in fact, a Verma module.
Definition.The irreducible quotient V λ,k is called generalized Weyl module. ✷ There arises the exact sequence
0 → M → M λ,k → V λ,k → 0.(6)
A simple property of Kac-Kazhdan equations [25] is that, given (6) Deodhar-Gabber-Kac [9].)
It is an exercise on Kac-Kazhdan equations to derive that the highest weight (λ, k) of a generalized Weyl module V λ.k belongs to either the line
λ = −it + j − 1, k = t − 2,(7)
for some i ≥ 0, j ≥ 1, or to the line
λ = it − j − 1, k = t − 2,(8)
for some i, j ≥ 1; in both cases t is regarded as a parameter. Formula (7) cooresponds to the case when V λ,k is obtained from M λ,k by quotienting out the singular vector S 0 i,j ; analogously, (8) cooresponds to the case when V λ,k is obtained from M λ,k by quotienting out the singular vector S 1 i,j . We see that for a fixed level k generalized Weyl modules are parametrized by the triples consisting of a pair of nonnegative numbers, i, j in the formulas above, and an element taking one of the 2 values needed to distinguish between (7) and (8). To be more precise,
denote by V i the i + 1−dimensional irreducible representation of g. Notation. Assign to V λ,k either the symbol (V 0 i , V j−1 ), i ≥ 0, j ≥ 1 if (λ, k) satisfies (7), or the symbol (V 1 i−1 , V j−1 ), i, j ≥ 1 if (λ, k) satisfies (8). ✷
This gives us a one-to-one correspondence between the set of generalized Weyl modules at a fixed generic level and the set of symbols (V ǫ i , V j ), where ǫ is understood as an element of Z/2Z.
Case (ii)
A Verma module contains infinitely many singular vectors and is embedded in finitely many other Verma modules. Among all singular vectors in M λ,k there are 2 independent ones and these generate the maximal proper submodule. Although formally all such Verma modules look alike a special role is played by those which can only embed (non-trivially) in themselves. Highest weights of such modules were called by Kac and Wakimoto admissible ( [28]) and are described as follows.
Let k + 2 = p/q, where p, q are relatively prime positive integers. The set of admissible highest weights at the level k = p/q − 2 is given by
Λ k = {λ(m, n) = m p q − n − 1 : 0 < m ≤ q, 0 ≤ n ≤ p − 1}.
What is said above about the structure of Verma modules implies that any Verma module appears in the exact sequence of the form
0 ← L λ 0 ,k ← M λ 0 ,k d 0 ← M λ 1 ,k ⊕ M µ 1 ,k d 1 ← M λ 2 ,k ⊕ M µ 2 ,k d 2 ← · · · ,(9)
where λ 0 is an admissible weight at the level k and L λ 0 ,k is the corresponding irreducible module. L λ 0 ,k is also called admissible. The exact sequence (9) is called Bernstein -Gel'fand -Gel'fand ( BGG ) resolution.
Again cohomological arguments show (see e.g. [28]) that if the composition series of aĝ−module only consists of admissible representations then the module is completely reducible.
The parametrization of the set of admissible representations we are going to use is as follows. Two different generalized Weyl modules project onto one and the same admissible representation: formula (9) implies that the the two modules projecting onto L λ 0 ,k are M λ 0 ,k /M λ 1 ,k and M λ 0 ,k /M µ 1 ,k . Therefore two different triples (V ǫ m , V n ) are related to the same admissible represenation. Introduce the equivalence relation ≈ by
(V ǫ m , V n ) ≈ (V ǫ+1 q−1−m , V p−2−n ), 0 ≤ m ≤ q − 1, 0 ≤ n ≤ p − 2.
Denote by (V ǫ m , V n ) ∼ the equivalence class of (V ǫ m , V n ). It easy to check that admissible representations are parametrized by the equivalence classes of the triples:
{ admissible representations } ⇐⇒ {(V ǫ m , V n ) ∼ }.(10)
2.5
Considerable part of the above carries over to the arbitrary Kac-Moody algebra case.
Here, for example, is the definition of an admissible representation. Drop the condition that g = sl 2 , let M λ,k be a Verma module overĝ and L λ,k be its irreducible quotient. Call (λ, k) admissible if M λ,k satisfies the following projectivity condition: if composition series of aĝ−module W contains L λ,k then M λ,k non-trivially maps in W .
Unfortunately we do not have a reasonable definition of a generalized Weyl module in the higher rank case. This is one of the reasons for which we have to confine mostly to the sl 2 −case.
Loop modules
We will also be usingĝ−modules different from Verma modules or corresponding irreducible ones.
Denote by F αβ a g− module with the basis F i , i ∈ Z and the action given by
eF i = −(α + i − β)F i+1 , hF i = (2α + 2i − β)F i , f F i = (−α − i)F i−1 . The space F C * αβ = F αβ ⊗ C[z, z −1 ]
is endowed with the naturalĝ−module structure. The elements F ij = F i ⊗ z j , i, j ∈ Z serve as a natural basis in it.
Recall (see 2.3) that S 1 nm , S 0 nm stand for a singular vector of degree n(−1, m) or n(1, m) resp. in a Verma module. The following formulas are proved by using (4,5):
S 1 nm F n,nm = { m i=1 n j=1 (−it − j − α + β)(−it − j − α)}{ n s=1 (α + i)}F 00(11)S 2 nm F −n,nm = { m i=1 n j=1 (it + j − α + β)(−it − j − α)}{ n s=1 (α − β − i)}F 00 ,(12)
where t = k + 2.
3 Construction of the modular functor.
Although most of our results have to do with sl 2 , up to some point it is no extra effort to work in greater generality. So until sect.4, g will stand for sl n unless otherwise stated.
3.1
Algebraĝ A and categories ofĝ A -modules 3.1.1
Let C be a smooth compact algebraic curve and ρ : E → C be a rank n vector bundle with a flat connection. The connection relates to a section s of any bundle A associated with E the section dc of Ω ⊗ A where Ω is the sheaf of differential forms over C. A typical example of A is the bundle EndE of fiberwise endomorphisms of E. The sheaf of sections of EndE is naturally a sheaf of Lie algebras over C.
For a point P ∈ C let g P be the algebra of sections of EndE over the formal neighborhood of P . For a finite subsetĀ = {P 1 , P 2 , . . . , P m } ⊂ C set gĀ = ⊕ m i=1 g P i . DefineĝĀ to be the central extension of gĀ by the cocycle
< x, y >= m i=1
Res P i Trdx · y.
In particular, we obtain the splittingĝĀ
= gĀ ⊕ C · c.(13)
Consider a finite set A = {(P 1 , b 1 ), . . . , (P m , b m )} where P i ∈ C are pairwise different and b i is a Borel subalgebra of the algebra of traceless linear transformations of the fiber
ρ −1 P i (1 ≤ i ≤ m).
LetĀ be the projection of A on C. Setĝ A =ĝĀ.
3.1.2
Given A as above, set
n i = [b i , b i ].
Denote byĝ A > the subalgebra consisting of sections x(.) such that x(P i ) ∈ n i , 1 ≤ i ≤ m, and byĝ A ≥ the subalgebra spanned by the space of sections x(.) such that x(P i ) ∈ b i , 1 ≤ i ≤ m, and the central element c. These are analogues of the maximal "nilpotent" and maximal "solvable" subalgebras forĝ A , c.f.2.2.
Denote by O A k , k ∈ C, the category of finitely generatedĝ A -modules satisfying the conditions:
(i) c acts as multiplication by k;
(ii) the action of the subalgebraĝ A > is locally finite. In much the same way as in 2.2 one defines Verma and generalized Weyl modules over g A :
Definition.
(i) We will say that (λ, k) is a highest weight ofĝ A if λ is a functional on ⊕ i b i /n i and k is a number.
(ii) A highest weight (λ, k) naturally determines a character ofĝ A ≥ sending c to k and annihilatingĝ A > . Denote by C λ,k the corresponding 1-dimensional representation. (iii) Define the Verma module M A λ,k to be the induced representation
Indĝ Â g A ≥ C λ . ✷
There is an isomorphism
M A λ,k ≈ ⊗ m i=1 M P i ,b i λ i ,k .
Suppose now that each M P i ,b i λ i ,k has at least one singular vector. If k ∈ C \ Q then this singular vector is unique for each i. Quotienting out all of them one obtains the generalized Weyl module V A λ,k . As above there is an isomorphism
V A λ,k ≈ ⊗ m i=1 V P i ,b i λ i ,k .
If k is not a rational number then any generalized Weyl module is irreducible. Denote byÕ k the full subcategory of O k consisting of allĝ A −modules whose composition series consist of generalized Weyl modules. Again if k is not a rational number thenÕ k is semisimple.
If k is rational then there arises the admissible representation L A λ,k if (λ, k) is admissible. If the composition series of a module V A consists only of admissible representations, then V A is completely reducible.
Remark 3.1.1 There is a canonical isomorphism b 1 /n 1 ≈ b 2 /n 2 for any 2 Borel subalgebras b 1 , b 2 . Therefore if Borel subalgebras appearing in A evolve, so does the projectivization of the module M A λ,k or its quotients.
3.1.3
Let A be as in 3.1.1. Let g(C, A) be the Lie algebra of meromorphic sections of EndE holomorphic outsideĀ. The maps of restriction to formal neighborhoods give rise to the Lie algebra morphism
g(C, A) → g A(14)
The splitting (13) provides us with the section s A : g A →ĝ A . Composition of (14) with s A gives the linear morphism
g(C, A) →ĝ A .(15)
The residue theorem implies that (15) is a Lie algebra morphism (even though s A is not!).
By (15), the standard pullback makes each object of M A ∈ O A k into a g(C, A)-module. Hence there arises the space of coinvariants The natural map Base(W ) → F (W ) is a principal (C * ) ×n -bundle.
(M A ) g(C,A) = M/g(C, A)M.
Localization ofĝ
Similar arguments applied to b show that the module V A = V P,b depends on the quadruple (P, b, j, x) such that j is a 1-jet of parameter at P and x ∈ Base(C n ) belongs to the preimage of b.
One concludes that we do get a locally trivial vector bundle after pull-back to the space of pairs "1-jet of parameter at P , element of the maximal torus of the Borel group related to b". Let us be more precise now.
3.2.2
Letπ : C S → S be a family of smooth projective curves and ρ S : E S → C S be a rank n vector bundle. There arise 2 more bundles:
(i) the bundle Base(ρ S ) : Base(E S ) → C S with the fiber over any x ∈ C S equal to the base affine space of the vector space ρ −1 S x; (ii) the C * −bundle J (1) (C S ) → C S of 1-jets of coordinates along fibers ofπ.
Consider the fibered product Base(E S ) × C S J (1) (C S ) and the natural map we can further associate a vector space, that is the space of coinvariants
π : Base(E S ) × C S J (1) (C S ) → S.(M A S (s) ) g(π −1 S,A S (s)) , see 3.1.3.
Theorem 3.2.1 Suppose the collection ψ = (M A , π, A S ), satisfying the conditions imposed above, is given. Then there is a twisted D−module (that is a sheaf of modules over a certain algebra of twisted differential operators) on S such that its fiber over s ∈ S is
(M A S (s) ) g(π −1 S,A S (s) .
This theorem is an immediate consequence of [4] and [5,7] . Briefly the construction is as follows. Take a vector field ξ on U ⊂ S. It lifts to a meromorphic vector field on C S − A S (S) over U, and further to a meromorphic vector field on π −1 (U) ⊂ Base(E S ) × C S J (1) (C S ); denote this vector field by ξ * . Trivializing the infinitesimal neighborhood of A S (U) ⊂ C S by chosing, locally with respect to U ⊂ S, coordinates in the fibers, one In the case when M A is an admissible representation the following result is valid.
gets vertical components {ξ * vert;i }, so that ξ * vert;i is the vertical component in the formal neighborhood of the i− section. Projecting ξ * vert;i on Base(E S ) one gets some element of U(g), say u i ; projecting ξ * vert;i on J (1) (C S ) one gets some vector field, say v i . Both u i , v i act on ourĝ A -module M A : u i naturally, v i
Proof.
To prove this theorem essentially means to show that the spaces (M A S (s) ) g(π −1 S,A S (s) , s ∈ S, are all finite dimensional. That will be done in 4.3.3, Proposition 4.3.2 in the higher genus case and in 4.4.2, Proposition 4.4.2 for CP 1 . . We will also give there a precise meaning to the phrase "almost any vector bundle" in Theorem3.2.2. ✷ Results of 4.7 will show that the standard combinatorial algorithm can be used to calculate the dimension of the fiber of our D−module using the dimensions of the spaces of coinvariants on a sphere with 3 punctures. The latter dimensions will be calculated in 4.5.4.
The spaces of coinvariants
In this section we will be concerned with the space of coinvariants (M A ) g(C,A) (or spaces closely related to it ) in the case when M A is either a generalized Weyl module or an admissible representation. The standard tool to get finiteness results about coinvariants is the notion of singular support.
Singular support and coinvariants
Let a be a Lie algebra. Universal enveloping algebra Ua is filtered in the standard way so that the associated graded algebra is Sa. One says that a filtration of a finitely generated a−module V is good if (i) it is compatible with the filtration of Ua, and (ii) the associated graded module Gr V is finitely generated as an Sa-module.
Definition Singular support, SSV , of V is the zero set of the vanishing ideal of the
Sa-module Gr V . ✷ Obviously, SSV is a conical subset of a * .
For a subalgebra n ⊂ a, call V an (a, n)-module if it is an a−module and n acts on V locally nilpotently. Typical example: any module from the O−category is a (ĝ,ĝ > )−module. Recall that from now on g = sl 2 unless otherwise stated.
Singular support ofĝ A −modules
Observe that there is an involution σ ofĝ sending f to e ⊗ z −1 and e ⊗ z −1 to f , see 2.2 for notations. There arises the involution, also denoted by σ, acting on the algebraŝ g A and their duals. This involution is not canonical but we do not have to care as our considerations here are purely local.
Denote by Ω A the space of g− valued differential forms on the formal neighborhoods of the points from A. There is a natural embedding Ω A ֒→ (g A ) * ("take the traces and then sum up all the residues!")
We will make use of 2 subspaces of Ω A : Ω A reg is all regular forms and Ω A nilp is all forms with values in the nilpotent cone.
Theorem 4.2.1 (i) If M A is a generalized Weyl module then SSM A = Ω A reg ∪ σΩ A reg . (ii) (E.Frenkel, B.F.) If M A is an admissible representation then SSM A = Ω A nilp ∪ σΩ A nilp .
Remark 4.2.2 It is easy to see that although σ is not determined uniquely the spaces σΩ A reg , σΩ A nilp are canonical. For example σΩ A reg is the space of forms such that: they have at most order 1 pole atĀ; their residue at each P i ∈Ā belongs to n i ; at each P i ∈Ā their constant term belongs to b i .
Finiteness of coinvariants -the higher genus case
4.3.1
Hitchin's theorem.
First recall a well-known result of Hitchin, [24]. With a vector bundle E → C associate the map For us, importance of Theorem 4.3.1 is in that generically a vector bundle does not allow a non-trivial global differential form with coefficients in nilpotent endomorphisms of the bundle.
H(E) : H 0 (C, Ω ⊗ EndE) → ⊕ n i=2 H 0 (C, Ω ⊗i ),(16)X → TrX i
4.3.2
Subtracting lines from rank 2 vector bundles.
An analogue of subtracting a point from a line bundle (or, better to say, from its divisor)
is an operation of subtracting a line from a rank 2 vector bundle.
To a rank 2 vector bundle E → C one can associate a module over the sheaf of regular functions -the sheaf of sections of E.; denote this sheaf by Sect(E). This establishes a one-to-one correspondence between rank 2 vector bundles and rank 2 locally free modules over the sheaf of regular functions. Now fix a line, l, in a fiber of E over some point P ∈ C.
Denote by S(l) a sheaf such that:
(i) S(l)| U = Sect(E)| U if P does not belong to U;
(ii) S(l)| U , P ∈ U, is the space of meromorphic sections of E over U regular outside P , having at most order 1 pole at P and such that their residue at P belongs to the fixed line l.
It is obvious that S(l) is a rank 2 locally free module. Therefore it defines a rank 2 vector bundle. Denote this vector bundle by E(l). If a collection of lines -l 1 , l 2 , ..., l m -is subtracted, then denote the corresponding vector bundle by E(l 1 + · · · + l m ). Proof. One extracts from definitions that the annihilator g(C, A) ⊥ of the algebra
g(C, A) is the space Ω C,A (E) of global meromorphic End(E)−valued differential forms reg- ular outsideĀ ⊂ C. By Theorem 4.2.1(ii) we get that SSM A ∩ g(C, A) ⊥ = Ω nilp (E) ∪ σΩ nilp (E),
where Therefore genericity condition also implies that σΩ nilp (E) = 0.
Ω nilp (E) is
Hence we get that SSM A ∩ g(C, A) ⊥ = 0. And as the spaceĝ A > + g(C, A) is of finite codimension inĝ A , application of Lemma 4.1.1, see 4.1, completes the proof. ✷ In order to study quadratic degenerations we will need the following stronger finiteness
result. Along with the set A = {(P 1 , b 1 ), ..., (P m , b m )}, consider the set A 2 = {(P m+1 , b m+1 ), (P m+2 , b m+2 )}
such that the points P 1 , ..., P m+2 ∈ C are different. Denote by g(C, A, A 2 ) the subalgebra of g(C, A) consisting of functions taking values in
n i = [b i , b i ] at point P i , i = m + 1, m + 2. Proposition 4.3.3 If (E, A A 2 ) is generic and M A is admissible, then dim (M A ) g(C,A,A 2 ) < ∞.
Proof. We are again going to apply Lemma 4.1.1. Observe that g(C, A, A 2 ) ⊥ consists of meromorphic forms on C with values in End(E), regular outside {P 1 , ..., P m+2 } ⊂ C, having at most order 1 poles at P m+1 , P m+2 , their residues at the latter points lying in b 1 ( b 2 resp.).
By Theorem 4.2.1(ii), g(C, A, A 2 ) ⊥ ∩ SSM A consists of forms with values in nilpotent endomorphisms, satisfying the above listed global conditions. This implies, in particular, that actually residues of our forms belong to n m+1 , n m+2 at P m+! , P m+2 resp.. Given an element ω ∈ g(C, A, A 2 ) ⊥ ∩ SSM A , subtract some lines from E so as to make ω be everywhere regular. Genericity condition implies then that ω = 0, and application of Let O(n) be the degree n line bundle over CP 1 . It is known, e.g. [38], that any rank 2 vector bundle over CP 1 is a direct sum O(r) ⊕ O(s) for some r, s.
As there are no moduli, it is hard to speak about generic vector bundles. Nevertheless
we will call O(r) ⊕ O(s) exceptional if |r − s| > 1.
Here is a justification.
E(l 1 + · · · + l m ) = O(p + 1) ⊕ O(p) if r + s − m = 2p + 1 O(p) ⊕ O(p) if r + s − m = 2p.
Lemma 4.4.1 seems to be common knowledge, although we failed to find a reference with its proof.
Proceed just like we did in 4.3.2: call (E, l 1 , ..., l m ) generic if E(l i 1 + · · · + l is ) is not exceptional for any subset {i 1 , ..., i s } ⊂ {1, 2, ..., m}.
Finiteness of coinvariants
A specific feature of the genus zero case is that we do not necessarily have to consider admissible representations -generalized Weyl modules, see 2.4.1, will also do.
Along with A = {(P 1 , b 1 ), . . . , (P m , b m )} consider 2 sets A 1 = {(P m+1 , b m+1 )} and A 2 = {(P m+1 , b m+1 ), (P m+2 , b m+2 )} such that P 1 , ..., P m+2 are different points in C.
With As to (i), its proof is again application of the same technique in a slightly different form: one has to take a form ω ∈ g(CP 1 , A, A 2 ) ⊥ ∩ SSM A and to subtract lines from E so as to make ω into a form with either one pole (at P m+1 ) or 2 poles (one of them is again at and ω. In both cases it is easy to prove that ω = 0 using the fact that any differential form with trivial coefficients has at least 2 poles. ✷
Holonomic D-module on (C × C) ×m
We will now get rid of twisted differential operators in Corollary 4.4.3 under the assumption that the vector bundle E → CP 1 is trivial. Consider the set
A ′ = A (P ∞ , b ∞ ).
Attach to the point (P ∞ , b ∞ ) the module (V 0 0 , V 0 ) known as the vacuum representation, see 2.4.1 for notations. (P ∞ , b ∞ ) can be redefined as the module induced from the trivial representation (see also 2.4.1) and therefore there is
an isomorphism M A g(CP 1 ,A) ≈ M A ′ g(CP 1 ,A ′ )
. Now consider the twisted D−module with fiber
M A ′ g(CP 1 ,A ′ ) on the space (J (1) (CP 1 ) × J (1) (CP 1 )) ×m+1 . Restrict it to the space (J (1) (CP 1 ) × J (1) (CP 1 )) ×m by having the point (P ∞ , b ∞ ) fixed.
The result of this operation is that the bundles in ques- . In particular, we get a bundle with flat connection over an open subset of (C × C) m .
tion trivialize: CP 1 − b ∞ = CP 1 − P ∞ = C and J (1) (C) = C * × C.
Notation. Denote the constructed in this way bundle with flat connection by ∆(M A ).
✷
We are unable to describe this open subset explicitly at present. It follows from the requirement that (E, A) be generic in all our finiteness results however that the diagonals should be thrown away meaning that P i = P j and b i = b i = b j for all i = j.
One may want to write down differential equations satisfied by horizontal sections of this bundle. We will show in 5.3 that horizontal sections satisfy a system of 2m differential Everything said here holds true for an admissible representation. It is easy to see that the bundle associated with an admissible representation is a quotient of the just constructed bundle for the corresponding generalized Weyl module.
Calculation of the dimensions of coinvariants. Fusion algebra
Let E → CP 1 be the rank 2 trivial vector bundle and M A be aĝ A −module. Here we will calculate the dimension of the space (M A ) g(CP 1 ,A) , ♯A = 3, in the following 2 cases: (i) the level k is not rational and M A is a generalized Weyl module; (ii) k + 2 = p/q, p and q being positive integers, and M A is an admissible representation. Without loss of generality we can:
fix a coordinate z on CP 1 ; assume that A = {(0, b 0 ), (1, b 1 ), (∞, b ∞ )}, where b 0 = Ce ⊕ Ch, b ∞ = Cf ⊕ Ch and b 1 = C(e − h − f ) ⊕ C(h + 2f ).
(In fact, for any b 0 = b ∞ we can always choose a basis of g so that b 0 , b ∞ are as above.
As to b 1 , there really is some freedom but it is easy to see that all the calculations below are independent of the choice. We have
set b 1 = (exp f )b 0 (exp −f ).) 4.5.1
The generic level case So by 3.1.2 we are given three irreducible generalized Weyl modules V 0 λ 0 ,k , V 1 λ 1 ,k , V ∞ λ∞,k . Recall, see 2.4.1, that generalized Weyl modules are parametrized by symbols (V ǫ m , V n ), where m, n are nonnegative integers, ǫ ∈ Z/2Z and V m is an m + 1-dimensional g−module.
Therefore we can and will assume that we have
(V ǫ i m i , V n i ), i = 0, 1, ∞.
It is convenient to interpret the result of calculation of dim (⊗ i (V ǫ i m i , V n i )) g(CP 1 ,A) in terms of the fusion algebra. The latter is defined as follows. Suppose that for any pair of generalized Weyl modules, say (V α i r i , V s i ), i = 0, 1, there is only finite number of (V α∞ r∞ , V s∞ ) such that
dim (⊗ i=0,1,∞ (V α i r i , V s i )) g(CP 1 ,A) = 0.
Now view the symbols (V ǫ m , V n ) as generators of a free abelian group. Then there naturally arises an algebra (over Z) with the operation of multiplication • defined by
(V α 0 r 0 , V s 0 ) • (V α 1 r 1 , V s 1 ) = (r∞,s∞,α∞) dim {(⊗ i=0,1,∞ (V ǫ i m i , V n i )) g(CP 1 ,A) }(V α∞ r∞ , V s∞ ).
The algebra defined in this way is called fusion algebra. Of course structure constants of the fusion algebra determine the dimensions of the spaces of coinvariants.
One last piece of notation: in the following theorem we formally set (X ⊕ Y, Z) = (X, Z) + (Y, Z) and (X, Y ⊕ Z) = (X, Y ) + (X, Z). Recall also that in the category of g−modules one has V r ⊗ V s ≈ V r+s ⊕ V r+s−2 ⊕ · · · ⊕ V |r−s| .
Theorem 4.5.1 (i) For any triple of generalized Weyl modules the space (V ǫ i m i , V n i )) g(CP 1 ,A) is finite dimensional.
(ii) The fusion algebra is well-defined, multiplication being given by the following for-
mula (V α r 1 , V s 1 ) • (V β r 2 , V s 2 ) = (V α+β r 1 +r 2 , V s 1 ⊗ V s 2 ) + (V α+β+1 r 1 +r 2 −1 , V s 1 ⊗ V s 2 ) + (V α+β r 1 +r 2 −2 , V s 1 ⊗ V s 2 ) + · · · + (V α+β |r 1 −r 2 | , V s 1 ⊗ V s 2 ) .
4.5.2
Proof of Theorem4.5.1. Proof of this lemma is essentially the same as that of Theorem 4.4 in [18] and mostly consists of solving a system of 2 equations related to 2 singular vectors -one in (V α r 1 , V s 1 ), another in (V β r 2 , V s 2 ). We will discuss it in 4.5.3. Derivation of Theorem 4.5.1 from Lemma 4.5.2 is again very similar to that of Theorem 3.2 from Theorem 4.4 in loc. cit and uses Verma modules as follows.
Throughout the proof
A will stand for {(0, b 0 ), (1, b 1 )}, A 1 -for {(∞, b ∞ )}.(M A ⊗ M) g(CP 1 ,A∪A 1 ) → (M A ⊗ M ∞,b∞ µ,k ) g(CP 1 ,A∪A 1 ) → (M A ⊗ W ) g(CP 1 ,A∪A 1 ) → 0. Since M ∞,b∞(ii)SSM A = Ω A reg ∪ σΩ A reg ; (iii) For any Verma module W A 1 dim(M A ⊗ W A 1 ) g(CP 1 ,A∪A 1 ,A 2 ) < ∞.
4.5.3
Here we sketch the proof of
⊕ i,j∈Z CF ij ≈ F C * αβ (17) where α = λ ∞ − λ 1 − λ 0 − 2 2 , β = λ 1(18)
The functional F factors through the projection M 0,b 0 λ 0 ,k → V 0,b 0 λ 0 ,k if and only if it vanishes on the singular vector of M 0,b 0 λ 0 ,k . In other words, if this singular vector, say S, has degree (i, j) then the following equation holds
SF ij = 0.
The latter equation can be written down and solved explicitly using formulas (11 or 12).
Similar arguments go through for the module M 1,b 1 λ 1 ,k giving another equation, say
S ′ F i ′ j ′ = 0.
Simultaneous solutions to these 2 equations give the desired result. By the way, as (11,12) show, each of the expressions SF ij , S ′ F i ′ j ′ splits in a product of linear factors; therefore geometrically the solution is a collection of intersection points of 2 families of lines in the plane.
(V ǫ m , V n ), 0 ≤ m ≤ q − 1, 0 ≤ n ≤ p − 2 modulo the relation (V ǫ m , V n ) = (V ǫ+1 q−1−m , V p−2−n )
. Denote by (V ǫ m , V n ) ∼ an equivalence class of (V ǫ m , V n ). We assume that (V ǫ m , V n ) ∼ satifies the same bilinear condition (V ǫ m , V n ) in Theorem4.5.1 does. The definition of the fusion (Verlinde) algebra in this case repeats word for word that in 4.5.1.
Recall finally that Kazhdan-Lusztig fusion functor [29] gives
V r⊗k V s = V |m−n| ⊕ V |m−n|+2 · · · ⊕ V min{2k−r−s,r+s} .
The following theorem was proved in [18] in an equivalent but much less illuminating form.
V ǫ i m i , V n i )) g(CP 1 ,A) is finite dimensional.
(ii) The fusion algebra is well-defined, multiplication being given by the following for-
mula (V α r 1 , V s 1 ) ∼ • (V β r 2 , V s 2 ) ∼ = (V α+β |r 1 −r 2 | , V s 1⊗ p−2 V s 2 ) ∼ + (V α+β |r 1 −r 2 |+1 , V s 1⊗ p−2 V s 2 ) ∼ + (V α+β |r 1 −r 2 |+2 , V s 1⊗ p−2 V s 2 ) ∼ + · · · + (V α+β N , V s 1⊗ p−2 V s 2 ) ∼ , where N = min{2q − 2 − r − s, r + s}.
It is an easy exercise to derive this theorem from Theorem 4.5.1. For future purposes, however, we now sketch its original proof.
Set A = {(∞, b ∞ }, A 2 = {(0, b 0 ), (1, b 1 )}.
In addition to the algebras g(CP 1 , A), g(CP 1 , A, A 2 ) as in 4.4, we introduce an algebrā A). The latter consists of all functions whose values at the points 0 (1 resp.) belong to b 0 (b 1 resp.). Obviously g(CP 1 , A, A 2 ) ⊂ḡ(CP 1 , A, A 2 ) is an ideal and the quotient algebraḡ(CP 1 , A, A 2 )/g(CP 1 , A, A 2 ) is commutative and 2-dimensional.
g(CP 1 , A, A 2 ) ⊂ g(CP 1 ,
This algebra naturally operates on the space (M A ) g(CP 1 ,A,A 2 ) . Leth 0 ,h 1 be a basis of g(CP 1 , A, A 2 )/g(CP 1 , A, A 2 ). "Inserting" Verma modules and using BGG resolution one derives Theorem 4.5.5 from Lemma4.5.6 in a way similar to that we used in 4.5.1.
Another important corollary of Lemma4.5.6 is as follows. (ii)SSM A = Ω A nilp ∪ σΩ A nilp ; (iii) dim(M A ) g(CP 1 ,A,A 2 ) < ∞.
Classical and quantum osp(1|2). Fusion algebra as a Grothendieck ring
A. osp(1|2) is a rank 1 superalgebra -one of the superanalogues of sl 2 . It can be defined as an algebra on 2 odd generators, x + , x − , one even generator, h, and relations
[x + , x − ] = h, [h, x ± ] = ±x ± .
Even part of this algebra is sl 2 and is generated by x 2 ± ; odd part is V 1 as an sl 2 − module, its basis is x + , x − . From this it is easy to obtain the following classification of all simple finite dimensional osp(1|2)−modules. (It is even simpler to do this in the way modelling the sl 2 −case -by starting with Verma modules and then quotienting out a singular vector; for details see [31]). Each osp(1|2)−module W is a sum of an even and odd part W = even W ⊕ odd W ; each · W is an sl 2 −module, i.e. direct sum of V n 's. These are generalities. But in reality each irreducible osp(1|2)−module is of one of the 2 following types:
V 0 n such that even V 0 n = V n , odd V 0 n = V n−1 ; V 1 n such that even V 0 n = V n−1 , odd V 0 n = V n .
The fact that the dimensions of the even and odd parts are different by 1 is a consequence of the fact that odd part of the algebra is V 1 .
We see that each irreducible osp(1|2)−module is odd-dimesional; further V 0 n and V 1 n are isomorphic as modules and obtained from each other by the change of parity. This is the category of finite dimensional representations of osp(1|2); denote it Rep(osp(1|2)). As in the sl 2 −case, one proves that Rep(osp(1|2)) is semisimple.
The universal enveloping algebra Uosp(1|2) is in fact a Hopf algebra, for example the comultiplication is given by the standard formula g → g ⊗ 1 + 1 ⊗ g, g ∈ osp(1|2). This Lemma 4.5.8
V α r 1 ⊗ V β r 2 = V α+β r 1 +r 2 + V α+β+1 r 1 +r 2 −1 + V α+β r 1 +r 2 −2 + · · · + V α+β |r 1 −r 2 | .
Proof. Direct calculations show that V α r 1 ⊗ V β r 2 contains one and only one singular (annihilated by x + ) vector of each weight from |r 1 − r 2 | to r 1 + r 2 and that the submodules generated by these vectors are irreducible. Proof is completed by counting dimensions. ✷ Theorems 4.5.1 and 4.5.5 provide us with 2 commutative algebras. Here we interprete these algebras as Grothendieck rings of certain categories. Start with the algebra of Theorem 4.5.1 and denote it A gen . Obviously A gen = A 0 ⊗ A, where A is the Grothendieck ring of the category of finite-dimensional representations of g (its multiplication law is defined by the formula preceding Theorem 4.5.1) and A 0 is the algebra with basis V α i , i ≥ 0, α ∈ Z/2Z, multiplication being given by
V α r 1 • V β r 2 = V α+β r 1 +r 2 + V α+β+1 r 1 +r 2 −1 + V α+β r 1 +r 2 −2 + · · · + V α+β |r 1 −r 2 | .(19)
Comparing (19) with Lemma 4.5.8 we get the following. Appearance of osp(1|2) here, although artificial as it may seem to be, has deep reasons behind it. To see this we will analyze the rational level case using quantized enveloping algebras. B. Both Usl 2 and Uosp(1|2) admit quantization, U t sl 2 and U t osp(1|2) resp.. Let us remind the relevant formulas. The Drinfeld-Jimbo (see [8,27]) algebra U t sl 2 , t ∈ C is defined to be an associative algebra on generators E, F, K ±1 and relations
EF − F E = K − K −1 t − t −1 , KEK −1 = t 2 E, KF K −1 = t −2 F. U t osp(1|2)
is similarly defined [31] as an associative algebra on generators X + , X − , K ±1
and relations
X + X − + X − X + = K − K −1 t − t −1 , KX ± K −1 = t ±1 X ± .
The representation theory of sl 2 and osp(1|2) "deforms to" the representation theory of U t sl 2 and U t osp(1|2) resp. We will continue denoting by V m the m + 1−dimensional module over U t sl 2 , and by V 0 m , V 1 m the 2 (2m + 1)− dimensional modules over U t osp(1|2). For generic t these modules are irreducible, the categories of finite dimensional representations, Rep(U t sl 2 ) and Rep(U t osp(1|2)), generated by these modules are semisimple.
The deformations U t sl 2 and U t osp(1|2) are especially remarkable in that they afford simultaneous deformation of the Hopf algebra structure. We get 2 tensor categories Rep(U t sl 2 ) and Rep(U t osp(1|2)). What has been said implies that the Grothendieck rings of Rep(U t sl 2 ) and Rep(U t osp(1|2)) are isomorphic to the Grothendieck rings of the corresponding classical objects.
If however t is a root of unity, things change dramatically. Suppose for simplicity that t is a primitive l-th root of unity, l being odd. Then Lemma 4.5.11 Let t be a primitive l-th root of unity, l being odd, m, n < l. Then
(i) V m ⊗ V n = V |m−n| ⊕ V |m−n|+2 · · · ⊕ V min{2(l−1)−m−n,m+n} ⊕ W,
where W is not semisimple.
(ii) V α m ⊗ V β n = V α+β |m−n| ⊕ V α+β+1 |m−n|+1 · · · ⊕ V α+β min{2(l−1)−m−n,m+n} ⊕ W,
where W is not semisimple.
Sketch of Proof. (i) is well-known, see [40]. We will however review both cases as at our level of brevity there will no difference between them. First, direct calculations as in the proof of Lemma 4.5.8 show that regardless of t at each weight space there can always be only one singular vector. Now decomposition of Lemma 4.5.8, statements (20,21) and this uniqueness result show that the submodules V l−1+i and V l−1−i (or V ǫ l−1+i and V ǫ l−1−i ), i ≤ m + n − l + 1 are non-trivially tangled. Other V j coming from generic t are still irreducible and appear as direct summands. ✷
Definition.
(i) Define Rep(U t sl 2 ) (l) and Rep(U t osp(1|2)) (l) to be subcategories of Rep(U t sl 2 ) and
Rep(U t osp(1|2))resp. consisting of direct sums of irreducible modules V m (or V α m resp.), m < l.
(ii) Define functors
Rep(U t sl 2 ) (l) × Rep(U t sl 2 ) (l) → Rep(U t sl 2 ) (l) , A, B → A⊗B, Rep(U t osp(1|2)) (l) × Rep(U t osp(1|2)) (l) → Rep(U t osp(1|2)) (l) , A, B → A⊗B,
by taking the usual tensor product and then throwing away W in the right hand side of formulas in Lemma 4.5.11. ✷
We get tensor categories Rep(U t sl 2 ) (l) and Rep(U t osp(1|2)) (l) .
C. It is easy now to interpret the fusion algebra at the rational level in terms of the Grothendieck rings of Rep(U t sl 2 ) (l) and Rep(U t osp(1|2)) (l) . In view of Lemma 4.5.11 and Definition above, Theorem 4.5.5 reads as follows.
Proposition 4.5.12 Fusion algebra at the level k + 2 = p/q is a quotient of the tensor product of the Grothendieck rings of the categories Rep(U t 1 sl 2 ) (p−1) and Rep(U t 2 osp(1|2)) (q) .
Further, the fusion algebra always contains the Grothendieck ring of Rep(U t 2 osp(1|2)) (q) via the classes of symbols V α m , V 0 .
Kac-Moody vs. Virasoro
Virasoro algebra, V ir, is defined to be a vector space with basis {L i , z, i ∈ Z} and bracket
[L i , L j ] = (i − j)L i+j + δ i,−j i 3 − i 12 z.
Representation theory of Virasoro algebra is to a great extent parallel to that ofĝ. We will confine to essentials, making reference to [14]. It has hardly been written anywhere, but is nevertheless known that the V ir− analogue of the fusion algebra from 4.5.1, i.e. at a generic level, is as follows:
(V n 1 , V m 1 ) • (V n 2 , V m 2 ) = (V n 1 ⊗ V n 2 , V m 1 ⊗ V m 2 ).(22)
(The interested reader can prove this result using methods of [15]; our treatment of thê g-fusion algebra in 4.5.1 is also a direct analogue of these.) There is a functor sendingĝ−modules to V ir−modules -quantum Drinfeld-Sokolov reduction. One of the prerequisites for it is a choice of a nilpotent subalgebra of sl 2 . The two obvious possibilities are Ce and Cf . Denote the corresponding functors φ e and φ f . It can be extracted from [12] that both functors send generalized Weyl modules to generalized Weyl modules. In our terminology one gets
φ e : (V 0 m , V n ) → (V m , V n ) (V 1 m , V n ) → (V m−1 , V n ), φ f : (V 0 m , V n ) → (V m−1 , V n ) (V 1 m , V n ) → (V m , V n ),
where the symbol V −1 , if arises, is understood as zero.
The easily calculate the fusion algebra. We will not write down the relevant formulas here and confine to mentionaing that the algebra is related to the product of Grothendieck rings of 2 quantum U t (sl 2 ) at appropriate roots of unity in much the same way as the fusion algebra forĝ is related to the product of Grothendieck rings of U t (osp(1|2)) and U t (sl 2 ).
Recall also that the V ir-fusion algebra was calculated in [6]; mathematically acceptable exposition can be found in [15].
Another property of the Drinfeld-Sokolov reduction is that both φ e and φ f send admissible representations at the level k = 2 − p/q ofĝ to minimal representations of V ir at the level c pq , see [17] .
Fusion functor.
This part is an announcement, proofs will appear elsewhere Suppose we have a trivial vector bundle E → CP 1 ,
A = {(P 1 , b 1 ), (P 2 , b 2 )}, B = {(P 3 , b 3 )}, so that (E, A B) isF A→B (M A ) = ∪ N ≥1 M A N .
One can show that the vector space F A→B (M A ) affords in a natural way a structure of anĝ B −module at the same level; this is easy to show in the spirit of [29,4]. Using our methods one can show that (i) if M A is from the O−category, or further a generalized Weyl module, or further an admissible representation, then F A→B (M A ) is also as aĝ B −module;
(ii) the arising in this way Grothendieck rings coincide with those in Theorem 4.5.1 or Theorem 4.5.5 if the level is generic or rational resp..
This generalizes the statement for the integrable representations, see [10].
Problem. Describe the arising tensoring in the spirit of Kazhdan-Lusztig.
Quadratic degeneration 4.7.1
The setup here will the following version of 3.2.2:
(i)π : C S → S be a family of curves over a formal disk S, such that the fiber over the generic point of S ("outside origin") is a smooth projective curve, and over the origin, O, the fiber is a curve C O . with exactly one quadratic singularity;
(ii) ρ S : E S → C S is a rank 2 vector bundle.
As in 3.2.2, we complete these data to the localization data with logarithmic singularities, sayψ. In the standard way, Theorem 3. Along with the familyπ : C S → S consider the familyπ ∨ : C ∨ S → S, obtained from π : C S → S by replacing the singular fiber C O with its normalization C ∨ O (i.e. be tearing C O apart at the self-intersection point). There is a projection C ∨ O → C O and the preimage of the self-intersection point a ∈ C O consists of 2 points a 0 , a ∞ ∈ C ∨ O . It is obvious that the datum E → C S is equivalent to the data "ρ ∨ S :
E ∨ S → C ∨ S , equivalence (ρ ∨ S ) −1 (a 0 ) ≈ (ρ ∨ S ) −1 (a ∞ )".
The localization data with logarithmic singularities ψ rewrites to give a "normalized" localization data ψ ∨ .
In addition fix 2 different lines l 0 , l ∞ in the fiber of E S over the point a ∈ C O . This determines 2 Borel subalgebras, b 0 , b ∞ operating in the fiber over a.
After normalization these additional data determine the line l 0 and the Borel subalgebra b 0 operating in the fiber of E ∨ S over a 0 , as well as the line l ∞ and the Borel subalgebra b ∞ operating in the fiber over a ∞ . We also get a distinguished Cartan subalgebra
h = b 0 ∩ b ∞ . Set A ∨ = A {(a 0 , b 0 ), (a ∞ , b ∞ )}.
Now with aĝ A −module M A at the level k and an admissible weight λ ∈ h * we as-
sociate theĝ A ∨ −module M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k
. We get a D−module for the "normalized"localization data:
⊕ λ ∆ ψ ∨ (M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k )
.
Proposition 4.7.1 Generically with respect to l 0 , l ∞ , if ∆ψ(M A ) is smooth then ⊕ λ ∆ ψ ∨ (M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k
) is also and there is an isomorphism of D−modules
∆ψ(M A ) ≈ ⊕ λ ∆ ψ ∨ (M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k
).
Proof
(i) Begin with the genus zero case.Observe that the algebra of regular functions on the
neighborhood of the point a is C[t 0 , t ∞ ][[t]]/ < t 0 t ∞ = t > where t is a coordinate on S;
C ∨ O in this case is just a union of 2 spheres. Therefore the set A splits in two: A ′ and A ′′ , each of which has to do with one of the spheres.
Hence the algebra g(π −1 (s), A) can be degenerated into the following one as s "approaches" O:
(g(CP 1 , A ′ , (P 0 , b 0 )) + h) ⊕ h (h + g(CP 1 , A ′′ , (P ∞ , b ∞ )).
Meaning of the last expression is as follows: recall, see 4.4, that g(CP 1 , A ′ , (P 0 , b 0 )) consists of functions regular outsideĀ and sending P 0 to n 0 ; g(CP 1 , A ′′ , (P ∞ , b ∞ )) is defined similarly with P 0 , n 0 replaced with P ∞ , n ∞ ; further the algebra g(CP 1 , A ′ , (P 0 , b 0 )) + h is the algebra of functions sending P 0 to b 0 , the same is true for h + g(CP 1 , A ′′ , (P ∞ , b ∞ )); finally "⊕ h " means direct product over h.
Therefore the coinvariants degenerate into the space
((M A ′ ) g(CP 1 ,A ′ ,(P 0 ,b 0 )) ⊗ (M A ′′ ) g(CP 1 ,A ′′ ,(P∞,b∞)) ) h ,
where h acts by means of the diagonal embedding; this makes sense as the fibers are identified.
By Proposition 4.4.4, the space
(M A ′ ) g(CP 1 ,A ′ ,(P 0 ,b 0 )) ⊗ (M A ′′ ) g(CP 1 ,A ′′ ,(P∞,b∞))
is finite dimensional. It is easy to extract from Lemma 4.5.2 that as an h-module this space is semisimple and therefore is isomorphic to
⊕ λ ((M A ⊗ M P 0 ,b 0 λ,k ⊗ M P∞,b∞ λ,k ) g(C ∨ O ,A ∨ )
. By Lemma 4.5.6, in the last formula λ can be chosen to be admissible and the Verma modules can be replaced with the corresponding admissible representations.
This proves that ⊕ λ ∆ ψ ∨ (M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k
) is smooth and gives a morphism
∆ψ(M A ) → ⊕ λ ∆ ψ ∨ (M A ⊗ L P 0 ,b 0 λ,k ⊗ L P∞,b∞ λ,k ).
That this is an isomorphism can be shown in the standard way constructing the inverse map using the formal character of L λ,k , see [4].
(ii) The higher genus case is not much different. For example, pinching makes a torus into a sphere. Therefore in this case proof is literally the same. It also proves an analogue of Lemma 4.5.6 for a torus. This provides a basis for induction. (iii) Quadratic degeneration for generalized Weyl modules on the sphere allows to write horizontal sections of the corresponding bundle as a product of vertex operators. This will be explained in sect.5.
Screening operators and correlation functions
In this section we will study in detail the situation described in 4.4.3: we have the trivial rank 2 bundle E → CP 1 , a generalized Weyl module M A , and a holonomic D−module A) . For the reasons which will become clear later we replace this bundle with the dual one, its fiber being ((M A ) * ) g (CP 1 ,A) .
∆(M A ) on the space C m × C m with fiber (M A ) g(CP 1 ,
Denote the corresponding D−module by ∆(M A ) * . Using our results on quadratic degeneration we rewrite horizontal sections of the corresponding bundle with flat connection as matrix elements of vertex operators, which serves the two-fold purpose: we find that the differential equations satisfied by horizontal sections are provided by the singular vectors of the corresponding Verma module and write down integral representations for solutions to these differential equations.
Vertex operators and corelation functions
An alternative to the language of coinvariants in the genus zero case is the language of vertex operators.
Definition. A vertex operator is aĝ−morphism
Y : F C * αβ ⊗ V 1 → V 2 ,(23)
where F C * αβ is a loop module (see 2.6) and V 1 , V 2 ∈ O k are highest weight modules. ✷
In other words, a vertex operator is an embedding F C * αβ ֒→ Hom C V 1 → V 2 . The space F C * αβ has the basis {F ij = F i ⊗ z j , i, j ∈ Z}, where {F i , i ∈ Z} is a basis in F αβ , see 2.6. Given a vertex operator Y , consider the generating function
Y (x, z) = x ∆ 1 z ∆ 2 ∞ i,j=−∞ F ij x −i z −j ,
the "monodromy coefficients" ∆ 1 , ∆ 2 are defined by:
∆ 1 = −λ 2 + λ 1 + β 2 , ∆ 2 = −C(λ 2 ) + C(λ 1 ) + C(β) 2 ,
where λ i is the highest weight of V i and C(λ) = λ(λ + 2)/2. (∆ 1 , ∆ 2 will later appear as genuine monodromy coefficients of a certain flat connection.)
The formal series Y (x, z) is, of course, an element of Hom C (V 1 ,
V 2 ⊗x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]]).
Further, for any g ∈ g the commutator [g ⊗ z n , Y (x, z)] is also a well-defined element of
Hom C (V 1 , V 2 ⊗ x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]])
. For the standard basis of g, see 2.2, one derives from the definition of a vertex operator that
[e ⊗ z n , Y (x, z)] = z n (−x 2 ∂ ∂x + βx)Y (x, z),(24)[f ⊗ z n , Y (x, z)] = z n ∂ ∂x Y (x, z),(25)[h ⊗ z n , Y (x, z)] = z n (2x 2 ∂ ∂x − βx)Y (x, z).(26)
We conclude that for any g ∈ g there is a differential operator D g (x) in x such that
[g ⊗ z n , Y (x, z)] = z n D g (x)Y (x, z),(27)
Suppose now we are given a collection of vertex operators
Y i : F C * λ i µ i ⊗ V i−1/2 → V i+1/2 , 1 ≤ i ≤ m.
The product of the corresponding generating functions
Y m (x m , z m ) · · · Y 2 (x 2 , z 2 )Y 1 (x 1 , z 1 ) is a well-defined element of Hom C (V 1/2 , V m+1/2 ⊗ i x ∆ i,1 i z ∆ i,2 i C[[x ±1 1 , . . . x ±1 m , z ±1 1 , . . . z ±1 m ]]). The matrix element < v * , Y m (x m , z m · · · Y 2 (x 2 , z 2 )Y 1 (x 1 , z 1 )v >, v ∈ V 1/2 , v * ∈ V * m+1/2 is, therefore, a formal Laurent series in x i , z i , 1 ≤ i ≤ m.
Definition Suppose Y i (x i , z i ), 1 ≤ i ≤ m are as above. Then the matrix element
Ψ(x 1 , . . . , x m , z 1 , . . . z m ) =< v * , Y m (x m , z m ) · · · Y 2 (x 2 , z 2 )Y 1 (x 1 , z 1 )v >(28)
is called correlation function if V 1/2 , . . . , V m+1/2 are irreducible generalized Weyl modules, V 1/2 is the vacuum module, v is the highest weight vector of V 1/2 and v * is the dual to the highest weight vector of V m+1/2 . (The latter condition is meaningful in view of the weight space decomposition of a highest weight module.) ✷ A correlation function has been understood as a formal power series. We will show that, in fact, it is a holomorphic function satisfying a certain holonomic system of partial differential equations. In order to do that we will interpret vertex opeartors as horizontal sections of a line bundle with a flat connection provided by three modules on CP 1 × CP 1 .
From coinvariants to vertex operator algebra
5.2.1
We return to the setup of 4.5.1. In the cartesian product CP 1 × CP 1 fix coordinate system (x, z). Attach to the point x in the first factor the Borel subalgebra b x spanned by the vectors e x = e − xh − x 2 f, h x = h + 2xf . This means, in particular, that b 0 is the standard Borel subalgebra Ce ⊕ Ch (see 2.2 ) and b ∞ is the opposite one. Set A) . By Theorem 4.5.1 this space is either 0-or 1-dimensional. Suppose the latter possibility is the case. Then by Theorem 3.2.2 we get a line bundle with flat connection over C * × C * whose fiber over the A) . There arises an embedding
A = {(0, 0), (x, z), (∞, ∞)}. Let V A = V b 0 ,0 0 ⊗ V bx,z 1 ⊗ V b∞,∞ ∞ be a generalized Weyl module overĝ A . Consider the space of invariants ((V A ) * ) g(CP 1 ,point (x, z) ∈ C * × C * is ((V A ) * ) g(CP 1 ,V bx,z 1 ֒→ Hom C (V b 0 ,0 0 ⊗ V b∞,∞ ∞ , C).
The dual space (V b∞,∞ ∞ ) * as aĝ−module is isomorphic to the contragredient module
(V b∞,∞ ∞ ) c , see 2.2.
As the level is generic, the latter module is irreducible and is, therefore, isomorphic to a certain generalized Weyl module V b 0 ,0 ∞ . Hence we get an embedding
V bx,z 1 ֒→ Hom C (V b 0 ,0 0 , V b 0 ,0 ∞ ⊗ x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]]),
where ∆ 1 , ∆ 2 are monodromy coefficients of the flat connection. We conclude that any w ∈ (ii) any vertex operator is obtained in this way.
Proof is a direct and simple calculation using definitions, see also 4.5.3 formula (17).
⊗ m i=1 V z i ,bx i i .
Consider all possible correlation functions
< v * , v m (x m , z m ) · · · v 1 (x 1 , z 1 )v >, where v i (x i , z i )< v * , v m (x m , z m ) · · · v 1 (x 1 , z 1 )v > .
Proof. Intertwining properties of vertex operators imply a correlation function is a horizontal section of ∆(M A ) * in a formal sense. This give a map in one direction. A map in the opposite direction in provided by quadratic degeneration, see Proposition 4.7.1. ✷
5.2.2
By Lemma 5.2.1 coinvariants recover vertex operators. In fact they give us much more:
the collection of generating functions w(x, z), w ∈ V bx,z 1 affords a kind of vertex operator algebra structure. We will not discuss the latter in detail (see [21]) and only explain how one can get exlicit formulas for w(x, z), w ∈ V bx,z 1 in terms of the vertex operator v 1 (x, z) related to the highest weight vector v 1 .
For any g ∈ g set g(i) = g⊗z i ∈ĝ. Define the current g(z) to be g(z) = i∈Z g(i)z −1−i ∈ g ⊗ C[[z ±1 ]]. Define g(z) (l) to be the l−th (formal) derivative of g(z) with respect to z. For any g(z) (l) set
(g(z) (l) ) + = ( d dz ) l ∞ i=0 g −i−1 z i , (g(z) (l) − = g(z) (l) − (g(z) (l) ) + .
Observe that for any w(x, z) ∈ Hom C (V b 0 ,0
0 , V b 0 ,0 ∞ ⊗x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]]
) and any g ∈ g, the products (g(z) (l) ) − w(x, z), w(x, z)(g(z) (l) ) + are also well-defined elements of Hom
C (V b 0 ,0 0 , V b 0 ,0 ∞ ⊗ x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]]).
Define for any g ∈ g, w(x, z) ∈ Hom C (V b 0 ,0
0 , V b 0 ,0 ∞ ⊗ x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]])
: g(z) (k) w(x, z) := (g(z) (k) ) − w(x, z) + w(x, z)(g(z) (k) ) + .
Lemma 5.2.3 Let g ∈ g, w ∈ V bx,z 1 , w(x, z) the corresponding element of Hom C (V b 0 ,0 0 , V b 0 ,0 ∞ ⊗ x ∆ 1 z ∆ 2 C[[x ±1 , z ±1 ]]). Then (i) (g · w)(x, z) = [g, w(x, z)]; (ii) (g(−l) · w)(x, z) = (1/(l − 1)!) : g(z) (l−1) w(x, z) :, l > 0;(29)
Proof is a direct calculation of matrix elements of the operator (g(−l) · w)(x, z) based on the definition of the space of coinvariants. ✷
Differential equations satisfied by correlation functions
We return to the setup of 5.1 and consider a correlation function
Ψ(x 1 , . . . , x m , z 1 , . . . z m ) =< v * , Y m (x m , z m ) · · · Y 2 (x 2 , z 2 )Y 1 (x 1 , z 1 )v >,
coming from the product of vertex operators
Y i : F C * λ i µ i ⊗ V i−1/2 → V i+1/2 , 1 ≤ i ≤ m.
Using Lemma 5.2.1 we assume that there are generalized Weyl modules
V i , 1 ≤ i ≤ m with highest weight vectors v i , 1 ≤ i ≤ m such that Y i (x, z) = v i (x, z
). An advantage of this point of view is that for any collection of elements w i ∈ V i , 1 ≤ i ≤ m we can consider the matrix element
< v * , w m (x m , z m ) · · · w 2 (x 2 , z 2 )w 1 (x 1 , z 1 )v > . Lemma 5.3.1 For any w i ∈ V i , 1 ≤ i ≤ m < v * , w m (x m , z m ) · · · w 2 (x 2 , z 2 )w 1 (x 1 , z 1 )v >= D · Ψ(x 1 , . . . , x m , z 1 , . . . z m ),
where D is a differential operator in x ′ s with coefficients in rational functions in z ′ s.
Proof. Start with the function
< v * , v m (x m , z m ) · · · v i+1 (x i+1 , z i+1 )(g(−l)v i )(x i , z i )v i−1 (x i−1 , z i−1 ) · · · v 1 (x 1 , z 1 )v >, l > 0.
By Lemma 5.2.3 (ii) it rewrites as
< v * , v m (x m , z m ) · · · v i+1 (x i+1 , z i+1 )(g(z) (l−1) − v i (x i , z i )−v i (x i , z i )g(z) (l−1) + )v i−1 (x i−1 , z i−1 ) · · · v 1 (x 1 , z 1 )v >, l >
Then commute all g i , i < 0 through to the right and all g i , i ≥ 0 to the left in a standard way, c.f. [20] and use commutation relations (24,25,26
) S i ∈ U(ĝ) such that S i v i = 0, 1 ≤ i ≤ m.
On the other hand, by Lemma 5.3.1 there are differential operators
D i , 1 ≤ i ≤ m such that D i < v * , v m (x m , z m ) · · · v 2 (x 2 , z 2 )v 1 (x 1 , z 1 )v >= < v * , v m (x m , z m ) · · · v i+1 (x i+1 , z i+1 )(S i v i )(x i , z i )v i−1 (x i−1 , z i−1 ) · · · v 1 (x 1 , z 1 )v >, 1 ≤ i ≤ m.
We arrive to the following result. for that, see (24,25,26).) For any A = s a i ⊗ b s ∈ g ⊗ g denote by A ij , 1 ≤ i, j ≤ m an operator acting on the m−fold tensor product of g-modules by the formula
Lemma 5.3.2 The correlation function
Ψ(x 1 , . . . , x m , z 1 , . . . z m ) =< v * , v m (x m , z m ) · · · v 2 (x 2 , z 2 )v 1 (x 1 , z 1 )v >A ij · w 1 ⊗ · · · w m = s w 1 ⊗ a s w i ⊗ · · · b s w j ⊗ · · · w m .
The formula (27) implies that A ij is a differential operator in x i , x j . Set Ω = ef +f e+h 2 /2.
Lemma 5.3.3 ([30])
The correlation function Ψ = Ψ(x 1 , . . . , x m , z 1 , . . . z m ) satisfies the system of Knizhnik-Zamolodchikov equations
(k + 2) ∂ ∂z i Ψ = j =i Ω ij z i − z j Ψ, 1 ≤ i ≤ m.(31)
There is no need to prove this lemma here as one can repeat word for word the known proofs. However we point out that if one considers a highest weight module V as a module over the semi-direct product ofĝ and the Virasoro algebra V ir then V is annihilated by
Ψ old =< v * , π • Y (x m , z m ) · · · Y 1 (x 1 , z 1 )v o >, Y i : F C * λ i µ i ⊗ V i−1/2 → V i+1/2 , 1 ≤ i ≤ m,
satisfying the same conditions as the expression in (28), see5.1, except that instead of assuming that V m+1/2 is a generalized Weyl module we assume that V m+1/2 is a contragredient Verma module, see 2.2. (Why "old" will become clear in a moment.) It is easy to see that Ψ old = Ψ(x 1 , . . . , x m , z 1 , . . . z m ) satisfies the same system of equations (30,31). Suppose in addition that there is a projection π : V m+1/2 → W onto another contragredient Verma module W . Denoting by w * an element dual to the highest weight vector w ∈ W one can consider the matrix element Ψ new =< w * , π • Y (x m , z m ) · · · Y 1 (x 1 , z 1 )v 0 > .
We again observe that Ψ old is a solution to the same system (30,31). This new solution can be calculated as follows.
There arises the dual map π * : W * → V * m+1/2 and by definition there is an element S of U(ĝ > ) such that π * (w * ) = Sv * . We now take the definition of Ψ new , replace in it π * (w * )
with Sv * and get
Ψ new =< S · v * , •Y (x m , z m ) · · · Y 1 (x 1 , z 1 )v 0 > .(32)
Then we commute S through to the right. The intertwining properties of vertex operators tell us that
Ψ new = S t · Ψ old ,(33)
where t signifies the canonical antiinvolution an a Lie algebra (g 1 g 2 · · · g n → g n g n−1 · · · g 1 ) and the action is determined by the following condition: if g ∈ g then
(g ⊗ z n ) · Ψ old = m i=1 D g (x i )z n i Ψ old ,
see (27).
We intend to use (33) in the case when π and therefore S do not exist!
Screening operators
Let V λ∞,k be a highest weight module and v ∈ V λ∞,k a highest weight vector. If the obvious integrality conditions are satisfied then the vectors f λ∞+1 v, (e ⊗ z −1 ) k−λ∞+1 v are singular and give rise to embeddings of the type W ֒→ V λ∞,k . Now take 3 highest weight modules V λ i ,k , i = 0, 1, ∞ attach them to 3 point in CP 1 and consider the space of coinvariants
(⊗ i=0,1,∞ V b i ,i λ i ,k ) g(CP 1 ,{0,1,∞}) .
Of course an embedding W ֒→ V λ∞,k gives rise to a map
(W b∞,∞ ⊗ i=0,1 V b i ,i λ i ,k ) g(CP 1 ,{0,1,∞}) ֒→ (⊗ i=0,1,∞ V b i ,i λ i ,k ) g(CP 1 ,{0,1,∞}) .
It is remarkable that even if the embedding W ֒→ V λ∞,k does not exist the last map still does. In the language of vertex operators this phenomenon was explained in great detail in [19].
Therefore with each of the formal singular vectors -f λ∞+1 v or (e ⊗ z −1 ) k−λ∞+1 v -we have associated an operator acting on coinvariants. Call these operators screenings and denote them R 1 and R 0 respectively.
Let us calculate the action of the screenings explicitly. By definition R j (⊗ i=0,1,∞ V b i ,i λ i ,k ) g(CP 1 ,{0,1,∞}) only depends on V λ∞,k so we will be simply writing R j (V λ∞,k ). Now formulas for the related singular vectors (f λ∞+1 v, (e⊗z −1 ) k−λ∞+1 v) and a very simple calculation using the formulas (7,8) give the following result:
R 1 ((V 0 m , V n )) = (V 1 m−1 , V n )(34)R 1 ((V 1 m , V n )) = (V 0 m+1 , V n ) (35) R 0 ((V 0 m , V n )) = (V 1 m+1 , V n )(36)R 0 ((V 1 m , V n )) = (V 0 m−1 , V n )(37)
Suppose we are given 2 generalized Weyl modules and a vertex operator acting between them. Suppose in addition that this vertex operator is related to a highest weight in the third generalized Weyl module, say (V ǫ m , V n ). Theorem 4.5.1 tells us that given such a vertex operator our screenings give us all the others of the type (V α i , V n ) -we cannot only change the value of n. But then there is the standard screening operator -S -which takes care of n, see e.g. [13]. So these three -R 1 , R 2 , S -provide us with all vertex operators.
This has an important application to the calculation of correlation functions.
Start with a simple correlation function given by the product of vertex operators, each of which is characterized by the condition m = 0. Then applying S an appropriate number of times one gets all vertex operators and, hence, all correlation functions in spirit of Varchenko-Schechtman, see [2]. Now take a Varchenko-Schechtman correlation function Ψ old . It comes from a product of vertex operators:
Ψ old =< v * , Y (x m , z m ) · · · Y 1 (x 1 , z 1 )v o >, Y i : F C * λ i µ i ⊗ V i−1/2 → V i+1/2 , 1 ≤ i ≤ m.
Let W i , 0 ≤ i ≤ m, be words on 2 letters R 1 and R 2 . Replacing V i+1/2 with W i (V i+1/2 ) we get a new correlation function Ψ new . Doing this with all Ψ old and sufficiently many W i , 0 ≤ i ≤ m we get all solutions to (30,31). In principle all these solutions can be written down explicitly. It is especially simple to do so in the case when we keep V i+1/2 , 0 ≤ i ≤ m − 1, and only change V m+1/2 .
So assume that Ψ old is as above and replace V m+1/2 with R j (V m+1/2 ), j = 0, 1. Then by (32) one is to expect that
Ψ new =< X α · v * , •Y (x m , z m ) · · · Y 1 (x 1 , z 1 )v 0 >,
where X is either e or f ⊗ z if j = 1 or 0 resp., and α is either λ + 2 or k − λ + 2 resp.,
where λ is the highest weight of V m+1/2 .
Of course if α is not a nonnegative integer then the last formula does not make much sense. Nevertheless using it and (33) as a motivation we arrive to
Ψ new = X α Ψ old .
Now the left-hand side of the last equality does make sense: X is a first order differential operator, see 5.4.1, therefore we can set in a rather straightforward manner
X α Ψ old = t −α−1 {exp(−Xt)Ψ old }dt
and get a nice integral operator, for details see [19].
This procedure can be easily iterated to provide the functions
n i=1 t −α i −1 i {exp(−X 1 t 1 ) exp(−X 2 t 2 ) · · · exp(−X n t n )Ψ old } n i=1 dti,(38)
where X 1 , X 2 , ... is either e, f ⊗ z, e, ... or f ⊗ z, e, f ⊗ z, .... Proof is same as the proof of the analogous statement in [18]. In fact it is an easy exrcise to make the heuristic arguments which have lead us to the formula (38) into a precise proof. ✷
Integrating functions (38) with respect to x ′ s (or doing something similar but more esoteric) one is supposed to get the Dotsenko-Fateev correlation functions for the Virasoro algebra. It would be interesting to do this explicitly and compare the result with the calculations in [23]. We conjecture that in this case formulas (38) actually give horizontal sections of the latter bundle.
Observe that the conventional Weyl module of the level k is defined to be the inducedrepresentation Indĝ g[[z]]⊕Cc V n , where g[[z]] operates on V n via the evaluation map g[[z]] → g and c → k. From our point of view the Weyl module is a quotient of the Verma module M n,k by the submodule generated by the singular vector f n+1 v λ,k . In other words, Weyl modules are associated to the symbols (V 0 0 , V n ). This partially explains appearance of g−modules in our notations.
recall that with an n−dimensional vector space W one associates the flag manifold F (W ) = GL(n, C)/B and the base affine space Base(W ) = GL(n, C)/N, where B is a Borel subgroup and N unipotent subgroup of B. The natural map Base(W ) → F (W ) is a principal (C * ) ×n -bundle. Now return to aĝ A −module V A and suppose for simplicity that A consists of 1 element (P, b). Consider a family of the data {P, E → C} -let us not care about Borel subalgebras for the moment. One expects that the corresponding family of vector spaces arranges then in a locally trivial vector bundle. An obstacle to get this is that we have defined V P up to an isomorphism but have not specified any such isomorphism. For example, an attempt to choose a basis in V P requires to choose (in particular) a local coordinate z at P , such that z(P ) = 0. Different choices of z are essentially different as the group Diff(P ) of diffeomorphisms of the formal neighborhood of P does not in general act on V P . However the subgroup Diff(P ) 1 ⊂ Diff(P ) of diffeomorphisms preserving the 1-jet of parameter does act on V P . We see that V P , in fact, depends on the 1-jet of parameter at P . To take care of Borel subalgebras, let us recall that with an n−dimensional vector space W one associates the flag manifold F (W ) = GL(n, C)/B and the base affine space Base(W ) = GL(n, C)/N, where B is a Borel subgroup and N unipotent subgroup of B.
a non empty finite set A S of sections of π satisfying the condition: for any s ∈ S the natural projection of the set A S (s) = {a(s), a ∈ A S } onπ −1 (s) is an injection. Pick an arbitrary curve, say C s 0 , from our family. Consider a highest weight module M A overĝ A , where we write A instead of the lengthy A S (s 0 ); what follows is obviously independent of the choice of s 0 . By 3.1.2, remark 3.1.1, and 3.2.1, we get aĝ A S (s) −module M A S (s) for any s ∈ S and the collection {M A S (s) , s ∈ S} arranges in a locally trivial vector bundle. With each s ∈ S
by means of the Sugawara construction. Going over definitions one gets that this well defines a twisted D−module with the fiber as in the theorem. ✷ Denote the constructed D−module by ∆ ψ (M A ).
If n = 2 and M A is an admissibleĝ A −module then ∆ ψ (M A ) is holonomic for almost any vector bundle E S (i.e.as a sheaf ∆ ψ (M A ) is isomorphic to a sheaf of sections of a certain finite rank vector bundle over some open set in S ).
Lemma 4.1.1 ( see[4]) Let a be a Lie algebra and p ⊂ a be its subalgebra. Denote by p ⊥ the annihilator of p in a * Let V be an (a, n)-module. If SSM ∩p ⊥ = {0} and dim a/n⊕p < ∞ then dimM p < ∞.
a bundle E exceptional if kerH(E) = 0. Obviously kerH(E) is exactly the space of global differential forms with values in nilpotent endomorphisms of the vector bundle E.
Theorem 4.3.1 (Hitchin[24]) Zero set of the map (16) is a maximal Lagrangian submanifold in the cotangent bundle of the moduli space of vector bundles over C. In particular, exceptional vector bundles form a positive codimension algebraic subset of the moduli space of vector bundles.
Suppose we have a moduli space of rank 2 vector bundles with parabolic structure with fixed determinant. Elements of such a space are isomorphism classes of the data (vector bundle E, fixed lines l 1 , ..., l m in some fibers.) It is rather clear that the map (E, l 1 , ..., l m ) → (E(l 1 + · · · + l m ), l 1 , ..., l m ) is a homeomorphism of 2 moduli spaces with different determinants. Definition. Call the data (E, l 1 , ..., l m ) generic if E(l i 1 + · · · + l is ) is not exceptional for any subset {i 1 , ..., i s } ⊂ {1, 2, ..., m}. ✷ It follows from Theorem 4.3.1 that the set of generic vector bundles is open and every-Suppose we are in the situation of 3.1.2: we have an admissibleĝ A −module M A on the curve C with a vector bundle E → C. As A is a collection of borel subalgebras b 1 , ..., b m operating in fixed fibers, we have parabolic structure -lines l 1 , ..., l m in the corresponding fibers preserved by the b i 's. Call the data (E, A) generic if the data (E, l 1 , ..., l m ) is generic in the sense of 4.3.2 above. Recall that we are interested in the space of coinvariants M A g(C,A) , where g(C, A) is an algebra of endomorphisms of the bundle E regular outside points from the correspondinḡ A, see 3.1.3 and 3.2.2, Theorem 3.2.1.
the space global nilpotent transformations of E, and σ is the twist introduced in 4.2. Genericity condition means that Ω nilp (E) = 0, see 4.3.1 and 4.3.2. On the other hand it is easy to see that the operation of subtracting a line generates the twist σ on endomorphisms. (In fact one has to compose subtracting of a line with a reflection in the fiber, but this does not change the isomorphism class of the bundle.)
Let E = O(r) ⊕ O(s) and (E, l 1 , ..., l m ), m ≥ |r − s|, a vector bundle with parabolic structure. Then generically with respect to l 1 , ..., l m the bundle E(l 1 + · · · + l m ) is not exceptional:
Let us again consider a vector bundle E over CP 1 and aĝ A −module M A . As in 4.3.3,A determines a parabolic structure on E, say (E, l 1 , ..., l m ). Call the data (E, A) generic if (E, l 1 , ..., l m ) is also.
Proposition 4.4. 2
2If (E, A) is generic and M A is either admissible or generalized Weyl module, thendim (M A ) g(CP 1 ,A) < ∞.Proof is a simplified version of the proof of Proposition 4.); instead of the Hitchin's theorem one uses the "observation" thatO(n) has no non-zero global sections if n < 0. ✷ Corollary 4.4.3 If M A is a generalized Weyl module then there is a holonomic twisted D-module living in the space (J (1) (CP 1 ) × J (1) (CP 1 )) ×m with the fiber M A g(CP 1 ,A)).Proof. Repeating word for word proof of Theorem3.2.2 one derives from Proposition 4.4.2 existence of a twisted D−module on the space (Base(C 2 ) × J (1) (CP 1 )) ×m . But for sl 2 , the flag manifold is CP 1 and the base affine space (Base(C 2 ) is also the space of 1-jets of parameter J (1) (CP 1 )) ×m . ✷ As in 4.3.3, we want to prove a generalization of Proposition 4.4.2 in order to prepare grounds for studying quadratic degeneration.
A 1 and A 2 associate the following 2 subalgebras of g(CP 1 , A): g(CP 1 , A, A 1 ) consists of all functions taking values in n m+1 = [b m+1 , b m+1 ] at the point P m+1 ; g(CP 1 , A, A 2 ) consists of all functions taking values in n i = [b i , b i ] at the point P i , i = m + 1, m + 2.
Proposition 4.4. 4
4Let (E, A 2 ) be generic. Then(i) If M A is a generalized Weyl module overĝ A , then dim(M A ) g(CP 1 ,A,A 1 ) < ∞; (ii) If M A is an admissible representation ofĝ A , then dim(M A ) g(CP 1 ,A,A 2 ) < ∞.Proof. of (ii) repeats almost word for word that ofProposition 4.3.3 in 4.3.3 with simplifications analogous to those indicated in the proof of Proposition 4.4.2.
P
m+1 ) in such a way that the bundle obtained is O(n) ⊕ O(n). The 2 cases are of course distinguished by the parity of the difference between the degrees of the determinant of E
Further pushing forward by "integrating along C * " one gets a D−module over the space (C × C) m . Observing that it is appearance of the bundle J (1) (CP 1 ) → CP 1 which was responsible for the twisting of the D−module, one argues that we get a usual holonomic D-module on (C × C) m with fiber M A g(CP 1 ,A)
equations of which mequations are Knizhnik-Zamolodchikov equations and the other m are obtained from singular vectors of the Verma module projecting onto M A .
Along with the algebras g(CP 1 , A), g(CP 1 , A, A 1 ) (see 4.4) introduce the algebraḡ(CP 1 , A, A 1 ) ⊂ g(CP 1 , A) consisting of all functions taking values in b ∞ at the point ∞.Ofcourse g(CP 1 , A, A 1 ) ⊂ḡ(CP 1 , A, A 1 ) is an ideal and dimḡ(CP 1 , A, A 1 )/g(CP 1 , A, A 1 ) = 1. Defineh ∞ to be a basis element of dimḡ(CP 1 , A, A 1 )/g(CP 1 , A, A 1 ).It is a standard (and simple) fact of Lie algebra cohomology theory thath ∞ acts on (M A ) g(CP 1 ,A,A 1 ) .Lemma 4.5.2 Let M A be a generalized Weyl module. The elementh ∞ has a simple spectrum as an operator acting on (M A ) g(CP 1 ,A,A 1 ) . Further, if M A = (V α r 1 , V s 1 )⊗(V β r 2 , V s 2 ) then the set of eigenvalues ofh ∞ is the set of the highest weights of the modules appearing in the right-hand side of Theorem4.5.1(ii).
∈
(M A ) g(CP 1 ,A,A 1 ) be the eigenspace related to the eigenvalue µ ofh ∞ . Then (M A ) µ g(CP 1 ,A,A 1 ) ≈ (M A ⊗ M ∞,b∞ µ,k ) g(CP 1 ,A∪A 1 ) . (ii) Projection of a Verma module M ∞,b∞ µ,k onto a generalized Weyl module W induces an isomorphism of the coinvariants(M A ⊗ M ∞,b∞ µ,k ) g(CP 1 ,A∪A 1 ) ≈ (M A ⊗ W ) g(CP 1 ,A∪A 1 ). algebra of functions on the formal disk whose value at the point belong to the corresponding Borel subalgebra. Therefore (i) follows from Frobenius duality.(ii) Consider the resolution of W by Verma modules (see 2.it with M A . There arises the long exact sequence of homology groups of which we consider the following part:
a Weyl module, the Verma module M does not, see 2.4.1. Lemma4.5.2 and now give that (M A ⊗ M) g(CP 1 ,A∪A 1 ) = {0}. ✷ To complete the proof of Therorem4.5.1 observe that Lemma 4.5.2 and Lemma 4.5.3 together is a reformulation of Therorem4.5.1. ✷ Corollary 4.5.4 Let ♯A = 1 and let A 1 and A 2 be as in 4.4. The following conditions are equivalent (i) M A is a direct sum of generalized Weyl module;
Lemma 4.5.2. First of all replace M A with the corresponding Verma module -M A . Then pass from the space(M A ) g(CP 1 ,A,A 1 ) to its dual, that is to the space of g(CP 1 , A, A 1 )−invariant functionals onM A . Choose h ⊗ (1 − z −1 ) to be a representative ofh ∞ . Let Ψ be the eigenvector of h ⊗ (1 − z −1 ). By definition Ψ is a linear functional on M 0,b 0 λ 0 ,k ⊗ M 1,b 1 λ 1 ,k .It is an excersise on Frobenius duality to show that such a functional exists and unique.Define F to be the following linear functional on M 0,b 0 λ 0 ,k : F (w) = Ψ(w ⊗ v λ 1 ), where, as usual, v λ 1 ) is the vacuum vector of M 1,b 1 λ 1 ,k . As M 0,b 0 λ 0 ,k is Z + × Z + −graded (see 2.2), we denote by F ij the restriction of F to the (i, j)−component. Direct calculations show that with respect to the natural action ofĝ on M 0,b 0 λ 0 ,k :
rational level case Suppose k + 2 = p/q, p and q being positive integers. Now instead of 3 generalized Weyl modules sitting at 3 points in CP 1 we are given 3 admissible representations sitting at 3 points on CP 1 . Recall, see 2.4.2, that admissible representations are parametrized by symbols
any triple of admissible representations the space (
dim(M A ) g(CP 1 ,A,A 2 ) < ∞ if and only if M A is an admissible representation. (ii) Let M A be an admissible representation. The elementsh 0 ,h 1 have simple spectra as operators acting on (M A ) g(CP 1 ,A,A 2 ) . Their eigenvalues recover the structure constants of the fusion algebra.
Corollary 4 .5. 7
47Let ♯A = 1 The following conditions are equivalent (i) M A is a sum of admissible representations;
makes
Rep(osp(1|2)) a tensor category: Rep(osp(1|2)) × Rep(osp(1|2)) → Rep(osp(1|2)), A, B → A ⊗ B, where the osp(1|2)−module structure on A ⊗ B is determined through the comultiplication (and the rule of sign!). Decomposing the tensor product of 2 irreducible modules one gets the Grothendieck ring of Rep(osp(1|2)).
Proposition 4.5.9 A 0 is the Grothendieck ring of the category of finite-dimensional representations of the superalgebra osp(1|2).
Remark 4.5.10 It follows from Lemma 4.5.8 that the functor Rep(osp(1|2)) → Rep(sl 2 ), V α m → V m ⊕ V m−1 induces an epimorhpism of the Grothendieck rings.
(i) V m is irreducible if and only if m < l;(20)(ii) V ǫ m is irreducible if and only if m < l.(21)(Both statements are proved by direct computations.)What is even more important is that the categories Rep(U t sl 2 ) and Rep(U t osp(1|2)) are no longer semisimple. For example, tensor product of 2 irreducible representations is not semisimple. Things, however, are still very much under control.
One defines the Verma module M h,c , where (h, c) is a highest weight, i.e. eigenvalues of L 0 , z resp. determined by the vacuum vector; c is sometimes referred to as level. A Verma module is reducible if and only if it contains a singular vector. M h,c generically has no singular vectors.By the Kac determinant formula, there is a family of hyperbolas labelled by pairs of positive integers m, n in the plane with coordinates (h, c) such that if M h,c contains a singular vector, then (h, c) belongs to one of these hyperbolas; generically along hyperbolas the singular vector is unique. Denote the singular vector arising in M h,c as (h, c) gets on hyperbola with the label m, n by S mn . There arises the V ir-analogue of the generalized Weyl module M h,c / < S mn = 0 >. Attach to M h,c / < S mn = 0 > the symbol (V n−1 , V m−1 ). Further, for c fixed there arises a one-to-one correspondence between the V ir-analogues of generalized Weyl modules and symbols (V n−1 , V m−1 ). This has all been in precise analogy with 2.4.1.
V ir−analogue of admissible representations is the celebrated minimal representations. The latter can be defined as quotients of generalized Weyl modules by repeating word for word definition of admissible representations from 2.4.2. It is known that minimal representations arise only when c = c pq = 1 − 6(p − q) 2 pq , where p, q are relatively prime positive integers. There are again 2 generalized Weyl modules projecting on a given minimal representation. Therefore minimal representations are labelled by equivalence classes of symbols (V m , V n ). It can be shown that the equivalence relation is as follows: (V m , V n ) ≈ (V q−2−m , V p−2−n ) for c = c pq . From this and (22) one can
Proposition 4.5.13 The functor φ e ⊕ φ f determines an epimorphism of theĝ−fusion algebra onto the V ir−fusion algebra at both generic and rational levels.Proof. The generic level case follows from Remark 4.5.10 and formula (22) above. In the rational level case, the statement follows from the fact that both,ĝ− and V ir-, fusion algebras are obtained from their generic level counterparts by imposing the equivalence relations and the 2 equivalence relations agree with each other. ✷
generic. There is a construction which to aĝ A −module associates aĝ B −module. This construction is a natural adjustment of the Kazhdan-Lusztig tensoring [29] to our needs. Denote by g(CP 1 , A, B) the subalgebra of g(CP 1 , A) consisting of functions taking values in n 3 = [b 3 , b 3 ] -just like we did in 4.4.2. For aĝ A −module M A , denote by M A N the subspace of (M A ) * annihilated by g(CP 1 , A, B) N . Obviously M A N ⊂ M A N +1 , N ≥ 1. Set
2.1 rewrites to give a D−module over S with logarithmic singularities at O; call it ∆ψ(M A ). This is because Spec(S) is C[[t]] and vector fields vanishing at q = 0 are exactly those which can be lifted to C S .
In genus ≥ 2 at an appropriate place instead of Proposition 4.4.4 one has to make reference to Proposition 4.3.3 and then use induction. ✷ 4.7.3 Remarks (i) Meaning of Proposition 4.7.1 is that the dimension of the generic fiber of the D−module ∆ ψ (M A ) can be calculated by the usual combinatorial algorithm: by pinching the surface and further inserting inserting all possible representations the problem is reduced to the case of a sphere with three punctures and in the latter case the complete results are available. (ii) In the genus 0 case the analogue of Proposition 4.7.1 for generalized Weyl modules is valid. To see this it is enough to examine part (i) of the proof and convince oneself that the only requirement on M A used there was that M A be generalized Weyl module; in fact at an appropriate place instead of Lemma 4.5.6 one has to use Lemma 4.5.3.
1 .
1looked upon as a certain generating function w(x, z) = x ∆ 1 −n z ∆ 2 −l i,j∈Z w ij x −i z −j of a family of operators {w ij ⊂ Hom C (V b 0 ,0 0 , V b 0 ,0 ∞ ),where (n, l) is a bidegree of w as an element of V bx,z Lemma 5.2.1 Suppose v 1 ∈ V bx,z 1 is the highest weight vector. Then (i) v 1 (x, z) is a generating function of a certain vertex operator as in 5.1;
✷
Let us now relate correlation functions to horizontal sections of the bundle built on the generalized Weyl module M A , ∆(M A ) * , see beginning of sect.5 for notations. Suppose that M A is the tensor product of "individual" generalized Weyl modules
is a generating function of a vertex operator related to the highest weightvector v i ∈ V z i ,bx i i .Corollary 5.2.2 Let M A be as above. Over a suitable open contractible subset U of C m × C m , there is an isomorphism between the space of horizontal sections of the bundle ∆(M A ) * and the space of correlation functions
satisfies the system of equationsD i Ψ(x 1 , . . . , x m , z 1 , . . . z m ) = 0, 1 ≤ i ≤ m.(30)Observe that, although there are in general no explicit formulas for D i , the fact that[D i , D j ] =0 is an obvious consequence of the definition. We have obtained m equations our function of 2m variables satisfies. The rest is, of course, the Knizhnik-Zamolodchikov equations. Let us write them down explicitly. Recall that we can look upon Ψ(x 1 , . . . , x m , z 1 , . . . z m ) as a function of z 1 , . . . z m with coefficients in a completed tensor product of m g−modules. (The variables x 1 . . . x m are responsible
the element d/dz − L −1 , where L −1 is one of the Sugawara elements. One then shows thatthe singular vectors (d/dz −L −1 )v i , where v is a highest weight vector of V i ,give rise to the equations (31) in exactly the same way the singular vectors S i gave rise to the equations (30). An immediate consequence of this proof is that the system of equations (31,Suppose a function Ψ old = Ψ(x 1 , . . . , x m , z 1 , . . . z m ) is the matrix element of the product of vertex operators
Lemma 5 .4. 1
51Functions(38) are solutions to(30,31).
Conjecture 5.4.2 (i) Formulas(38) provide all solutions to the system(30,31).(ii) If the level k is rational, then there arises a subbundle of the bundle in question, the one with fiber ((L A ) * ) g(CP 1 ,A , where L A is the corresponding admissible representation.
Choose a basis e, h, f of g satisfying the standard relations [h, e] = 2e, [h, f ] = −2f, [e, f ] = h. We say that
). Outside the critical level (k = −2) a Verma module is generically irreducible. M λ,k happens to be reducible if and only if it contains a singular vector. If M λ,k is reducible then the following 2 cases arise:
, the module M is irreducible and does not project on any generalized Weyl module. Note that if the composition series of aĝ−module only consist of generalized Weyl modules then this module breaks into a direct sum of its components. (This can be proved by methods of
). The case l = 0 is treated in a similar and simpler way using Lemma 5.2.3 (i). Further argue by induction using again Lemma 5.2.3. ✷ By definition each V i is a quotient of a Verma module and therefore there are elements, singular vectors in the corresponding Verma module (see 2.2
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. V Schechtman, A Varchenko, Invent.Math. 106Schechtman V., Varchenko A., Invent.Math. 106(1991), 139-194
. A Tsuchiya, K Ueno, Y Yamada, Adv. Studies in Pure Math. 19Tsuchiya A., Ueno K., Yamada Y., Adv. Studies in Pure Math. 19 (1989) 459-566
. E Verlinde, Nucl.Phys. 300360Verlinde E. Nucl.Phys. B300 (1988) 360
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[
"THE WEDDERBURN B-DECOMPOSITION FOR ALTERNATIVE BARIC ALGEBRAS",
"THE WEDDERBURN B-DECOMPOSITION FOR ALTERNATIVE BARIC ALGEBRAS"
] |
[
"Bruno L M Ferreira [email protected] \nTechnological Federal University of Paraná\nProfessora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil\n",
"Ruth Nascimento [email protected] \nTechnological Federal University of Paraná\nProfessora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil\n"
] |
[
"Technological Federal University of Paraná\nProfessora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil",
"Technological Federal University of Paraná\nProfessora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil"
] |
[] |
In this paper we deal with the Wedderburn b-decomposition for alternative baric algebras.
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"https://arxiv.org/pdf/1410.7078v1.pdf"
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1410.7078
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09b576bfe6fea12b5d141d04865302f05307e454
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THE WEDDERBURN B-DECOMPOSITION FOR ALTERNATIVE BARIC ALGEBRAS
26 Oct 2014
Bruno L M Ferreira [email protected]
Technological Federal University of Paraná
Professora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil
Ruth Nascimento [email protected]
Technological Federal University of Paraná
Professora Laura Pacheco Bastos Avenue, 80085053-510GuarapuavaBrazil
THE WEDDERBURN B-DECOMPOSITION FOR ALTERNATIVE BARIC ALGEBRAS
26 Oct 2014Alternative algebrasBaric algebrasWedderburn b-decomposition AMS Subject Classification: 17D0517D9217A65
In this paper we deal with the Wedderburn b-decomposition for alternative baric algebras.
Baric algebras
Baric algebras play a central role in the theory of genetic algebras. They were introduced by I. M. H. Etherington, in [1], in order to give an algebraic treatment to Genetic Populations. Several classes of baric algebras have been defined, such as: train, Bernstein, special triangular, etc.
In this paper F is a field of characteristic = 2, 3, 5. Let U be an algebra over F not necessarily associative, commutative or finite dimensional. If ω : U −→ F is a nonzero homomorphism of algebras, then the ordered pair (U, ω) will be called a baric algebra or b-algebra over F and ω its weight function or simply its weight. For x ∈ U, ω(x) is called weight of x.
When B is a subalgebra of U and B ⊂ kerω, then B is called a bsubalgebra of (U, ω). In this case, (B, ω B ) is a b-algebra, where ω B = ω| B : B −→ F. The subset bar(B) = {x ∈ B | ω(x) = 0} is a two-side ideal of B of codimension 1, called bar ideal of B. For all b ∈ B with ω(b) = 0, we have B = F b ⊕ bar(B). If bar(B) is a two-side ideal of bar(U ) (then by [2,Proposition 1.1], it is also a two-sided ideal of U ), then B is called normal b-subalgebra of (U, ω). If I ⊆ bar(B) is a two-side ideal of B, then I is called b-ideal of B.
Let (U, ω) be a b-algebra. A subset B is called maximal (normal) bsubalgebra of U if B is a (normal) b-subalgebra of U and there is no (normal) b-subalgebra C of U such that B ⊂ C ⊂ U. A subset I is called maximal b-ideal of U if I is a b-ideal of U, I = bar(U ) and there is no b-ideal J of U such that I ⊂ J ⊂ bar(U ).
A nonzero element e ∈ U is called an idempotent if e 2 = e and nontrivial idempotent if it is an idempotent different from multiplicative identity element. If (U, ω) is a b-algebra and e ∈ U is an idempotent, then ω(e) = 0 or ω(e) = 1. When ω(e) = 1, then e is called idempotent of weight 1. Let (U, ω) be a b-algebra. We define the bar-radical or b-radical of U, denoted by rad(U ), as:
rad(U ) = (0), if (U, ω) is b-simple, otherwise as rad(U ) = bar(B), where B runs over the maximal normal b-subalgebra of U. Of course, rad(U ) is a b-ideal of U.
We say that U is b-semisimple if rad(U ) = (0).
Alternative algebras
In this section, we present some definitions and properties of alternative algebras and prove some results which will be used later.
An algebra U over a field F is called alternative algebra if it satisfies the identities:
(x, x, y) = (y, x, x) = 0,(1)
for all x, y ∈ U , where the (x, y, z) = (xy)z − x(yz) is the associator of the elements x, y, z. Let U be an alternative algebra over F . Then, U is a power-associative algebra and if U has an idempotent e, then U is the vector space direct sum U = U 11 ⊕ U 10 ⊕ U 01 ⊕ U 00 , where U ij = {x ij ∈ U | ex ij = ix ij and x ij e = jx ij } (i, j = 0, 1) satisfying the multiplicative relations U ij U jl ⊂ U il , U ij U ij ⊂ U ji and U ij U kl = 0, j = k, (i, j, l = 0, 1), see [2].
A set of idempotents {e 1 , . . . , e t }, in an (arbitrary) alternative algebra, is called pairwise orthogonal in case e i e j = 0 for i = j. Note that any sum e = e 1 + · · · + e t , of pairwise orthogonal idempotents (t ≥ 1), is an idempotent. Also, ee i = e i e = e i , (i = 1, . . . , t).
A more refined Peirce decomposition for an alternative algebra than the one given above is the following decomposition relative to a set {e 1 , . . . , e t }, of pairwise orthogonal idempotents in U : U is the vector space direct sum
U = i,j U ij (i, j = 0, 1, . . . , t), where U ij = {x ij ∈ U | e k x ij = δ ki x ij and x ij e k = δ jk x ij for (k = 1, . . . , t)} (i, j = 0, 1, . . . , t)
, satisfying the multiplicative relations:
U ij U jl ⊂ U il (i, j, l = 0, 1, . . . , t),(2)U ij U ij ⊂ U ji (i, j = 0, 1, . . . , t),(3)U ij U kl = 0 j = k, (i, j) = (k, l) (i, j, k, l = 0, 1, . . . , t),(4)
where δ jk (j, k = 0, 1, . . . , t) is the Kronecker delta. An nonzero ideal I of an alternative algebra U is called minimal if for any ideal of U such that (0) ⊂ J ⊂ I, then J = (0) or J = I.
Let U be a finite dimensional alternative algebra over F , since U is a power-associative algebra, then by [2] U has a unique maximal nilideal, we define nilradical R(U ) of U as the maximal nil ideal of U . Let us say that U is simple when its only ideals are the trivial ideals and U is not a zero algebra. If R(U ) = 0, then U is called semisimple.
Lemma 2.1. Let U be a finite dimensional alternative algebra over F with a non trivial idempotent e. If U = i,j U ij (i, j = 0, 1), relative to e, then
R(U ii ) = R(U ) ∩ U ii (i = 0, 1).
Proof. See [2, Corollary 3.8] .
Proposition 2.1. Let U be a finite dimensional alternative algebra. If I is a minimal ideal of U , then either I 2 = 0 or I is simple.
Baric alternative algebra
In this section, we introduce a notion of Wedderburn b-decomposition of a b-alternative algebra and we present conditions for which it has such decomposition.
If (U, ω) is a b-algebra and I is a b-ideal of U , then (U/I,ω) is a balgebra, whereω(u + I) = ω(u). Definition 3.1. Let (U, ω) be b-alternative algebra over a field F . We say that U has a Wedderburn b-decomposition if we can decompose U as a direct sum U = S⊕V ⊕rad(U ) (vector space direct sum), where S is a b-semisimple b-subalgebra of U and V is a vector subspace of bar(U ) such that V 2 ⊂ rad(U ). Proof. Let us consider the quotient algebra U/I = {x | x ∈ U } and the application ω : U/I → F defined by ω(x) = ω(x), for all x ∈ U . Then ω is a nonzero algebra homomorphism and therefore (U/I, ω) is a b-algebra such that U/I = F 1 ⊕ bar U/I), where bar U/I) = bar(U )/I. Next, since u 1 is an idempotent of bar U/I , then any representative u 1 of u 1 is non nilpotent and belongs to bar(U ). It follows that the subalgebra generated by the element u 1 is a non nil subalgebra of bar(U ). This implies that bar(U ) has an idempotent
e 1 = i α i u i 1 , α i ∈ F, verifying e 1 = αu 1 , α ∈ F, α = i α i . Since e 1 / ∈ R(U )
, then e 1 / ∈ I and it follows that e 1 = 0 and e 1 = αe 1 . Hence α = 1 and e 1 = u 1 . Moreover, if bar(U ) is an algebra with a multiplicative unity f and f = u 1 , then f = e 1 which implies
f − e 1 ∈ I. Since (f − e 1 ) 2 = f − e 1 , then f = e 1 .
Lemma 3.2. Let U be a finite dimensional b-alternative algebra over F with unity element 1 and J a b-ideal of U such that J ⊂ R(U ). If {u 1 , . . . , u t } is a set of nonzero pairwise orthogonal idempotents of bar U/J , then there are a set of nonzero pairwise orthogonal idempotents {e 1 , . . . , e t } of bar(U ) verifying e i = u i (i = 1, . . . , t). Moreover, if e is any idempotent of bar(U ) such that e = t i=1 u i , we may choose by e i such that e = t i=1 e i . Proof. To prove this lemma we use the principle of mathematical induction. For t = 1, the result is true, by Lemma 3.1. Now, suppose that for a positive integer t ≥ 1, the lemma is true. Then for the set of nonzero pairwise orthogonal idempotents {u 1 , . . . , u t+1 } of bar U/J , there is a set of nonzero pairwise orthogonal idempotents {e 1 , . . . , e t }, of bar(U ), verifying e i = u i (i = 1, . . . , t), by the principle of mathematical induction. Let us consider the Peirce decompositions:
bar(U ) = i,j bar(U ) ij and bar U/J = i,j bar U/J ij (i, j = 0, 1),
relative to idempotents e = t i=1 e i and e = t i=1 e i , respectively. It follows that:
(i) e i ∈ bar(U ) 11 (i = 1, . . . , t);
(ii) e i ∈ bar U/J 11 (i = 1, . . . , t);
(iii) u t+1 ∈ bar U/J 00 .
Let us define P = F 1 ⊕ bar(U ) 00 . Then (P, ω P ) is a finite dimensional b-subalgebra of (U, ω) with unity element 1 and bar(P ) = bar(U ) 00 , where
ω P := ω| P . Let us define K = J ∩ bar(U ) 00 . It is easy to check that K is a b-ideal of P such that K ⊂ R(P ), because R(P ) = R bar(P ) = R bar(U ) 00 , by [6, Proposition 4.1] and the fact that R bar(U ) 00 = R bar(U ) ∩ bar(U ) 00 = R(U ) ∩ bar(U ) 00 , according to Lemma 2.1, where R bar(U ) = R(U ), again by [6, Proposition 4.1].
Let us consider the quotient algebra
P/K = { x | x ∈ P } and ω : P/K → F , defined by ω P ( x) := ω P (x) for all x ∈ P. Certainly, (P/K, ω P ) is a b- algebra such that bar(P/K) = bar(P )/K = bar(U ) 00 / J ∩ bar(U ) 00 .
Also observe that bar U/J 00 = bar(U ) 00 + J /J ∼ = bar(U ) 00 / J ∩ bar(U ) 00 .
Since u t+1 ∈ bar U/J 00 , we can assume that u t+1 ∈ bar(U ) 00 . In fact, let us write u t+1 = a 11 + a 10 + a 01 + a 00 , where a ij ∈ bar(U ) ij (i, j = 0, 1). Then u t+1 = a 11 + a 10 + a 01 + a 00 which implies 0 = e u t+1 = a 11 + a 10 and 0 = u t+1 e = a 11 + a 01 . Thus, a 11 = a 10 = a 01 = 0 implying u t+1 = a 00 .
From the isomorphism, in (5), and the assumption on the element idempotent u t+1 , we have u t+1 ∈ bar(P/K) which implies that there is an idempotent e t+1 ∈ bar(P ) such that e t+1 = u t+1 , by Lemma 3.1. From the isomorphism, in (5), we conclude that e t+1 ∈ bar(U ) and e t+1 = u t+1 . Since e t+1 ∈ bar(U ) 00 , then the elements idempotent e 1 , . . . , e t , e t+1 are pairwise orthogonal.
Finally, suppose that e is an arbitrary idempotent of bar(U ) such that e = t i=1 u i . Let us consider the Peirce decompositions bar(U ) = i,j bar(U ) ij and bar U/J = i,j bar U/J ij (i, j = 0, 1), relative to idempotents e and e, respectively. It follows that u i ∈ bar U/J 11 (i = 1, . . . , t).
Let us define the vector subspace of Q = F 1 ⊕ bar(U ) 11 of U . Naturally, Q is an subalgebra of U such that Q ⊂ ker(ω). It follows that (Q, ω Q ), where ω Q := ω| Q , is a finite dimensional b-subalgebra of U with unity element 1 and bar(Q) = bar(U ) 11 . Let us define L = J ∩ bar(U ) 11 . As in the
previous definitions, certainly L is a b-ideal of Q such that L ⊂ R(Q), because R(Q) = R bar(Q) = R bar(U ) 11 ,
Since u i ∈ bar U/J 11 and u i = e u i e (i = 1, . . . , t), we can take a representative u i of u i in bar(U ) 11 (i = 1, . . . , t).
From the isomorphism, in (6), and the assumption on the element idempotent u i , we have that u i ∈ bar(Q/L) which implies that there is a set of idempotents e 1 , . . . , e t , in bar(Q), pairwise orthogonal, such that e i = u i (i = 1, . . . , t).
As the idempotent e is a multiplicative unity in the subalgebra bar(U ) 11 . . , f tt }, respectively. It follows that: (i) f ii ∈ bar(U ) ii (i = 1, . . . , t); and (ii) f ii ∈ bar U/J ii (i = 1, . . . , t).
and e = t i=1 e i , then e − t i=1 e i ∈ J ⊂ R(U ) and e − t i=1 e i 2 = e − t i=1 e i which implies e = t i=1 e i .
Now, let us observe that for every index i = 2, . . . , t, we can take the representative u i1 , of u i1 ∈ M t , in bar(U ) i1 . For a i = 1, let us take u 11 = f 11 . Similarly, for every index j = 2, . . . , t, we can take the representative u 1j , of u 1j ∈ M, in bar(U ) 1j .
Yet, since u 1j u j1 = f 11 (j = 1, . . . , t), then u 1j u j1 = f 11 + a j , where a j ∈ J ∩ bar(U ) 11 is a nilpotent element. Let us consider m the smallest positive integer such that a m j = 0 and let us define b j = m−1 i=1 (−a j ) i . Then:
(i) b j ∈ J ∩ bar(U ) 11 ; (ii) b j a j = − m−1 i=2 (−a j ) i ; and (iii) a j + b j + b j a j = 0. It follows that (f 11 + b j )(f 11 + a j ) = f 11 + a j + b j + b j a j = f 11 .
Let us define f i1 = u i1 and f 1j = (f 11 + b j )u 1j (i, j = 2, . . . , t).
Then
f 1j f j1 = (f 11 + b j )u 1j u j1 = (f 11 u 1j )u j1 + (b j u 1j )u j1 = f 11 (u 1j u j1 ) + b j (u 1j u j1 ) = f 11 (f 11 +a j )+b j (f 11 +a j ) = f 11 . Next, let us define f ij = f i1 f 1j (i = j; i, j = 2, . . . , t)
. From a direct calculus, we have f ij = u ij and f ij f kl = δ jk f il (i, j, k, l = 1, . . . , t). Thus, the set {f ij | i, j = 1, . . . , t} is a basis for a total matrix algebra M of degree t, in bar(U ), with identity element f such that M = M t . Lemma 3.4. Let (U, ω) be a finite dimensional b−alternative algebra with unity element 1 and J a b-ideal of U such that J ⊂ R(U ). If bar(U/J) contains a direct sum of b-ideals M t 1 ⊕ · · · ⊕ M ts , where each M t i is a total matrix algebra of degree t i (i = 1, . . . , s), then bar(U ) contains a direct sum of pairwise orthogonal subalgebras M 1 ⊕ · · · ⊕ M s , where each M i is a total matrix algebra of degree t i (i = 1, . . . , s), such that M i = M t i and
M 1 ⊕ · · · ⊕ M s ∼ = M t 1 ⊕ · · · ⊕ M ts .
Proof. Let e t i be the unity element of M t i (i = 1, . . . , s). By Lemma 3.2, bar(U ) has a set of idempotents e 1 , . . . , e s , pairwise orthogonal, such that e i = e t i (i = 1, . . . , s). This implies that bar(U ) contains a total matrix algebra M i of degree t i with identity element e i such that M i = M t i , by Lemma 3.3.
Let us consider the Peirce decomposition bar(U ) = i,j bar(U ) ij (i, j = 1, . . . , s), relative to set of idempotents {e 1 , . . . , e s }. For all element x i ∈ M i (i = 1, . . . , s), we have x i = e i x i . But in an alternative algebra each associator (x, e j , e l ) = 0 and (e j , e l , x) = 0 (j, l = 1, · · · , s), which implies 1, . . . , s). Since the subalgebras bar(U ) ii (i = 1, . . . , s) are pairwise orthogonal, then the sum M 1 ⊕ · · · ⊕ M s , is a direct sum, pairwise orthogonal, such that M 1 ⊕ · · · ⊕ M s ∼ = M t 1 ⊕ · · · ⊕ M ts . Lemma 3.5. Let (U, ω) be a finite dimensional b−alternative algebra with unity element 1 and J a b-ideal of U such that J ⊂ R(U ). If bar(U/J) contains a direct sum of b-ideals J 1 ⊕ · · · ⊕ J r such that J 2 i = 0 (i = 1, . . . , r), then bar(U ) contains a vector subspace V such that V ∼ = V = J 1 ⊕ · · · ⊕ J r and V 2 ⊂ rad(U ).
e k x i = e k (e i x i ) = (e k e i )x i = δ ki x i . Similarly, we show x i e k = δ ik x i . Thus, M i ⊂ bar(U ) ii (i =
Proof. Let v i,1 , v i,2 , . . . , v i,r i be a basis of vector subspace J i (i = 1, . . . , r) and v i,j a representative of the class v i,j (i = 1, . . . , r; j = 1, . . . , r i ), in bar(U ). From a direct calculus, we have: k = 1, . . . , r) and (j = 1, . . . , r i ; l = 1, . . . , r k ).
(i) the set r i=1 v i,1 , v i,2 , . . . , v i,r i is linearly independent; and (ii) v i,j v k,l ∈ rad(U ) (i,
Let us define V the vector subspace generated by the set
r i=1 v i,1 , v i,2 , . . . , v i,r i .
It follows that V ∼ = V = J 1 ⊕ · · · ⊕ J r and V 2 ⊂ rad(U ).
Lemma 3.6. Let (U, ω) be a finite dimensional b-alternative algebra with unity element 1 and J a b-ideal of U such that J 2 = 0. If bar(U/J) contains a direct sum of b-ideals I 1 ⊕ · · · ⊕ I r such that I i is a split Cayley algebra (i = 1, . . . , r), then bar(U ) contains a subalgebra C ∼ = I 1 ⊕ · · · ⊕ I r .
Proof. We may take I k = F 2 + w k F 2 , w k 2 = 1 by [2, Lemma 3.16] where F 2 is the algebra of all 2 × 2 matrices over F , k = 1, · · · , r. By Lemma 3.4, bar(U ) contains a total matrix algebra D ∼ = F 2 such that D contains an identity element and the matric basis {e ij } of D yields the matric basis
{e ij } of F 2 . Let ι : D −→ D the involution in D. We know x + ι(x) = t(x)1
for all x ∈ D, where t(x) is the trace of x and 1 is the identity element of D by [2, page 45]. Note that ι(x) = ι(x), a(w k b) = w k (ι(a)b), (w k a)b = w k (ba) and (w k a)(w k b) = bι(a) for x, a, b ∈ D, we have indicated to the reader [2, Chapter III, Sec. 4] for Cayley Algebras. In order to prove the lemma, it is sufficient to show the existence of v k / ∈ D satisfying v 2 k = 1, v k = w k and xv k = v k ι(x) for all x ∈ D.
Consider f ij = w k e jj for i = j (i, j = 1, 2). Using the Peirce decomposition of bar(U ) relative to e 1 = e 11 , e 2 = e 22 , we may take f ij ∈ bar(U ) ij (i = j). In fact e ii (f ij e jj ) = e ii (w k e jj 2 ) = w k ι(e ii )e jj = w k e jj = f ij . Now e ji f ij = e ji (w k e ij ) = −w k (e ji e jj ) = 0, implying that
e ji f ij = c j , c j ∈ J ∩ bar(U ) jj (i = j; i, j = 1, 2).
Write h ij = f ij − e ij c j . Then h ij ∈ bar(U ) ij , h ij = f ij , and e ji h ij = h ij e ji = 0 (i = j; i, j = 1, 2).
In fact by Lemma 3.3 we know e ji e ij = e jj , so e ji h ij = c j − e ji (e ij c j ) = c j − (e ji e ij )c j = 0. Also e ij c j = e ij (e ji f ij ) = (e ij e ji )f ij − (e ij , e ji , f ij ) =
f ij + (f ij , e ji , e ij ) = f ij + (f ij e ji )e ij − f ij = (f ij e ji )e ij , so that h ij e ji = f ij e ji − [(f ij e ji )e ij ]e ji = 0. Now h ij h ji = f ij f ji = e ii ι(e jj ) = e ii implies that h ij h ji = e ii + a i , a i ∈ J ∩ bar(U ) ii (i = j; i, j = 1, 2).
Then a 2 i = 0 since J 2 = 0, and
(e ii − a i )(e ii + a i ) = e ii = (e ii + a i )(e ii − a i ) (i = 1, 2).
Write p 12 = (e 11 − a 1 )h 12 , p 21 = h 21 . Then p ij ∈ bar(U ) ij , p ij = f ij , and we shall prove p ij p ji = e ii (i = j; i, j = 1, 2).
In fact,
p 12 p 21 = [(e 11 − a 1 )h 12 ]h 21 = (e 11 − a 1 )(h 12 h 21 ) = (e 11 − a 1 )(e 11 + a 1 ) = e 11 .
But Proof. The same inductive argument based on the dimension of U which is used for associative algebras suffices to reduce the proof of the theorem to the case rad(U ) 2 = 0. Let us take the quotient b-algebra U/rad(U ). By [5,Corollary 3.1], we have rad U/rad(U ) = 0 which implies that U/rad(U ) is b-semisimple, by [5,Theorem 4.2]. So, bar U/rad(U ) is a sum of minimal b-ideals I 1 ⊕ · · · ⊕ I k ⊕ I k+1 ⊕ · · · ⊕ I s ⊕ J s+1 ⊕ · · · ⊕ J r , of U/rad(U ), where I i are simple associative algebras (i = 1, . . . , k), I i are Cayley algebras (i = k + 1, . . . , s) and J 2 j = 0 (s + 1 ≤ j ≤ r), by Proposition 2.1 and [3, Corollary 1, page 151]. Since every ideal I i (i = 1, . . . , k) is a total matrix algebra M t i of degree t i (i = 1, . . . , k), by [4, Corollary b, §3.5], then bar(U ) contains a direct sum of pairwise orthogonal total matrix algebras M i of degree t i (i = 1, . . . , k) such that M 1 ⊕ · · · ⊕ M k ∼ = M t 1 ⊕ · · · ⊕ M t k , by Lemma 3.4. On the other hand, bar(U ) contains a vector subspace V such that V ∼ = J s+1 ⊕ · · · ⊕ J r and V 2 ⊂ rad(U ), by Lemma 3.5. Moreover bar(U ) contains also a subalgebra C ∼ = I 1 ⊕ · · · ⊕ I k , by Lemma 3.6.
a i h ij = (h ij h ji − e ii )h ij = h ij (h ji h ij ) − h ij = h ij (e jj + a j ) − h ij = h ij a j ,
Let us define S = F 1⊕M 1 ⊕· · ·⊕M k ⊕C. Certainly, S is a b-subalgebra of U such that bar(S) is a semisimple algebra and which yields S semisimple, by [
⊕ · · · ⊕ M k ) ∩ V = (0) and C ∩ (M 1 + · · · + M k + V + rad(U )) = (0), then U = S ⊕ V ⊕ rad(U ).
Finally, let us show that the subspace V ⊕rad(U ) is a nil ideal of bar(U ). In fact, for arbitrary elements x ∈ bar(U ) and y ∈ V ⊕ rad(U ), we have x = k i=1 a i + s i=k+1 a i + r j=s+1 b j , where a i ∈ I i (i = 1, . . . , s) and b j ∈ J j (j = s+1, . . . , r), and y = r j=s+1 c j , where c j ∈ J j (j = s+1, . . . , r). Hence xy = x y ∈ J s+1 ⊕ · · · ⊕ J r which implies xy ∈ V ⊕ rad(U ). Similarly, we prove yx ∈ V ⊕ rad(U ). Thus, V ⊕ rad(U ) is an ideal of bar(U ). Since y 2 ∈ rad(U ), then y is a nilpotent element and therefore we can conclude that V ⊕ rad(U ) is a nil ideal of bar(U ). Proof. Let us consider a principal idempotent e and take U = U 11 ⊕ U 10 ⊕ U 01 ⊕ U 00 , the Peirce decomposition of U , relative to e. We know that: (i) U 11 is a subalgebra with unity element e; (ii) U 10 ⊕ U 01 ⊕ U 00 ⊂ R(U ) and (iii) R(U ) = R U 11 ⊕ U 10 ⊕ U 01 ⊕ U 00 . Moreover, as the idempotent e is principal in U , then it has weight one. This implies that U 11 is a b-subalgebra of U . Thus, U and U 11 admit the decompositions U = F e ⊕ bar(U ) and U 11 = F e ⊕ bar U 11 , respectively. From Theorem 3.1, we can decompose U 11 as a direct sum U 11 = S ⊕ W 11 ⊕ rad U 11 , where S is a b-semisimple b-subalgebra of U 11 such that bar(S) is a semisimple algebra, W 11 is a vector subspace of bar U 11 satisfying W 2 11 ⊂ rad U 11 and W 11 ⊕ rad U 11 is a nil ideal of bar U 11 . It follows that, S is a bsemisimple b-subalgebra of U , by [6, Proposition 4.1. and Lemma 4.1.]. Now, let us observe that rad(U ) U 11 ⊂ R(U ) U 11 = R U 11 ⊂ bar U 11 , by [6, Teorema 4.1.] and Lemma 2.1, and bar U 11 = bar S ⊕ W 11 ⊕ rad U 11 . Hence, rad(U ) U 11 ⊂ W 11 ⊕ rad U 11 , because S R U 11 = {0}. Let us take V 11 an complementary subspace of rad(U ) U 11 , in W 11 ⊕ rad U 11 . Then W 11 ⊕ rad U 11 = V 11 ⊕ rad(U ) U 11 . Since rad U 11 = bar U 11 2 R U 11 ⊂ bar(U ) 2 R(U ) = rad(U ), by [6,Theorem 4.2.], then V 2 11 ⊂ rad(U ). Thus U 11 = S ⊕ V 11 ⊕ rad(U ) U 11 , where V 2 11 ⊂ rad(U ). Next, let us consider the complementary subspaces V 10 , V 01 and V 00 , in U 10 , U 01 and U 00 , respectively, such that U 10 = V 10 ⊕ rad(U ) ∩ U 10 , U 01 = V 01 ⊕ rad(U ) ∩ U 01 and U 00 = V 00 ⊕ rad(U ) ∩ U 00 and take the vector subspace V = V 11 ⊕ V 10 ⊕ V 01 ⊕ V 00 . Certainly, V is a vector subspace of bar(U ) and U = U 11 ⊕ U 10 ⊕ U 01 ⊕ U 00 = S ⊕ V 11 ⊕ rad(U ) U 11 ⊕ U 10 ⊕ U 01 ⊕ U 00 = S ⊕ V 11 ⊕ rad(U ) U 11 ⊕ V 10 ⊕ rad(U ) U 10 ⊕V 01 ⊕ rad(U ) U 01 ⊕ V 00 ⊕ rad(U ) U 00 = S ⊕ V ⊕ rad(U ). Now, V 11 and V 10 are vector subspaces of bar(U ) and R(U ) respectively which implies V 11 V 10 ⊂ bar(U ) 2 and V 11 V 10 ⊂ R(U ). This yields
V 11 V 10 ⊂ R(U ) bar(U ) 2 = rad(U ),
by [6,Teorema 4.2]. Similarly, we can show that the products V 10 V 01 , V 10 V 00 , V 01 V 11 , V 01 V 10 , V 00 V 01 and V 2 00 are subsets of rad(U ). As all remaining products are zeros, then we can conclude that V 2 ⊂ rad(U ).
Final remarks
Importantly, the concept of heredity to b-algebras can not be extended to alternative algebras, we can see this through an example found in [7].
A b-algebra (U, ω) is called b-simple if for all normal b-subalgebra B of U, bar(B) = (0) or bar(B) = bar(U ). When (U, ω) has an idempotent of weight 1, then (U, ω) is b-simple if, and only if, its only b-ideals are (0) and bar(U ).
Proof. [ 3 ,
3Chap. VIII, Theorem 10].
Lemma 3 . 1 .
31Let U be a finite dimensional b-alternative algebra over F with unity element 1 and I a b-ideal of U such that I ⊂ R(U ). If u 1 is a nonzero idempotent of bar U/I , then there is an idempotent e 1 in bar(U ) such that e 1 = u 1 . Moreover, if bar(U ) is an algebra with a unity f and f = u 1 , then f = e 1 .
by [ 6 ,
6Proposition 4.1] and the fact that R bar(U ) 11 = R bar(U ) ∩bar(U ) 11 = R(U )∩bar(U ) 11 , according to Lemma 2.1, where R bar(U ) = R(U ), again by [6, Proposition 4.1]. Let us take the quotient algebra Q/L = { x | x ∈ Q} and ω : Q/L → F , defined by ω Q ( x) := ω Q (x) for all x ∈ Q. Again, we have that (Q/L, ω Q ) is a b-algebra such that bar(Q/L) = bar(Q)/L = bar(U ) 11 / J ∩ bar(U ) 11 . Now, let us observe that bar U/J 11 = bar(U ) 11 + J /J ∼ = bar(U ) 11 / J ∩ bar(U ) 11 .
Lemma 3. 3 .
3Let (U, ω) be a finite dimensional b-algebra of (γ, δ) type with unity element 1 and J a b-ideal of U such that J ⊂ R(U ). If bar(U/J) contains a total matrix algebra M t of degree t with identity element u and f is an idempotent of bar(U ) such that f = u, then bar(U ) contains a total matrix algebra M of degree t with identity element f such that M = M t .Proof. Let M t be a total matrix algebra M t of degree t with identity element u. By hypothesis we have M t = {u ij | i, j = 1, . . . , t}, with the familiar multiplication table u ij u kl = δ jk u il (i, j = 1, . . . , t).By Lemma 3.2, there exist pairwise orthogonal idempotents f 11 , . . . , f tt , in bar(U ), such that f ii = u ii (i = 1, . . . , t) and f = t i=1 f ii . Let us consider the Peirce decompositions bar(U ) = i,j bar(U ) ij and bar U/J = i,j bar U/J ij (i, j = 0, 1, . . . , t), relative to the sets of idempotents {f 11 , . . . , f tt } and {f 11 , .
so that p 12 = h 12 − a 1 h 12 = h 12 − h 12 a 2 = h 12 (e 22 − a 2 ) and p 21 p 12 = h 21 [h 12 (e 22 −a 2 )] = (h 21 h 12 )(e 22 −a 2 ) = e 22 . Also we have e ij p ji = p ji e ij = 0 (i = j; i, j = 1, 2). Thus write v k = p 12 + p 21 . Then v k = f 12 + f 21 = w k , implying v k / ∈ D. Also v 2 k = (p 12 + p 21 ) 2 = e 11 + e 22 = 1.Writing b = α 1 e 11 + α 2 e 12 + α 3 e 21 + α 4 e 22 , we have ι(b) = α 4 e 11 − α 2 e 12 − α 3 e 21 + α 1 e 22 , bv k = α 1 p 12 + α 2 e 12 p 12 + α 3 e 21 p 21 + α 4 p 21 = α 1 p 12 − α 2 p 12 e 12 − α 3 p 21 e 21 + α 4 p 21 = v k ι(b)which completes the proof of the lemma.
Theorem 3. 1 .
1Let F be an algebraically closed field and (U, ω) be a finite dimensional b-alternative algebra over F with unity element 1. Then U has a Wedderburn b-decomposition U = S ⊕ V ⊕ rad(U ). Furthermore, bar(S) is a semisimple algebra and V ⊕ rad(U ) is a nil ideal of bar(U ).
Theorem 3.2. (Main) Let F be an algebraically closed field and (U, ω) a finite dimensional b−alternative algebra over F . Then U has a Wedderburn b-decomposition.
6, Proposition 4.1.]. Hence, S is a b-semisimple, by [6, Lemma 4.1.]. Moreover, since (M 1
On non-associative combinations. I M H Etherington, Proc. Roy. Soc. Edinburgh. 59I. M. H. Etherington, On non-associative combinations, Proc. Roy. Soc. Edinburgh, 59 (1939) 153-162.
An introduction to nonassociative algebras. R D Schafer, Academic PressNew YorkR. D. Schafer, An introduction to nonassociative algebras, Academic Press, New York, 1966.
K A Zhevlakov, A M Slin'ko, I P Shestakov, A I Shirshov, Rings that are nearly associative. New YorkAcademic PressK. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov and A. I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982.
Associative Algebras, Associative algebras. R S Pierce, Graduate Texts in Mathematics. 88Springer-VerlagR. S. Pierce, Associative Algebras, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York (1982).
The bar-radical of baric algebras. H GuzzoJr, Arch. Math. 67H. Guzzo Jr, The bar-radical of baric algebras, Arch. Math., 67 (1996) 106-118.
The radical in alternative baric algebras. M A Couto, H GuzzoJr, Arch. Math. 75M. A. Couto and H. Guzzo Jr, The radical in alternative baric algebras, Arch. Math., 75 (2000) 178-187.
Aditividade de aplicações e b−decomposição de Wedderburn. B L M Ferreira, 87Universidade de São PauloInstituto de Matemática e Estatística. TeseB.L.M. Ferreira, Aditividade de aplicações e b−decomposição de Wedderburn, Instituto de Matemática e Estatística, Universidade de São Paulo, (2013) 87f. Tese
|
[] |
[
"A 52 hours VLT/FORS2 spectrum of a bright z ∼ 7 HUDF galaxy: no Lyα emission",
"A 52 hours VLT/FORS2 spectrum of a bright z ∼ 7 HUDF galaxy: no Lyα emission"
] |
[
"E Vanzella \nINAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1I-40127BolognaItaly\n",
"A Fontana \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly\n",
"L Pentericci \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly\n",
"M Castellano \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly\n",
"A Grazian \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly\n",
"M Giavalisco \nDepartment of Astronomy\nUniversity of Massachusetts\n710 North Pleasant Street01003AmherstMAUSA\n",
"M Nonino \nINAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34131TriesteItaly\n",
"S Cristiani \nINAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34131TriesteItaly\n",
"G Zamorani \nINAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1I-40127BolognaItaly\n",
"C Vignali \nDipartimento di Fisica e Astronomia\nUniversita' degli Studi di Bologna\nViale Berti-Pichat 6/240127BolognaItaly\n"
] |
[
"INAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1I-40127BolognaItaly",
"INAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly",
"INAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly",
"INAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly",
"INAF -Osservatorio Astronomico di Roma\nvia Frascati 3300040MonteporzioItaly",
"Department of Astronomy\nUniversity of Massachusetts\n710 North Pleasant Street01003AmherstMAUSA",
"INAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34131TriesteItaly",
"INAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34131TriesteItaly",
"INAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1I-40127BolognaItaly",
"Dipartimento di Fisica e Astronomia\nUniversita' degli Studi di Bologna\nViale Berti-Pichat 6/240127BolognaItaly"
] |
[] |
Aims. We aim to determine the redshift of GDS_1408, the most solid z ∼ 7 galaxy candidate lying in the Hubble Ultra Deep Field. Methods. We have used all the VLT spectra of GDS_1408 collected by us and two other groups with FORS2 at VLT in the last five years, for a total integration time of 52hr. The combined spectrum is the deepest ever obtained of a galaxy in the Reionization epoch. Results. We do not detect any emission line or continuum over the whole wavelength range, up to 10100Å. Based on an accurate set of simulations, we are able to put a stringent upper limit of f (Lyα) < 3 × 10 −18 erg/s/cm 2 at 3-9 sigma in the explored wavelength range, corresponding to a rest-frame equivalent width EW < 9 Å. Combining this limit with the SED modelling we refine the redshift to be z = 6.82±0.1 (1-sigma). The same SED fitting indicates that GDS_1408 is relatively extinct (A 1600 1) with a dust corrected star formation rate of 20M yr −1 . The comparison between the un-attenuated equivalent width predicted by the case-B recombination theory and the observed upper limit, provides a limit on the effective Lyα escape fraction of f e f f esc (Lyα) < 8%. Even though we cannot rule out a major contribution of the inter/circum galactic medium in damping the line, a plausible interpretation is that G2_1408 is moderately evolved and contains sufficient gas and dust to attenuate the Lyα emission, before it reaches the intergalactic medium.Conclusions. The redshift confirmation of even the best z 7 candidates is very hard to achieve (unless the Lyα or unusually strong rest-UV nebular emission lines are present) with the current generation of 8-10m class telescopes. We show that both JWST and E-ELT will be necessary to make decisive progresses. Currently, the increased redshift accuracy obtained with this kind of analysis makes ALMA an interesting option for the redshift confirmation.
|
10.1051/0004-6361/201424285
|
[
"https://arxiv.org/pdf/1407.3787v1.pdf"
] | 58,901,828 |
1407.3787
|
534e768ac72e8ed3d34d36ea99966492ed13a623
|
A 52 hours VLT/FORS2 spectrum of a bright z ∼ 7 HUDF galaxy: no Lyα emission
July 16, 2014
E Vanzella
INAF -Osservatorio Astronomico di Bologna
via Ranzani 1I-40127BolognaItaly
A Fontana
INAF -Osservatorio Astronomico di Roma
via Frascati 3300040MonteporzioItaly
L Pentericci
INAF -Osservatorio Astronomico di Roma
via Frascati 3300040MonteporzioItaly
M Castellano
INAF -Osservatorio Astronomico di Roma
via Frascati 3300040MonteporzioItaly
A Grazian
INAF -Osservatorio Astronomico di Roma
via Frascati 3300040MonteporzioItaly
M Giavalisco
Department of Astronomy
University of Massachusetts
710 North Pleasant Street01003AmherstMAUSA
M Nonino
INAF -Osservatorio Astronomico di Trieste
via G. B. Tiepolo 11I-34131TriesteItaly
S Cristiani
INAF -Osservatorio Astronomico di Trieste
via G. B. Tiepolo 11I-34131TriesteItaly
G Zamorani
INAF -Osservatorio Astronomico di Bologna
via Ranzani 1I-40127BolognaItaly
C Vignali
Dipartimento di Fisica e Astronomia
Universita' degli Studi di Bologna
Viale Berti-Pichat 6/240127BolognaItaly
A 52 hours VLT/FORS2 spectrum of a bright z ∼ 7 HUDF galaxy: no Lyα emission
July 16, 2014Received -; accepted -Astronomy & Astrophysics manuscript no. vanz8_52Galaxies: high -redshiftGalaxies: formationGalaxies: distances and redshifts
Aims. We aim to determine the redshift of GDS_1408, the most solid z ∼ 7 galaxy candidate lying in the Hubble Ultra Deep Field. Methods. We have used all the VLT spectra of GDS_1408 collected by us and two other groups with FORS2 at VLT in the last five years, for a total integration time of 52hr. The combined spectrum is the deepest ever obtained of a galaxy in the Reionization epoch. Results. We do not detect any emission line or continuum over the whole wavelength range, up to 10100Å. Based on an accurate set of simulations, we are able to put a stringent upper limit of f (Lyα) < 3 × 10 −18 erg/s/cm 2 at 3-9 sigma in the explored wavelength range, corresponding to a rest-frame equivalent width EW < 9 Å. Combining this limit with the SED modelling we refine the redshift to be z = 6.82±0.1 (1-sigma). The same SED fitting indicates that GDS_1408 is relatively extinct (A 1600 1) with a dust corrected star formation rate of 20M yr −1 . The comparison between the un-attenuated equivalent width predicted by the case-B recombination theory and the observed upper limit, provides a limit on the effective Lyα escape fraction of f e f f esc (Lyα) < 8%. Even though we cannot rule out a major contribution of the inter/circum galactic medium in damping the line, a plausible interpretation is that G2_1408 is moderately evolved and contains sufficient gas and dust to attenuate the Lyα emission, before it reaches the intergalactic medium.Conclusions. The redshift confirmation of even the best z 7 candidates is very hard to achieve (unless the Lyα or unusually strong rest-UV nebular emission lines are present) with the current generation of 8-10m class telescopes. We show that both JWST and E-ELT will be necessary to make decisive progresses. Currently, the increased redshift accuracy obtained with this kind of analysis makes ALMA an interesting option for the redshift confirmation.
Introduction
Understanding the process of reionization of the intergalactic medium in the early Universe and the nature of the first galaxies responsible for that process are among the most important goals of modern cosmology (Robertson et al. 2010, and references therein). Thanks to the deep and panchromatic data, such as GOODS, CANDELS and Hubble Ultra Deep Field (HUDF), great progress has been made in our ability to identify and subsequently confirm (spectroscopically) galaxies at z < 7. The most prominent spectral feature in the UV rest-frame wavelengths probed by optical/NIR spectroscopy at z 7 is the Lyα emission line (e.g., Vanzella et al. 2011, V11;Pentericci et al. 2011;Pentericci et al. 2014, P11, P14;Schenker et al. 2012;Schenker et al. 2014, Ono et al. 2012, Shibuya et al. 2012. However, at z > 7 the situation is still challenging and at present only a handful objects are spectroscopically secured. There are at least two main reasons: (1) Physical processes in the galaxies: the Lyα emission is a resonant atomic transition very sensitive to dust attenuation, and can be used for a diagnostic of the physical processes occurring within the galaxy (Giavalisco et al. This work is based on data collected at ESO VLT (prog.ID 084.A-0951(A), 086.A-0968(A), 088.A-1013(A) and 088.A-1008(A)), and at NASA HST. 1996; Atek et al. 2013), since its strength and velocity profile depend on the instantaneous star formation rate, gas and dust content, metallicity, kinematics, and geometry of the interstellar medium. Therefore an evolution of the average galaxy properties with time (gas and/or dust properties and/or ionizing emission) can make it intrinsically absent at specific cosmic epochs. (2) Reionization: the Lyα emission line may suffer of damping effect due to an increase of the neutral gas fraction in the intergalactic and/or circum galactic media (Miralda-Escudé et al. 2000;Dijkstra et al. 2011;Treu et al. 2012) or a increasing incidence of optically thick absorption systems (Bolton & Haehnelt 2013) as the ending phase of the reionization is approached. Another limiting effect was instrumental: efficient near infrared spectrographs (λ 1µm) with multiplexing capabilities are needed to capture UV rest-frame features for many targets at once, and only recently they are becoming available (e.g., VLT/KMOS, Keck/MOSFIRE, LBT/LUCI).
While point (1) is alleviated by the fact that (on average) the frequency of Lyα line emitters appear to increase as redshift increases (3 < z < 6.5, Ouchi et al. 2008;Schaerer et al. 2011;Stark et al. 2011), the damping due to reionization may severely limit our current ability to confirm galaxies at 7 < z < 10 (but see Dijkstra & Wyithe 2010). Finkelstein et al. (2013) find only one galaxy at z = 7.51 out of 43 candidates at z > 6.5 with Article number, page 1 of 8 arXiv:1407.3787v1 [astro-ph.GA] 14 Jul 2014 A&A proofs: manuscript no. vanz8_52 Fig. 1. Two dimensional signal to noise spectra and sky spectra of the galaxies discussed in the text. Top: spectra of the i-band dropout GDS J033229.41-274537.8 observed in F18 (upper part) and B27 (lower part). The Lyα line is marked with a dotted circle. Bottom: the stacked 52hr spectrum of the z-band dropout G2_1408 is shown, with the individual spectra of F18, B27, B7 (lower part). The expected position of the Lyα continuum break is marked with dotted horizontal lines (see text). In the B27 spectrum the redshift values are reported above the sky as a reference. In the F18 spectrum the dotted circle marks the older (here revisited) feature discussed in Fontana et al. (2010). B7 spectrum shows also the Hα emission from the secondary object J033242.32-274651.1 at z=0.535, useful here as an example of a still performing FORS+600z configuration beyond one micron.
the Keck/MOSFIRE and similarly Schenker et al. (2014) find only one possible Lyα line at z = 7.62 in the their sample of 19 z ∼ 8, remarking that some process is making the Lyα difficult to detect. Therefore, there is a large fraction of non-Lyα emitters lying well within the first Gyr after the Big-Bang, whose nature is worth to investigate. Clearly, all these explainations assume that the efficiency of the color selection based on the Lyα-break remains extremely high at z ≥ 7, i.e. that most of current z ≥ 7 candidates are indeed at their estimated redshifts.
In this work, we report on an ultradeep spectrum (52 hours integration) of the brightest z 7 galaxy in the Hubble Ultra Deep Field (F125W = 26.1 ± 0.02). Being by far the brightest z 7 candidate in this well-studied region, it has been continuously selected as a high redshift candidate from the earliest NIC-MOS to the current ultradeep HUDF data, at variance with other targets (Yan & Windhorst 2004;Bouwens et al. 2004;Labbé et al. 2006;Bouwens et al. 2008;Oesch et al. 2010;Fontana et al. 2010;McLure et al. 2010;Bunker et al. 2010;Yan et al. 2010;Finkelstein et al. 2010;Castellano et al. 2010;Wilkins et al. 2011;Bouwens et al. 2011;Grazian et al. 2011;McLure et al. 2013;Bouwens et al. 2014). We focus here on the combination of ultradeep spectroscopy and photometry to derive new insights about its nature.
Errors are quoted at the 1σ confidence level, unless otherwise stated. Magnitudes are given in the AB system (AB ≡ 31.4 − 2.5 log f ν /nJy ). We assume a cosmology with Ω tot , Ω M , Ω Λ = 1.0, 0.3, 0.7 and H 0 = 70 km s −1 Mpc −1 . (Fontana,F18), 27 (Bunker,B27), 7 (Bouwens, B7) and 8 (Carollo, C8) hours on target, respectively, for a total of 60 hours integration time. Unfortunately, in the program 283.A-5063 the G2_1408 source was placed at a position where the CCD has a defect and no dithering has been performed, therefore we exclude the 283.A-5063 program from the following analysis. The total usable exposure time is 52 hours. The median seeing was ∼0.8 arcsecond in all runs. The F18 data were presented in Fontana et al. (2010) where we reported a tentative detection of a Lyα line at z=6.972. The other runs, B27 and B7, have been obtained subsequently in many dif-ferent nights (especially B27), and particular care has been devoted to the alignments of the frames by using bright sources and sky emission lines. The B27 program has also been presented in Caruana et al. (2013), where they reported a S /N 3.2 at the location of the putative Lyα at z = 6.972. We discuss in very detail any possible presence of Lyα emission by combining all the available programs.
FORS2 Observations and Data Reduction
Data reduction has been performed as in Vanzella et al. (2011), with particular care to the sky subtraction. The classical "A-B" dithering scheme that combines the partial frames (A-B) and (B-A) is performed, with an additional treatment that equalizes for local differences in the number counts between frames (e.g., due to time variation of sky lines, distortion, etc.). The algorithm implements an "A-B" sky subtraction joined with a zero (e.g., median) or first order fit of the sky along columns that regularized possible local differences in the sky counts among the partial frames before they are combined. Finally, the twodimensional spectra have been combined with a weighted average, and the subsequent resulting spectrum has been flux and wavelength calibrated. The two dimensional sky-subtracted partial frames are also combined (in the pixel domain) to produce the weighted RMS map, associated to the final reduced spectrum. This allows us to calculate the two dimensional signal to noise spectra, useful to access the reliability of the spectral features (as we address below with simulations).
Further checks have been performed on other targets placed in the masks and on those in common between F18 and B27. In particular, Caruana et al. (2013) targeted the same faint i−band dropout we confirmed previously with F18 (Fontana et al. 2010), GDS J033229.41-274537.8 at z = 5.927. While in the F18 spectrum the continuum is not detected, in our B27 spectrum we find a very faint trace redward of the line and a Lyα with estimated flux of 3.8 × 10 −18 erg/s/cm 2 at S/N 8, about a factor 1.25 higher than F18, whose flux is estimated to be 3.0 × 10 −18 erg/s/cm 2 with S/N 6.5. Both flux and error estimates are compatible within the flux calibration accuracy and different integration time. The top part of Figure 1 shows the S/N spectra of the z = 5.927 galaxy (i.e., the reduced spectrum divided by its RMS map).
Following the above approach, we have combined the 52 hours of observations of GDS_1408. The three individual S/N spectra and the conbined one are shown in the bottom part of Figure 1. The 52 hours spectrum is the deepest spectroscopic observation of a z ∼ 7 galaxy obtained to date.
Results
Before discussing any feature in the spectrum we note that the available photometry already constrains the redshift in the range (6.5 < z < 7.0) where FORS2 is an efficient instrument for the detection of the corresponding Lyα, both in terms of wavelength coverage and response. First, we recall that globally the high redshift nature (z > 6.5) is guaranteed by the large observed break between the ultradeep optical and near infrared bands (∆m 4 magnitudes) and the well determined flat behaviour of the SED in the near infrared bands, detected with a high SNR (HST-WFC3, S/N 20-50), as shown in Figure 2 and 3. This makes GDS_1408 one of the most robust z ∼ 7 candidates. Second, a reliable upper limit on the redshift is provided by the clear detections in the z 850 band (z < 7.3) and narrow band filter, NB973 (z < 7), centered at λ = 9755Å and dλ = 200Å (Figure 3). The limit provided with the 3σ detection in the NB973 is z < 6.94 if the entire NB973 filter is capturing the galaxy con- The well detected signal in the z 850 -band is reproducible with large EW(Lyα) at higher redshift (z > 7) or very faint Lyα emission at the lower redshift, z < 7. Upper and lower horizontal dotted lines mark the limits provided by the NB973 detection and noncontinuum detection below z = 6.6 in the deep spectrum, respectively. The vertical dotted line marks the upper limit of the Lyα rest-frame equivalent width. The transparent gray region underlines the most probable redshift interval. Bottom: SED fitting performed with BC03 libraries with (thin black) and without (blue) nebular emission lines are shown. The fit to the HST bands only is shown with the thick black line. Blue, yellow and red points represent photometry from HST, VLT (U and K bands) and Sptizer/IRAC, respectively. Upper limits at one sigma are marked with arrows. tinuum, z MAX = (9755 − 100)/1215.7 − 1 = 6.94 (see inset of Figure 3).
A deep upper limit to the Lyα flux
Given the above upper limit z MAX , the FORS2+600z configuration provides a safe constraint on the Lyα line flux. 1 Indeed, there are no obvious spectral features in the three S/N spectra at the position of the G2_1408 (marked by arrows), neither in the combined one (see Figure 1).
We note that these deeper data do not confirm the tentative detection of a weak line located at λ = 9691.5 ± 0.5Å, corresponding to a redshift of 6.972 that we reported in Fontana et al. (2010) with S /N < 7 (and shown with a circle in the F18 spectrum of Figure 1). Exploiting the RMS map we derived, the reliability of the spectral feature in the F18 spectrum turns out to have S /N 4.5. In the combined (52h) spectrum, however, the S/N at the same wavelength position is even smaller, 3, and suggests that the earlier tentative detection was most probably a noise spike.
To assign a statistical significance to our 52h non-detection, we estimated the minimum line flux reachable with the deep FORS2+600z spectrum by computing simulations as in V11 and P14. Two dimensional asymmetric Lyα lines have been inserted A&A proofs: manuscript no. vanz8_52 in the science raw frames moving the line from z = 5.7 to 7.3 with dz = 0.0013 (i.e., one pixel at the given spectral resolution), including the dithering pattern, varying the FWHM and the line flux, convolving with spectral resolution along the dispersion and with the seeing along the spatial direction (extracted from the header of each science frame). Knowing the exact position of the inserted lines in the raw frames (that by definition include also the cosmic rays), and including the full reduction pipeline process and the response curve, we can reliably access the limits attainable by the instrument. The resulting S/N of the simulated lines are fully compatible with those we observed at redshift 6 and 7 (P11, P14 and V11).
The upper limit we derive from the combined 52h spectrum is f (Lyα) < 3 × 10 −18 erg/s/cm 2 at 3-sigma (in the skylines) and up to 9-sigma (in sky-free regions) in the whole wavelength range, i.e., up to z = 7.0 (see Figure 4). Adopting the F125W=26.10±0.02 as the estimation of the continuum under the Lyα line, and given the relatively flat UV slope in the near infrared bands, the limit on the Lyα flux corresponds to an upper limit on the equivalent width EW < 9 Å, with the same statistical accuracy. It represents the faintest limit on the Lyα flux ever derived at z > 6.5.
While absorption lines are clearly impossible to detect in our spectra, it is worth to investigate whether the non detection of the continuum in the 52h spectrum is compatible with the expected magnitude 26.1. To this aim, a set of two-dimensional continua (with flat UV slope) without any absorption line have been added to the raw frames as performed for line simulations, adopting a Gaussian profile in the spatial direction (consistent with the observed seeing). They have been added to the raw frames with dithering and dimmed in magnitude, from 23.0 to 27.0 with dm = 0.25 (in all the process the response curve has been taken into account). Figure 4 shows the results. The S/N decreases accordingly with the magnitude dimming and in the presence of sky emission lines, becoming impossible to detect at magnitude 26.0 unless the object is at redshift below 6.6 (see Figure 4). Both simulations on Lyα line and continuum show the clear decrease of the S/N at the position of sky emission lines, i.e., it is lower where the sky emission is stronger. The results of the continuum simulation are in line with the observed faint galaxies, in particular with those we confirmed in P11 at z ∼ 6 based on continuum-break only.
Refined redshift
Depending on its EW, the Lyα emission can contribute significantly to the broad band photometry and as a consequence to the photometric redshift determination. In particular, the flux observed in the z 850 band (see Figure 3) depends on the position of the Lyα-line/break (i.e., the redshift), the IGM attenuation, the EW of the possible Lyα emission (EW(Lyα)), and mildly on the UV slope of the source, β (F λ = λ β ). With the depth and photometric quality available for this galaxy and the upper limit derived above, the redshift value can be refined. To this aim we performed the SED fitting with Bruzual & Charlot (2003)(BC03, hereafter) templates by including different equivalent widths of the Lyα (0-200Å rest-frame) and focusing solely on the break between the optical and near infrared bands, i.e., B 435 , V 606 , i 775 , z 850 , F105W, F125W, F140W and F160W. We have not included the NB filter in this exercise since its photometry is not as accurate as the HST data. The templates adopted a range of e-folding star-formation timescales ranging from 0.1 to 15 Gyr (the latter being in practice a constant-star formation history), metallicities from Z = 0.2Z to Z = Z , and a Calzetti attenuation curve with A V = 0 − 3. Figure 2 shows the plane (redshift-EW(Lyα)), in which the degeneracy is clear, i.e., a progressively stronger Lyα emission is required at increasing redshifts to compensate for the increasing absorption by the IGM. The degeneracy with the Lyα line is broken by including the result from the ultradeep 52h spectrum discussed in Sect. 3.1, that forces the EW(Lyα) to be smaller than 9Å rest-frame. Once the Lyα constraints are included, the redshift of G2_1408 is constrained within 6.7 and 6.95 at 1σ and 6.6 and 7.1 at 2σ. The top panel of Figure 2 shows the resulting region where the redshift would lie, that is z = 6.82 +0.1 −0.1 (at one sigma). We note that the 2σ range is consistent with the continuumonly simulations described above and the non detection in the sky-free region of the spectrum at 9000Å< λ < 9280Å, that both suggest a lower limit of the redshift, z MIN = 6.6. Less significant, but consistent with z MIN , is the slightly brighter 3.6µm magnitude to respect the 4.5µm channel in the data we have, that could suggest a flux boost of the [OIII]4959-5007+Hβ structure in the first channel, while the Hα line is outside (redward) the second one, respectively. Indeed, this is confirmed with deeper IRAC data, in which a clear break has been measured, [3.6 − 4.5]µm = 0.66 ± 0.2 (Smit et al. in preparation, Labbé et al. (2013)). This is fully consistent with our redshift estimation.
The SED fitting with and without the nebular component has then been performed including the IRAC and ground based photometry. We have included the full treatment of the nebular emission (both in lines and continuum), computed using the Schaerer & de Barros (2009) model, as described in Castellano et al. (2014). The resulting SED fits with HST-only bands and the whole photometry are shown in the bottom panel of Figure 2. The derived physical quantities in the various cases agree within a factor of two. This exercise provides an estimate of the stellar mass of M = 5 +3 −2 × 10 9 M , an age 0.1 +0.15 −0.05 Gyr, a dust attenuation E(B-V) 0.1 ± 0.05 and a dust corrected star formation rate of 21 +20 −10 M yr −1 .
Discussion
The combination of excellent photometry and ultradeep spectroscopic data lead us to conclude that the lack or extreme weakness of a Lyα emission is a real feature of this object. It is therefore worth investigating the reasons why the Lyα line is not present.
The effective Lyα escape fraction
Dijkstra & Jeeson-Daniel (2013) differentiate between the term 'escape' fraction and 'effective escape' fraction of Lyα photons, f esc (Lyα) and f e f f esc (Lyα), respectively. The former being related to the transport of photons out of the galaxy's interstellar medium, while the latter includes further damping by the IGM, giving rise to a low-surface-brightness Lyα glow around galaxies. Therefore, f e f f esc (Lyα) can be much smaller than f esc (Lyα). If we assume "normal" metallicities and IMFs, and that on average star formation is ongoing at equilibrium (age > 100Myrs with constant SFR), an estimation of the Lyα luminosity can be obtained imposing SFR(UV)/SFR(Lyα) 1 (Hayes et al. 2011;Verhamme et al. 2008). Assuming the SFR(UV) corrected by dust extinction is representative of the total ongoing star formation activity, the intrinsic Lyα flux turns out to be 4 × 10 −17 ergs −1 cm −2 , i.e., EW(Lyα) rest-frame 120Å. The upper limit on the observed Lyα line flux derived above corresponds to an observed EW(Lyα) < 9Å rest-frame (at 3 − 9σ), 13 times smaller than the expected intrinsic emission, that gives an upper limit of f e f f esc (Lyα) < 8%.
What is attenuating the Lyα emission ?
The escape of Lyα photons from a gaseous and dusty interstellar medium is a complex process which depends sensitively on a number of physical properties, such as dust and gas mass, star formation rate, metallicity, kinematic, escaping ionizing radiation, as well as the gas geometry and filling factor and the galaxy orientation (Neufeld 1991;Verhamme et al. 2008;Verhamme et al. 2012;Hayes et al. 2011;Yajima et al. 2014;Laursen et al. 2013;Dijkstra et al. 2014). Additionally, the neutral hydrogen in the IGM can scatter part of the Lyα photons and decrease the line luminosity. For example, Laursen et al. (2011) suggested that the average IGM transmission could be 20% at z = 6.5. Also in this case the transmission depends sensitively on the viewing angle and the environments of the galaxy, as it is affected by the inhomogeneous filamentary structure of IGM. At z > 6.5 the IGM damping could be the dominant effect. It is not possible to investigate in detail the above quantities with the current information. Therefore we have to rely on global properties, e.g., by performing a comparative analysis with lower redshift galaxies at z > 3 − 4. First, it has been shown that 2 < z < 6 UV-bright star-forming galaxies (L > L ) show on average a deficit of Lyα emission with respect the fainter UV counterparts (e.g., Ando et al. 2006;Vanzella et al. 2009;Lee et al. 2013;Balestra et al. 2010), explained by the fact that on average galaxies that are intrinsically brighter in the UV are also more star forming and more massive, more chemically enriched and thus more likely to be obscured by dust, especially the Lyα resonant transition that is expected to be efficiently absorbed.
A decreasing trend of f esc (Lyα) emission with increasing E(B-V) has also been observed in several studies (Giavalisco et al. 1996;Atek et al. 2008;Verhamme et al. 2008;Kornei et al. 2010;Hayes et al. 2011). The observed UV slope (β = −2) of GDS_1408 suggests there is a dust attenuation A1600 ∼ 1.3 (Castellano et al. 2014;de Barros et al. 2014). The inferred S FR 21M yr −1 and assuming the Schmidt law (Schmidt 1959;Kennicutt 1998a), implies a non negligible amount of gas, M gas 3 × 10 9 M , comparable to the stellar mass, and would favor a more efficient Lyα photon destruction operated by dust. The presence of dust and gas would also suggest that the escape fraction of ionizing radiation is very low, as it has been observed in L star-forming galaxies at z 3 (Vanzella et al. 2010a;Boutsia et al. 2011).
Second, GDS_1408 is one of the more extended sources among the z ∼ 7 candidates (half light radius of 0.26 , i.e., 1.4 proper Kpc, Grazian et al. 2011, with an elongated morphology of 4.8 × 2.5 proper Kpc, see Figure 3). It has been shown that the Lyα equivalent width and the size observed at the 1500Å restframe of the stellar continuum anti-correlate, such that on average the emitters appear more compact and nucleated than the non-emitters, with average half light radius 1kpc (Law et al. 2007;Vanzella et al. 2009;Pentericci et al. 2010).
GDS_1408 is moderately star-forming and contains a non negligible amount of dust, it is UV bright and spatially extended. While we cannot exclude that the CGM/IGM plays an additional role, the properties of GDS_1408 are consistent with those of others lower-z L star-forming galaxies that show faint Lyα emission, and is not a "smoking gun" of an increased neutral IGM at z = 7. It is worth noting that we are discussing a single galaxy case which clearly cannot be considered as representative of a population. The deficit of Lyα lines from redshift 6 to 7 recently observed is significant and is based on a statistical analysis that compares tens of star-forming galaxies selected with very similar color techniques, indepenently from the presence of the Lyα line. However, as noted also in Schenker et al. (2014), galaxies like GDS_1408 are not the best tracers of an IGM damping. The higher is the probability the Lyα emission is internally absorbed, the lower is the power of tracing the neutral gas fraction of the circum/inter-galactic medium.
Nonetheless, it is important to assess the nature of "continuum-only" star-forming galaxies at z > 6.5, still an unexplored line of research. As reported in this work, the tentative spectroscopic investigation of "continuum-only" galaxies at z > 6.5 shows all the limitations of the 8-10 meter class telescopes coupled with optical spectroscopy (λ < 1.1µm). Different and future facilities are needed to shed light on their nature.
Future prospects: ALMA, JWST and ELT
We have shown the current limits of 8-10m class telescopes in the spectroscopic characterization and redshift measurement of non-Lyα emitters at 6.6 < z < 7.3. If z ∼ 7 is a critical value above which the visibility of Lyα lines decreases drastically, then future facilities are necessary to capture the UV continuumbreak and the ultraviolet absorption lines and/or optical nebular emission lines at 7 < z < 10. JWST-NIRSPEC will probe the typical nebular emission lines, e.g., [OII]3727,Hβ,[OIII]4959-5007 up to z 9 (5µm) and the extremely large telescopes (ELT, 30-40m diameter) will allow us to cover the mid-infrared part (e.g., [8][9][10][11][12][13][14] and to probe the UV continuum with S/N=10 down to J 27 and therefore to study in detail the ultraviolet absorption lines, not to mention the possibility to perform high spatial resolution analysis.
In particular, the case of GDS_1408 is shown in Figure 5, where a schematic view of the limits on the continuum at S/N=10 are reported for the ELT, JWST and VLT telescopes. A good characterization of the ultraviolet absorption lines will be feasible with the ELT. JWST would marginally identify the trace of the continuum (S /N < 5), but will open for the measure of optical emission lines (up to .
While JWST and ELT will definitely perform these kind of studies, at present the measure of the redshift with 8-10m telescopes could be achieved looking at emission lines different from Lyα. Though unusual, lines like OIII]λλ1661-1666 and CIII]λλ1907-1909 could be identified by means of near infrared spectrographs. Moreover, depending on the source of ionizing photons, other lines like NV 1240, NIV]λλ1483-1486, CIV 1550, HeII 1640 can also be measured (e.g., Osterbrock & Ferland 2006;Vanzella et al. 2010b;Raiter et al. 2010;Steidel et al. 2014).
Another promising facility that might be able to determine the spectroscopic redshift of this galaxy is ALMA. Indeed, a spectral scan of the [CII]158µm line encompassing the full 2sigma range of uncertainty could be covered with two ALMA spectral bands. Assuming that the local SFR-[CII]158µm relation of Sargsyan et al. (2012) holds also at these high redshifts, based on the best-fitting SFR=20M /yr −1 we expect a flux of 0.24 Jy km/s, that can be secured in a relatively short time (< 6 hours in the Cycle2 sensitivity) with ALMA. A molecular line scan that covers the 3mm window (89-115 GHz) can also reveal multiple CO transitions in GDS_1408. In particular at z > 6.5 the transitions J = 7 → 6 and J = 6 → 5 (where J is the rotational quantum number) are observable. Given the UV slope and the star formation activity, and assuming the relation between CO line luminosity and SFR (see Carilli & Walter 2013;Decarli et al. 2014), the expected CO transitions can be detected (S /N 3) with an ALMA 3mm scan with sensitivity limit of 0.1mJy beam −1 .
Conclusions
We have reported the combined VLT/FORS2 spectroscopy of one of the most reliable z 7 galaxy candidates in the Hubble Ultra Deep Field. Three different programs at VLT targeted GDS_1408 in the last five years, for a total integration time of 52h. We have retrieved from the ESO archive all these data, and re-analyzed them in a consistent way. Our main results are:
1. An upper limit on the Lyα emission: We are able to place a stringent upper limit of f (Lyα) < 3 × 10 −18 erg/s/cm 2 at 3-sigma (in the sky-lines) and up to 9-sigma (in sky-free regions) in the whole wavelength range, i.e., up to z = 7.0 (see Figure 4). This corresponds to an upper limit on the equivalent width EW < 9 Å, with the same statistical accuracy.
With the deeper data used here we do not confirm the tentative detection of a weak line located at λ = 9691.5 ± 0.5 Å, corresponding to a redshift of 6.972, that we claimed in F10. 2. Redshift refinement: The combination of ultradeep spectroscopy, superb HST information and narrow band imag- ing, have allowed us to refine the photometric redshift value of GDS_1408, that turns out to be 6.82 ± 0.1. The same analysis indicate that GDS_1408 has a nearly solar metallicity and is relatively dust attenuated (A 1600 1) galaxy. 3. Lyα escape fraction: We derived a f e f f esc (Lyα) < 8%. The Lyα attenuation can be a combination of internal (ISM) and external effects (IGM).
Even though we cannot rule out a major contribution of the IGM in damping the line, the most plausible interpretation is that G2_1408 is a star-forming galaxy at z 6.82 moderately evolved and containing sufficient gas and dust to attenuate the Lyα emission, before it reaches the intergalactic medium. If compared with fainter (0.05L < L < 0.2L ), young, less massive and less evolved counterparts that show Lyα emission and much steeper UV slopes β < −2.5 (e.g., Balestra et al. 2013;Vanzella et al. 2014), G2_1408 appears more massive and evolved (β −2).
An absence of the line due to dust absorption is not in contrast with recent results on the deficit of Lyα lines ascribed to a possible increase of the neutral gas fraction of the IGM between z ∼ 6 and 7 (e.g., P11, P14, Finkelstein et al. 2013). In fact, the apparent drop of lines is an evidence based on statistical samples, in which the properties of redshift 6 and 7 galaxies are compared in a differential way, and applying similar color selection techniques.
Redshift 7 sources like GDS_1408, and in general even fainter ones at J = 27, will be well studied spectroscopically with ELT and JWST telescopes, in particular for what concerns the UV absorption lines and nebular (optical) emission lines, respectively. As an example, JWST will provide the systemic redshift from rest-frame optical nebular emission lines (Oxigen, Balmer lines) and the ELT will capture the signature of the ISM in absorption in the rest-frame ultraviolet. This will be feasible not before 2018-2020. Currently, a facility that can provide redshift measures at z 7 and characterize the properties of the ISM and stellar population is ALMA. Sources like GDS_1408 with a reliable guess on the redshift value are suitable candidates for the scanning mode of ALMA.
HUDF-J033242.56-274656.6 galaxy (G2_1408 in Castellano et al. 2010 and hereafter) has been observed through four different VLT/FORS programs collected in the period 2009-2012, 084.A-0951(A) (P.I. A. Fontana), 086.A-0968(A) -088.A-1013(A) (P.I. A. Bunker), 088.A-1008(A) (P.I., R. Bouwens), and 283.A-5063 (P.I., M. Carollo) with exposure times of 18
Fig. 2 .
2Top: The projected Redshift-EW(Lyα) solutions of the SED fitting are shown. Black, blue and red symbols represent 3-,2-, and 1-sigma solutions.
Fig. 3 .
3The cutouts (3 × 3 ) from U to H-band of G2_1408 are shown with the magnitudes (limits are at one-sigma). The green contour marks the galaxy's shape as a guidance. The inset image placed between the z 850 and F105W refer to the narrow band filter NB973 in which the source is detected (3-sigma). The bottom-right cutout shows the clear detection in the z 850 band (S /N ∼ 8), for which there is a good coverage of the ultradeep FORS2 spectrum (also reported in the bottom).
Fig. 4 .
4Simulation of the expected signal to noise ratio for emission Lyα line and continumm with 52 hours integration with FORS2. From top to bottom: the two dimensional spectra of the sky and RMS map are shown as an example and guidance for the eye. Simulated signal to noise 2d-spectra of line emission (snapshot) and continuum emissions are shown, with indicated the line flux and magnitude at the (flat) continuum on the right. It is clear that the S/N is lower where the sky lines are stronger. In the bottom part the S/N ratio of the Lyα feature versus redshift (solid black line) and the continuum (red dotted line) are reported for the case of GDS_1408. The transparent grey region marks the z MIN < z < z MAX interval, see text for more details. Grey horizontal dotted lines mark S /N = 1.5 and 3.0.
Fig. 5 .
5Expected magnitude limits in the continuum (S/N=10) with 30-50h integration time are shown for ELT, (orange, Evans et al. 2013), JWST (black, available in tabular form http://www.stsci.edu/jwst/instruments/nirspec/sensitivity/) and VLT-FORS (green, this work). The estimated limit for KMOS is magnitude 21.5 in the YJ band with 50h integration (rescaling from the KMOS manual, Sect. 2.3.4). The schematic position of GDS_1408 is reported (blue), with the continuum magnitude 26.1, the typical ultraviolet absorption lines from the Lyα to CIV 1550 (grey region, UV-abs) and the basic emission lines, e.g., CIII]1909, [OII]3727, Hβ, [OIII]4959-5007 and Hα.
As an exampleFigure 1shows the secondary object in B7 spectrum in which the Hα line is detected at z=0.535 consistently with the zphot = 0.5, and showing that the FORS2+600z configuration is still performing well at ∼10100Å.
Acknowledgements. We thanks R. Bouwens for providing information about the deep IRAC magnitudes of the source and P. Oesch for providing information about the GDS_1408 target in the spectroscopic program 088.A-1008(A). We thank F. Vito, M. Mignoli, A. Cimatti and S. de Barros for useful discussions. Part of this work has been funded through the INAF grants (PRIN INAF 2012).
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|
[] |
[
"Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model",
"Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model"
] |
[
"Pierre Emmanuel Berche ",
"Christophe Chatelain ",
"Bertrand Berche ",
"\nLaboratoire de Physique des Matériaux\n\n",
"\nUniversité Henri Poincaré\nB.P. 239F-54506Nancy 1, Vandoeuvre les Nancy CedexFrance\n"
] |
[
"Laboratoire de Physique des Matériaux\n",
"Université Henri Poincaré\nB.P. 239F-54506Nancy 1, Vandoeuvre les Nancy CedexFrance"
] |
[] |
to appear in Phys. Rev. Lett.)We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using FSS techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second order phase transition is found. The exponents β/ν and γ/ν are obtained at the new fixed point.
|
10.1103/physrevlett.80.297
|
[
"https://export.arxiv.org/pdf/cond-mat/9711205v1.pdf"
] | 119,332,772 |
cond-mat/9711205
|
c50465fa84ca9d46ccda601c4b699d3a733bf1e1
|
Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model
20 Nov 1997 (November 18, 1997,
Pierre Emmanuel Berche
Christophe Chatelain
Bertrand Berche
Laboratoire de Physique des Matériaux
Université Henri Poincaré
B.P. 239F-54506Nancy 1, Vandoeuvre les Nancy CedexFrance
Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model
20 Nov 1997 (November 18, 1997,
to appear in Phys. Rev. Lett.)We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using FSS techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second order phase transition is found. The exponents β/ν and γ/ν are obtained at the new fixed point.
The study of the influence of bond randomness on phase transitions is a quite active field of research, motivated by the importance of disorder in real experiments [1]. According to the Harris criterion [2], quenched randomness is a relevant perturbation at a second order critical point when the specific heat exponent α of the pure system is positive.
The analogous situation when the pure system exhibits a first order phase transition was also studied. Imry and Wortis first argued that quenched disorder could induce a second order phase transition [3], and it was shown that in two dimensions, an infinitesimal amount of randomly distributed quenched impurities changes the transition into a second order one [4]. The first large-scale Monte Carlo study of the effect of disorder at a temperaturedriven first order phase transition is due to Chen, Ferrenberg, and Landau. These authors studied the 2D 8state Potts model (which is known to exhibit a first order phase transition when the number of states q is larger than 4 [5]). They first showed that the transition becomes second order in the presence of bond randomness, and obtained critical exponents very close to those of the pure 2D Ising model at the new critical point [6]. On the other hand, drastically different results were obtained for random lattices [7].
The essential properties of random systems are governed by disorder fluctuations. All physical quantities depend on the configuration of disorder, and the study of the influence of randomness requires an average over disorder realisations. Among the systems where the presence of fluctuations is also of primary importance, aperiodic systems have been of considerable interest since the discovery of quasicrystals [8]. They are built in a deterministic way, making any configurational average useless, and their critical properties have been intensively studied (for a review, see Ref. [9]). In layered systems, aperiodic distributions of the exchange interactions between successive layers in the Ising model have been considered [10], leading to unchanged universal behavior or to modified critical properties, depending on the aperiodic series under consideration. The major result was obtained when Luck, generalizing the Harris criterion to layered perturbations, proposed a relevance criterion for the fluctuating interactions [11]. According to Luck's criterion, aperiodic modulations may be relevant, marginal, or irrelevant, depending on the correlation length exponent ν of the unperturbed system and on a wandering exponent ω which characterizes the fluctuations of the couplings around their average [12]. Systematic studies of the critical properties for irrelevant, marginal, and relevant aperiodic perturbations have then been achieved in the extreme anisotropic limit [13].
In this letter, we report results of Monte Carlo simulations of the two-dimensional 8-state Potts model with an aperiodic modulation of exchange couplings between nearest-neighbor layers. Our aim is to study the effect of such a distribution on the nature of the phase transition. In particular, we ask if the fluctuations are able to induce a second order phase transition. The Hamiltonian of the system with aperiodic interactions can be written
− βH = (i,j) K ij δ σi,σj(1)
where the spins σ i , located at sites i, can take the values σ = 1, 2, . . . , q and the sum goes over nearest-neighbor pairs. The coupling strengths are allowed to take two different values K 0 = K and K 1 = Kr. They are distributed according to a layered structure i.e. the distribution is translation invariant in one lattice direction, and follows an aperiodic modulation {f k } of digits f k = 0 or 1 in the other direction: In layer k, both horizontal and vertical couplings take the same value Kr f k (Fig. 1). The sequence of digits {f k } is generated through iteration of substitution rules. The Thue-Morse (TM) sequence is obtained by substitutions on digits: 0 → S(0) = 01, 1 → S(1) = 10, while the so-called paper folding sequence (PF) is generated through substitutions on pairs of digits: 11 → S(11) = 1101, 10 → S(10) = 1100, 01 → S(01) = 1001, 00 → S(00) = 1000. After 3 iterations initiated by 0 and 11 respectively, we get the following sets {f k } TM : 01101001, PF : 1101100111001001 (2) Most of the properties of a sequence are obtained from the substitution matrix [12]. The asymptotic density ρ ∞ of 1, the length L n of the sequence after n iterations, but also the fluctuations of the f k 's at a length scale L n around their average values are related to the substitution matrix. For the fluctuations, one has
Ln k=1 (f k − ρ ∞ ) ∼ L ω n(3)
where ω, the wandering exponent, discriminates between bounded and unbounded fluctuations. In the case of TM and PF sequences, the fluctuations are respectively nondivergent (ω TM = −∞), and logarithmically divergent (ω PF = 0).
f k 0 1 1 0 1 0 0 1 K 0 K 0 K 1 K 1 K 1 K 0 K 1 K 0 FIG. 1.
Layered aperiodic modulation of the coupling strengths on the square lattice and dual system.
Our particular choice of coupling distribution makes it possible to determine exactly the critical point by duality arguments. Consider a system of L layers with a distribution {f k }, made from a succession of verticalhorizontal (V-H) bonds when read from left to right (Fig. 1), and let us write its singular free energy density f s (K 0 , K 1 ; {f k }). Under a duality transformation, the strong and weak couplings K i are replaced by weak and strong dual couplingsK i respectively. Since a vertical bond on the original lattice becomes horizontal on the dual system, the same V-H bond configuration is recovered for the transformed system when the distribution is read from right to left. One thus gets the same type of system, but a reverse distribution {f L+1−k }, so that the free energies of the two systems are the same:
f s (K 0 , K 1 ; {f k }) = f s (K 0 ,K 1 ; {f L+1−k }).
The sequences considered here have the interesting property that the reverse distribution corresponds to the original one if one interchanges perturbed and unperturbed couplings
K 1 ↔ K 0 : f s (K 0 ,K 1 ; {f L+1−k }) = f s (K 1 ,K 0 ; {f k }).
The system being self-dual the critical point, if unique, is exactly given by the critical line K 0c =K 1c of the usual anisotropic model [14]:
(e Kc − 1)(e Kcr − 1) = q.(4)
One should mention there that the required symmetry property of the sequences holds for odd iterations in the case of TM, and works in the case of PF if one omits the last digit, which simply introduces an irrelevant surface effect in the simulations. We performed extensive simulations of L × 2L lattices (16 ≤ 2L ≤ 512) with periodic boundary conditions in one direction (vertical) and free boundaries in the other (2L columns). The Swendsen-Wang cluster flipping method [15] was used. Between 2 × 10 5 (smaller lattice sizes) and 6 × 10 5 (larger lattice sizes) Monte Carlo steps (MCS) per spin were performed (this is always larger than 10 4 times the correlation time, and seems sufficient in order to produce reliable thermal averages).
The order parameter is defined by the majority orientation of the spins [16]:
M = m = qρ max − 1 q − 1 .(5)
Here, ρ max = max σ (ρ σ ) , where ρ σ is the density of spins in the state σ and . . . denotes the thermal average over the Monte Carlo iterations. The susceptibility is given by χ = KV ( m 2 − m 2 ). Although local ordering mechanisms are not yet clarified in aperiodic systems, we expect a unique transition temperature for all the columns, so we used average quantities in order to reduce fluctuations. The first task is to identify the order of the transition. For that purpose, we made some preliminary runs at several temperatures first of all in order to confirm numerically the location of the critical point, and then to have a general picture of the phase transition. The examination of the energy autocorrelation time shows that it is diverging in the case of TM sequence, while it seems to remain more or less bounded for PF. It is consistent with a first order transition in the first case, and a second order one for the latter. We have further estimated temperature-dependent effective exponents for the average magnetization and suceptibility. This can be done by comparing the data at two different sizes L and L ′ = L/2: Assuming the following scaling form for the average magnetization M L (t) = L −β/ν M(Lt ν ), where t =| K − K c |, we define the quantity [17] X L (t) = ln M L /M L ′ ln L/L ′ .
Close to K c , this can be expanded in powers of Lt ν , leading to
X L (t) ≃ − β ν + Lt ν 2 ln 2 M ′ (Lt ν ) M(Lt ν ) + O(L 2 t 2ν )(7)
which defines an effective exponent. As the critical point is approached and in the thermodynamic limit, it evolves towards − β ν . The analogous quantity can be computed for the susceptibility. The results are shown for TM on Fig. 2. The successive estimates of β ν = d − y t and γ ν = 2y h − d clearly evolve towards the values 0 and 2, characteristic of a first order phase transition. The scaling dimensions associated to the temperature and magnetic field, y t and y h , indeed take a special value equal to the dimension d of the system [18]. In the case of the PF sequence, the behavior is drastically different, and this first analysis does not allow any conclusion.
Once the qualitative description of the phase transition was made, our strategy was to use finite-size scaling (FSS) techniques in order to get more accurate results. We made runs for systems of larger sizes, and in a L × 4L geometry (4L in the aperiodic direction, going from 8 to 1024), for which we estimated the number of MCS/spin from the preliminary runs. We have moreover studied a periodic system (PS) with alternate couplings K 0 and K 1 (i.e. the same critical point given in Eq. 4), and which is a "first order reference" system. This is illustrated in Fig. 3 where the energy autocorrelation time τ is plotted for the three samples. In the case of TM, τ diverges exponentially as expected for a first order phase transition, although it is always quite small compared to the periodic system. For PF, the data are compatible with a power law with a very small dynamical exponent as expected for cluster algorithm simulations at a second order phase transition. The crude data for M and χ furthermore show that, in the case of the TM sequence, a cross-over appears between small sizes where the data more or less follow the same behavior than PF, and large sizes where the first order regime analogous to PS is well established (Fig. 4). Certainly, a careful procedure is needed for a reliable determination of the critical exponents. From the log-log curves between 4L min and 4L max = 1024, one determines an effective exponent x(L min ) for each quantity, then the smaller size is cancelled from the data and the whole procedure is repeated until the three largest sizes only remain. The effective exponent is then plotted against L −1 (Fig. 5). The critical exponent is finally deduced from the extrapolation at infinite size. The numerical results are given in Table I. All of them are in agreement with the scaling law d = 2β/ν + γ/ν, within the precision of the results.
To summarize, we have shown from numerical simulations that the fluctuations introduced by an aperiodic modulation of exchange interactions is liable to induce a second order phase transition in a system which originally exhibits a first order transition. From Monte Carlo simulations, we have strong evidences in favor of a second order regime for PF sequence. This type of effect was already known since the work of Imry and Wortis in the case of a random distribution where the fluctuations are unbounded and can be characterized by a wandering exponent ω rand = 1/2. Here, the same type of behavior is induced by a smoother perturbation, namely by the PF sequence which exhibits only logarithmic fluctuations (ω = 0) while the bounded fluctuations generated by the TM sequence (ω = −∞) are not strong enough to destroy the first order transition. The same type of problem on a quasicrystal is currently under investigation by Ledue et al and the transition seems, in this case also, to remain of first order [19]. We may thus infer that Luck's criterion can probably be applied to first order phase transition. Here, we can replace ν by 1/d [18,20] in the criterion in order to compare thermal fluctuations to those introduced by the distribution of couplings. Luck's cross-over exponent then becomes φ = 1 + (ω − 1)/d, and the aperiodicity can induce a second order phase transition when φ > 0. This is in agreement with the results of our simulations. We can finally mention that the study of local order parameter is currently under investigation.
We are indebted to L. Turban, W. Janke, and D. Ledue for valuable discussions. This work was supported by the Ciril and the Centre Charles Hermite in Nancy. * To whom correspondence should be addressed, Electronic address: [email protected]
FIG. 2 .(
2Temperature-dependent effective exponents for the average magnetization and suceptibility (TM sequence) estimated from the ratios XL(t) = ln M L /M K1/K0 = r = 5). The sizes vary from 2L = 16 to 2L = 256, and the arrows indicate the crossings between successive sizes which give rough estimates for the critical exponents.
FIG. 3 .
3Energy autocorrelation time τ at K c (r = 5). For TM, the dashed line is a fit to an exponential behavior.
FIG. 4 .
4Log-log plots of M and χ vs L for a periodic reference, and for TM and PF sequences (r = 5). For M , error bars are smaller than the symbol sizes.
FIG. 5 .
5Effective size-dependent exponents for the two aperiodic sequences (r = 5).
TABLE I .
IExponents associated to the magnetization and susceptibility for the three examples considered in the text. The uncertainties given there are rough estimations corresponding to the standard deviation for the fit of the data in the whole range of sizes.PS
TM
PF
β/ν
0.05 ± 0.03
0.05 ± 0.04
0.48 ± 0.03
γ/ν
1.99 ± 0.08
1.96 ± 0.06
1.01 ± 0.04
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|
[] |
[
"Spacetime picture for entanglement generation in noisy fermion chains",
"Spacetime picture for entanglement generation in noisy fermion chains"
] |
[
"Tobias Swann \nRudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nParks RoadOX1 3PUOxfordUK\n",
"Denis Bernard \nLaboratoire de Physique de l'École Normale Supérieure\nCNRS\nENS & Université PSL\nSorbonne Université\nUniversité Paris Cité\n75005ParisFrance\n",
"Adam Nahum \nRudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nParks RoadOX1 3PUOxfordUK\n\nLaboratoire de Physique de l'École Normale Supérieure\nCNRS\nENS & Université PSL\nSorbonne Université\nUniversité Paris Cité\n75005ParisFrance\n"
] |
[
"Rudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nParks RoadOX1 3PUOxfordUK",
"Laboratoire de Physique de l'École Normale Supérieure\nCNRS\nENS & Université PSL\nSorbonne Université\nUniversité Paris Cité\n75005ParisFrance",
"Rudolf Peierls Centre for Theoretical Physics\nClarendon Laboratory\nParks RoadOX1 3PUOxfordUK",
"Laboratoire de Physique de l'École Normale Supérieure\nCNRS\nENS & Université PSL\nSorbonne Université\nUniversité Paris Cité\n75005ParisFrance"
] |
[] |
Studies of random unitary circuits have shown that the calculation of Rényi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a 1D chain of Majorana modes with nearest neighbour hoppings, random in both space and time. We analyze the N th Rényi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an SO(2N ) Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the N = 2 case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with this domain wall relaxing diffusively in the time direction. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture, reflecting a crossover from diffusive to ballistic spreading of information.
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"https://export.arxiv.org/pdf/2302.12212v1.pdf"
] | 257,102,550 |
2302.12212
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6dfbfa2913f91cffa84bd43e1c50abda1a6bca15
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Spacetime picture for entanglement generation in noisy fermion chains
Tobias Swann
Rudolf Peierls Centre for Theoretical Physics
Clarendon Laboratory
Parks RoadOX1 3PUOxfordUK
Denis Bernard
Laboratoire de Physique de l'École Normale Supérieure
CNRS
ENS & Université PSL
Sorbonne Université
Université Paris Cité
75005ParisFrance
Adam Nahum
Rudolf Peierls Centre for Theoretical Physics
Clarendon Laboratory
Parks RoadOX1 3PUOxfordUK
Laboratoire de Physique de l'École Normale Supérieure
CNRS
ENS & Université PSL
Sorbonne Université
Université Paris Cité
75005ParisFrance
Spacetime picture for entanglement generation in noisy fermion chains
(Dated: February 24, 2023)
Studies of random unitary circuits have shown that the calculation of Rényi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a 1D chain of Majorana modes with nearest neighbour hoppings, random in both space and time. We analyze the N th Rényi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an SO(2N ) Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the N = 2 case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with this domain wall relaxing diffusively in the time direction. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture, reflecting a crossover from diffusive to ballistic spreading of information.
Studies of random unitary circuits have shown that the calculation of Rényi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a 1D chain of Majorana modes with nearest neighbour hoppings, random in both space and time. We analyze the N th Rényi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an SO(2N ) Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the N = 2 case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with this domain wall relaxing diffusively in the time direction. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture, reflecting a crossover from diffusive to ballistic spreading of information.
I. INTRODUCTION
The aim of this paper will be to develop a spacetime picture for entanglement generation [1,2] in random free-fermion systems, analogous to the scaling picture in terms of an entanglement "membrane" in spacetime that holds for various random and non-random interacting systems [3][4][5], and also to explore the crossover between the two "universality classes" when weak interactions are switched on. We find that in the free case, the key spacetime structures are smooth field configurations in a simple effective field theory, in contrast to the membrane that is relevant in the interacting case (which in 1+1 dimensions is a spacetime trajectory). Turning on interactions leads to field configurations with localized domain wall structures that may be identified with the entanglement membrane.
We will focus on free-fermion systems which are otherwise completely random, meaning there are no additional symmetries which would give rise to conserved charges. To study the crossover in entanglement dynamics from the non-interacting case to the interacting case, we add weak random interactions to the free-fermion model and see how the effective theory is modified.
Our approach will be to calculate the N th Rényi entropy of entanglement using a replica formalism, with N forward and N backward Feynman trajectories. In this replica formalism, an effective model is obtained by averaging over randomness. For generic systems, this introduces a symmetry under discrete permutations of the replicas, and there is a corresponding effective degree of freedom that takes discrete values [5][6][7][8][9][10][11][12][13][14]. By contrast for free fermions there is a larger symmetry under continuous rotations in replica space, which in the most generic case is SO (2N ). This leads to a qualitatively different spacetime picture, with domain walls enforced by the boundary conditions relaxing continuously in the time direction.
To develop the spacetime picture for free fermions, we will study a specific, simple model, which is a one-dimensional chain of Majorana modes with random nearest-neighbour couplings. There is only one possible form of such couplings, proportional to product of the two Majorana operators. The coupling strength (hopping amplitude) on each bond is simply a real value, which we take to be random in both space and time. (We consider the limit where the evolution is in continuous time.) This model can be thought of as a noisy version of the Kitaev chain [15] (or equivalently a noisy version of the transverse field Ising model). A discrete-time version of this model has previously been studied in Ref. [16], in which the authors identified the key SO(2N ) symmetry as well as the N = 2 effective Hamiltonian, although there the focus was on the effect of measurements (see Refs. [17][18][19][20][21][22][23][24] for free fermions with measurement).
Under the nickname of Q-SSEP, charge-conserving noisy Hamiltonians have been addressed using a formalism based on a stochastic formulation in [25,26]. There, the dynamics of the N replica theory admits an SU(2N ) symmetry [27]. Because of the local charge conservation, these model systems allow for transport and can be driven out of equilibrium [28][29][30].
The model we study however is a model of noisy free fermions with no local conserved quantities. This is the simplest setting: since no conserved local densities exist, the only fields that will appear in the continuum theory are those encoding entanglement. The only conserved quantity in our model is global fermion parity, which is conserved in any local fermionic model.
After applying the replica formalism to this Majorana chain, we find that the resulting model is equivalent to an SO(2N ) ferromagnetic Heisenberg chain evolving in imaginary time. When calculating the purity of a section of the Majorana chain (N = 2), the resulting SO (4) Heisenberg chain can be reduced to an ordinary SU (2) chain, giving an intuitive picture of an (initially sharp) domain wall relaxing under imaginary time evolution.
This also allows us to reformulate the problem as a spin coherent state path integral in terms of a complex field z(x, τ ). By applying a saddle-point approximation, z(x, τ ) essentially becomes a classical field, giving us a spacetime picture of entanglement generation, in which the entanglement is identified with the classical action of a smooth "domain wall" configuration in spacetime.
However, an interesting feature of this picture, not anticipated from the interacting case, is that the field z(x, t) and its conjugate fieldz(x, t) must be treated independently. Both fields exhibit a kind of diffusive relaxation, but the field z relaxes in one time direction, while the fieldz relaxes in the opposite time direction (z andz also interact). In Figure 1 we show contour plots of z(x, τ ) andz(x, τ ) for the saddle-point solution relevant to calculating the purity (bipartite entanglement) of one half of the chain, where the boundary condition enforces a sharp domain wall in z(x) at the final time.
Once we have established this picture for free fermions, we include weak interactions, which induce a crossover to a distinct universality class for entanglement generation. In the replica approach, the free-fermion model has a continuous SO(2N ) symmetry which leads "diffusive" growth of entanglement. However, if we add weak 4-Majorana terms to our model (changing the freefermion system into an interacting fermion system), the continuous symmetry is explicitly broken down to a discrete symmetry. This may be taken into account in the saddle-point treatment. The initially sharp domain wall enforced by the boundary conditions relaxes to some extent, but reaches a steady state at a finite length scale l int , set by the strength of the interactions. At length scales l l int this is indistinguishable from a sharp domain wall similar to those found for strongly interacting systems, indicating a crossover from free behaviour to interacting behaviour in entanglement generation.
The physical consequence of this is a crossover from "diffusive" to "ballistic" spreading of information. A bipartite entanglement entropy equal to S implies (by counting degrees of freedom in an interval of a given size) that correlations exist over a length scale l that is also of order S. The diffusively relaxing domain wall leads to S ∼ √ t, so the length scale of correlations grows diffusively in the free case, whereas a sharp domain wall leads to S ∼ t, meaning the length scale grows ballistically, as in generic strongly interacting systems. This "ballistic" spreading of information should not be confused with ballistic motion of quasiparticles (which are of course not well-defined in the interacting system).
In many non-random free systems it has been possible to understand the scaling of entanglement via an intuitive picture in terms of the trajectories of quasiparticles [1,[31][32][33]. The picture that we obtain here for the noisy system is different, but we will make a heuristic connection with the quasiparticle picture in Section VIII. Contour plot of the two independent fields z andz involved in calculating the purity of rightmost half of the Majorana chain after time t using a saddle point approximation of the coherent states path integral. The coordinate x is the physical spatial coordinate. τ is a reversed time coordinate: i.e. τ = 0 at the final time t, and τ = t at physical time 0 when the system is prepared in a state of short range entanglement. The field z begins as a sharp domain wall at τ = 0 but relaxes diffusively as τ increases. The fieldz matches z at τ = t, but relaxes diffusively for decreasing τ . There are also interactions between the two fields. We focus on a 1D chain of Majorana modes interacting randomly with their nearest neighbours (as in [16], but without measurements). We wish to study how different regions of the chain become entangled over time, starting from an initial state with short-range entanglement. Our main focus will be on the dynamics of the purity Tr[ρ 2 A ] (see Sec. II C) for a region A which we will usually take to be the right half of the chain.
CONTENTS
There is a Majorana operator γ i for each site i, where i ranges from 1 to L (L even). The Majorana operators are Hermitian, γ † i = γ i , and obey the anti-commutation
relations {γ i , γ j } = 2δ ij .
The Hilbert space H has dimension 2 L/2 and is spanned by the simultaneous eigenstates of the operators iγ 2j−1 γ 2j , which have eigenvalues ±1. If we define L/2 pairs complex fermion operators c = (γ 2j−1 +iγ 2j )/2 and c † = (γ 2j−1 − iγ 2j )/2, we find iγ 2j−1 γ 2j = 2c † c − 1, so these are simply occupation number states. Alternatively, we can map the Majorana chain to a spin chain with L/2 sites and identify iγ 2j−1 γ 2j = Z j and iγ 2j γ 2j+1 = X j X j+1 (where X and Z are Pauli matrices).
We take our time-dependent Hamiltonian to be
H γ (t) = −i i η i (t)γ i γ i+1(1)
where the η i on each site are independent Gaussian noise terms with zero mean and with correlation func-
tion η i (t)η j (t ) = ∆ 2 δ ij δ(t − t ).
The time-evolution operator for a given realization of the noise is U η = T e −i dtHγ (t) , where T denotes timeordering. For convenience we will Trotterize the time evolution to get a "brick wall" unitary circuit as shown in Figure 2, where the gates are independent random two-site unitaries of the form
u(η) = exp(−ηγ i γ i+1 )(2)
and η is a Gaussian random variable chosen independently for each gate with zero mean and variance ∆ 2 δt. The number of double layers of the brickwall circuit is t/δt.
B. Replicas
For any given realization of noise terms η, the system undergoes unitary evolution U η . We will be interested in the averages of physical quantities over realizations.
The simplest quantity we can study is the expectation of some observableÔ. If we prepare our system in the initial state |ψ and this evolves over time into some new state U η |ψ , then the averaged expectation value is Ô = ψ|U † ηÔ U η |ψ , where the overline denotes the average over noise.
Since U η and U † η are not independent, it is useful to "fold" this expression as shown in Figure 3 to get a unitary circuit with two layers, one being the original circuit U η and the other being its complex conjugate U * η (with respect to some orthonormal basis |ψ n ). This double-layer circuit acts on two replicas of the original system, or in other words on a ket |ψ ⊗ |ψ * in the replicated Hilbert space H ⊗ H. The expectation value Ô is then a linear function of this ket, so that the oper-atorÔ is effectively a bra Ô |, and ψ|Ô|ψ becomes Ô |(|ψ ⊗ |ψ * ). For example, the identity operator 1 becomes 1| = n ψ n | ⊗ ψ n |, where |ψ n are the basis vectors. (Note that we will use Dirac notation |• for kets in both physical and the replicated Hilbert spaces.)
The double-layer circuit is made up of gates of the form u ⊗ u * . Taking the average over disorder is now straightforward because these gates are independent from one another for different points in spacetime. To find the expectation value of some operatorÔ averaged over noise, we simply start with the replicated initial state |ψ ⊗|ψ * , evolve deterministically using the averaged replicated gates u ⊗ u * , and then find the overlap with Ô |.
We can use a similar trick to calculate the average over noise of the N th moment of the expectation value, Ô N , by considering a circuit with 2N layers acting on 2N replicas of the system. We can label these replicas with the index a = 1, . . . , 2N , where the original unitary circuit U η is applied for odd a and its complex-conjugate U * η for even a. In this case, the identity operator acting on every pair of replicas is
1| = n ψ n | ⊗ ψ n | ⊗N .
(3)
C. Entanglement entropy
Our main focus will be the way in which quantum correlations are generated between different parts of the system by the random unitary dynamics. We quantify this by the entanglement entropy of a subsystem A -for example, the rightmost half of the chain i > L/2. If we prepare the chain in a state with short-range entanglement, this entropy will initially be small, but it will grow over time as the couplings entangle different parts of the chain. Here we recall the expression for the Rényi entanglement entropies in the folded representation (readers familiar with this may wish to skip ahead).
For any integer N > 1, the N th Rényi entanglement entropy for a subsystem A is defined as
S N = − 1 N − 1 ln Tr[ρ N A ](4)
where ρ A is the reduced density matrix of the subsystem (the system as a whole is in a pure state). The limit N → 1 gives the von Neumann entanglement entropy. The quantity Tr[ρ N A ] does not correspond to the expectation value of any conventional observable because it is not linear in the reduced density matrix ρ A . However, it does correspond to the expectation value of an operator acting on N replicas of the system [2,34]. This operator, which we denote by C A , performs a cyclic permutation of the replicas within the subsystem A, while leaving everything outside the subsystem unchanged. (For simplicity, we will always assume that A and B each contain an even number of Majorana modes so that there is a well-defined Hilbert space in each subsystem.)
In the "folded" representation described above, we now consider a 2N -layer unitary circuit that acts on 2N replicas of the system. The expectation value of C A becomes a bra C A |, with Tr[ρ N A ] = C A |(|ψ ⊗ |ψ * ⊗ · · · ⊗ |ψ * ). The bra C A | is a product state of the single-site identity bra 1| on every site in B, and the single-site bra C| which cycles replicas on every site in A:
C A | = i 1| for i / ∈ A C| for i ∈ A,(5)
where (taking ψ n | to be an orthonomal single-site basis)
C| = {ni} ψ n N | ⊗ ψ n1 | ⊗ ψ n1 | ⊗ ψ n2 |⊗ · · · ⊗ ψ n N −1 | ⊗ ψ n N |(6)
We can alternatively think of the operator C A as performing an index contraction, as illustrated in Fig. 4 for the case N = 3. Outside of the subsystem A, the state C A | acts like the identity state 1|, which simply contracts each replica a = 2m − 1 with its complex conjugate a = 2m for all m = 1 . . . N (generalizing the contractions at the top of Fig. 3). Inside subsystem A, the operators C A permutes the odd indices in a cycle before Each row of sites, labelled by the replica index a, represents one the 2N = 6 replicas arising from ρ ⊗N . Outside subsystem A (in its complement B), each replica is contracted with its conjugate. This takes the partial trace of each of the N copies of the density matrix to give ρ ⊗N A . Inside A, every even replica is contracted with the "next" replica to give ρ N A and the last replica is contracted with first to give Tr[ρ N A ]. In the "folded" representation, this entire index contraction becomes a bra CA|.
contracting the replicas in these pairs, so it contracts a = 2m with a = 2m + 1 for m = 1 . . . N − 1 and contracts a = 2N with a = 1: see Fig. 4.
Finally, it is useful to characterize these states in the Majorana language. In Appendix A we show that if the bra C A | contracts two replicas a and b of a Majorana chain at site i, this simply means that C A | is an eigenstate of −iγ a i γ b i with eigenvalue +1 (when a < b). Given that there are N such independent stabilizers at each site, and the local Hilbert space dimension is 2 N , these stabilizers uniquely determine the state C A | up to a phase (the phase is then fixed by the fact that Trρ N A must be real and positive).
III. HEISENBERG MAPPING
We now return to the noisy Majorana Hamiltonian. We show that taking the average over noise in the folded representation with 2N replicas gives an SO(2N ) Heisenberg model evolving in imaginary time, similarly to [16].
A. Average over noise
First we need the form of the averaged gates. Using lower indices for sites and upper indices for replicas, the replicated gates take the form
u(η) = exp −η N m=1 γ 2m−1 i γ 2m−1 i+1 − γ 2m i γ 2m i+1(7)
where the minus sign comes from taking the complex conjugate on even replicas, in a basis where iγ a i γ a i+1 are all real. (The choice of basis is a matter of convention, but this choice was natural given the standard Ising mapping iγ 2j−1 γ 2j = Z j and iγ 2j γ 2j+1 = X j X j+1 .) However, this minus sign is inconvenient, so we will absorb it into the definition of the Majorana operators. We redefine the Majorana operators on even sites and even replicas via γ 2m 2j → −γ 2m 2j . This us gives us
u(η) = exp −η 2N a=1 γ a i γ a i+1(8)
This change reveals an SO(2N ) symmetry under replica rotations which we discuss in section III C. This symmetry does not depend on the fact that the lattice is bipartite: 1 SO(2N ) symmetry arises for any quadratic Majorana Hamiltonian, and therefore for any free-fermion Hamiltonian, since complex fermions may always be expressed in terms of Majoranas. If the Hamiltonian has additional symmetries, such as charge conservation, the replica symmetry may be still larger. (If the evolution becomes non-unitary, say due to measurements, then the symmetry is reduced [16] unless an appropriate bipartite structure is preserved [24].) Taking the average over noise gives
u = exp ∆ 2 δt 2 2N a=1 γ a i γ a i+1 2 (9) = exp −∆ 2 δt N − 2N a<b A ab i A ab i+1(10)
where we have defined the single-site Hermitian operators
A ab i ≡ − i 2 [γ a i , γ b i ](11)
These operators can be viewed as generators of the so(2N ) Lie algebra (Sec. III C). They commute on different lattice sites, [A ab i , A cd j ] = 0 if i = j, and square to the identity, (A ab i ) 2 = 1. In order to recover our original continuous-time model (1), we take the limit δt → 0. In this limit, the unitary gates acting during a given time-step effectively commute, so the (replicated and averaged) system evolves deterministically in imaginary time with the effective Hamiltonian
H = ∆ 2 i N − 2N a<b A ab i A ab i+1 .(12)
This is a ferromagnetic Heisenberg model, as we discuss in Sec. III C. Before that, we briefly describe an alternative formulation of the averaged dynamics.
B. Alternative stochastic description
The dynamics generated by the Hamiltonian (1) and its discretized version can be viewed as a stochastic dynamics on the quantum many-body Hilbert space, as in the model systems discussed in [25]. Let us elaborate briefly on this connection.
The stochastic unitary evolution U t;t+dt = U t+dt U −1 t between time t and t + dt on the Majorana Hilbert space generated by the Hamiltonian (1) reads U t;t+dt = e −idHt with Hamiltonian increment dH t ,
dH t = j E j dB j t ,(13)
with E j = −iγ j γ j+1 and where dB j t are Brownian increments normalized to dB j t dB k t = δ j,k ∆ 2 dt (formally dB j t = t+dt t η j (s)ds). The quantum state or density matrix ρ t evolves during the time interval [t, t + dt] according to ρ t+dt = e −idHt ρ t e +idHt , so that its increment dρ t := ρ t+dt − ρ t is, up to order O(dt 3/2 ),
dρ t = −i[dH t , ρ t ] − 1 2 [dH t , [dH t , ρ t ]].(14)
This way of writing the equation of motion implicitly assumes that the Brownian increments dB j t are sampled from time t to t + dt, independently of the evolution of the density matrix ρ t up to time t. This is the Itô convention for stochastic calculus. This is also the convention taken in the discretized random unitary circuit of Fig. 2 in which the Gaussian random variables η j , with variances ∆ 2 δt and independently updated at each step, represent the Brownian increments dB j t . The first term in (14) is a purely noisy contribution, since the Hamiltonian increment (13) contains no deterministic part, whereas by the Itô rule dB j t dB k t = δ jk ∆ 2 dt mentioned above, the second term is a purely deterministic drift. Since dH t and ρ t are independent and dH t has zero mean, the first term in (14) vanishes in average, and the time evolution of the mean density state ρ t possesses a standard Lindblad form and reads
dρ t = − 1 2 [dH t , [dH t , ρ t ]] =: L(ρ t ) dt,(15)
where the last equality defines the (one-replica) Lindblad operator L. A similar structure also emerges when looking at higher moments of the density matrix, and hence at higher replicas. The N -times replicated state ρ ⊗N t evolves randomly but unitarily with
ρ ⊗N t+dt = e −idH (N ) t ρ ⊗N t e +idH (N ) t ,(16)
with N th replica Hamiltonian increments dH
(N ) t = j E (N ) j dB j t , where E (N ) j
= α E α j , and α (which runs over N values, in contrast to the indices a, b in in Eq. (12) which run over 2N values) indexes the replicas of the density matrix. 2 Each of the replicas is coupled identically to the noise through the operators E j , and the replicas are non-interacting before any average.
As a consequence, the replicated density matrix ρ ⊗N t evolves according to the SDE (14) but with dH t replaced with dH
(N ) t
. Taking then the average, we learn that
dρ ⊗N t = − 1 2 [dH (N ) t , [dH (N ) t , ρ t ]] =: L (N ) (ρ ⊗N t ) dt. (17)
The last equality defines the N -replica Lindblad operator:
L (N ) (•) = − ∆ 2 2 j α,α E α j , E α j , • .(18)
By construction, L (N ) coincides with the Hamiltonian (12) which is thus a Lindblad operator (but acting on density matrices on the physical Hilbert space replicated N times). Actually, the equation of motion (14) is simply a classical stochastic differential equation (SDE) but on manybody quantum density matrices. Like any SDE, it is associated with a Fokker-Planck operator. Due to the linearity of (14), this Fokker-Planck operator preserves polynomials in ρ of any fixed degree, and its restriction to monomials of order N is the Lindblad operator L (N ) , or equivalently H as in (12).
By an argument similar to that used in the case of noisy complex fermionic chains [26], this description leads to a simple understanding of the invariant probability distribution, reached at large time (for finite L). Indeed, the operators E j generate the Lie algebra so(L) by iterative commutation relations. These multiple commutators form a representation of so(L) on the Majorana Fock space which decomposes into two irreducible so(L) representations depending on the chirality (L is even). As a consequence, iterations of multiple products e −idHt k , for series of times t 1 , t 2 , · · · , t p , that is
e −idHt 1 e −idHt 2 · · · e −idHt p ,
span the group SO(L). 3 Since the invariant measure is, by definition, stable under the dynamics generated by dH t , it is SO(L) invariant. Since the SO(L) action is faithful on its spin representations, this invariant measure is thus identical to the measure induced by the SO(L) Haar measure on each of the two irreducible SO(L) components of the Majorana Fock space. 2 Alternatively, E (N ) j = E j ⊗ 1 · · · ⊗ 1 + · · · + 1 ⊗ · · · 1 ⊗ E j , with the tensor structure reflecting the replicas. 3 More precisely, they form a dense set of operators on the image of SO(L) by the representation map on the Majorana Fock space.
C. Symmetry group
Let us return to the effective Hamiltonian (12). The operators A ab are a generalization of Pauli matrices in that we can define the operators J ab ≡ 1 2 A ab which have eigenvalues ± 1 2 and have angular momentum commutation relations
[J ab , J cd ] = i(δ ac J bd + δ bd J ac − δ ad J bc − δ bc J ad ). (19)
The J ab therefore function as angular momentum operators in 2N -dimensions, i.e. they form a representation of an so(2N ) Lie algebra. The operator J ab generates rotations in the ab-plane.
The Hamiltonian in Eq. (12) is a ferromagnetic Heisenberg model, with SO(2N ) global symmetry [16]. The constant term ensures that the ground state energy of the model is zero. This is necessary: as we will see below, expectation values in the Majorana chain map to transition amplitudes in the ferromagnet, and the existence of a state with negative energy would imply that expectation values could grow indefinitely over time, even in a finite system. We also know that the state 1| corresponding to the identity operator satisfies 1|e −Ht = 1|, meaning it is an energy eigenstate with zero energy and therefore a ground state. The SO(2N ) symmetry mentioned above means that for N > 1 the number of ground states increases with L, and these ground states approach a continuous manifold of states in the thermodynamic limit. The existence of this continuous manifold of ground states is in stark contrast to generic interacting systems, where the discrete replica symmetry guarantees only a discrete family of ground states, as we discuss in Sec. VII B.
For each site on the original Majorana chain, the replicated system has 2N Majorana operators (one for each replica), so the local Hilbert space dimension is 2 N . Viewed as matrices on this space, the generators A ab therefore form a 2 N -dimensional representation of so(2N ). This is the bispinor representation of so(2N ) (the Euclidean equivalent of a Dirac spinor in 2N dimensions). This representation is reducible: it is the direct sum of two chiral spinor representations (the equivalent of Weyl spinors). These 2 N −1 -dimensional representations are irreducible. (As representations of SO(2N ), they are projective.) We can define the chirality operator on a site
χ = (−i) N 2N a=1 γ a(20)
which has eigenvalues ±1 and commutes with the generators J ab (here we multiply the Majorana operators from left to right γ 1 γ 2 γ 3 . . . ). States with χ = +1 transform under one of the spinor representations, while states with χ = −1 transform under the other representation.
D. Special cases
We can understand the implications of the mapping to the ferromagnetic chain by focussing on some special cases:
1. N = 1: Expectation values
The simplest case is N = 1. This tells us about the expectation values of observables averaged over noise. In the folded representation, the unitary circuit has two layers labelled a = 1, 2, and the ket for the replicated system represents the physical density matrix.
The local Hilbert space dimension of the chain is 2 N = 2 and there is only one generator A 12 i ≡ A i at each site, with eigenvalues ±1. We can think of there being a spin-1 2 at each site i, with A i acting like the Pauli matrix σ z i . The Hamiltonian
H = ∆ 2 i [1 − A i A i+1 ](21)
then simply describes an Ising model with no applied fields. Say we prepare our Majorana chain in the initial state |ψ and want to calculate the average over noise of some operator Ô (t) . By the discussion in Sec. II B, this is given by Ô |e −Ht |ψ , where |ψ = |ψ ⊗ |ψ * is the replicated initial state (and |ψ * is complex conjugated with respect to our fixed basis, in which iγ i γ i+1 is real).
Instead of thinking about this in the "Schrodinger picture", where we evolve the replicated initial state |ψ to get e −Ht |ψ , it is more convenient to think about it in the "Heisenberg picture", in which we evolve the bra, Ô | → Ô |e −Ht . This has the advantage that once we have calculated Ô |e −Ht , we can easily calculate Ô (t) for any initial state. For this to be useful, we need a way of expressing Ô | in terms of Ising configurations (i.e. simultaneous eigenstates of {A i }). In Appendix B, we show that if the operatorÔ is a product of distinct Majorana operators, then Ô | corresponds to a single Ising configuration, with
A i = −1 if γ i appears inÔ, and A i = +1 otherwise.
From this it is clear that all such states Ô | decay exponentially over time except the two ground states of the Ising model, which have zero energy and are therefore unaffected by time evolution. The ground state with all A i = +1 corresponds to the identity operator 1, which simply measures the normalization of the state ψ|ψ . The ground state with all A i = −1 corresponds to the fermion parity operator U p = (−i) L/2 i γ i , which is also conserved by the dynamics.
All other operators decay exponentially over time. For example the expectation value iγ i γ i+1 corresponds to the Ising state ↑ · · · ↑↓↓↑ · · · ↑| which has energy 4∆ 2 , so its noise average will decay over time from its initial value as iγ i γ i+1 t = e −4∆ 2 t iγ i γ i+1 0 . The expectation value of iγ i γ i+2 corresponds to ↑ · · · ↑↓↑↓↑ · · · ↑| which has energy 8∆ 2 so iγ i γ i+2 t = e −8∆ 2 t iγ i γ i+2 0 .
N = 2: Purity
A more interesting case, and the focus of the rest of this paper, is N = 2. This will allow us to calculate the averaged purity of a subsystem
Tr[ρ 2 A ] = e −S2(22)
where S 2 is the second Rényi entropy of entanglement.
To simplify the notation, let us define the quantity
S 2 = − ln Tr[ρ 2 A ].(23)
Here the average of the purity is taken before taking the log, which meansŠ 2 = S 2 in general. However, we fill find empirically that fluctuations in S 2 are sufficiently small that the two kinds of average coincide to leading order (see comment at the end of this section). The unitary circuit now has 2N = 4 layers, and the local Hilbert space dimension of the replicated system is 2 N = 4. The generators A ab i form a representation of the Lie algebra so(4) at each site, which is equal to su(2) ⊕ su(2). The "spins" at each site are Dirac spinors, composed of two Weyl spinors with chirality χ = ±1. These 2-component Weyl spinors are nothing but regular spin-1 2 spins. In other words, the four-dimensional Hilbert space at a site splits into two doublets, each of which is a spin-1 2 under one of the two SU(2) groups, and invariant under the other SU(2) (see Appendix C).
To calculate the purity of a subsystem A after time t, we can start with the state C A | (corresponding to the operator which swaps replicas within the subsystem), evolve for imaginary time t using the Hamiltonian (12), and finally overlap the resulting state with the some replicated initial state |ψ . Let us consider these three elements one at a time.
The state C A | corresponds to two different local operations: outside the subsystem A, it acts like the identity operator, while inside A it swaps replicas. In Appendix D we show that both of these local operations map to spins with χ = +1. Given that such spins transform non-trivially only under so(3) generators, we can think of them as having directions in 3-dimensional space. We choose generators such that the identity operator maps to spin "up" ↑|, while the swap operator maps to spin "right" →|.
The Hamiltonian in eq. (12) commutes with the chirality operator χ i on every site, so each χ i is individually conserved. This means that if the initial state of the replicated system has χ i = +1 on every site, then this will be true at all times, and we can restrict the Hilbert space to these states of positive chirality. These spins have a local Hilbert space dimension of 2 and transform non-trivially under only one set of so(3) generators. The Hamiltonian (12) becomes:
H = 2∆ 2 i [1 − σ i · σ i+1 ](24)
where σ i = (σ x i , σ y i σ z i ) are the Pauli matrices for the positive chirality spins. This is simply a standard ferromagnetic su(2) Heisenberg model.
Finally, we need the replicated initial state |ψ . For simplicity, we will choose a specific initial state with short-range entanglement. We choose the state which is a simultaneous eigenstate of iγ 2j−1 γ 2j with eigenvalue +1 for all j. In this state, the Majorana modes are maximally entangled in pairs. In Appendix E we show that replicating this state, and projecting onto the subspace with χ i = +1, gives a product state of entangled pairs of the form
|ψ + ≡ (|↑↑ + |↓↓ ) ⊗L/2 .(25)
Note that this state is invariant under rotations in the xz-plane, i.e. rotations generated by i σ y i . Putting this all together we get
Tr[ρ 2 A ] = e −Š2 = C A |e −Ht |ψ +(26)
where C A | is a product state of ↑| outside subsystem A and →| inside subsystem A. If we take the subsystem A to be the rightmost half of the system, we can write this schematically as C A | = ↑ · · · ↑→ · · · →|.
Higher replicas
As mentioned above, the quantityŠ 2 is not generally equal to the average second Rényi entropy S 2 , given that we are taking the average over disorder of e −S2 before taking the log. To learn more about the distribution of S 2 we could calculate higher moments of the purity, of the form
e −kS2 = Tr[ρ 2 A ] k .(27)
This is the generating function for S 2 . For integer k ≥ 1, it can be written using 2kN replicas of the Majorana chain, which can be mapped to the SO(2kN ) Heisenberg model in Eq. (12). However, as explained later, numerical evidence sug-gestsŠ 2 and S 2 are in fact equal to leading order in t. This means that many of the results we derive forŠ 2 will also apply to S 2 . The replica approach could also be used to compute Rényi entropies S n for other n, but in this paper we will stick to S 2 .
IV. ENTANGLEMENT PURITY: SADDLE POINT APPROXIMATION
Eq. (26) tells us that the growth in entanglement of a subsystem (or the decrease in its purity) is given by an imaginary-time transition amplitude in the Heisenberg chain.
In the state C A |, the Heisenberg chain is locally in a ground state both inside and outside of subsystem A, but with sharp domain walls at the boundaries of A. Imaginary-time evolution suppresses high energy states, so after sufficiently long times the evolved state C A |e −Ht should be locally close to a ground state everywhere (but no longer normalized). The ground states of a Heisenberg ferromagnet are product states of identical spins (or "cat" states formed from superpositions of such states). We therefore expect that, after sufficiently long times, the state is close to a product state, with the spin direction slowly varying with position. Over time, this product state should slowly relax to become smoother and smoother, and in a finite system should approach a uniform spin direction.
This intuitive picture suggests that, for long times ∆ 2 t 1, quantum fluctuations are small for most of the trajectory. This motivates applying a saddle-point approximation to Eq. (26).
In fact, it turns out that thinking in terms of C A |e −Ht as a product state evolving semiclassically is too simplistic. A saddle-point calculation is indeed valid in the regime ∆ 2 t 1, but it requires us to consider both "forward" and "backward" evolving fields as discussed below.
A. Coherent state path integral
In order to use a saddle-point approximation, we formulate the right-hand side of (26) as a coherent state path integral following Refs. [35,36]. We define singlespin coherent states in terms of the complex variable z
|z = (1 + z * z) − 1 2 z 1 (28)
which is just a stereographic projection from the Bloch sphere to the complex plane. We have chosen a basis such that the "south pole" of the Bloch sphere corresponds to z = 0 and the xz-plane corresponds to real z. A spin with azimuthal angle θ and polar angle φ is described by z = e −iφ cot θ 2 . We will first recall [35] how the coherent state path integral for a single spin is used to compute the coherent state propagator:
K(z I , z * F , t) = z F |e −Ht |z I .(29)
The single-spin identity operator may be resolved as
1 = dµ|z z|,(30)
where the measure
dµ = 2 π d 2 z (1 +zz) 2(31)
is uniform over the Bloch sphere. Here d 2 z is dadb where z = a + bi andz = a − bi. By inserting many such resolutions into the right-hand side of (29), we get a path integral form for the propagator
K(z I , z F , t) = z(t)=z * F z(0)=z I Dµ exp(−S)(32)
with an action S given by
S = − 1 2 dτż z −zż 1 +zz + dτ H(z,z) + S bdry(33)
This action contains a Berry phase term, a term resulting from the Hamiltonian H(z,z) ≡ z|H|z , and a term S bdry from the initial and final-time boundaries given by
S bdry = − 1 2 ln (1 + z * F z(t))(1 +z(0)z I ) (1 + z * F z F )(1 + z * I z I ) .(34)
When this path integral is treated via saddle-point, it is crucial to treat z andz as independent in searching for saddle-point trajectories, so that in generalz = z * [35] (or equivalently a and b as defined below Eq. (31) become complex). Note that boundary conditions are fixed at opposite boundaries for the two fields: z is fixed only at τ = 0 by the initial state |z I , so that z(0) = z I , whereasz is fixed only at τ = t by the final state z F |, so thatz(t) = z * F . The above formulation is for a single spin, but can be extended immediately to multiple spins. We simply sum over Berry phase terms and boundary terms in the action, and replace the single-spin Hamiltonian with the full many-body Hamiltonian from eq. (24). In the continuum limit (taking the lattice spacing to be 1 and replacing i with dx) this gives us (35) where the continuum Hamiltonian H(z,z) is given by
S = − 1 2 dxdτż z −zż 1 +zz + dτ H(z,z) + S bdryH(z,z) = 4∆ 2 dxz z (1 +zz) 2(36)
(with denoting a spatial derivative, and suppressing arguments whenever possible) and the boundary term is now
S bdry = − 1 2 dx ln (1 + z * F z(t))(1 +z(0)z I ) (1 + z * F z F )(1 + z * I z I ) .(37)
B. Equations of motion
In general, this path integral cannot be evaluated exactly. However, the intuitive argument above suggests that when ∆ 2 t 1, we should be able to make use of a saddle-point approximation, and indeed this will be justified below (Sec. V D). In the saddle-point approach we consider the single trajectory for which action is stationary and approximate
K(z I , z * F , t) ≈ K fl (z I , z * F , t)e −S(38)
where S is the action of the stationary trajectory and K fl comes from integrating over quadratic fluctuations around this stationary trajectory. Here the arguments z I and z F that specify the coherent states at the initial and final times are functions of position, z I (x) and z F (x). If we vary the action S while keeping the boundary conditions fixed and set this variation to zero, we find that the trajectory must obey the equations of motioṅ
z = +4∆ 2 z − 2z(z ) 2 1 +zz ,(39)z = −4∆ 2 z − 2z(z ) 2 1 +zz ,(40)
with the boundary conditions z(x, 0) = z I (x) and z(x, t) = z * F (x). These equations look like diffusion equations with extra terms. Note that, due to the opposite signs in the two equations, z diffuses in the positive time direction, whilez diffuses in the negative time direction.
C. Boundary conditions and the classical trajectory
The equations of motion (39) and (40) enforce local constraints on the trajectory, but to find the unique classical trajectory we must consider the initial and final boundaries.
Given that C A |e −Ht |ψ + = ψ+ |e −Ht |C A , we are free to consider our "initial" state at τ = 0 to be |C A and our "final" state at τ = t to be |ψ + . That is, we will use a coordinate τ that is reversed with respect to physical time. This is convenient because the state |C A = |↑ · · · ↑→ · · · → will impose a sharp domain-wall boundary condition for z(x, τ ) at τ = 0.
Since the symmetry between |↑ and |→ is not immediately apparent in the coherent states parameterization (28), it will be convenient to make a symmetry rotation of our coherent states basis so that the domain wall boundary condition at τ = 0 becomes z = − tan(π/8) on the left and z = + tan(π/8) on the right.
While the initial state |C A is already a product of coherent states, and translates directly into a boundary condition for z(x, τ ) at τ = 0, the final state |ψ + is not (it is a product of entangled pairs of spins). Therefore we cannot immediately employ the above result for the propagator (29).
However, we can evaluate the overlap ψ+ |e −Ht |C A by writing it as dµ w ψ+ |w w|e −Ht |C A , where |w denotes a product of coherent states with parameters w i , and dµ w is the product of the measures for these states. Treating w i , w i as extra degrees of freedom in the path integral, we see that they become the boundary values of the fields,
w i = z i (t) andw i =z i (t)
. We can absorb the extra factor from the overlap with ψ+ | into the action via
ψ + |z(t) = exp Ψ(z(t))− 1 2 i ln(1+z i (t)z i (t)) ,(41)
where the function Ψ(w) = Ψ(w 1 , . . . w L ) depends only on w (notw), and the second term comes from the normalization of the coherent state.
Varying the action including these terms, the bulk saddle point equations (39,39) are supplemented by the boundary equation
∂Ψ(z(t)) ∂z i (t) =z i (t) 1 +z i (t)z i (t) ,(42)
which comes from variation with respect toz i (t). This equation provides a relationship between the boundary values z(t) andz(t) for the two fields at τ = t, given the function Ψ(w), which is in turn determined by the boundary state |ψ .
In the present case the boundary condition at τ = t (which corresponds to the physical initial time) becomes remarkably simple. In Appendix G we show that our specific initial state |ψ has the effect of swapping the spins on neighbouring sites at the τ = t boundary. More precisely, the boundary constraint (42) is solved by settingz
2j−1 (t) = z 2j (t),z 2j (t) = z 2j−1 (t).(43)
The swapping of the spatial indices here is significant at microsopic timescales. But at late times ∆ 2 t 1 the spins are slowly varying with position, and it is sufficient to approximate the equations above as 4
z(t) = z(t).(44)
(We are suppressing spatial arguments, so this really meansz(x, t) = z(x, t).) Together with the domain wall boundary condition on z(x, 0), Eq. 44 for the boundary values of z(x, τ ) and z(x, τ ) at τ = t is sufficient to fully specify a solution to the saddle point equations.
Note that, if we were to neglect the nonlinear terms in the equations of motion Eqs. (39), (40), then Eq. (44) would convert the problem into a diffusion process of duration 2t for a field φ(τ ), with φ(τ ) = z(τ ) for 0 ≤ τ ≤ t, and φ(τ ) =z(2t − τ ) for t ≤ τ ≤ 2t. That is, the field z(τ ) that is diffusing in the "forward" time direction is reflected back from the boundary condition as the field z(t) that diffuses as we progress "backward". To find S, the equations of motion are solved numerically, then these solutions are substituted into (35). As expected, S initially grows rapidly, before eventually approaching a constant.
One important consequence of the boundary condition in Eq. (44) is that z andz can be chosen to be real. If z(x, τ = 0) is real for all x, then eqs. (39), (40) imply that z andz are real everywhere.
We have converted the problem from one with both initial and final boundary conditions to one with just an initial boundary condition. If we fix z I and then minimize the action S while allowing z F to vary, we find that S is minimized when z * F = z(t). When this condition is satisfied, the boundary action (including the contribution from ψ+ |z(t) ) becomes
S bdry = − 1 2 dx ln 1 +z(0)z I 1 + z * I z I(45)
which depends only on the initial configuration z I (given by C A |) and the value ofz at τ = 0. Once the saddlepoint is found, the total action is computed using Eq. (35) with boundary term Eq. (45). This action then gives the saddle-point approximation toŠ 2 , by Eq. (26) .
V. APPLICATIONS OF THE SADDLE POINT APPROXIMATION
The saddle point approximation introduced in the previous section can be used to calculate the purity of arbitrary subsystems as a function of time, in chains with either open or periodic boundary conditions. For concreteness, we will focus on a chain of length L = L A +L B with open boundaries, where the subsystem A is the rightmost L A sites. We will normally fix L A = L B for greater symmetry. This will allow us to study how bipartite entanglement grows over time and eventually saturates, on a timescale L 2 /∆ 2 , if the system is finite.
In Figure 5 we plot the saddle-point prediction for S against ∆ 2 t for the sharp domain wall boundary condition on z I (x) described above, with L A = L B = 50 sites either side of the discontinuity. We calculate S by numerically solving the spatially discrete versions of the equations of motion (39) and (40) and substituting into the action (35).
The discretization we use is the natural one given by the lattice Hamiltonian. To numerically solve the equations, we initially guessz = 0 everywhere and evolve z forward to generate a guess for z. We then evolvez backwards using this guess for z, to generate a new guess for z, and repeat this procedure until the guesses are consistent.
In this section we will discuss in turn: the equilibrium entanglement of the finite chain and the approach to this equilibrium as ∆ 2 t L 2 ; the infinite chain and ∆ 2 t L 2 ; the interpolation between these two limits and a post hoc justification for the saddle point approximation. In the next section, we compare the predictions of the saddle point approximation to direct numerical simulations of the Majorana chain.
A. Equilibrium entanglement
As a sanity check we can use the saddle-point approximation to calculate the averaged purity in an equilibrium "infinite temperature" pure state. To do this, we take a finite system of length L and take t → ∞. We take L to be a multiple of four and take the subsystem to be rightmost L/2 Marojanas. In the continuum picture, we take our system to lie in the interval −L/2 < x < L/2 and our subsystem to lie in the interval 0 < x < L/2. The initial boundary condition is that z I (x) is a step function with azimuthal angle θ = 0 for x < 0 and θ = π 2 for x > 0 (where the azimuthal angle is defined by z = e −iφ cot θ 2 ). For t → ∞, z andz relax to a trivial steady state as τ → ∞. The only steady state consistent with the boundary conditions (including the spatial boundaries) is one in which z =z and they are spatially uniform, that is, they both approach some constant value. We can always choose a basis such that this constant is zero by rotating around the y-axis (this will leave z andz real because real z corresponds to the xz-plane). In the symmetrical case L A = L/2, we rotate by 3π 4 so that our initial domain wall has θ = 3π 4 for x < 0 and θ = 5π 4 for x > 0. This means taking z I (x) = sgn(x)c where c ≡ tan π 8 = √ 2 − 1. Ifz(t) = 0, then the equation of motion (40) means thatz is also zero at all earlier times.
Switching to a basis wherez = 0 everywhere means that both the Berry phase term and the Hamiltonian term in eq. (35) are zero. The action is determined solely by the boundary term (45). Substituting inz(0) = 0 and z * I z I = c 2 gives
S eq = L 2 ln(1 + c 2 )(46)
The saddle-point prediction equilibrium purity is therefore
Tr[ρ 2 A ] ≈ K ∞ fl (1 + c 2 ) −L/2(47)
where K ∞ fl comes from integrating over fluctuations around the stationary path (note that these quantum fluctuations are not directly related to the fluctuations in S 2 over different realizations of the noise η). The predicted purity is exponentially small in L as expected.
In the Appendix F, we calculate the equilibrium purity in the thermodynamic limit more carefully for L A = L/2 by including fluctuations around the saddle point solution. We find that the equilibrium purity approaches
Tr[ρ 2 A ] = (1 + c 2 ) −(L+1)/2 (48) corresponding to K ∞ fl = (1+c 2 ) −1/2 , so K fl remains O(1) as L → ∞.
We can generalize the saddle-point prediction eq. (47) to arbitrary subsystem size L A (and L B = L − L A )
S eq = L A 2 ln(1 + c 2 − ) + L B 2 ln(1 + c 2 + )(49)
where c 2
± = (3 − q 2 − 2 2 − q 2 )/(1 ∓ q) 2 with q = (L A − L B )/L.
B. Approach to equilibrium
Having computed the steady state entanglement, we consider the approach to it.
In the limit t → ∞, we foundz → 0 (in the natural basis). We should expect that for large but finite t,z will be small but non-zero, and that we can expand the action (35) in powers ofz. To leading order inz this gives
S ≈ − 1 2 dτ dx [żz −zż] + 4∆ 2 dτ dxz z + S bdry ,(50)S bdry = − 1 2 dx ln 1 +z(τ = 0)z I 1 + z * I z I .(51)
If we vary this action with respect to z andz and set it to zero, we get the following diffusion equations:
z = +4∆ 2 z ,(52)z = −4∆ 2z .(53)
As discussed around Eq. (44), for late times ∆ 2 t 1 we take the boundary condition at τ = t to be z(t) =z(t). Therefore, z simply "reflects" off this boundary asz: The initial domain wall z(x, 0) diffuses forwards for time t as z(x, τ ), then backwards for a further time t asz(x, τ ). That is, diffusion for a total time 2t transforms z(x, 0) intoz(x, 0). Assuming these equations of motion hold, we can substitute them into the action (50). Integrating by parts, the Berry phase term perfectly cancels the Hamiltonian term, so the action is once again determined solely by the boundary terms.
By solving the diffusion equation exactly for the sharp domain wall, keeping only the slowest mode, and substituting into eq. (50), we get
S ≈ S eq − 4Lc 2 π 2 exp −8π 2 ∆ 2 t L 2(54)
where S eq = L 2 ln(1 + c 2 ) and we have used the fact thatz is small to approximate the boundary term ln(1 +z(0)z(0)) ≈z(0)z(0). The terms we have dropped are of orderz 2 and are therefore exponentially smaller in t than the last term in eq. (54).
In Figure 6, we compare eq. (54) to numerical solutions of the full equations of motion (39) and (40) substituted into the full action (35).
We can understand this exponential approach to S eq by conisdering the energy spectrum of the spin chain. As the imaginary time t → ∞, the approach to the ground state is dominated by the excited state with the lowest energy. The spin chain is a Heisenberg ferromagnet and therefore has quadratic dispersion, meaning it has an energy gap δE ∝ 1/L 2 and this contribution decays as exp(−δE · t) with δE = 8π 2 ∆ 2 L 2 (similar to [27]).
C. Entanglement growth at early times
In the two previous subsections, we assumed that the system size L was finite, and that ∆ 2 t L 2 so that correlations had been able to spread throughout much of the system (the entanglement was comparable to the equilibrium entanglement). This allowed us to assume thatz is small everywhere. As L becomes larger, it takes indefinitely long to reach this regime, and if L → ∞ then we cannot use this approximation at all. On the other hand, as L → ∞, the length-scale L drops out of the problem, because for ∆ 2 t L 2 , the physical boundaries of the system are not important. We can use scaling arguments to understand how entanglement grows in this regime.
Let's focus on the case where the system is an infinite chain and we are interested in the purity of the semi-infinite subsystem x > 0. We will also assume that ∆ 2 t 1 so that we can take the τ = t boundary condition to be simply z(t) =z(t).
In this case, we rewrite the problem in terms of dimensionless coordinatesτ = τ /t andx = x/(∆ √ t). The solutions z(x,τ ) andz(x,τ ) obey the dimensionless equations of motion
∂τ z = +4 ∂ 2 x z − 2z(∂xz) 2 1 +zz (55) ∂τz = −4 ∂ 2 xz − 2z(∂xz) 2 1 +zz (56)
which are independent of t. The boundary conditions are z(x, 0) = sgn(x)c and z(x, 1) =z(x, 1) which are also independent of t. This means that z(x,τ ) andz(x,τ ) have a single solution in this regime for any t.
If we rewrite the action (35) using the dimensionless coordinatesτ andx, we therefore end up with something of the form
S = κ · ∆ √ t(57)
where κ is a numerical constant independent of t and ∆, equivalent to calculating the action with t = ∆ = 1. We compare (57) to the a full numerical solution for S in Figure 6. This means that, in the regime 1 ∆ 2 t L 2 , the average purity decays as 5
Tr[ρ 2 A ] ∝ e −κ∆ √ t ,(58)
that isŠ
2 ∼ κ · ∆ √ t.(59)
In Figure 7 we plot numerical calculations of ln S against ln(∆ 2 t) for increasing values of L (we take L A = L B = L/2). We see an approach to the asymptotic form ln S = ln κ + 1 2 ln(∆ 2 t). To connect this with the limit ∆ 2 t L 2 , we can write a more general scaling form for the saddle-point action S 5 We have neglected time dependence of the O(1) correction due to K fl . where g(α) → κ as α → ∞ but g(α) ∼ 1 2 s eq α as α → 0 (where we have used the equilibrium action density s eq ≡ S eq /L A ).
S = g L ∆ √ t ∆ √ t(60)
In Appendix H, we plot the function g(α). By plotting g(α) for increasing values of L we can approximate κ ≈ 0.49855 . . . .
D. Validity of the saddle-point approximation
The saddle-point approximation assumes that the path integral is dominated by paths close to the classical trajectory. That is, there should be a large parameter that suppresses deviations from this trajectory. In our case, this large parameter is given by the characteristic timescale for the evolution.
Consider for definiteness the regime discussed above where the total time t is large, and L is of order √ t. Rewriting the continuum action (35) in terms of the dimensionless coordinates of the previous section,τ = τ /t andx = x/(∆ √ t), we find that ∆ √ t stands outside the action as a large parameter, with all the other quantities that appear in the action generally being of order g(α) (where α ≡ L/(∆ √ t)). For any fixed value of α > 0, the limit ∆ √ t → ∞ leads to an indefinitely large prefactor in the action, justifying a saddle-point treatment with large parameter ∆ √ t.
A standard idea for the ferromagnetic chain is that we can understand this as being similar to a large S limit [37]. Heuristically, in the low-energy states that are relevant at large times, large blocks of spins of size M 1 are very well aligned, so they act as individual spins with S = M/2. Fig. 9).
VI. NUMERICS
To test the validity of the saddle-point approximation, we compared the predictions of the approximation with direct simulations of the Trotterized noisy Majorana chain.
To simulate the Majorana chain we don't need to keep track of the entire many-body wave function (which has 2 L/2 complex coefficients). Because it describes free fermions, we only need to keep track of the timedependence of the two-point correlation functions
G ij ≡ i 2 [γ a i , γ b i ] ,(61)
(which are L(L − 1)/2 real numbers) for a given realization of the noise. Entanglement entropies can be computed from G using standard techniques [1,[38][39][40][41]. This approach allows much larger system sizes. Figure 8 showsŠ 2 /L A plotted against ∆ 2 t/L 2 A for increasing values of L A . The average was taken over 500 trials. These simulations are consistent with a collapse to the saddle-point prediction.
From the discussion of the effective large parameter in the saddle-point treatment in Sec. V D, corrections toŠ 2 from fluctuations around the saddle point are of relative order 1/L compared to the saddle-point result, for any fixed value of the scaling variable ∆ 2 t/L 2 A . See for example the resultŠ 2 = L+1 2 ln(1 + c 2 ) for the late-time entanglement in Sec. V A. To test this, Fig. 9 shows the result forŠ 2 /L A found from simulations, plotted against L A , for a fixed value of the scaling variable ∆ 2 t/L 2 A = 1/4. We see that the dependence on L A is indeed consistent A ] given that e −S 2 ≤ e −S 2 , but the difference does not appear to grow with LA. This suggests that S2 andŠ2 are equal to leading order in t. The data was generated by simulating the Majorana chain and averaging over 500 trials.
with a 1/L convergence to the saddle point prediction as L → ∞.
A. Small size of statistical fluctuations
Next we examine the size of statistical fluctuations of S 2 arising from the noise in the physical Hamiltonian. Empirically, we find these are surprisingly small, such thatŠ 2 as defined by Eq. (23) coincides at leading order with S 2 . Fig. 10 shows difference between these two kinds of average, as a function of the scaling variable ∆ 2 t/L A , for various system sizes (with L = 2L A ). Notably, this figure strongly suggests that this difference remains bounded as L → ∞. (This is in contrast to interacting random circuits, where the two quantities have slightly different growth rates [8].)
The difference between the two quantities may expressed using the cumulant expansion:
e −Š2 = e −S2 = exp ∞ k=1 (−1) k k! S k 2 c ,(62)
where S k 2 c denotes the kth cumulant. Therefore the fact that the difference betweenŠ 2 and S 2 is order 1 at large size requires that the cumulants with k > 1 are at most of order 1 size.
As mentioned in Sec. III B, the distribution of the pure state at asymptotically late times (t → ∞ for fixed L) is invariant under SO(L) transformations [42][43][44][45][46]. This means that the L × L correlation matrix G ij (Eq. (61)) is given by
G = OΣO T , Σ = 0 1 −1 0 ⊕ . . . ⊕ 0 1 −1 0 ,(63)
where O is a Haar-random SO(L) matrix, and Σ is the correlation matrix for the initial state. For the case of fermions with a conserved U(1) charge, the analogous expression has been used to compute the probability distribution of the entanglement in the late-time steady state using random matrix techniques, and the cumulants (at large time) have the scaling [42] S k 2 c ∼ L 2−k (64) (the variance was also considered in [44]). It is expected that the same scaling will generalize to the Majorana case due to the link between the invariant measure and random matrix theory [47]. If so, it is consistent with Fig. 10, with the fluctuations (from the second cumulant) being of order 1. The result in this Section is perhaps surprising: it means that for these free-fermion (unitary [24]) dynamics we obtain asymptotically exact results for the second Rényi entropy S 2 in a typical realization without needing taking a replica limit, again in contrast to random interacting circuits.
VII. EFFECT OF INTERACTIONS
So far, we have been considering the free-fermion model given by eq. (1). We have seen that this leads us to a Heisenberg model with a continuous SO(2N ) symmetry, in stark contrast with the discrete permutation symmetry which emerges from interacting models. We might ask, how does adding weak interactions to our freefermion model qualitatively change the emergent Heisenberg model?
A. An interacting model Let's focus on a particular weakly interacting model. The simplest terms we can add to our Hamiltonian are terms involving four Majorana operators, so let's consider adding a term to the Hamiltonian of the form
H int γ (t) = − i η i (t)γ i γ i+1 γ i+2 γ i+3 (65)
where η i (t) is Gaussian noise with zero mean and
η i (t)η j (t ) = ∆ 2 I δ ij δ(t−t )
. This is equivalent to adding gates to the unitary circuit of the form
u I (η ) = exp(iη γ i γ i+1 γ i+2 γ i+3 ) (66)
where η is a Gaussian random variable with zero mean and variance ∆ 2 I δt. In order to study weak interactions, we will assume ∆ 2 I ∆ 2 . When we perform the mapping from 2N -replicas of this model to a Heisenberg model, this gives us noiseaveraged replicated gates of the form
u I = exp − ∆ 2 I δt 2 2N + 2 2N a<b (−1) a+b A ab i A ab i+1 A ab i+2 A ab i+3
(67) which, in the limit δt → 0, is equivalent to adding an interaction Hamiltonian H I to our original Hamiltonian given by
H I = ∆ 2 I i N + 2N a<b (−1) a+b A ab i A ab i+1 A ab i+2 A ab i+3(68)
This interaction Hamiltonian explicitly breaks the continuous SO(2N ) symmetry of the original model down to a discrete symmetry. Interacting random unitary circuits generically have an N -permutation symmetry in the replica formalism, leading to N ! distinct ground states. However, the interacting model introduced above has additional symmetries, which we discuss in the next section. These additional symmetries lead a spin model with 2 N −1 N ! relevant ground states.
B. Symmetries
We can understand heuristically why 2N -replicated generic interacting models have N ! ground states by considering the path integral formulation. When summing over trajectories in the 2N -replicated system, we are summing over N forward trajectories and N backward trajectories. Each forward trajectory contributes a factor of "e iS " (where S is the action of the trajectory) and each backward trajectory contributes a factor of "e −iS ". 6 After averaging over noise, the sets of trajectories which dominate at late times are ones in which every forward trajectory is paired with a backward trajectory, so that the phases cancel [5]. There are N ! different ways of doing this, corresponding to the N ! distinct ground states of the emergent ferromagnet.
However, the interacting model introduced in the previous section is not a generic interacting model. For one thing, it is a fermion model, which means it has a global Z 2 symmetry generated by the fermion parity operator U p = (−i) L/2 i γ i . This symmetry means that for any trajectory with action S, there is a parity-transformed trajectory which also has action S. When pairing forward and backward trajectories, we can either pair any trajectory with itself or with its parity-transformed version. This gives us 2 N N ! possible pairings, instead of N !.
This may be made clearer in the language of states. First, let us recall why, for a generic noisy interacting Hamiltonian H γ (t), the corresponding replica Hamiltonian has N ! paired ground states. We start with the identity
e −i dtHγ (t) ⊗ (e −i dtHγ (t) ) * |1 = |1(69)
for a single forward and a single backward replica, where |1 is the paired state between the replicas (Sec. III C). This equation is a restatement of the unitarity of the time-evolution operator prior to averaging over the noise. Eq. (69) generalizes to a larger number of replicas: for any pairing σ of the forward with the backward layers, we can form the corresponding paired state |σ by taking the tensor product of |1 states for each pair, and
e −i dtHγ (t) ⊗ . . . ⊗ (e −i dtHγ (t) ) * |σ = |σ . (70)
Averaging over disorder shows that |σ is a ground state 7 of the replica Hamiltonian H,
e − dtH |σ = |σ .(71)
In the presence of a Z 2 symmetry generated by a unitary U Z2 that commutes with the physical Hamiltonian, then the state |1 − ,
|1 − ≡ (1 ⊗ U Z2 ) |1 ,(72)
is also invariant under e −i dtHγ (t) ⊗ (e −i dtHγ (t) ) * (cf. Eq. (69)). After replication, this allows us to form 2 N ground states for a given pairing pattern, since for each pair we can use |1 or |1 − . A second property of our model is a "statistical gauge symmetry"
γ i → −γ i(73)
that holds for each site. This is not a symmetry of any individual realization of the noisy Hamiltonian H γ (t), but the fact that the probability distribution of the noise is invariant under η i (t) → −η i (t) for each bond means that (73) is a local symmetry of the replica Hamiltonian. In other words, the replica Hamiltonian contains only even numbers of Majorana operators on each individual lattice site. This leads to the conversation of chirality
χ i = (−i) N 2N
a=1 γ a i on every site i. The boundary conditions we are interested in have χ i = +1 everywhere, so only ground states with positive chirality are relevant to us. Of the 2 N N ! ground states, half of them have positive chirality (the ones with an even number of Z 2transformed pairings), meaning there are 2 N −1 N ! relevant ground states. Below we will see this more explicitly for the case N = 2, where there are 4 such states.
There is a third property, which is true for the freefermion Hamiltonian, but which is lost for the interacting Hamiltonian given above. This is that changing the sign of Majorana operators on even sites γ 2j → −γ 2j gives minus its complex conjugate, i.e.
U S H γ U † S = −H * γ + const.(74)
where U S implements this sign change. This property allowed us to make the sign change to get Eq. (8), revealing the larger SO(2N ) symmetry, as opposed to just SO(N ) × SO(N ). This kind of sign change is always possible for free-fermion systems (we can choose an ordering of the Majorana operators γ i such that iγ i γ i+1 is always real, then let U S change of the sign of γ i when i is even; then iγ i γ i+n is real for odd n and changes sign under U S , and imaginary for even n but doesn't change sign under U S ). In a general interacting system there is no corresponding symmetry. However, if instead of the interaction in Eq. (65) we pick a six-Majorana interaction, for example
H 6 γ (t) = i i η i (t)γ i γ i+1 γ i+2 γ i+3 γ i+4 γ i+5 ,(75)
then the property (74) is preserved. In this case, the interactions still break the continuous replica symmetry down to a discrete one, but a larger discrete symmetry than in the generic case. This corresponds to fact that it is possible to pair forward trajectories with forward trajectories and backward trajectories with backward trajectories. Algebraically, this is because, if we define a paired state between two forward replicas of the form
|S = (1 ⊗ U S ) |1 ,(76)
then this is invariant under e −i dtHγ (t) ⊗ e −i dtHγ (t) .
In models of this type, the number of ways of pairing replicas is (2N )!/(2 N N !) which, when combined with 2 Nfold degeneracy from the fermionic Z 2 symmetry results in (2N )!/N ! paired states, and there is an enlarged S 2N permutation symmetry of the replicas. This property holds for any model where the interaction terms only involve 4n + 2 distinct Majorana operators for integer n ≥ 0 (and we can sum terms with different values of n). This symmetry is therefore broken by any four-Majorana interaction, including (65) 8 .
C. Crossover in entanglement scaling due to interactions
To make the discussion of interactions more concrete, let's focus on the dynamics of the averaged purity by fixing N = 2. The chirality on each site χ i is still conserved so, like in the free-fermion case, we can restrict the states of the spin chain to states with χ i = +1 everywhere, leaving a single spin-1 2 degree of freedom at each site.
The interaction Hamiltonian (68) becomes
H I = 2∆ 2 I i (1 − Σ x i + Σ y i − Σ z i ) ,(77)
where
Σ α i = σ α i σ α i+1 σ α i+2 σ α i+3 .(78)
This interaction breaks the continuous SO(3) symmetry of the free-fermion case down to the dihedral group D 4 . Instead of a continuum of ground states as L → ∞, only 4 survive, namely uniform product states of |↑ ,|→ ,|↓ or |← .
To compute the purity we must start with the sharp domain-wall state C A | = ↑ · · · ↑→ · · · →| and evolve in imaginary time using the Hamiltonian H = H free + H I . We will again address this using the semiclassical equations of motion. Given that H free punishes sudden changes in the spin direction, at short times the domain wall in z(t) will relax in a way similar to the free-fermion case, becoming wider and smoother at larger timescale. However, H I is an anisotropy term in spin space which penalizes spin directions away from the x and z axes, and as a result the domain wall cannot relax to become arbitrarily wide. Instead, the domain wall will relax only to a finite width (computed below), as a compromise between minimising the contribution to the action from spin gradients and from the anisotropy term. This is in analogy to magnets with weakly broken continuous symmetry. This domain wall width is a crossover lengthscale associated with a renormalization group flow from the universality class of the Gaussian problem to that of the interacting one (there is also an associated crossover timescale).
The finite domain wall width also implies that the action cost becomes extensive in time t, rather than scaling as √ t as in the free case. This implies "ballistic" spreading of information: recall that the linear growth of entanglement entropy implies that correlations exist over a ballistically growing distance, of order t, at late times. 9 This ballistic spreading of correlations should not be confused with ballistic transport of quasiparticles (there is no notion of a quasiparticle in the interacting system). Similarly it should be borne in mind that these correlations need not be detectable by two-point functions of simple operators.
To analyze the finite-width domain wall in more detail, let us consider the limit ∆ 2 I ∆ 2 . In this limit, the steady state is nearly polarized everywhere, although the spin directionn(x) varies slowly with position (i.e. the state is approximately a coherent state with spatial derivatives 1). In this limit we may continue to use a continuum approximation.
The equations of motion take the general forṁ
z = −(1 +zz) 2 δH δz ,ż = (1 +zz) 2 δH δz .(79)
The difference between the free case and the interacting case is that in the latter these equations permit a non-trivial steady-state solution. As we will show below, there exist solutions (satisfying the required boundary conditions as x → ±∞) withż =ż = 0. Unlike the complexified solutions that we required in previous sections, these solutions havez(x) = z * (x) (soz is simply the complex conjugate of z). This means that the spin polarization vector
n(x) = 1 1 +zz (z +z, −i(z −z), 1 −zz)(80)
is simply a real unit vector, and the solution can be interpreted straightforwardly as a domain wall in this polarization. For a time-independent solution with z * =z, Eqs. (79) give simply
δH[n] δn(x) = 0.(81)
In the limit ∆ I ∆ we may approximate That is, the growing lengthscale as ∆ I /∆ → 0 (quantified below) means that we can ignore spatial derivativesthey are an indefinitely small contribution to the already small H I . Let us now find the coherent state which minimizes the Hamiltonian H. Let us parameterize the coherent state using the angle θ(x) such that the spin direction iŝ n(x) = (sin θ(x), 0, cos θ(x)). Substituting this into the Hamiltonian H gives a continuum Hamiltonian
Σ α i ≈ [n α (x)] 4 + · · · .(82)H ≈ dx ∆ 2 (∂ x θ) 2 + 2∆ 2 I (1 − cos 4 θ − sin 4 θ) (83) = dx ∆ 2 (∂ x θ) 2 + ∆ 2 I sin 2 2θ(84)
Minimising this Hamiltonian with the usual boundary conditions θ(x) → 0 as x → −∞ and θ(x) → π/2 as x → +∞ yields the solution
θ(x) = arctan e 2Kx(85)
where K ≡ ∆ I /∆ (we have chosen a domain wall at the origin, but any translation of this solution is also a solution). We can immediately infer the length scale of this domain wall (see Fig. 11) to be l int ≡ K −1 , or if we restore the lattice spacing a (previously set equal to 1)
l int ≡ a∆ ∆ I .(86)
Combining this with the diffusive dynamical scaling on shorter lengthscales (cf. Eq. (39)) gives the characteristic timescale for relaxation of the domain wall:
t crossover ∼ 1 ∆ 2 I .(87)
A more detailed analysis of the crossover could be obtained by solving the equations of motion (79) with interaction terms included, to determine how z andz approach the above steady-state solution.
The entanglement growth rate is given by the bulk action per unit time, cf. Eq. (35). Since the time derivatives vanish, this is simply the value of the Hamiltonian for the domain wall state above, which is H = 2∆ I ∆. This means that at late times t t crossover ≡ ∆ −2 I , the average purity will decay exponentially in ť
S 2 ∼ 2∆ I ∆ t.(88)
Note the contrast with the √ t growth in the free model. Eq. (88) is consistent with linear growth of entanglement entropy expected for generic interacting systems [48].
It would be very interesting to study quantum fluctuations on top of this classical solution, given that it is only unique up to translations. At the largest length scales, we expect that the centre of mass executes a random walk, whose "entropic" fluctuations will correct the growth rate in Eq. (88) by an amount that is subleading at small ∆ I . 10
VIII. OUTLOOK
In contrast with the case of interacting systems, which have a discrete replica symmetry, free-fermion systems have a continuous replica symmetry. This leads to smooth domain walls which relax over time, as opposed to the sharp domain walls which give rise to the "entanglement membrane" picture in interacting systems.
Despite the original problem of calculating entanglement growth being purely quantum mechanical, the equivalent calculation for the Heisenberg chain can be tackled using a semi-classical approximation. The only large parameter required to justify the semiclassical limit is the large timescale for the evolution.
An interesting feature of the effective theory for freefermion systems is that we must describe not one classical configuration of our Heisenberg chain, but two: one relaxing forwards in time, and the other relaxing backwards.
If we introduce interactions into our model by adding quartic terms into the Hamiltonian, the continuous SO(2N ) replica symmetry is explicitly broken down to a discrete symmetry. At sufficiently short times t t crossover , the sharp domain wall created by the boundary between the subsystems A and B relaxes as in the freefermion case, but over longer times it approaches a finite thickness l int which depends on the strength of the interactions. Below this length scale, the domain wall appears smooth, but above this length scale, the domain wall resembles a sharp domain wall similar to those found in strongly interacting systems, indicating a crossover from free-fermion behaviour to interacting behaviour.
This gives for the first time an explicit picture of the renormalization group flow between two different universality classes for the entanglement dynamics -one associated with Gaussian systems and one associated with interacting ones.
Although we have focussed on the case N = 2 in order to calculate the averaged purity, a similar qualitative picture should hold for a general number N of replicas of the density matrix, allowing calculation of, for example, the von Neumann entropy (the limit N → 1 of the Rényi entropy). In principle, extending the present approach would require a generalization of the SU(2) coherent states parameterization that proved useful above.
Interestingly, though, numerics (see Fig. 10) suggest that the averageŠ 2 that we have computed is also equal at leading order to the standard mean S 2 . This suggests that certain calculations for a larger number of replicas in fact reduce (at the saddle-point level) to the calculation presented here. It will be interesting to study how this happens. A natural guess is that the saddle-point solutions needed for the higher moments of the purity in Eq. 27 can be "factorized" into k copies of the saddle point solution in Sec. V (see also [24]). If the saddle-point action is simply proportional to k, then the generating function (27) becomes the generating function of a deterministic variable, implying that fluctuations in S 2 (t) -arising from the randomness in the couplings η(t)are strongly subleading compared to the mean. It would be interesting to interpret these subleading fluctuations in the effective theory.
The effective theory could be used to compute many other quantities: it remains to be seen which quantities are most naturally understood via the spacetime picture developed here (i.e. at the level of the action) and which through other tools (e.g. integrability [27]). In the number-conserving case [28], the probability distribution of "quantum coherences" c † i c j has non-trivial combinatorial structure [27,29,30,42,49]. In the Majorana model, the variance iγ i γ j 2 maps to an overlap similar to the one we have considered, but with a different boundary state with two flipped spins: . . . ↑↓ i ↑ . . . ↑↓ j ↑|e −Ht |ψ + . The out-of-time order correlation function could similarly be computed.
The effective theory we have used throughout this paper seems very different from the quasiparticle picture which has been used successfully to describe entanglement spreading in (non-random) conformal field theories and lattice fermion systems [1,[31][32][33]. One possible way of seeing a connection between these two pictures is to use the fact that the Heisenberg Hamiltonian can also be viewed as the transition operator for a classical Markov process. This Markov process is the classical Symmetric Simple Exclusion Process [36], except that we must retain some "pairing" information about the particles as discussed below. We write the effective SO(3) Hamiltonian H in terms of the operator ς i,j which swaps the Heisenberg spins on sites i and j,
H = 4∆ 2 i [1 − ς i,i+1 ].(89)
We can then reinterpret the spin wavefunction e −Ht |ψ as a probability distribution, 11 where H randomly swaps 11 If we write the wavefunction in terms of simultaneous eigenstates neighbouring spins at a rate of 4∆ 2 while conserving total probability. The initial state |ψ , which lives in the N = 2 replicated Hilbert space, consists of entangled pairs (not pairs of physical Majoranas, but Bell pairs of spin-1 2 s). Time evolution under H causes the members of these pairs to hop randomly to the left or right while preserving the entanglement between the two members of the pair.
Contracting with C A | at the final time gives a factor of 1 for every pair in which both members end up in A or both end up in B, but gives a factor of 1/ √ 2 for every pair which has one member in A and one in B. We find
e −S2 = C A |e −Ht |ψ = e − 1 2 n ln 2 (90)
where n is the number of pairs which end up with one member in A and one member in B, and where the expectation value is taken in this classical Markov process.
It should be borne in mind that this not a physical Markov process for the Majoranas. Despite this, the expression above coincides with the average of e −S2 that would be obtained for a physical Markov process in which we swapped Majoranas. That is a special feature of the average of e −S2 , and does not hold for the average of S 2 . Nevertheless, we see that for N = 2 there is a formal connection with a particle dynamics that resembles a diffusive version of the quasiparticle picture [1,[31][32][33].
The problem with the expression above is that the average on the right hand side is strongly biased towards small values of n. Therefore while n is easy to compute in the Markov process by standard diffusion arguments, this is not very useful. The semiclassical picture in previous sections is more useful, partly because it makes explicit the SU(2) symmetry that is hidden in the Markov representation.
Let us briefly mention some ways in which the effective theory could be extended. Non-unitarity due to measurement of Majorana bilinears [16,19,50] in this model will be addressed separately in Ref. [24], where nonlinear sigma model effective field theories will be provided for that case. (In contrast to the theory discussed here, those nonlinear sigma models have a relativistic dispersion.)
Most straightforwardly, the mappings of previous sections also carry over directly to a hypercubic lattice of Majoranas in higher dimensions, and could be used to obtain scaling forms for entanglement of regions of different shapes.
Four-fermion interactions have been discussed above, but a different way to break the Gaussian structure is to explicitly break the Z 2 symmetry of the Ising model (recall that the original Majorana chain is dual to a noisy Ising model). This is a non-local perturbation in the Majorana language. It would be interesting to explore how the number of effective ground states is reduced of σ z i , then the coefficients (not their absolute squares) can be chosen to be real and positive and can be interpreted as probabilities.
(Sec. VII B). We believe that this process can be understood as a confinement phenomenon in the replica theory (we will discuss this elsewhere).
Retaining a non-random hopping in addition to the random one we have considered would mean that, at least at short scales, it was necessary to retain fermionic degrees of freedom in the effective theory; it would be interesting to study the reduction to a bosonic effective theory at large scales. It would also be interesting to study the case of static disorder (on lengthscales where Anderson localization effects are weak or in dimensionalities where localization is avoided). With standard kinds of approximations [51], this leads to an effective model where the interactions are long-range in time. However, the replica symmetry is still continuous, and the saddlepoint approach developed here could be extended to this case. 1 • If A i is a stabilizer, then A i+1 is also a stabilizer.
The second point follows from the relations
A 2j−1 = (X 1 j−1 X 2 j−1 )(X 1 j X 2 j )A 2j−2 (A7) A 2j = (Z 1 j Z 2 j )A 2j−1 (A8)
which follow from equations (A3) and (A4) and the definition of A i . This logic extends to a larger number of replicas. Let us consider the state |C A that we need for the computation of the purity. Region B consists of the first Ising sites (or the first 2 Majorana sites) and region A consists of the remainder. In the Ising language, the state at a given site is as follows. In B, we form a maximally entangled state of the form |1 between layers a = 1 and a = 2, and separately also between layers a = 3 and a = 4. In A, we similarly entangle a = 1 with a = 4 and a = 2 with a = 3 (Fig. 4). As a result, the stabilizers are
Z 1 Z 2 , X 1 X 2 , Z 3 Z 4 , X 3 X 4 (region B), (A9) Z 2 Z 3 , X 2 X 3 , Z 1 Z 4 , X 1 X 4 (region A). (A10)
In analogy to the above, we connect the chains at both the left and the boundaries by defining
X 1 1 X 2 1 = iγ 2 1 γ 1 1 , X 3 1 X 4 1 = iγ 4 1 γ 3 1 , X 2 L X 3 L = iγ 3 2L γ 2 2L .(A11)
Then we can show that in the Majorana language we have the stabilizers (with eigenvalue +1)
iγ 2 i γ 1 i , γ 4 i γ 3 i (i ∈ B),(A12)iγ 3 i γ 2 i , γ 4 i γ 1 i (i ∈ A) (A13)
at a site i. These can be determined in the same way as for N = 1. For the stabilizers in region B we start at i = 1 (where the above equations hold by Eq. (A11)) and propagate to the right by multiplying with the stabilizers in Eq. (A9). For the stabilizers of the form iγ 3 γ 2 in region A we start at the right and propagate to the left similarly. The stabilizers of the form iγ 4 i γ 1 i in region A we write as a snake-like product of all the Majoranas in the 4 layers between site 2 and site i inclusive (note that i > 2 since i is in region A). Then we may check that the terms in this product all cancel by using either the stabilizers in Eqs. (A9), (A10), or by using the alreadyknown stabilizers iγ 3 i γ 2 i , iγ 2 2 γ 1 2 , iγ 4 2 γ 3 2 .
Appendix B: Expectation values of operators as Ising configurations
For N = 1 we get an Ising spin chain with spin operators A i ≡ −iγ 1 i γ 2 i . We can work in the basis of simultaneous eigenstates of these operators.
We can write any operator acting on the original Majorana chain as a linear combination of products of γ operators, so we can choose our basis of operators to be the set of products of the γ operators. Given that (γ i ) 2 = 1, there are 2 L such operators (including the identity operator).
How can we write these operators as bras in the replicated system? Given that φ|Ô|ψ = φ|1(Ô|ψ ), if we know how to write the identity operator 1 as a bra 1|, then we can generate the bra for any other operatorÔ using Ô | = 1|Ô 1 where the operatorÔ 1 acts only on the first replica, and the bras are now states in the doubled representation.
As explained in the previous appendix, the identity state |1 satisfies A i |1 = +|1 for all i. In other words, the Ising spin on every site i is in the spin up state.
Let us now consider how acting with Majorana operators affects arbitrary Ising configurations. Take some Ising configuration |{s i } where A i |{s i } = s i |{s i } . Now consider the state γ 1 j |{s i } . The new values of s i can be determined from A i (γ 1 j |{s i } ) = (−1) δij γ 1 j A i |{s i } = (−1) δij γ 1 j s i |{s i } = s i (γ 1 j |{s i } ) so s i = (−1) δij s i . This means that acting with γ 1 j simply flips the Ising spin at j. (This doesn't tell us anything about the overall phase of the state. Acting with γ 1 j then γ 1 k gives the same state as acting with γ 1 k then γ 1 j but with the opposite sign.) If we act on the state |1 (all up spins) with some arbitrary product of Majorana operatorsÔ, the resulting state |Ô is therefore a single Ising configuration with s i = −1 if γ i appears in the this product, and s i = +1 otherwise. This tells us how to convert from products of Majorana operators to states in the replicated system, up to complex number prefactors which drop out of the final result. To see this, write a general operatorÔ of interest as a sumÔ = α c αÔα of products of Majoranas. After the Ising mapping, each Ô α | has a definite Ising energy E α , whose computation is explained in Sec. III D 1, so that the physical expectation value at time t is Ô (t) = α c α e −Eαt Ô α (0).
(B1)
Appendix C: Reduction from an SO(4) spin to two SU(2) spins
For N = 2, we have six operators A ab with a, b = 1, . . . , 4, such that J ab ≡ 1 2 A ab satisfy the angular momentum commutation relations (19) and therefore form a representation of so (4).
We can define a set of Pauli operators σ i which act on spins with χ = +1
σ x = 1 2 (A 23 + A 14 ) (C1) σ y = 1 2 (A 31 + A 24 ) (C2) σ z = 1 2 (A 12 + A 34 )(C3)
These operators commute with χ and satisfy σ i 1−χ 2 = 0 (so they annihilate states with χ = −1). They also satisfy [σ i , σ j ] = 2i ijk σ k and (σ i ) 2 = 1+χ 2 so act as Pauli operators on the subspace of states with χ = +1.
Similarly we can define a set of Pauli operatorsσ i which act on spins with χ = −1
σ x = 1 2 (A 23 − A 14 ) (C4) σ y = 1 2 (A 31 − A 24 ) (C5) σ z = 1 2 (A 12 − A 34 )(C6)
which again commute with χ but satisfy σ i 1+χ 2 = 0 (so they annihilate states with χ = +1). They satisfy [σ i ,σ j ] = 2i ijkσ k − but (σ i ) 2 = 1−χ 2 so act as Pauli operators on the subspace of states with χ = −1.
We have reduced the SO(4) spin down to a superposition of two conventional SU(2) spin-1 2 s. With σ i acting on the first type andσ i on the second type. The Heisenberg interaction term in (12) becomes which clearly also satisfies χ|C = +|C .
Using the Pauli operators defined in Appendix C, the fact that σ z |1 = +|1 and σ x |C = +|C allows us define
|1 =|↑ (D5) |C =|→ (D6)
In other words, the state |1 corresponds to an "up" spin, and the state |C corresponds to a "right" spin. The state |C A is a product state of |C on every site inside the subsystem A and |1 on every site outside the subsystem A, so in the Heisenberg model it is similarly a product state of |→ on every site inside A and |↑ on every site outside A.
So we need to find the overlaps ψ+ |z and z|C A .
To calculate z|C A , take the state |C A to have z = −c in the left side of the chain and z = +c in the right side of the chain, where c = To calculate ψ+ |z , take the state |ψ + to be a product of pairs in the state | →→ + | ←← . In terms of single-spin coherent states this is 1 2 |+1 |+1 + 1 2 |−1 |−1 . Therefore we get
ψ + |z = 1 2 +1|z 2 + 1 2 −1|z 2 L/2 (F6) = 1 2 (1 + z) 2 + 1 2 (1 − z) 2 L/2 (F7) = (1 + z 2 ) L/2 (F8) = (1 + z 2 ) j (F9)
This gives us the exact expression
ψ + |P |C A = 2j + 1 π (1 + c 2 ) −j × d 2 z (1 +zz) 2j+2 (1 + z 2 ) j (1 − c 2z2 ) j (F10)
To study the thermodynamic limit we take j → ∞. But before we can do this we need to change variables to ζ = z j + 1/2 (F11) ζ =z j + 1/2 (F12)
If we do this and use (1 + x/n) n ≈ e x for large n we get
ψ + |P |C A ≈ 2 π (1 + c 2 ) −j d 2 ζe −2ζζ+ζ 2 −c 2ζ2 × 1 + 1 j + 1/2 ζ 2 −1/2 1 − c 2 j + 1/2ζ 2 −1/2 (F13)
For large j the bracketed terms on the right are close to one for typical values of ζ and we can approximate the integral as a basic Gaussian integral
ψ + |P |C A ≈ 2 π (1 + c 2 ) −j d 2 ζe −2ζζ+ζ 2 −c 2ζ2(F14)
Rewriting ζ = x 1 + ix 2 we get
ψ + |P |C A ≈ 2 π (1 + c 2 ) −j d 2 xe −x T Mx (F15) where M = 1 + c 2 −i(1 + c 2 ) −i(1 + c 2 ) 3 − c 2 (F16)
and we get det(M) = 4(1 + c 2 ). Performing the integral we get ψ + |P |C A ≈ 2 π (1 + c 2 ) −j · π 2 4(1 + c 2 ) (F17)
= (1 + c 2 ) −(j+1/2) (F18) = (1 + c 2 ) − L+1 2 (F19)
so the log of the averaged purity is proportional to L + 1 as L becomes large. . As α → ∞, g(α) approaches a constant value κ, corresponding to behaviour in a infinite system. As α → 0, g(α) is linear in α, given that the entanglement has saturated and is independent of t. The slope as α → 0 is half the equilibrium density seq = Seq/LA.
For each value of L A , we can fit a value of κ L A , this being the value that g L A (α) approaches when α becomes large, so that ∆ 2 t L 2 (but not so large that ∆ 2 t ∼ 1 and the lattice become important).
For fixed α 1, we find that g L A (α) ≈ g(α) − const./L 2 .
Taking α 1 gives κ ≈ 0.49855 . . ..
FIG. 1. Contour plot of the two independent fields z andz involved in calculating the purity of rightmost half of the Majorana chain after time t using a saddle point approximation of the coherent states path integral. The coordinate x is the physical spatial coordinate. τ is a reversed time coordinate: i.e. τ = 0 at the final time t, and τ = t at physical time 0 when the system is prepared in a state of short range entanglement. The field z begins as a sharp domain wall at τ = 0 but relaxes diffusively as τ increases. The fieldz matches z at τ = t, but relaxes diffusively for decreasing τ . There are also interactions between the two fields.
FIG. 2 .
2The unitary circuit corresponding to the Trotterized time evolution. Each grey rectangle is an independent twosite unitary of the form u(η) = exp(−ηγiγi+1).
OFIG. 3 .
3The folded unitary circuit for ψ|Ô|ψ . The front layer is the original unitary circuit Uη and the back layer is its complex conjugate U * η . At the top, we show how a singlesite observableÔ acts on the replicated system. At every other site, we have the single-site identity operator 1 denoted by an undecorated line.
FIG. 4 .
4An example of the index contractions involved in calculating the N th Rényi entropy of entanglement, here with N = 3.
FIG. 5 .
5An example of S against ∆ 2 t for LA = LB = 50.
Small z approximation FIG. 6. A comparison of the full numerical solution for S (labelled "Saddle point") against the smallz approximation (54), and the √ t growth (57) found at early times. Here LA = LB = 50. The full solution approaches the smallz approximation when ∆ 2 t is comparable to L 2 . The full solution grows approximately as κ · ∆ √ t when ∆ 2 t L 2 .
FIG. 7 .
7Plot of ln S against ln(∆ 2 t) for LA = LB = 10, 20, 30, 40, 50. The asymptotic √ t behaviour is shown by the dashed line. The y-intercept gives the value of ln κ.
/LA against ∆ 2 t/L 2 A for LA = 10, 20, 30, 40, 50 from direct simulations of the Majorana chain, compared with the saddle-point prediction. The averages are calculated over 500 random realizations of the random unitary circuit with ∆ = 0.1. For finite LA, we expecť S2/LA to match to the saddle point prediction up to a correction of order 1/LA (see
FIG. 9 .
9Values ofŠ2/LA against LA for ∆ 2 t/L 2 A = 0.25. The values were found by averaging over 500 trials, and the error bars show the standard error. The O(1/L) correction to the saddle point prediction forŠ2/LA is approximated as − ln K ∞ fl /L, where K ∞ fl is the pre-exponential factor K fl in the limit t → ∞. LA = 20 LA = 30 LA = 40 LA = 50 FIG. 10. Plot of S2 −Š2 against ∆ 2 t/L 2 A for LA = LB = 10, 20, 30, 40, 50. The mean second Rényi entropy S2 is larger thanŠ2 = − ln Tr[ρ 2
9
For example, the mutual information I between the regions A = [−∞, 0] and B = [x, ∞] is given in a pure state by Ix = S A + S B − S C , where C = [0, x]. We have S A = S B ∼ Γt (where Γ is computed below) while S C is straightforwardly bounded by S C ≤ x 2 ln 2 for a Majorana chain. Therefore Ix ≥ ln 2 2 (vt − x), where v = 4Γ/ ln 2. This shows that correlations between A and B are significant if x is within the ballistic lightcone x ≤ vt.lint FIG. 11. Cartoon of the spin polarizationn(x) in the nontrivial steady state. To the left, all spins are pointing up, and to right, all spins are pointing right. The length scale lint corresponds to the thickness of the domain wall. (Note that in reality we are assuming lint a.)
i A ab i+1 = 2 [σ i · σ i+1 +σ i ·σ i+1 ](C7)so two neighbouring spins with the same chirality simply experience a Heisenberg interaction, while neighbouring spins of opposite chirality do not interact.Appendix D: The state |CA in the SO(4) Heisenberg modelLet us consider a single Majorana site. As explained in Appendix A, the state |1 which acts like the identity at a given Majorana site can be defined by−iγ 1 γ 2 |1 = +|1 (D1) −iγ 3 γ 4 |1 = +|1 (D2)Clearly this state satisfies χ|1 = −γ 1 γ 2 γ 3 γ 4 |1 = +|1 . Similarly, the state |C which swaps replicas at a given Majorana site can be defined by−iγ 2 γ 3 |C = +|C (D3) −iγ 1 γ 4 |C = +|C (D4)
c 2 )
2−j (1 − c 2z2 ) j (F5)
FIG. 12 .
12Plot of the scaling function g(α)
If we added next-nearest-neighbour interactions iγ i γ i+2 , these terms would not be affected by the above sign change, but iγ i γ i+2 is pure imaginary so the SO(2N ) symmetry is apparent even before the sign change.
The boundary condition(44) simply neglects the specific pattern of correlations found in the initial state |ψ , which we have arbitrarily chosen to be iγ 2j−1 γ 2j |ψ = +|ψ . This does not affect the asymptotic entanglement growth at late times, which will be the same for any initial state with only short-range correlations.
More precisely, in the discrete circuit "e iS " becomes a product of local unitary amplitudes.7 In a generic model the replica Hamiltonian could have complex eigenvalues. However unitarity prior to averaging implies that the real part of the eigenvalue is non-negative. It is equal to zero for the ground states.
If we had only these four-Majorana terms, we could recover the symmetry by doing a different sign change, flipping every fourth Majorana operator on even replicas.
It would also be interesting to compute the action cost for a domain wall whose centre of mass travels at a non-zero speed: this will give the entanglement membrane tension E(v) as a function of v, as well as the butterfly velocity v B .
Acknowledgments: We are extremely grateful to John Chalker for valuable discussions throughout this project, and to Fabian Essler for extremely useful discussions that helped us clarify the coherent states path integral. TS was supported by a James Buckee Scholarship and by Royal Society Enhancement Award. AN was supported in part by a Royal Society University Research Fellowship and by CNRS and the ENS. DB was in part supported by CNRS, by the ENS, and by the ANR project "ESQuisses", contract number ANR-20-CE47-0014-01.Appendix A: Stabilizers for |CA in terms of Majorana operatorsWe wish to find more explicit expressions for the bras corresponding to index contractions on the replicas. We do this by determining the stabilizers of such states in terms of Majorana operators.It will be convenient to use the following mapping from the original Majorana chain to an Ising spin chain:where X j and Z j are Pauli matrices. The Pauli matrices act on spins j formed from pairs of Majorana modes. When we create 2N replicas of the chain and carry out the sign change γ 2n 2j → γ 2n 2j this becomes:The sign change effectively changes the order of the gammas on even replicas with respect to the mapping to the Ising chain. How do we write index contractions as bras? Take the example of a single spin with N = 1 (so there are two replicas). How do we write the identity operator as a state 1|?We insist that 1|(|ψ ⊗ |φ * ) = φ|ψ where |ψ and |φ are arbitrary spins. This means that |1 =|↑↑ + |↓↓ (where the left and right spins refer to replicas 1 and 2 respectively).The state |1 can be uniquely defined by its stabilizerswhere X a and Z a are once again Pauli matrices acting on replica a. For a chain of Majoranas with 2 replicas, we similarly get a pair of stabilizers for every pair of Majorana sites, assuming we are applying the identity contraction everywhere. The stabilizer Z 1 j Z 2 j can be straightforwardly written in terms of Majorana operators using equation (A3) to give γ 1 2j−1 γ 1 2j γ 2 2j−1 γ 2 2j . The stabilizer X 1 j X 2 j is not as simple. We can write products of different X matrices on the same replica by stringing together products of neighbouring X operators e.g. X j X j+3 = (X j X j+1 )(X j+1 X j+2 )(X j+2 X j+3 ) and then use equation (A4). However, we cannot do this for products of X operators in different chains because there are no relations of the form (A4) which relate Majoranas from different replicas.We can fix this by "connecting" the two chains on the left-hand side by defining iγ 2 1 γ 1 1 = X 1 1 X 2 1 . This essentially means we use the mapping in equations (A1) and (A2) but treating both replicas as part of a single chain.We can now iteratively find a simple set of stabilizersAppendix E: The state |ψ in the Heisenberg modelWe wish to find the state |ψ + , that is, the replicated initial state |ψ projected onto the subspace where all spins have positive chirality χ i = +1. To do this, we will first consider a single pair of spins.Given that the initial state |ψ is defined by iγ 1 γ 2 |ψ = +|ψ , the replicated initial state |ψ can be defined (up to a phase) by the stabilizerswhere the minus signs are a result of the redefinition γ 2n 2j → −γ 2n 2j .This immediately tells us that χ 1 χ 2 |ψ = +|ψ , so the chiralities of the two spins are perfectly correlated.Using the Pauli matrices σ i 1 and σ i 2 which act only on the positive chirality parts of the spins (as defined in Appendix C) we can writeacting with both sides on |ψ givesBut the fact that χ 1 χ 2 |ψ = +|ψ means that projecting onto χ 1 = +1 also projects onto χ 2 = +1 soThese stabilizers determine |ψ + up to a phase and normalization. The fact that quantum states must have norm 1 means that ψ+ |1 = 1 (where |1 =|↑↑ ) which fixes the state to beIt is worth noting that if we had chosen the orthogonal state iγ 1 γ 2 |ψ = −|ψ we would still get the same |ψ + . This implies that |ψ + is not sensitive to the specific initial state |ψ we choose, but rather to the entanglement of this state.For the general case of L/2 pairs, the state |ψ + is simply a product of the above for each pair.Appendix F: Equilibrium purity including fluctuationsWe wish to calculate ψ+ |e −Ht |C A in the limit t → ∞. Here |C A is the sharp domain wall state and |ψ + is a product of entangled pairs. We can write this as ψ+ |P |C A whereP is the projector onto ground states of the Heisenberg chain.Instead of using spin coherent states with spin j = 1 2 , we can writeP as an integral over coherent states of spin j = L/2, where L is the number of sites:Let's first consider a single pair of spins. The state ψ+ | is then simplyTaking the z values of the spins to be w 1 and w 2 respectively, the un-normalized state isSubstituting this into eq. (42) giveswhich givesw 1 = w 2 andw 2 = w 1 . So the boundary condition simply "swaps" w 1 and w 2 to givew 1 andw 2 .In general, when we have L/2 pairs of spins, the state ψ+ | is simply a product state of entangled pairs, each in the state ↑↑| + ↓↓|. The function f is simply a sum of the f s for each pair of spins, so the above argument applies to each pair of spins individually, and we get w 2j−1 = w 2j andw 2j = w 2j−1 .Appendix H: Scaling function g(α) and numerical approximation of κWe can numerically approximate the scaling function g(α) by numerically finding S for a given L A at different values of t, then dividing by ∆ √ t. As L A → ∞ this will give the scaling function g(α) exactly. A plot of g(α) using L A = 50 is shown inFigure 12.
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Page curve for fermionic gaussian states. Eugenio Bianchi, Lucas Hackl, Mario Kieburg, Physical Review B. 103241118Eugenio Bianchi, Lucas Hackl, and Mario Kieburg, "Page curve for fermionic gaussian states," Physical Re- view B 103, L241118 (2021).
Volume-law entanglement entropy of typical pure quantum states. Eugenio Bianchi, Lucas Hackl, Mario Kieburg, Marcos Rigol, Lev Vidmar, PRX Quantum. 330201Eugenio Bianchi, Lucas Hackl, Mario Kieburg, Marcos Rigol, and Lev Vidmar, "Volume-law entanglement en- tropy of typical pure quantum states," PRX Quantum 3, 030201 (2022).
Phase transitions in the distribution of bipartite entanglement of a random pure state. Celine Nadal, N Satya, Massimo Majumdar, Vergassola, Physical review letters. 104110501Celine Nadal, Satya N Majumdar, and Massimo Vergas- sola, "Phase transitions in the distribution of bipartite entanglement of a random pure state," Physical review letters 104, 110501 (2010).
Statistical distribution of quantum entanglement for a random bipartite state. Celine Nadal, N Satya, Massimo Majumdar, Vergassola, Journal of Statistical Physics. 142Celine Nadal, Satya N Majumdar, and Massimo Ver- gassola, "Statistical distribution of quantum entangle- ment for a random bipartite state," Journal of Statistical Physics 142, 403-438 (2011).
. Denis Bernard, Andrea De Luca, Lorenzo Pirolo, Denis Bernard, Andrea De Luca, and Lorenzo Pirolo, "unpublished," .
Ballistic spreading of entanglement in a diffusive nonintegrable system. Hyungwon Kim, David A Huse, Physical review letters. 111127205Hyungwon Kim and David A Huse, "Ballistic spread- ing of entanglement in a diffusive nonintegrable system," Physical review letters 111, 127205 (2013).
Bernoulli variables, classical exclusion processes and free probability. Michel Bauer, Denis Bernard, Philippe Biane, Ludwig Hruza, arXiv:2211.01710arXiv preprintMichel Bauer, Denis Bernard, Philippe Biane, and Ludwig Hruza, "Bernoulli variables, classical exclu- sion processes and free probability," arXiv preprint arXiv:2211.01710 (2022).
Entanglement transitions with free fermions. Joseph Merritt, Lukasz Fidkowski, Physical Review B. 10764303Joseph Merritt and Lukasz Fidkowski, "Entanglement transitions with free fermions," Physical Review B 107, 064303 (2023).
Nonlinear sigma model for normal and superconducting systems: A pedestrian approach. V Igor, Lerner, cond-mat/0307471arXiv preprintIgor V Lerner, "Nonlinear sigma model for normal and superconducting systems: A pedestrian approach," arXiv preprint cond-mat/0307471 (2003).
|
[] |
[
"arXiv:quant-ph/0508180v2 28 Sep 2005 Secret parameters in quantum bit commitment",
"arXiv:quant-ph/0508180v2 28 Sep 2005 Secret parameters in quantum bit commitment"
] |
[
"Chi-Yee Cheung \nInstitute of Physics\nAcademia Sinica Taipei 11529\nTaiwan, Republic of China\n"
] |
[
"Institute of Physics\nAcademia Sinica Taipei 11529\nTaiwan, Republic of China"
] |
[] |
The no-go theorem of unconditionally secure quantum bit commitment depends crucially on the assumption that Alice knows in detail all the probability distributions generated by Bob. We show that if a protocol is concealing, then the cheating unitary transformation is independent of any parameters (including probability distributions) secretly chosen by Bob, so that Alice can calculate it without knowing Bob's secret choices. Otherwise the protocol cannot be concealing. Our result shows that the original impossibility proof was based on an incorrect assumption, despite the fact that its conclusion remains valid within the adopted framework. Furthermore, our result eliminates a potential loophole in the no-go theorem.
| null |
[
"https://export.arxiv.org/pdf/quant-ph/0508180v2.pdf"
] | 117,019,954 |
quant-ph/0508180
|
7a173485a93f83a490d47dec58a383509d34df4b
|
arXiv:quant-ph/0508180v2 28 Sep 2005 Secret parameters in quantum bit commitment
Chi-Yee Cheung
Institute of Physics
Academia Sinica Taipei 11529
Taiwan, Republic of China
arXiv:quant-ph/0508180v2 28 Sep 2005 Secret parameters in quantum bit commitment
numbers: 0367Dd0367Hk0367Mn Keywords: quantum bit commitmentquantum cryptography
The no-go theorem of unconditionally secure quantum bit commitment depends crucially on the assumption that Alice knows in detail all the probability distributions generated by Bob. We show that if a protocol is concealing, then the cheating unitary transformation is independent of any parameters (including probability distributions) secretly chosen by Bob, so that Alice can calculate it without knowing Bob's secret choices. Otherwise the protocol cannot be concealing. Our result shows that the original impossibility proof was based on an incorrect assumption, despite the fact that its conclusion remains valid within the adopted framework. Furthermore, our result eliminates a potential loophole in the no-go theorem.
The security of quantum bit commitment (QBC) is an important issue in quantum cryptography because QBC is a primitive which can be used as the building block of other important two-party cryptographic protocols [1].
A QBC protocol involves two parties customarily named Alice and Bob. Alice secretly commits to a bit b (0 or 1) which is to be revealed to Bob at a later time. In order to bind Alice to her commitment, the two parties execute a series of quantum and/or classical procedures, so that at the end of the commitment phase, Bob is in possession of a quantum mechanical state |ψ (b) B . The idea is that, with additional classical information from Alice in the unveiling phase (when she unveils the value of b), Bob can use |ψ
(b) B
to check whether Alice is honest. A QBC protocol is said to be binding if Alice cannot change her commitment or Bob will find out. Furthermore it is concealing if Bob can obtain no information about the value of b before it is unveiled, which implies that the encoding density matrix ρ
(b) B of the state |ψ (b) B is independent of the value of b, i.e., ρ (0) B = ρ (1) B .(1)
A QBC protocol is secure if and only if it is both binding and concealing. Moreover, if a protocol is secure even if Alice and Bob had unlimited computational power, then it is said to be unconditionally secure.
In 1997, Lo and Chau [2,3] and Mayers [4,5] proved that unconditionally secure QBC is impossi- * Electronic address: [email protected] ble. In a nutshell, the proof goes as follows. It is observed that the commitment process, which may involves any number of rounds of quantum and classical exchange of information between Alice and Bob, can always be represented by an unitary transformation U (b) on some initial state |φ AB in the combined Hilbert space H A ⊗ H B of Alice and Bob:
|Ψ (b) AB = U (b) |φ (b) AB ,(2)
Without loss of generality, we can take |φ AB to be pure states. In this approach, Alice and Bob do not fix their undisclosed classical parameters in the commitment phase, but leave them undetermined at the quantum level instead. This is called quantum purification. In general it requires that Alice and Bob have access to quantum computers with unrestricted capacities, which is consistent with the assumption that they have unlimited computational power.
Therefore instead of honestly following the original protocol, Alice can always follow a modified protocol as described above, so that at the end of the commitment phase, there exists a pure state |Ψ
(b) AB in H A ⊗ H B .
As long as the reduced density matrix on Bob's side is unchanged, i.e.,
Tr A |Ψ (b) AB Ψ (b) AB | = ρ (b) B ,(3)
Bob has no way of knowing what Alice has actually done. Then it follows from Schmidt decomposition theorem [6,7] that,
|Ψ (0) AB = i √ λ i |e i A ⊗ |ψ i B ,(4)
and
|Ψ
(1)
AB = i √ λ i |e ′i A ⊗ |ψ i B ,(5)AB = U A |Ψ (0) AB .(6)
The existence of U A implies that Alice has a sure-win cheating strategy (called EPR attack): Alice always commits to b = 0 in the beginning. Later on, if she wants to keep her initial commitment, she unveils as prescribed. However if she wants to switch to b = 1 instead, she just needs to apply the unitary transformation U A to the particles in her control, and then proceeds as if she had committed to b = 1 in the first place. The crucial point is that, because of Eq. (1), it is impossible for Bob to find out what Alice actually did, and he would conclude that she is honest in either case. Hence if a QBC protocol is concealing, it cannot be binding at the same time. This is the conclusion of the "no-go theorem" of unconditionally secure QBC. Note that the no-go theorem only proves the existence of the cheating unitary transformation U A in a QBC protocol which is concealing, but there is no proof that U A is always known to Alice. The point is, at the end of the commitment phase, the overall state |Ψ
B (ω) = Tr A |Ψ (b) AB (ω) Ψ (b) AB (ω)|(7)
is independent of b, then in principle a cheating transformation U A (ω) exists, so that
|Ψ (1) AB (ω) = U A (ω)|Ψ (0) AB (ω) .(8)
However without the knowledge of ω, Alice cannot calculate U A (ω) by herself. As a result unconditionally secure QBC may be possible. This is a potential loophole of the no-go theorem. The no-go theorem emphasizes that one should purify all undisclosed classical variables in analyzing the security issues. Even so, the question remains: What if Bob is allowed to choose probability distributions secretly? To this question, the authors of the no-go theorem state that "In order that Alice and Bob can follow the procedures, they must know the exact forms of all unitary transformations involved" [2,3], and "It is a principle that we must assume that every participant knows every detail of the protocol, including the distribution of probability of a random variable generated by another participant" [5]. In other words the no-go theorem asserts without proof that in any QBC protocol the overall state |Ψ (b) AB cannot contain any unknown parameters. This assertion is in fact not correct, and it has caused confusion among researchers. Without clarifying this issue, the impossibility proof is not complete and the no-go theorem will continue being challenged [8]. In any case, as long as it does not jeopardize the security of a protocol, there is no reason why a party has to disclose the values of any secret parameters he/she might have chosen in the commitment phase.
To settle this issue, we prove the following theorem. The secret parameter ω will be taken to be a probability distribution, because in a fully quantum description, probability distributions are the only unknowns left. Except for the issue of secret parameters, we shall stay within the QBC framework adopted by the no-go theorem.
Theorem 1 If a QBC protocol is concealing, then the cheating unitary transformation is independent of any probability distributions (ω's) secretly chosen by Bob.
Proof If Bob is allowed to choose ω in secret, he can always postpone his choice with the help of a quantum computer. That means, instead of picking a particular ω = ω i and keeping it secret, he can purify his choices with a probability distribution π = {p i }. The resulting overall state is given by
|Ψ ′(b) AB (π) = i √ p i |Ψ (b) AB (ω i ) |χ i ,(9)
where {|χ i } is a set of orthonormal ancilla states in Bob's Hilbert space H B . The new density matrix is given by
ρ ′(b) B (π) = Tr A |Ψ ′(b) AB (π) Ψ ′(b) AB (π)|.(10)
Since the protocol is concealing, we have
ρ ′(0) B (π) = ρ ′(1) B (π)(11)
for all possible π. Consider the case where p i = 0, for all i. According to the no-go theorem there exists a cheating unitary transformation U ′ A , such that
|Ψ ′(1) AB (π) = U ′ A |Ψ ′(0) AB (π) .(12)
It is easy to see that this same U ′ A also transforms |Ψ
AB (ω i ) = U ′ A |Ψ (0) AB (ω i ) , ∀ω i .(13)
The reason is that Bob can obtain |Ψ
(b) AB (ω i ) from |Ψ ′(b)
AB (π) by collapsing the ancilla states {|χ i } on the right hand side of Eq. (9). Since U ′ A acts on Alice's Hilbert space H A only, it commutes with any operations executed on Bob's Hilbert space H B . Consequently Eq. (12) holds independent of whether the ancilla has been measured or not, and Eq. (13) follows. Hence U ′ A is independent of ω. To avoid a circular argument, we need to show that U ′ A is also independent of the probability distribution π = {p i }. As shown in the Appendix, any superposition of probability distributions can be rewritten as a single effective distribution. That is, by a redefinition of the ancilla states on Bob's side, we can rewrite |Ψ
′(b) AB (π) of Eq. (9) as |Ψ ′(b) AB (π) = |Ψ (b) AB (ω j (π)) .(14)
Substituting Eq. (14) into Eq. (12), we see that U ′ A might depend on π through ω j (π). But that is not possible since we have already proved that U ′ A is independent of ω i for all i [see Eq. (13)]. Hence the EPR cheating transformation U ′ A does not dependent on π. QED. Therefore in a concealing QBC protocol, the cheating unitary transformation U A is independent of any secret probability distributions chosen by Bob, and Alice can calculate U A without knowing Bob's particular choices. In fact, according to the corollary proven in the Appendix, U A cannot depend on any probability distribution (specified or secret) generated by Bob. This contradicts the claim that, to be able to cheat, Alice must know every detail of the protocol, including all the probability distributions generated by Bob, so that no unknown parameter is allowed in |Ψ [2,3,4,5]. Conversely, if in any protocol the cheating unitary transformation is claimed to depend on a secret parameter ω chosen by Bob, then the protocol must be non-concealing under closer scrutiny. Thus our result eliminates a potential loophole in the no-go theorem.
(b) AB
In summary, we find that there is nothing wrong with secret parameters in QBC. We prove that in a concealing protocol, the cheating unitary transformation is independent of any parameters secretly chosen by Bob. Our result shows that the original proof of the no-go theorem [2,3,4,5] was based on an incorrect assumption. Nevertheless, even with secret parameters, unconditional security remains impossible within the framework adopted by the no-go theorem.
where
|ξ ′ k ≡ |χ ′ k |ξ k(24)
are the new orthonormal ancilla states. Comparing with Eq. (16), we obtain the desired result
|Ψ ′ AB (π) = |Ψ AB (ω j (π)) ,(25)
where ω j (π) ≡ {q ′ k } is the new effective probability distribution.
Using the above result, we can prove the following corollary:
Corollary It is in general not meaningful to specify a probability distribution to an untrustful party in any quantum protocol, because he/she can always cheat.
Proof Suppose the protocol specifies that Bob should take certain action V k on each qubit (or group of qubits) in his possession according to a probability distribution ω j = {q ′ k }. According to the theorem just proven, he can always generate a superposition of distributions with appropriately chosen p i 's, such that the effective distribution is ω j [see Eqs. (17, 25)]. Obviously Bob would have no problem passing any checks concerning ω j . In general some qubits are measured and discarded in the checking procedure. For each of the remaining qubits, Bob could either stay with ω j , or he could measure the ancilla states {|χ i } in Eq. (17) to obtain a new distribution ω i which is not necessarily equal to ω j . For a large number of qubits, the probability that ω j is obtained for every qubit is exponentially small. Hence it is not meaningful to specify a probability distribution to an untrustful Bob, because one can never be sure that he is honest.
AB
(ω) may depend on some unknown parameter ω secretly chosen by Bob. If the reduced density matrixρ (b)
( 0 )
0AB (ω i ) to |Ψ
( 1 )
1AB (ω i ) for all possible ω i , i.e.,|Ψ(1)
AppendixSuppose Bob chooses an unitary operator in {V k } with a probability distribution ω i = {q ik } and applies it to a state |φ AB , such thatAs is well known, if V k is not disclosed, Bob can postpone (or purify) his decision by entangling with a set of orthonormal ancilla states {|ξ k }, so that instead of |ψ AB , he generatesLikewise, if ω i is not disclosed, Bob can also purify his choices with another probability distribution π = {p i }, such thatwhere {|χ i } is another set of orthonormal ancilla states.Theorem 2 Purifying a probability distribution [in Eq. (17)] is equivalent to picking a new effective one [in Eq. (16)].Proof Defineso thatOn the right hand side of Eq. (18), we writewhereNote that the |χ ′ k 's are normalized but not necessarily orthogonal.Substituting Eq. (21) into Eq. (18), we get
. G Brassard, C Crépeau, SIGACT News. 27and references thereinG. Brassard and C. Crépeau, SIGACT News 27 (1996) 13, and references therein.
. H. -K Lo, H F Chau, Phys. Rev. Lett. 783410H. -K. Lo and H. F. Chau, Phys. Rev. Lett. 78 (1997) 3410.
. H. -K Lo, H F Chau, Physica D. 120177H. -K. Lo and H. F. Chau, Physica D 120 (1998) 177.
. D Mayers, Phys. Rev. Lett. 783414D. Mayers, Phys. Rev. Lett. 78 (1997) 3414.
. G Brassard, C Crépeau, D Mayers, L Salvail, arXiv:quant-ph/9712023G. Brassard, C. Crépeau, D. Mayers, and L. Salvail, arXiv:quant-ph/9712023.
. L P Hughston, R Jozsa, W K Wootters, Phys. Lett. A. 18314L. P. Hughston, R. Jozsa, and W. K. Wootters, Phys. Lett. A 183 (1993) 14.
. E Schmidt, Math. Ann. 63433E. Schmidt, Math. Ann. 63 (1906) 433.
. H P Yuen, arXiv:quant-ph/0207089;/0505132H. P. Yuen, arXiv:quant-ph/0207089; /0505132.
|
[] |
[
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes",
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes",
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes",
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes",
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes",
"Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes"
] |
[
"Tim B Miller \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Pieter Van Dokkum \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Lamiya Mowla \nDunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada\n",
"Tim B Miller \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Pieter Van Dokkum \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Lamiya Mowla \nDunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada\n",
"Tim B Miller \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Pieter Van Dokkum \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n",
"Lamiya Mowla \nDunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada\n"
] |
[
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Dunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada",
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Dunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada",
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA",
"Dunlap Institute for Astronomy and Astrophysics\nUniversity of Toronto\n50 St George StreetM5S 3H4TorontoONCanada"
] |
[] |
Recent studies have indicated that the ratio between half-mass and half-light radii, r mass /r light , varies significantly as a function of stellar mass and redshift, complicating the interpretation of the ubiquitous r light − M * relation. To investigate, in this study we construct the light and color profiles of ∼ 3000 galaxies at 1 < z < 2 with log M * /M > 10.25 using imcascade, a Bayesian implementation of the Multi-Gaussian expansion (MGE) technique. imcascade flexibly represents galaxy profiles using a series of Gaussians, free of any a-priori parameterization. We find that both star-forming and quiescent galaxies have on average negative color gradients. For star forming galaxies, we find steeper gradients that evolve with redshift and correlate with dust content. Using the color gradients as a proxy for gradients in the M/L ratio we measure half mass radii for our sample of galaxies. There is significant scatter in individual r mass /r light ratios, which is correlated with variation in the color gradients. We find that the median r mass /r light ratio evolves from 0.75 at z = 2 to 0.5 at z = 1, consistent with previous results. We characterize the r mass − M * relation and we find that it has a shallower slope and shows less redshift evolution than the r light − M * relation. This applies both to star-forming and quiescent galaxies. We discuss some of the implications of using r mass instead of r light , including an investigation of the size-inclination bias and a comparison to numerical simulations.
|
10.3847/1538-4357/acbc74
|
[
"https://export.arxiv.org/pdf/2207.05895v1.pdf"
] | 250,493,214 |
2207.05895
|
67ba9fc001ae8a4c870726553275156884a95aa5
|
Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes
July 14, 2022
Tim B Miller
Department of Astronomy
Yale University
52 Hillhouse Ave06511New HavenCTUSA
Pieter Van Dokkum
Department of Astronomy
Yale University
52 Hillhouse Ave06511New HavenCTUSA
Lamiya Mowla
Dunlap Institute for Astronomy and Astrophysics
University of Toronto
50 St George StreetM5S 3H4TorontoONCanada
Color gradients and half-mass radii of galaxies out to z = 2 in the CANDELS/3D-HST fields: further evidence for important differences in the evolution of mass-weighted and light-weighted sizes
July 14, 2022(Received XXX; Revised YYY; Accepted ZZZZ) Submitted to ApJDraft version Typeset using L A T E X twocolumn style in AASTeX631Galaxy Structure (622)Galaxy Radii (617)High-redshift galaxies (734)
Recent studies have indicated that the ratio between half-mass and half-light radii, r mass /r light , varies significantly as a function of stellar mass and redshift, complicating the interpretation of the ubiquitous r light − M * relation. To investigate, in this study we construct the light and color profiles of ∼ 3000 galaxies at 1 < z < 2 with log M * /M > 10.25 using imcascade, a Bayesian implementation of the Multi-Gaussian expansion (MGE) technique. imcascade flexibly represents galaxy profiles using a series of Gaussians, free of any a-priori parameterization. We find that both star-forming and quiescent galaxies have on average negative color gradients. For star forming galaxies, we find steeper gradients that evolve with redshift and correlate with dust content. Using the color gradients as a proxy for gradients in the M/L ratio we measure half mass radii for our sample of galaxies. There is significant scatter in individual r mass /r light ratios, which is correlated with variation in the color gradients. We find that the median r mass /r light ratio evolves from 0.75 at z = 2 to 0.5 at z = 1, consistent with previous results. We characterize the r mass − M * relation and we find that it has a shallower slope and shows less redshift evolution than the r light − M * relation. This applies both to star-forming and quiescent galaxies. We discuss some of the implications of using r mass instead of r light , including an investigation of the size-inclination bias and a comparison to numerical simulations.
INTRODUCTION
The physical processes that affect galaxies shape their morphology. A crucial tool to understand the build-up of stellar mass within galaxies is the relation between galaxy size and stellar mass. Many studies, spanning a wide range of observational facilities and galaxy properties, have produced a consistent picture of the evolution of galaxy structure from high-to-low redshift (Shen et al. 2003;Ferguson et al. 2003;Trujillo et al. 2006;Williams et al. 2010;Ono et al. 2013;van der Wel et al. 2014;Lange et al. 2015;Mowla et al. 2019c;Kawinwanichakij et al. 2021). A galaxy's half-light radius, also known as the effective radius or r light , is most often used as a measurement of its size (although some alternatives have been proposed; see, e.g., Ribeiro et al. 2016;Miller et al. 2019;Trujillo et al. 2020). The slope of the r light -M * relation is generally observed to be positive, i.e., more massive galaxies are larger, and star-forming galaxies are seen to have have larger sizes than their quiescent counterparts in almost all circumstances, except at the highest masses (log M * /M 11 (Shen et al. 2003;Mowla et al. 2019c)).
The slope of the size-mass relation for starforming galaxies is measured to be fairly shallow, d log r light /d log M * ∼ 0.2, suggesting that growth is largely self-similar, where star formation proceeds equally at all radii. For quiescent galaxies, a much steeper slope is observed along with rapid evolution in size since z = 2. These observations have been linked to growth via minor mergers; by depositing remnants of galaxies at large radii and disturbing the central galaxy, minor mergers increase the radius of a galaxy proportionally more than the stellar mass (Naab et al. 2009;Bezanson et al. 2009;Newman et al. 2012;van Dokkum et al. 2015). A complication is progenitor bias, as galaxies that cease their star formation later introduce a bias into the observed evolution of the size evolution of quiescent galaxies (van Dokkum & Franx 2001;Carollo et al. 2013;Poggianti et al. 2013).
An important caveat to all of these studies is that they are not based on half-mass radii but on half-light radii, which are typically measured at rest-frame optical wavelengths. This produces a biased view of the underlying stellar mass distribution as galaxies are known to have color gradients. Gradients in the color are the manifestation of differing stellar populations or dust content which affect the mass-to-light ratios (M/L). Spatial variations in M/L can bias light weighted morphology away from the intrinsic stellar mass distribution within galaxies. To study the stellar mass build up of galaxies, the massweighted radius is a more fundamental measurement.
M/L gradients are traced by color gradients which have been observed in both star-forming and quiescent galaxies in the local universe since the adoption of the CCD camera in astronomical observations (Kormendy & Djorgovski 1989;Franx & Illingworth 1990;Balcells & Peletier 1994). For both types of galaxies color gradients are on average negative, i.e., going from a redder center to bluer outskirts. For star-forming galaxies this is typically due to the presence of distinct components, namely an old red bulge in the center and a young blue disk at larger radii. For quiescent galaxies the main cause is likely a lower metallicity in the outskirts (Wu et al. 2005; La Barbera & de Carvalho 2009;Tortora et al. 2011), with perhaps some contribution from variation in age and/or stellar initial mass function (Conroy et al. 2017;Domínguez Sánchez et al. 2019;van Dokkum & Conroy 2021).
Whatever the cause of the color gradients, negative color gradients imply negative M/L gradients, as variations in age, metallicity, and dust content all produce approximately the same variation in the plane of M/L ratio and color (Bell & de Jong 2001). As a result, consistently negative color gradients imply that on average galaxies' half-mass radii, r mass , are systematically smaller than their optical half-light radii. In the local universe the median r light at near-IR wavelengths, a more direct tracer of stellar mass than optical, is indeed observed to be smaller than at optical wavelengths (Möllenhoff et al. 2006;Kelvin et al. 2012;Lange et al. 2015). Similarly, studies directly investigating r mass have shown it to be smaller than r light at z ∼ 0 (La Barbera & de Carvalho 2009;Ibarra-Medel et al. 2022).
Initial studies at z 1 have mirrored these results, at least qualitatively. At high redshift average color gradients have been observed to be negative for all types of galaxies (Guo et al. 2011;Wuyts et al. 2013;Szomoru et al. 2013;Liu et al. 2016;Wang et al. 2017;Liu et al. 2017;Suess et al. 2019a). Furthermore, Szomoru et al. (2013) studied 177 massive galaxies (log M * /M > 10.7) and found r mass was on average 25% smaller than r light , which did not correlate significantly with redshift or any galaxy properties. Chan et al. (2016) find r mass is 41% smaller than r light for 36 quiescent galaxies in a z = 1.55 cluster. A crucial aspect of these studies is the finding that the ratio of r mass /r light did not depend significantly on galaxy properties, such as stellar mass or r light . In this scenario, the interpretations of the r light − M * relation would remain intact as converting to r mass would only entail a constant scaling from the optical half-light radii.
This situation changed with the results of Suess et al. (2019a), who measured half-mass radii for a large, uniform sample of ∼ 7000 galaxies in the CANDELS fields with a wide mass range of log M * = 9.5 − 11.5 at z = 1 − 2.5. With this sample, they find significant correlations between r mass /r light and many galaxy properties for both star-forming and quiescent galaxies. Importantly, they find that the median ratio of r mass /r light evolves significant between z = 1 − 2.5, such that the median half-mass radii of galaxies do not evolve strongly between z = 1 − 2.5. Mosleh et al. (2020) analyzed r mass − M * distribution for galaxies between 0.3 < z < 2 using an independent method to measure half-mass radii. They also find little evolution in the r mass − M * relation for either star-forming or quiescent galaxies across the redshift range. These recent results show that M/L gradients may mimic evolution in the r mass − M * and alter its appearance. The observed lack of evolution in the median r mass of galaxies is starkly different from the evolution of r light .
There is some tension between the earlier results of Szomoru et al. (2013) and Chan et al. (2016) and the later results of Suess et al. (2019a) and Mosleh et al. (2020), and to understand the structural evolution of galaxies it is imperative to come to a firm measurement of the evolution of M/L gradients and how they alter our interpretation of r light . The tension in the literature may be caused by differences in sample size and selection (see Suess et al. 2019a, for a discussion), by differences in methodologies or how they were applied, or a combination of effects. Specifically, all these studies rely on parametric fits to model the light profile and/or the M/L profile, and this can induce systematic errors if reality does not match the parametric form.
As an alternative to parametric methods we recently developed imcascade, a Bayesian implementation of the Multi-Gaussian expansion (MGE) formalism . MGE flexibly models galaxies as a mixture of Gaussians and therefore does not require the a-priori choice of parameterization. In we show that imcascade can accurately model faint and semi-resolved galaxies, like those at z ∼ 1.5 in HST images.
In this paper we use imcascade applied to data from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) and 3D-HST surveys (Koekemoer et al. 2011;Skelton et al. 2014) to measure color gradients and half-mass radii for galaxies at 1 < z < 2. By using imcascade to model the galaxy light distribution and account for the PSF, we measure intrinsic light and color profiles free of any parameterization. We then measure half-mass radii by using the color profile to account for variations in M/L. In this study we focus on the evolution of color gradients along with how they affect the relationship between r light and r mass .
The paper is organized as follows. In Section 2 we describe our galaxy sample, the 3D-HST and CANDELS data that are used, and how we model the images with imcascade. Section 3 contains results showing the evolution of observed color gradients, Section 4 discusses the relationship between r light and r mass and in Section 5 we analyze the resulting r mass −M * relation, focusing on how it compares to the well-studied r light version. Our method and the results are discussed in Section 6 along with our final conclusions in 7.
Throughout this paper we assume a cosmology with H 0 = 68 km/s/Mpc and Ω m = 0.31 following Planck Collaboration et al. (2016). Unless otherwise stated, "radius" in this paper represents the semi-major halflight (or half-mass) radius. Surface brightness, color, and surface density profiles are shown along the semimajor axis.
OBSERVATIONS AND MODELING
Galaxy Sample and Observations
We derive the galaxy sample for this study from the 3D-HST catalog (Brammer et al. 2012;Skelton et al. 2014). This catalog uses FAST to derive stellar masses and EAZY to derive rest frame colors (Kriek et al. 2018;Brammer et al. 2008). We select all galaxies in the catalog with log M * > 10.25, 1 < z < 2, SNR F125W > 25 and SNR F160W > 25, which is equivalent to m F 160W < 23 in the tests performed This sample is > 90% mass complete. The signal-to-noise ratio for each band is measured within a 0.7 arcsec aperture. This sample contains a total of 3481 galaxies. As a supplement we use the size catalog from van der Wel et al. (2012), who measured structural parameters of the galaxies using Sersic models. Galaxies are separated into star-forming and quiescent based on their rest frame U, V and J colors following the prescription in Muzzin et al. (2013).
The relatively high mass cutoff of our sample is due to imcascade requiring a higher SNR to accurately model galaxies than parameterized fitting techniques such as Sersic fitting with galfit, imfit, etc. This is a downside of the more flexible MGE method implemented in imcascade. In we perform injection-recovery tests and find reliable results down to m F 160W ∼ 23 for the depth of the CANDELS/3D-HST survey. This is contrasted with fitting single Sersic profiles to galaxies which has been shown to be reliable down to m F 160W ∼ 24 in CANDELS data (van der Wel et al. 2012).
In this study we utilize HST data from the five fields (UDS, AEGIS, COSMOS, GOODSS and GOODSN) studied in the CANDELS survey (Koekemoer et al. 2011;Grogin et al. 2011). We use the mosaics produced by the 3D-HST survey 1 (Skelton et al. 2014). We utilize the F160W and the PSF-matched F125W mosaic. We opt for the PSF-matched images in order to help reduce artificial wiggles in the brightness and color profiles that can occur when using a discrete number of Gaussians in the MGE techniques (see Fig. 2 in .
While HST imaging exists at shorter wavelengths, specifically F814W and F606W for most of these fields, we opt to focus on F125W and F160W for this study. This ensures uniformity of analysis for all galaxies in our sample. Essentially all (> 95%) galaxies that meet our SNR criterion in F160W are also bright enough in F125W. For comparison, only one third of galaxies in our sample with SNR F 160W > 25 meet that criterion in F814W. This is due to a combination of intrinsic variation of rest-frame UV luminosity for galaxies at this epoch and the slightly different coverage of the optical and near-IR filters.
imcascade Modeling
To model the light distibution of galaxies we use our recently developed method imcascade . imcascade is a Bayesian implementa- Linear fits that are used in this paper to convert the observed color to rest frame g − r color and mass-to-light ratio. Fits are performed to results from EAZY and FAST as derived in Skelton et al. (2014) for the integrated light of galaxies in the 3D-HST survey.
tion of the Multi-Gaussian Expansion (MGE) method (Emsellem et al. 1994a,b;Cappellari 2002). MGE models galaxies as a mixture of Gaussians, allowing for increased flexibility over traditional parameterized techniques. Additionally, using a Gaussian decomposition of the PSF, this allows for analytic convolution, increasing performance and accuracy. In imcascade, the widths (or standard deviations) of all the Gaussian components are fixed and the fluxes (or weights) are the free parameters that are fit for. Additionally the central position, position angle and axis ratio are fit for, but set to be the same for all components.
Bayesian inference with imcascade is performed in two steps. The first step is a least squares fit to the image, varying all the parameters: the fluxes of each component along with the central position, position angle and axis ratio. After this the structural parameters, central position, position angle, and axis ratio are fixed. In we show that the least squares fitting process consistently and accurately returns these parameters. With these fixed, images of each component can be pre-rendered. Then model generation is a simple sum of existing array which is orders of magnitude faster computationally compared to rendering each component individually. With this performance increase Bayesian inference can be reasonably performed on a population of galaxies. Inference is performed with nested sampling using dynesty (Speagle 2020), to derive the posteriors of the weights along with the Bayesian evidence.
Another technique we employ is Bayesian model averaging described in . Specifically since the choice of widths for the Gaussian components is not unique, we perform inference on the same galaxy using multiple sets of different widths. The derived posteriors are then combined according to the resulting Bayesian evidences. We show that this helps reduce artificial "wiggles" in the derived profiles resulting from the discrete number of Gaussian components used.
For a full description of the imcascade method and its implementation please see in which we introduce and fully describe it. The procedure we use here follows that described in Sec. 4.4 of . That paper also includes injection-recovery tests which show that we can accurately model realistic z ∼ 1.5 galaxies in CANDELS-like imaging.
To model our galaxies, we begin by preparing cutouts of each galaxy for both F160W and F125W with size 40 × r eff,F160W , taken from the van der Wel et al. (2012) size catalog. We then create a variance image, tak-
F125W
Step Step 4: Integrate to find COG and Mass Radii Half-mass radius P(x) Figure 2. An overview of our methods, described in Section 2, that we use to measure a galaxy's color profile and halfmass radius.
ing into account the instrumental response and Poissson noise. A mask for each galaxy is created using the segmentation map provided by the 3D-HST team, and derived using a combination of the F125W, F140W and F160W image.
Each galaxy image is modelled with imcascade using five sets of ten Gaussian widths. This initial set of Gaussian widths is logarithmically spaced from 0.5 pixel to 10 × r vdW2012 , utilizing the F160W size measured by van der Wel et al. (2012). The subsequent sets are then shifted in logarithmic space by integer factors of d log σ/5, one fifth the logarithmic spacing between widths in the initial set. This creates a set of widths which are uniformly spaced in log r. The same sets of widths for the Gaussian components are used for each galaxy to fit both the F160W and F125W images.
The posterior distributions for the weights (or fluxes) of each component and the Bayesian evidence are calculated through nested sampling using dynesty (Speagle 2020), as described above and in . For each image of each galaxy we then combine the posteriors of the observed quantities (flux, half-light radii and surface brightness profile) derived for each set of widths weighted by the relative evidences following the Bayesian model averaging method. These combined posteriors are then used in the further analysis.
Measuring mass-weighted radii
In order to calculate the mass-weighted radii and other parameters, we must convert the observed color profiles to physical quantities. We achieve this by deriving a relationship between the observed color, m F125W −m F160W and (M/L) F160W to convert brightness profiles to mass profiles. Additionally we convert the observed color to rest-frame (g − r) so we can effectively compare color gradients across redshifts. These methods are necessarily approximate: as we show below, there is considerable scatter in M/L ratio at fixed m F125W − m F160W color. As explained in the Introduction, this paper is concerned with broad trends, and for this purpose our methodology is sufficient. 2 In Figure 1 we show linear fits to log(M/L) F160W and (g − r) rest as a function of observed m F125W − m F160W color for galaxies in six redshift bins. These measurements are taken from the 3D-HST catalogs Skelton et al. (2014). The best-fitting relations are shown in this figure and also in Appendix A. We place a floor and ceiling of log(M/L) F160W = −1 and 0.7 respectively. This will limit the effect of uncertain or outlying colors of altering the resultant half-mass radii. We will use these relationships to convert the observed color profiles measured by imcascade to rest frame color and M/L profiles. Table 1. Measurements of half-light radii, half-mass radii and color gradients using imcascade (Skelton et al. 2014) ⊥ According to rest-frame UVJ colors following Muzzin et al. (2013) ‡ Median value from posterior. The uncertainty is half of the 16th-84th percentile range. The radii are half-(light or mass) radii measured along the semi-major axis. The color gradient is measured between 0.5 − 2 r light , see Sec. 3 for more details.
Field ID * RA * DEC * z * log Mstar/M † Quenched ⊥ r light (kpc) ‡ rmass (kpc) ‡ d (g − r) / d log r ‡ GOODSN
To measure r mass we begin by multiplying the M/L profile by the F160W surface brightness profile, measured using imcascade, to calculate the mass density profile. We numerically integrate the mass density profile to find the enclosed mass profile and calculate the half-mass radius, within 10 × r light . The resulting measurements are not sensitive to the precise choice for this cutoff. This process is repeated, using samples from the posterior distribution of the surface brightness profiles and color profiles to build the posterior distribution of half-mass radii. The reported half-mass radii and associated errors are calculated as the median and half the 16th-84th percentile range respectively. An overview of the procedure for an example galaxy is shown in Figure 2.
Our method to measure half-mass radii resembles that presented in Szomoru et al. (2013) and Method 3 presented in Suess et al. (2019a). Their method relies on the residual-corrected Sersic profile 3 in several bands and EAZY to construct a (u − g) rest profile which is then converted to a M/L profile using a linear relation. Other studies (Wuyts et al. 2013;Suess et al. 2019a;Mosleh et al. 2020) have instead opted to perform SED fitting on several bands in either pixels or annuli to improve the accuracy of the M/L ratio.
The difference between our method and previous ones is that we do not assume a parametric form for the luminosity profile. In previous studies color profiles and M/L profiles were derived using parameterized functions. In our method the color profile is derived directly, and can be studied meaningfully along with r mass . In appendix B we show a detailed comparison between our measurements and those presented in Suess et al. (2019a) and Mosleh et al. (2020). In general, our measurements appear largely consistent with both studies. This comparison is discussed further in Section 6 and Appendix B.
We make our measurements available online here 4 . We include salient information taken from the 3D-HST catalog along with r light , r mass and color gradient measurements as described in this section. For these measurement we include the mean of the posterior along with the uncertainty measured as half of the 16th-84th percentile range. We inspected random examples and found the posteriors to be roughly symmetric, making this a reasonable estimate of the width of the posterior. Table 1 displays the first few entries of the table to show its form and content. Figure 3 displays median color profiles measured by imcascade for galaxies in our sample as a function of stellar mass and redshift. Here, and throughout we separate galaxies into star-forming and quiescent based on their rest frame UVJ color. Before calculating the median we normalize each galaxy's color profile by its integrated color and plot the normalized color profile: 4 Found at this link: https://raw.githubusercontent. com/tbmiller-astro/tbmiller-astro.github.io/main/assets/ Miller2022 morph CANDELs.txt log M * = 10.9, z = 1.9 Figure 3. Median color profiles over a range of stellar masses and redshifts. Color profiles are normalized by each galaxy's total color before calculating the median. In each panel the blue line shows the median profile for star-forming galaxies and the red line for quiescent galaxies. Errorbars show the 16th -84th percentile range, which encapsulates both observational uncertainties and intrinsic scatter with each population. Stellar mass increases top to bottom and redshift increases left to right with the mid point of each stellar mass and redshift bin also displayed in each panel. The thin dotted line shows where ∆(g − r) = 0 and the grey region signifies where r < PSF HWHM. In general star-forming galaxies have larger color gradients which get steeper at lower redshift and higher masses. We find very mild color gradients in quiescent galaxies that do not appear to correlate strongly with stellar mass or redshift.
COLOR GRADIENTS
∆(g − r)(R) = (g − r)(R) − (g − r) int.
. This allows us to focus on color gradients, changes in color as a function of radius, and enables easier comparison between galaxies which have a large range of integrated colors. Each panel shows the median ∆(g − r) for both star-forming and quiescent galaxies as a function of radius.
We observe several interesting trends in the median color profiles of galaxies. In general galaxies of all types tend to have negative color gradients: redder in the center and bluer in the outskirts. This agrees with the well established results in both the local universe and at high redshift (Kormendy & Djorgovski 1989;Wu et al. 2005;Wuyts et al. 2013;Suess et al. 2019a). Starforming galaxies appear to have stronger color gradients, i.e. larger negative slopes, compared to quiescent galaxies. Additionally for star-forming galaxies the slope of the color profiles is steeper at higher mass and lower redshift. This is especially evident for the central region, within ∼ 2 kpc. Quiescent galaxies appear to have consistently negative but relatively shallower color profiles with no apparent trends with redshift and stellar mass.
Individual color profiles for galaxies at a fixed stellar mass and redshift (log M * /M = 10.25 − 10.75 and z = 1 − 1.25) are shown in Figure 4. For each galaxy the median normalized color at each radii is shown. For both star-forming and quiescent galaxies we see a large amount of scatter in the distribution of profiles. This scatter increases at small radii, r < 2 kpc, where some galaxies are much redder and others are significantly bluer in the center compared to the outskirts. At larger radii the scatter also increases, this is likely due to the lower S/N in the outskirts of galaxies.
We quantify the slope, or strength, of color gradients for our galaxy sample. The slope of the color profile, d(g − r)/d log r is measured simply using the change in a galaxy's median color between two radii. We choose We see a large amount of scatter in the color profiles of both types of galaxies which tends to increase at smaller radii. two sets of radii: a constant physical separation of 2 kpc to 8 kpc and between 0.5 r light and 2 r light , based on the F160W half light radius of each galaxy. Figure 5 displays the median color gradient as a function of stellar mass and redshift for star-forming and quiescent galaxies in our sample.
The trends qualitatively observed above are confirmed in Figure 3. Quiescent galaxies have negative but shallower color gradients (d(g − r)/d log r ∼ −0.15) compared to star-forming galaxies, that do not correlate strongly with either stellar mass or redshift. Whereas for star-forming galaxies the strength of the gradients depends strongly on redshift. The median slope changes rapidly from −0.3 to −0.5 around z = 1.5. The median color gradient as a function of stellar mass appears to be different for our two estimates (one based on the physical radius and the other based on the half-light radius). This is likely due to the steepening of the size mass relation at high masses (Mowla et al. 2019c,a).
Although with only two wavelength bands it is difficult to investigate the physical cause of color gradients we attempt to gain some insight in Figure 6 by calculating the median color gradients across the rest-frame U − V vs V − J plane. Here we focus on the gradient measured between 0.5 and 2 r light , but note that the results are similar if we instead use the definition based on physical radii. There is a clear trend where the color gradients are stronger moving from bottom left to the top right in the UVJ plane. This resembles the dust track, which follows the effect of increasing dust attenuation given a standard attenuation curve (Williams et al. 2009;Leja et al. 2019), additionally recently interpreted as part of an evolutionary track by Suess et al. (2021). We infer that the color gradients at 1 < z < 2 likely reflect gradients in optical depth due to dust, rather than the traditional old bulges + young disks that are seen in spiral galaxies in the local Universe. We note that the regular trends in Fig. 6 provide some evidence that our measurements are robust; if they were driven by noise or systematic errors then we would not expect to see systematic trends with the (unrelated, from a measurement perspective) integrated rest-frame U −V and V −J colors.
THE RELATIONSHIP BETWEEN MASS AND LIGHT WEIGHTED RADII
In this section we investigate how color gradients, which trace (or in our case directly map to) M/L gradients, interact with the light profile to produce the difference between half-light and half-mass radii. To begin we examine galaxies with similar properties, but different ratios of r mass /r light . Specifically we pick star-forming galaxies with similar (within 10%) stellar mass, redshift, observed axis ratio and r mass , but differing r light . Figure 7 displays the observed images and colors for two sets of galaxies. As expected from our derivation of r mass , the major difference between the galaxies is the strength of the color gradient. The centers of the galaxies with r mass r light are significantly redder than their outskirts, whereas the observed color of galaxies with r mass ≈ r light shows little variation as a function of radius. Figure 8 displays the redshift evolution of r mass /r light for our entire sample of galaxies. The median evolution is shown along with the distribution of individual galaxies in our sample. We also show results using measurements from Suess et al. (2019a), where we exclude galaxies below our mass limit (log M * > 10.25) to enable a direct apples-to-apples comparison.
We find similar results to Suess et al. (2019a), with a smooth decline in the average ratio of r mass /r light by The median color gradient within each hexbin is shown. Here we use the gradient measured between 0.5 − 2 r eff,l but the results are similar if we use the gradient based on physical radii. For star-forming galaxies we see that the median color gradient increases from bottom-left to top-right, parallel to the track which is thought to trace increasing dust attenuation.
about a factor of ∼ 1.5 from z = 2 to z = 1 for both starforming and quiescent galaxies. The median ratio we de-rive in this study is roughly 10% lower than that of Suess et al. (2019a) at all redshifts for both star-forming and quiescent galaxies, consistent with our measurements of the half-mass radius of galaxies being slightly lower on average (see Appendix B for a full comparison).
Looking at the distribution of individual galaxies we see a significant amount of scatter around the median evolution. For both star-forming and quiescent galaxies, the observed scatter of r mass /r light around the median, estimated using the bi-weight scale, is 0.3. This is much larger than the median 1 − σ observational uncertainty of r mass /r light , which is 0.09 for quiescent galaxies and 0.05 for star-forming galaxies, suggesting that a large amount observed scatter is driven by intrinsic variation within the galaxy population (reflecting the scatter in color profiles discussed earlier).
To help understand the differences between galaxies with r mass ≈ r light and those with r mass r light we plot color profiles of these two populations. Galaxies are selected with 1.4 < z < 1.6 and 10.4 < log M * /M < 10.6. We then select the top and bottom third of galaxies ranked by radii ratio r mass /r light . The top and bottom third for star-forming (quiescent) galaxies has a radii ratio of 0.98 and 0.33 (0.99 and 0.42) respectively. These two populations are denoted r mass ∼ r light and r mass r light respectively. Again there is a clear differ- ence between the two populations in the color profile. For star-forming galaxies with r mass r light , the color of the center is ∼ 0.75 mag redder than the outskirts, where for the population with r mass ≈ r light , the color difference is only ∼ 0.2 mag. For quiescent galaxies the difference is only apparent in the inner 1 kpc. Unfortunately we cannot resolve much of this behavior due to the PSF.
Another interesting observation is that the average ratio of r mass /r light for star-forming and quiescent galaxies looks very similar, yet their color gradients (shown in Figure 3) appear very different. These two populations have very different light profiles on average. Quiescent galaxies tend to have more centrally concentrated (i.e., high Sersic index) profiles compared the star-forming galaxies which often show exponential-like profiles (Mowla et al. 2019a).
To investigate the effect of the light profile on the conversion between r mass and r light we set up an experiment based on one dimensional profiles. We start with two Sersic profiles to represent the light profiles of galaxies: n = 1 to mimic star-forming galaxies and n = 4 to mimic quiescent galaxies. We then apply a mass-to-light ratio gradient based on an exponential decline, M/L (r) = 8 × exp [α (r/r eff,l )] + 2.
(1)
We choose this form to roughly match realistic M/L profiles, however the following conclusions are not sensitive to the details of the chosen profile.
The effect of changing the exponential slope α on the ratio r mass /r light is shown in Figure 10. For both profiles, the half-mass radius decreases as the slope of the M/L gradient gets steeper, as expected. At a fixed value of α the n = 4 profile has a smaller half-mass radius compared to the n = 1 profile. The n = 4 profile is more centrally concentrated and since applying the M/L is a multiplicative effect, the mass profile is even more centrally concentrated compared to an exponential profile. This offers an explanation as to why star-forming and quiescent galaxies can have differing color gradients yet similar evolution of r mass /r light .
THE GALAXY HALF-MASS SIZE -STELLAR MASS RELATION
In this section we study the r mass -M * relation, comparing it to the well-studied r light -M * relation. The r mass -stellar mass relation for galaxies in our sample at three different redshift bins is shown in Figure 11. To investigate the evolution of the relation and compare to previous results we fit a power-law to the r mass -stellar mass relation. This is a common parameterization of the size mass-relation (van der Wel et al. 2014; Mowla et al. 2019a). Other studies have used a broken power law or other functions with a changing slope at low mass to fit the size-mass relation (Shen et al. 2003;Lange et al. 2015;Mowla et al. 2019c;Mosleh et al. 2020), however since our sample only contains galaxies with log M * /M > 10.25 we cannot constrain the lowmass slope. The relations for star-forming and quiescent galaxies are fit separately in each redshift bin. Galaxies with median r mass less than 1 pixel (equal to 0.06 arcsec or ∼ 0.5 kpc at z = 1.5) are excluded from the fit. We fit the following functional form, log 10 (r mass /kpc) = s log 10 M * /(5 × 10 10 M ) + b,
(2) with s (the slope) and b (the intercept or average size at M * = (5 × 10 10 M ) are free parameters. The fit is performed by minimizing the least squares residuals utilizing five rounds of σ-clipping with a 4σ threshold. With 500 bootstrap samples of each population we take the median values of s and b as the best-fit value and the uncertainty as half of the distance between the 16th and 84th percentile. The best fit parameters for each population are shown in Table 2. We compare our best fit parameters for the r light − M * relation to previous works in Appendix C and show that they agree well.
The best fit lines for the r light and r mass relations for quiescent and star-forming galaxies are shown in each panel of Figure 11. Consistent with the results in Section 4 we find that the half-mass radii of galaxies are Table 2. The best fit parameters to the half-mass size-stellar mass relation for star-forming and quiescent galaxies of the form: log 10 (rmass/kpc) = s log 10 M * /(5 × 10 10 M ) + b z Star-forming Quiescent s b s b
1.1 0.17 ± 0.04 0.38 ± 0.01 0.53 ± 0.06 0.06 ± 0.02 1.3 0.09 ± 0.04 0.41 ± 0.01 0.22 ± 0.09 0.02 ± 0.02 1.5 0.14 ± 0.04 0.44 ± 0.01 0.30 ± 0.09 0.08 ± 0.03 1.7 0.10 ± 0.05 0.43 ± 0.01 0.40 ± 0.06 0.08 ± 0.02 1.9 0.01 ± 0.04 0.42 ± 0.01 0.42 ± 0.11 0.06 ± 0.04 smaller at all redshifts and masses compared to the halflight radii. We observe that the slope of the r mass − M * relation appears to be shallower for both star-forming and quiescent galaxies and the relations appear to evolve less with redshift. The best fit parameters for the r mass − M * relation are compared those of the r light − M * relation as a function of redshift in Figure 12. The slope of the half-mass relation is lower on average compared to the half-light relation. For quiescent galaxies, the half-mass slope is Figure 9. Comparing the properties of galaxies in the upper and lower third of rm / r l , denoted rm ∼ r l and rm r l respectively. These populations only contain galaxies with 1.4 < z < 1.6 and 10.4 < log M * < 10.6. As with previous figures the color profiles are normalized by each galaxy's total color and the light profile is normalized by its total flux. Starforming galaxies are shown by the blue line and quiescent galaxies by the red line. There is a clear difference between the two populations in the presence of color gradients. ≈ 0.4, compared to ≈ 0.6 for the half-light relation. The slope for star-forming galaxies of the half-mass slope roughly matches that for half-light sizes at z = 1 but decreases to higher redshift. At z = 2, the star-forming r mass − M * is consistent with being flat. The normalization evolves similarly for star-forming and quiescent galaxies. The best fit value for b remains constant over the redshift range of our sample while the average r light value increases steadily over the redshift range for both populations of galaxies.
The difference in the evolution of the average size between half-mass and half-light radii is highlighted in Figure 13. In this figure we normalize the evolution of b to that of the highest redshift bin (z = 1.9±0.1). This highlights the striking difference between r mass and r light . The ratio between the average r light at z=1 to z = 2 is by 35±3% (31±4%) for star-forming (quiescent). However, for r mass this ratio is much more consistent: −2±4% and The resulting ratio of rmass /r light is shown as a function of the steepness of the M/L profile, parameterized by α. It is observed that at any value of α the value of rmass /r light is smaller for the more centrally concentrated n = 4 profile compared to that with n = 1. This explains why quiescent galaxies can have weaker color gradients than star-forming galaxies but similar values of rmass /r light . 9 ± 2% for quiescent and star-forming galaxies respectively. This result is consistent with Suess et al. (2019a) and Mosleh et al. (2020) who find little evolution in the average half-mass size of galaxies between 1 < z < 2.5.
6. DISCUSSION
Comparison to previous work
In this study we use imcascade to measure color gradients for galaxies at 1 < z < 2. imcascade utilizes the MGE formalism to flexibly measure surface-brightness and color profiles. Half-mass radii are measured by converting the color profile into an M/L using an empirical relation. This is not the first such analysis, and here we discuss differences and similarities with previous work. Previous studies have relied on parametric representation for either the surface-brightness profile of the M/L profile. Suess et al. (2019a) test three methods, but the authors favor a forward modeling approach which uses a Sersic function for the light profile and a powerlaw to represent the M/L profile, which is compared to the observed M/L profile, measured by SED fitting in consecutive annuli. Mosleh et al. (2020) perform SED fitting directly to observed pixels and then use a Sersic profile to model the resulting stellar mass image.
The flexibility of the MGE method is the strength of our procedure. The light and color profiles, which are likely complex and multi-component, are measured directly without prescribing a parameterization. In Figure 11. The rmass -stellar mass relation for star-forming and quiescent galaxies in three redshifts bins. The distribution of star-forming galaxies is shown by blue points, with red points for quiescent galaxies. Solid lines show the best fit power-law relations in each redshift bin, described in Table 2. Dotted lines show the best fit relations to the r light -stellar mass relation for comparison.
ure 3 we find non-monotonic and multi-component color profiles. The other side of this coin is that we are using a simple, empirically derived relation, to convert from observed color to M/L. This is subject to systematic uncertainties and large scatter, as shown in Figure 1. The SED fitting techniques employed by Suess et al. (2019a) and Mosleh et al. (2020) likely result in a lower scatter estimate of the observed M/L. These are based on flux measurements from several bands which is not possible in this work. Given these major differences in the procedures used it is heartening to see that measurements all three methods generally agree. An additional concern beyond measuring the M/L profile is its interaction with the light profile. As we show in Figure 10 and is also discussed in Bernardi et al. (2022), the structure of the galaxy also impacts the measurement of r mass . Due to their central light concentration, for galaxies with a larger observed Sersic index, the same M/L gradient produces a much lower ratio of r mass /r light . Therefore to accurate measure r mass one needs to accurately measure both the M/L and light profiles.
Given the subtleties involved in measuring r mass it is encouraging that our measurements generally agree with the previous results of Suess et al. (2019a); Mosleh et al. (2020). A More detailed comparisons of these measurements is shown in Appendix B. The half-mass radii agree to within 10% between the three studies and the ratio of r mass /r light 20% on average. However the observed scatter, of roughly 0.2 dex, is larger than the combination of quoted uncertainties indicating systematic differences on the level of individual galaxies.
Physical drivers of the color gradients
The color gradients measured for quiescent galaxies in our sample match well with observations of earlytype galaxies (ETG) in the local universe. Wu et al. (2005), La Barbera & de Carvalho (2009) and Tortora et al. (2011) all find negative color gradients on average for ETGs in the local universe with mass greater than log M /M * 10.5. Each study finds mildly negative color gradients, d(g − r)/d log R ∼ 0.1, similar to our study, for their galaxy samples. Utilizing SED fitting of optical and NIR bands, all three studies attribute metallicity gradients as the dominant cause, as opposed to stellar age gradients, the other possibility considered. If the quiescent galaxy populations at z = 1−2 are related to those in the local universe, the fact that we measure similar color gradients supports the conclusion that they are driven by metallicity rather than age. La Barbera & de Carvalho (2009) finds a slight positive age gradient ( ∆ log(age)/∆ log R = 0.13), which can explain the observed evolution of the half-light sizes of ETG's. In their model, ETGs form at z ≈ 2, with younger and therefore brighter centers. The central stellar population then ages and dims, causing the half-light radius to increase. Using a simple burst SFH, this can explain the observed evolution of their half light sizes. This is consistent with our findings that the mass radii of quiescent galaxies do not evolve significantly with redshift.
An additional driver of M/L gradients in quiescent galaxies is variation in the stellar initial mass function (IMF). Recent studies have suggested that the IMF is more bottom heavy in the center of some massive elliptical galaxies in the local universe (Conroy et al. 2017;Domínguez Sánchez et al. 2019;van Dokkum & Conroy 2021). This effect is essentially impossible to measure log r e (kpc) at 5 × 10 10 M Figure 12. The evolution of best-fit parameters to the rmass -stellar mass relation. The evolution of the slope (top panel) and normalization (bottom panel) is shown for both starforming and quiescent galaxies. Also shown for comparison are the best fit parameters of the r light -stellar mass relation. The slopes of the rmass relations are shallower than their r light counterparts and we see much less evolution in the normalization.
from photometry alone, yet can still drastically alter M/L. Domínguez Sánchez et al. (2019) use the MANGA IFU survey to measure the systematic variations in the IMF of elliptical galaxies as function of radius. Bernardi et al. (2022) investigate how these variations manifest as M/L gradients. The authors find that the inclusion of an IMF gradient, in coordination with the centrally concentrated light profile of elliptical galaxies, can decrease half-mass size by up to 30%. Such changes would be additional to the results presented here. Previous studies of color gradients in star forming galaxies at cosmic noon suggest they are mostly due to variation in dust content. Liu et al. (2016) and Liu et al. (2017) use measured NUV and optical colors, combined with SED fitting, to infer that color gradients in galaxies along the star-forming sequence from 0.5 < z < 2 are predominantly due to variation in dust extinction. This
Evolution of Average Size
Half-Mass Half-Light Quiescent Star-forming Figure 13. The evolution of the normalization of the sizemass relation, b, is highlighted. For each curve we normalize the growth to the highest redshift bin at z = 1.9 ± 0.1. Each curve is offset slightly on the x axis for clarity. The average rmass does not evolve (or even gets smaller) for both star-forming and quiescent galaxies, whereas r light grows by roughly 30 %.
generically produces negative color gradients as the centers of galaxies have a dust higher optical depth than the outskirts. Wang et al. (2017) measure radial gradients in the UVI color plane, similar to the more common UVJ variant, and find gradients within galaxies are almost parallel to the expected dust extinction vector. These findings are supported by studies of the Balmer decrement, that have found that dust extinction is higher in the centers of galaxies (Nelson et al. 2016a) and recent studies showing the FIR sizes, which traces dust mass, is typically smaller than optical sizes (Tadaki et al. 2020;Gómez-Guijarro et al. 2022).
Our study provides additional evidence that color gradients in star-forming galaxies are caused by dust extinction. In Figure 6 we find the strongest color gradients in the region of the UVJ plane where dust extinction is expected to be the largest. We note that this trend has also been interpreted as an evolutionary track, in Suess et al. (2021). In the local universe Kennedy et al. (2015) find that for star-forming galaxies almost all the variation in Sersic index and half-light radius with wavelength can be attributed to changes in dust opacity and inclination angle. Similarly Patel et al. (2012) show that at 0.6 < z < 0.9 a galaxy's location along the dust vector in the UVJ plane is strongly correlated with inclination angle. This is again suggestive that dust opacity and geometry plays a large role in shaping the morphology and color gradients of star-forming galaxies.
Related to this is the issue of an inclination bias in the size-mass plane, as suggested by Mowla et al. (2019b). , finding an inclination bias in the half-light sizes of galaxies. Similar to their study we focus on star-forming galaxies at 1 < z < 1.25 and 10.5 < log M * /M < 11. We see a clear correlation between axis ratio and r light . We also plot the median color gradient and half-mass size as a function of q. There is a clear correlation between color gradient and q whereas rmass does not depend on the axis ratio. These results are consistent with a scenario where the stars and dust are present in a (relatively) thin disk, leading to a bias in r light for certain viewing angles.
The authors investigate a discrepancy between the observed line width and that expected by simple viral estimator for three compact star-forming galaxies. They find the most plausible explanation is that these galaxies are mostly face on disks. Additionally they show a clear trend between r light and axis ratio for these galaxies, suggesting that compact star-forming galaxies may simply be a face-on projection rather than a separate class. In Figure 14 we recreate this relationship using both the light and mass semi-major axis. We find a similar correlation to Mowla et al. (2019b) between the semi-major axis and axis ratio. For mass radii there is much less correlation. Additionally we show that there is a clear correlation, where galaxies with lower axis ratios have steeper color gradients. A simple explanation is that the stars and dust in these galaxies live in (relatively) thin disks. Therefore the dust attenuation is higher when looking edge-on. Similar to Mowla et al. (2019b),we find an inclination bias when using r light . Accounting for color gradients and measuring half-mass sizes, appears to mostly alleviate this bias.
Comparison to simulations
A benefit of using half-mass radii is that they offer a relatively direct comparison to size predictions in numerical galaxy formation simulations. In order to properly compare simulations to observations, it is common to forward model the simulation output and apply observational techniques. This is a non-trivial process that involves expensive radiative transfer calculations to properly account for dust extinction. Additionally simulations do not fully resolve the interstellar medium of galaxies and many do not self-consistently track the production and destruction of dust leading to additional uncertainties. This has been implemented successfully by several studies (Price et al. 2017;Rodriguez-Gomez et al. 2019;Bignone et al. 2020;Parsotan et al. 2021;de Graaff et al. 2022) and is an important consistency check. However the process of creating mock observations can make it difficult to investigate the underlying causes of discrepancies. Using half-mass radii provides a more direct way to compare observations to simulations.
The observed evolution of r mass is compared to results from hydrodynamical simulations of galaxies in Figure 15. For observations we use the results from this study along with catalogs published in Suess et al. (2019a) and Mosleh et al. (2020). To directly compare to simulations we convert the observed (projected) halfmass radius to a 3D radius based on the findings of van de Ven & van der Wel (2021). The authors use analytic calculations to investigate the mapping between projected and 3D radii. They find that 1.3 × r eff,circ , with r eff,circ the circularized half-mass (or light) radius, is an unbiased estimator of r med , the 3D effective radius (with a scatter of ≈ 30% from galaxy to galaxy). As we are analyzing the evolution of the median size in this figure we use this to convert from observed radii to 3D radii, which we directly compare to simulations.
We use previously published results from the EAGLE Schaye et al. 2015;Furlong et al. 2017) and TNG-100 Pillepich et al. 2018;Marinacci et al. 2018;Naiman et al. 2018;Springel et al. 2018;Genel et al. 2018) Simulations Eagle -Furlong+17 TNG100 -Genel+19
Simulations Eagle -Furlong+17 TNG100 -Genel+19 Figure 15. The sizes of galaxies at fixed mass from z = 1 − 2, comparing observations with predictions from hydrodynamical simulations. We show the median 3D half-mass sizes within a fixed stellar mass range of log M * /M = 10.5 − 11. For the TNG and Eagle simulations the measurements are taken from Genel et al. (2018) and Furlong et al. (2017), see text for details. The observed 3D radii are calculated using the circularized half-mass radii as prescribed in van de Ven & van der Wel (2021). The simulations and observations agree fairly well at z ∼ 2. However, the simulations predict significant change in the sizes of galaxies as a function of redshift whereas the observations suggest there is very little variation with redshift in either star-forming or quiescent galaxies.
plot the evolution of the 3D half-mass sizes of galaxies taken directly from Figure 3 in Furlong et al. (2017) for active and passive galaxies (which we plot as starforming and quiescent respectively). For TNG-100 we take the measurements of the median 3D half-mass sizes of galaxies at z = 1 and z = 2 from Figure 1 in Genel et al. (2018). We interpolate these median relations to log M * /M = 10.75 for main-sequence and quiescent galaxies, which we plot on Figure 15 as star-forming and quiescent respectively. We note that both Furlong et al. (2017) and Genel et al. (2018) directly compare 3D mass radii, or mock observations without accounting for dust, to observed half-light radii which we argue is not an apples-to-apples comparison and should be interpreted with caution. The measurements from simulations and observations appear consistent at z = 2, however their evolution is qualitatively different. As seen in Suess et al. (2019a), Mosleh et al. (2020) and we show in Section 5, the observed half-mass radii for both star-forming and quiescent galaxies are not significantly different over the redshift range z = 1 − 2. By contrast, for both the EA-GLE and TNG-100 simulations the mean size of galaxies changes 50% − 75% over the same epoch. This growth in radii of simulated galaxies roughly matches that observed in half-light radii (van der Wel et al. 2014;Mowla et al. 2019c), yet is starkly different when an apples-toapples comparison to r mass is made.
In Table 2, we provide the results of power-law fits to the r mass − M * relation. These measurements agree well with previous studies (Suess et al. 2019a;Mosleh et al. 2020) 5 . For quiescent galaxies the relation is still steeper than the star-forming relation however the slope is when using r mass (d log r mass / d log M * ∼ 0.4 ) is shallower than when using r light (d log r light / d log M * ∼ 0.6−0.8 ) (van der Wel et al. 2014;Mowla et al. 2019c). Combined with the lack of evolution in median size of quiescent galaxies, this seems inconsistent with the theory that quiescent galaxies grow by dry-minor mergers, which predicts a large increase in radius roughly following: d log r/d log M * ∝ 2 (van Dokkum et al. 2010). However it is possible that the time between z = 1 − 2, is not long enough to see significant evolution in the population due to mergers. Suess et al. (2019b) extend the previous study down the z ∼ 0 and find the median r mass of quiescent galaxies does evolve significantly below z = 1, consistent with a model of growth based on minor mergers (Newman et al. 2012). The flat slope and little evolution in the r mass − M * relation for starforming galaxies, favors self-similar or spatially coherent growth. This scenario, characterized by flat sSFR profiles, has been suggested by observational studies at high redshift (Liu et al. 2016;Nelson et al. 2016b) and by hydrodynamical simulations of galaxies (Nelson et al. 2021).
SUMMARY AND CONCLUSIONS
In this work we measure color gradients and half-mass radii of galaxies at 1 < z < 2. We utilize imcascade, a Bayesian implementation of MGE representation, to flexibly measure the surface brightness and color profiles of galaxies in our sample. By using an empirical relationship between color and M/L, we measure half-mass radii for galaxies as well. We characterize the r mass −M * relationship comparing it to the well studied r light − M * relationship.
We typically find negative color gradients, with the slope for star-forming galaxies steeper than for quiescent galaxies. The gradients roughly match those measured in the local universe. We find significant variation in the color gradients of galaxies at a fixed M * and z, leading to intrinsic scatter in r mass /r light .
We confirm previous results by Suess et al. (2019a) and show that r mass /r light decreases by ∼ 50% between z = 2 and z = 1. Even though the color gradients are shallower, quiescent galaxies have a similar ratio of r mass /r light , due to the steeper light profile.
By characterizing the r mass − M * relation, we find it is shallower compared to the r light − M * counterpart, and that the normalization changes less with redshift. The ratio of the average half-mass size between z = 1 and z = 2 is 9 ± 2% and −2 ± 4% for quiescent galaxies and star-forming galaxies respectively. This is in stark contrast with the average half-light size which grows by ≥ 35% for both populations. Our study adds to the growing consensus which indicates that half-mass radii evolve much slower than half-light radii. (Suess et al. 2019a;Mosleh et al. 2020) This challenges the traditional view of galaxy growth along with predictions from hydrodynamical simulations.
We emphasize here that evolution of relations is not the same as evolution of individual galaxies. Galaxies increase their mass over time, which means that a comparison at fixed mass (such as done in Fig 13) is not equivalent to the size evolution of any individual galaxy. These relations are a consequence of the evolution of a population of galaxies. Galaxies may evolve along the size mass relation yet decoding the growth of individual galaxies is much more difficult as one must account for their growing stellar mass and the transition from starforming to quiescent (or vice-versa). (see van Dokkum et al. 2015, for a further discussion of these issues) Alter-natively hydrodynamical simulations provide the ability to study the evolution of individual galaxies (e.g. Genel et al. 2018).
We also stress that measuring the half-mass radii of faint and barely resolved galaxies is pushing current data to its limit. One needs to measure both color and light profiles accurately, while also converting to M/L, in order to correctly measure the half-mass radii of galaxies. While it is encouraging that on average our results agree with both Suess et al. (2019a) and Mosleh et al. (2020) within ∼ 10% on average, there are still systematic uncertainties when comparing individual galaxies. With the coverage of longer wavelength and higher spatial resolution of the recently launched James Webb Space Telescope we will be able to study the rest-frame NIR morphology of galaxies at z > 1 for the first time. The NIR is a much more direct tracer of stellar mass, with less variation in M/L compared to optical wavelengths. This will provide a definitive answer to the evolution of half-mass radii.
With JWST at high redshift, the next generation multi-band ground based surveys, like HSC-SSP (Aihara et al. 2019) and the upcoming Rubin observatory (Ivezić et al. 2019), will be able to accurately measure half-mass radii at z < 1 from multi-band photometry. This combination will map the evolution of half-mass radii from galaxies at high-redshift to the local universe. Using r mass as a more fundamental tracer of the structure of galaxies, this will shed new light on the morphological evolution of galaxies.
ACKNOWLEDGMENTS
The authors would like the thank Arjen van der Wel for useful discussions and comments on the manuscript. TBM would like to thank Patricia Gruber and the Gruber foundation for their support of the work presented here. This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Table 3 we show the best fit parameters for the empirical relations used in Section 2.3 to convert from observed color to rest frame color or M/L. The best fit parameters were found by maximizing the likelihood, taking into account measurement uncertainties in both the x and y parameters following the prescription in Hogg et al. (2010). Separate fits were performed in six redshift bins with dz = ±0.1 and the central redshift listed in the table. We tested the alternative of fitting a smooth model, a = f (z) for e.g., instead of using separate redshift bins and found similar overall scatter. However when using linear or quadratic functions for the evolution we observed structure in the residuals as a function of z so we opted to perform fits in separate bins instead. Also listed in the table is the RMS scatter for the best fit line.
Residuals to the empirical fit to both the rest from color and M/L as a function of galaxy properties are shown in Figure 16. The stellar masses and SSFR's are measured using FAST (Skelton et al. 2014). For clarity we show all galaxies with SSFR < 10 −13 yr −1 at 10 −13 yr −1 . We observe that there are no large systematic offset as a function of galaxy properties.
B. COMPARISON OF HALF-MASS RADII TO PREVIOUS STUDIES
In this section we compare our measurements to those of Suess et al. (2019a) and Mosleh et al. (2020). Both studies test multiple methods, we opt to use Method 1 for each study as it is favored by each of the authors. In Method 1 of Suess et al. (2019a) the authors start by performing SED fitting to calculate the convolved M/L in concentric annuli and using GALFIT to find the best fit Sersic model to describe the light profile in the F160W image. They use a power-law parameterization for the M/L profile which they multiply by the intrinsic light model and convolve with the PSF to construct a convolved mass map which they compare to the image of the galaxy and measure the convolved M/L profile. This is then compared to the results of SED fitting to find the best fit MLR, which is multiplied by the Sersic light model to calculate the intrinsic mass map, where the half-mass radius is measured from. Mosleh et al. (2020) also perform SED fitting directly to pixels of the PSF matched CANDELS images. The result is the observed stellar mass map. This is used to measure a 1D mass density profile, which is fit with a 1D Sersic profile while accounting for the effect of the PSF. Figure 17 compares the half-mass radius measured using the different methods for overlapping galaxies in the GOODS-S field. This includes 247 galaxies in both our and the Suess et al. (2019a) sample and 347 in the Mosleh et al. (2020) sample. On average our measurements agree with the previous studies to within 10%. This is reassuring given that the three different methods use vastly different methodologies to measure the half-mass radius. However there is significant scatter of ∼ 0.2 − 0.25 dex between the methods. The median observed error for these galaxies is 0.1,0.1,0.15 for this study, Suess et al. (2019a) and Mosleh et al. (2020), respectively. This implies systematic uncertainties in these relations beyond the observational uncertainties. While the three methods agree on average, they do no appear to be entirely consistent on the level of an individual galaxy Also shown is the comparison of r mass /r light . In some sense this is the more important comparison as the M/L gradient is the key to measuring r mass yet is difficult to measure and prone to systematic biases. We see very similar results when comparing r mass . While all 3 methods agree on average the observed scatter is larger than the observational uncertainties.
C. COMPARISON OF THE r light -STELLAR MASS RELATION TO PREVIOUS STUDIES
The best fit relationship to the r light −M * found in this study is compared Mowla et al. (2019c) is shown in Figure 18. The authors use the COSMOS-DASH survey, which covers a larger area compared to CANDELS, to measure the sizes of massive galaxies using GALFIT assuming a single Sersic profile. This combined with data from van der Wel et al. (2014) to update the r light − M * relationship.
For star-forming galaxies, our two studies agree, within uncertainties, on the evolution of both the slope and normalization of the r light − M * relationship. For quiescent galaxies we see qualitatively consistent evolution however in detail there are a few discrepancies. Mainly at z = 1.5 we measure a shallower slope, by 0.2, and at z > 1.5 we find the half-light sizes of quiescent galaxies are ∼ 25% larger on average then Mowla et al. (2019c). It is possible these difference are due to the different galaxies sample, our study only use the CANDELS survey and therefore contains far fewer galaxies at log M * /M > 11. Another possibility is the difference between the MGE parameterization used in imcascade and the Sersic profile. For quiescent galaxies the measured Sersic index is often greater than 4 which produces profiles with very large central concentrations. This central peak is much greater than the MGE models Comparing rmass measurements from this study to Suess et al. (2019a) and Mosleh et al. (2020). The comparison of rmass (top) and the ratio rmass/r light (bottom) are shown. In each panel we show the average offset and scatter measured using the biweight location and biweight scale respectively. On average the three methods agree within ∼ 10% yet there are systematic biases when comparing individual galaxies.
which, for even the components with the smallest widths, become flat near the center. This would lead to the smaller sizes for the Sersic profiles. It is unclear which parameterization is more accurate for the central regions of these galaxies without higher resolution data; the half light radii are already on the order of the PSF FWHM. log r e (kpc) at 5 × 10 10 M Figure 18. The best fit-parameters of the r light − M * relationship in this study compared to Mowla et al. (2019c). For star-forming galaxies our measurements agree well. For quiescent galaxies, the slope is generally similar but the normalization differs, especially at z > 1.5. This is likely due to a difference in parameterization between imcascade's MGE models and the Sersic profiles us in Mowla et al. (2019c)
Figure 1 .
1Figure 1. Linear fits that are used in this paper to convert the observed color to rest frame g − r color and mass-to-light ratio. Fits are performed to results from EAZY and FAST as derived in Skelton et al. (2014) for the integrated light of galaxies in the 3D-HST survey.
Figure 4 .
4Normalized color profiles for individual galaxies with log M * /M = 10.25 − 10.75 at z = 1 − 1.25. The median normalized color from the posterior distribution at each radius is used to construct the color profile. The top panel shows star-forming galaxies in blue and the bottom panel shows quiescent galaxies in red. In each panel the black dashed line shown the median of the galaxy sample.
Figure 5 .Figure 6 .
56Measured color gradients shown as a function of redshift and stellar mass. Median color gradients for star-forming galaxies are shown in blue with quiescent galaxies in red. Additionally the gradient is measured in two separate ways: the solid line shows the gradient between 0.5 − 2r eff,l and the dashed lines show the gradient measured between a fixed physical radius of 2 − 8 kpc. Quiescent galaxies have weaker gradients (d (g − r)/d log r ∼ 0.15) which do not depend on stellar mass or redshift while star-forming galaxies have stronger gradients (d (g − r)/d log r ∼ 0.4) which get steeper at lower redshift. Color gradients across the U − V vs. V − J plane.
(a) z ≈ 1. 2 , 0 5 Figure 7 .
2057M * ≈ 5 × 10 10 , q ≈ ) z ≈ 1, M * ≈ 7 × 10 10 , q ≈ 0.Two sets of two galaxies which are matched based on z, log M * /M , q light and rmass but have different r light . We show color renditions of the galaxies based on the F160W and F125W images (top) and their observed color maps (bottom), obtained by dividing the images. Cyan and gold ellipses display the half-light and half-mass radius for each galaxy. The presence of color gradients affects the relationship between rmass and r light .
Figure 8 .
8The evolution of the ratio between rmass and r light . Star-forming galaxies are shown in blue in the left panel and quiescent galaxies in red in the right panel. In each panel the points show individual galaxies while the line shows the median as a function of redshift. There is significant scatter in rmass / r light but for both star-forming and quiescent galaxies we see a general trend where this ratio decreases at lower redshift. Also shown are results fromSuess et al. (2019a), where we have excluded galaxies in their sample below log M * /M = 10.25 to match our sample. Our results qualitatively agree with those ofSuess et al. (2019a).
Figure 10 .
10Results of a simple experiment of applying a varying M/L gradient to two different Sersic light profiles.
Figure 14 .
14We confirm the results ofMowla et al. (2019b)
Software: numpy (Van Der Walt et al. 2011), scipy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2018), dynesty (Speagle 2020), pandas (Wes McKinney 2010), asdf(Greenfield et al. 2015), SEP(Barbary 2016;Bertin & Arnouts 1996), imcascade
Figure 16 .
16Residuals of the empirical fit performed to calculate rest frame (g − r) (top) and M/L (bottom) from the observed color as a function of galaxy properties. We see there are no large systematic effects as a for different galaxy populations for either fit. The orange lines show the median and scatter calculated with the bi-weight location and bi-weight scale.
Figure 17 .
17Figure 17. Comparing rmass measurements from this study to Suess et al. (2019a) and Mosleh et al. (2020). The comparison of rmass (top) and the ratio rmass/r light (bottom) are shown. In each panel we show the average offset and scatter measured using the biweight location and biweight scale respectively. On average the three methods agree within ∼ 10% yet there are systematic biases when comparing individual galaxies.
simulations. For EAGLE, we1.00 1.25 1.50 1.75 2.00
z
2
3
4
5
6
r
m, 3D (kpc)
Star-forming
1.00 1.25 1.50 1.75 2.00
z
1
2
3
4
5
Quescient
Observations
This Study
Suess+19
Mosleh+20
Table 3 .
3The best-fit values for the empirical relations used to convert from the observed color to other quantities of the form:X = a (mF125W − mF160W) + b APPENDIX A. BEST FIT RELATIONSHIPS BETWEEN OBSERVED COLOR AND PHYSICAL QUANTITIESz
X = (M/L)F160W
X = (g − r)rest
a
b
r.m.s.
a
b
r.m.s.
1.1 2.52 −1.19 0.33
1.74
0.07
0.23
1.3 2.88 −1.63 0.48
1.49
0.01
0.24
1.5 3.15 −1.81 0.71
1.39 −0.02 0.21
1.7 1.31 −1.07 0.23
0.96
0.18
0.12
1.9 1.28 −1.13 0.22
0.99
0.14
0.21
2.1 1.16 −1.20 0.30
0.86
0.18
0.26
The mosaics were downloaded from the 3D-HST website: https://archive.stsci.edu/prepds/3d-hst/
We can expect great improvements in this area from JWST, owing to its ability to obtain high resolution images at rest-frame near-IR wavelengths.
The method of residual correcting the best fit Sersic profile, developed inSzomoru et al. (2010) is meant to alleviate biases caused by the choice of parameterization.
We note that inMosleh et al. (2020), the authors opt for a broken power law fit making comparison more difficult. However focusing on the high mass slope and normalization, our measurements agree within quoted uncertainties.
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|
[] |
[
"Discrepancy measures for sensitivity analysis",
"Discrepancy measures for sensitivity analysis"
] |
[
"Arnald Puy [email protected] \nSchool of Geography, Earth and Environmental Sciences\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom\n",
"Pamphile T Roy \nQuansight, ViennaAustria\n",
"Andrea Saltelli \nBarcelona School of Management\nPompeu Fabra University\nCarrer de Balmes 13208008BarcelonaSpain\n"
] |
[
"School of Geography, Earth and Environmental Sciences\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom",
"Quansight, ViennaAustria",
"Barcelona School of Management\nPompeu Fabra University\nCarrer de Balmes 13208008BarcelonaSpain"
] |
[] |
While sensitivity analysis improves the transparency and reliability of mathematical models, its uptake by modelers is still scarce. This is partially explained by its technical requirements, which may be hard to understand and implement by the nonspecialist. Here we propose a sensitivity analysis approach based on the concept of discrepancy that is as easy to understand as the visual inspection of input-output scatterplots. Firstly, we show that some discrepancy measures are able to rank the most influential parameters of a model almost as accurately as the variance-based total sensitivity index. We then introduce an ersatz-discrepancy whose performance as a sensitivity measure matches that of the best-performing discrepancy algorithms, is simple to implement, easier to interpret and orders of magnitude faster.
| null |
[
"https://export.arxiv.org/pdf/2206.13470v2.pdf"
] | 257,622,524 |
2206.13470
|
aae4f7090581da413333b0294faebe4b4928cb93
|
Discrepancy measures for sensitivity analysis
17 Mar 2023
Arnald Puy [email protected]
School of Geography, Earth and Environmental Sciences
University of Birmingham
B15 2TTBirminghamUnited Kingdom
Pamphile T Roy
Quansight, ViennaAustria
Andrea Saltelli
Barcelona School of Management
Pompeu Fabra University
Carrer de Balmes 13208008BarcelonaSpain
Discrepancy measures for sensitivity analysis
17 Mar 20231Design of experimentsuncertainty analysisscatterplotsmathematical models * Corresponding author
While sensitivity analysis improves the transparency and reliability of mathematical models, its uptake by modelers is still scarce. This is partially explained by its technical requirements, which may be hard to understand and implement by the nonspecialist. Here we propose a sensitivity analysis approach based on the concept of discrepancy that is as easy to understand as the visual inspection of input-output scatterplots. Firstly, we show that some discrepancy measures are able to rank the most influential parameters of a model almost as accurately as the variance-based total sensitivity index. We then introduce an ersatz-discrepancy whose performance as a sensitivity measure matches that of the best-performing discrepancy algorithms, is simple to implement, easier to interpret and orders of magnitude faster.
Introduction
Sensitivity Analysis (SA) studies how the uncertainty in a model output can be apportioned to uncertainty in the model inputs [1]. Modelers and system analysts use SA to explore how a range of different assumptions on the model parameters, boundary conditions or hypotheses affect a system of interest; to identify influential and non/influential inputs to guide further research and/or to simplify the dimensionality of the problem at hand; and to inform decision-making [2]. Due to its capacity to pry the model black box open and increase the transparency of the modeling process, SA is a cornerstone of responsible modeling [3,4]. It is also a recommended practice in the modeling guidelines of the European Commission, the Intergovernmental Panel on Climate Change or the US Environmental Protection Agency [5,6].
After more than 50 years of development, modelers dispose of several SA procedures and of a rich literature informing on which methods are most efficient in each specific SA setting [7,8]. We briefly mention here some of these routines without further description and direct the reader to existing references:
• Variance-based methods: they are well embedded in statistical theory (ANOVA), can treat sets of factors and can be used in problem settings such as "factor fixing" (which factor/s are the least influential and hence can be fixed to simplify the model) or "factor prioritization" (which factors convey the most uncertainty to the model output) [9]. • Moment-independent methods: they assess sensitivities based on the entire probability distribution of the model output and may be preferred over variance-based ones when the output is long-tailed [10]. • The Variogram Analysis of Response Surfaces (VARS): it computes sensitivities using variograms (functions to characterize the spatial covariance structure of a stochastic process) and is especially suited to inform about the local structure of the output [11,12]. • Shapley coefficients: it relies on Shapley values (the average marginal contribution of a given feature across all possible feature combinations). It is a good alternative to the SA approaches just mentioned when model inputs are correlated [13,14].
Despite this abundance of methods, there is still a scarce uptake of SA in mathematical modeling. When performed, it is often conducted by moving "One variable-At-a-Time" (OAT) to determine its influence on the output, an approach that only works in lowdimensional, linear models [15]. A key reason behind this neglect is that proper SA methods are grounded on statistical theory and may be hard to understand and implement by non-specialists [16].
Here we propose an SA measure whose use and interpretation requires little to no statistical training and that is as intuitive as the visual inspection of input-output scatterplots. Since the presence (absence) of "shape" in a scatterplot indicates an influential (non-influential) input, we build on the concept of discrepancy (the deviation of the distribution of points in a multi-dimensional space from the uniform distribution) to turn discrepancy into a sensitivity measure. We show that some discrepancy algorithms nicely match the behavior of the total-order sensitivity index, a variance-based measure which estimates the first-order effect of a given input plus its interactions with all the rest [17]. We also present a simple-to-implement "ersatz" discrepancy whose behavior as a sensitivity index approximates that of the best-performing discrepancy algorithms at a much more affordable computational cost. Our contribution thus provides modelers with a straight-forward SA tool by turning the concept of discrepancy upside down -from a tool to inspect the input space of a sample to an index to examine its output space.
Materials and methods
The link between scatterplots and discrepancy
Due to their ease of interpretation, scatterplots are widely used in SA as a preliminary exploration of sensitivities before embarking on more quantitative approaches [18]. To understand the rationale, let us first define Ω = [0, 1) d as a d-dimensional unit hypercube formed by N s sampling points and represented by the matrix X, such that
X = x (1) 1 x (1) 2 · · · x (1) d x (2) 1 x (2) 2 · · · x (2) d · · · · · · · · · · · · x (Ns) 1 x (Ns) 2 · · · x (Ns) d ,(1)
where x (i) k is the value taken by the k-th input in the i-th row, and
x (i) = (x (i) 1 , . . . , x (i) d ). Let y = y (1) y (2) . . . y (Ns) = f (x (1) ) f (x (2) ) . . . f (x (Ns) ) (2)
be the vector of responses after evaluating the function (model) f (.) in each of the N s rows in X. If y is sensitive to changes in x k , a scatterplot of y against x k will display a trend or shape, meaning that the distribution of y-points over the abscissa (over input x k ) will be non-uniform [9, p. 28]. Generally, the sharper the trend/shape, the larger the area without points and the stronger the influence of x k on y. In contrast, a scatterplot where the dots are uniformly distributed across the space formed by x k and y evidences a totally noninfluential parameter (Fig. 1a-c). This heuristic suggests that, in a 2-dimensional space, the deviation of points from the uniform distribution can inform on the extent to which y is sensitive to x k . [19]. In a), y is completely driven by x1 while x2 is non-influential. In b), x1 is more influential than x2 given its sharper trend. In c), x1 is more influential than x2 given the presence of larger areas where points are more rarefied.
There are different ways to assess the "uniformity" of a sample. Geometrical criteria such as maximin or minimax respectively maximize the minimal distance between all points or minimize the maximal distance between any location in space and all points of the sample [20]. These criteria are notably used in circle packing problems. In contrast, uniformity criteria measure how the spread of points deviates from a uniform spread of points (in the sense of a multi-dimensional uniform distribution): taking a subspace of the parameter space J x = [0, x), we count the number of points N s [0,x) in the subspace and compare it to the total number of the points N s of the sample. The resulting value is subtracted by the volume of the subspace Vol [0,x) ,
N s [0,x) N s − Vol [0,x) .(3)
The resulting quantity is known as the discrepancy at point x. Notice that with this description, the origin of the domain ([0] d ) is part of every subspace.
Several measures calculate the discrepancy over the whole domain, assumed to be the unit hypercube. The L p -discrepancy measure, for instance, takes the average of all discrepancies, as
D p = N s [0,x) N s − Vol [0,x) p dx 1/p .(4)
When p → ∞, the measure is known as the "star discrepancy", which corresponds to the Kolmogorov-Smirnov goodness-of-fit statistic [21].
D * = sup x∈X N s [0,x) n − Vol [0,x) .(5)
When p → 2, the measure is known as the "star L 2 discrepancy" [22], which corresponds to the Cramér-Von Mises goodness-of-fit statistic [21]. Its analytical formulation reads as
SL 2 (X Ns d ) = 3 −d − 2 1−d N s Ns i=1 d k=1 1 − (x (i) k ) 2 + 1 N s 2 Ns i=1 Ns j=1 d k=1 1 − max(x (i) k , x (j) k ) . (6)
The "modified discrepancy" M 2 slightly varies the "star L 2 discrepancy" [23], and reads as
M 2 (X Ns d ) = 4 3 d − 2 1−d N s Ns i=1 d k=1 3 − (x (i) k ) 2 + 1 N s 2 Ns i=1 Ns j=1 d k=1 2 − max(x (i) k , x (j) k ) .(7)
Fang et al. [24] proposed the following criteria to assess the quality of discrepancy measures:
1. They should be invariant under permuting factors and/or runs. 2. They should be invariant under coordinate rotation. 3. They should measure not only uniformity of the hypercube, but also of any subprojections. 4. They should have some geometric meaning. 5. They should be easy to compute. 6. They should satisfy the Koksma-Hlawka inequality. 7. They should be consistent with other criteria in experimental design, such as the aforementioned distance criteria.
Equations 4-7 do not satisfy these criteria as they lack sensitivity, vary after rotation and consider the origin to have a special role. To mitigate these issues, some modified formulations of the L 2 -discrepancy have been proposed. As we shall see, these methods treat the corners of the hypercube differently. The "centered discrepancy" C 2 , for instance, does not use the origin of the domain when selecting samples to create the volumes, but the closest corner points J x [25], as
C 2 (X Ns d ) = 13 12 d − 2 N s Ns i=1 d k=1 1 + 1 2 | x (i) k − 0.5 | − 1 2 | x (i) k − 0.5 | 2(8)+ 1 N 2 s Ns i,j=1 d k=1 1 + 1 2 | x (i) k − 0.5 | + 1 2 | x (j) k − 0.5 | − 1 2 | x (i) k − x (j) k | .
The "symmetric" discrepancy S 2 is a variation of the centered discrepancy that accounts for the symmetric volume of J x [25]:
S 2 (X Ns d ) = 4 3 d − 2 N s Ns i=1 d k=1 1 + 2x (i) k − 2 x (i) k 2 (9) + 2 d N 2 s Ns i,j=1 d k=1 1− | x (i) k − x (j) k | .
The "wrap-around discrepancy" W D 2 , on the other hand, does not use any corners nor the origin (hence it is also called "unanchored discrepancy") [26].
W D 2 (X Ns d ) = − 4 3 d + 1 N 2 s Ns i,j=1 d k=1 3 2 − | x (i) k − x (j) k | + | x (i) k − x (j) k | 2 .(10)
The centred discrepancy C 2 and the wrap-around discrepancy W D 2 are the most commonly used formulations nowadays. As per numerical complexity, these equations have a complexity of O(N 2 s d). The Koksma-Hlawka inequality implies that low discrepancy sequences reduce a bound on integration error [27]. Hence, averaging a function over samples of points with a low discrepancy would achieve a lower integration error as compared to a random sample [also called Monte Carlo (MC) sampling]. Quasi-Monte Carlo (QMC) methods are designed with that problem in mind. For well behaved functions, they typically achieve an integration error close to O(N −1 s ). There is an extensive body of literature covering methods to create a sample with a low discrepancy. Notably, the low discrepancy sequence of Sobol' [28] is one of the most widely used method and it's randomized version nearly achieves a convergence rate of O(N −3/2 s ).
An ersatz discrepancy
The discrepancies presented in Equations 4-10 are state-of-the-art measures whose understanding demands a modicum of statistical training. They are also computationally complex given their reliance on column-wise and row-wise loops, a feature that hampers their scalability to larger sample sizes and/or higher-dimensional settings. Here we propose an ersatz discrepancy that addresses these issues and builds on the link between scatterplots, discrepancy and sensitivity discussed in Section 2.1.
We suggest to split the x k , y plane into a uniform grid formed by √ N s × √ N s cells, and calculate the ratio between the number of cells with points and the total number of cells (Algorithm 1). The resulting value thus informs on the fraction of cells that are populated by at least one point: the closer this value is to 1, the more the design approaches a uniform distribution -and the less influential x k is. Note that our measure (S-ersatz hereafter) is very close to the definition of discrepancy, but differs from previous discrepancy measures in that larger values indicate better uniform properties. For Equations 4-10, smaller values are better because they reflect the difference between the distribution of sampling points and a distribution with an equal proportion of points in each explored sub-region of the unit hypercube.
Algorithm 1 The S-ersatz of discrepancy. Require: x k , y
Model input x k and model output y, both of length N s .
1: s ← √ N s 2: O ← Create a s × s zero matrix Each element in O is a grid cell. 3: M ← Column-bind x k and y. 4: for each k ∈ M do 5: M k ← M k × s 6: if M (i) k = 0 then
Round up. We graphically represent our approach in Figure 2 for both random (a) and quasirandom (b) numbers [Quasi-Monte Carlo (QMC), using the low discrepancy sequence of Sobol']. The latter are known to outperform the former in sampling the unit hypercube by leaving smaller unexplored volumes. This means that a design with QMC should display larger S-ersatz values. For N s = 2 2 (first column), the plane is partitioned into four cells given that . The behavior of the S-ersatz therefore nicely matches the well-known capacity of quasi-random sequences in covering the domain of interest more evenly and quicker than random numbers (Figure 2c).
√ N s × √ N s = 4. For N s = 2 b , b = 4,
The S-ersatz is very close to the definition of the discrepancy. Instead of counting the number of points in each cell and dividing by the number of samples, it counts 1 when a cell contains a point and 0 if not. Another difference with previous discrepancy measures is that cells are pre-defined as a uniform grid. This setting corresponds to the "property A" described by Sobol' [29], which is another way of assessing the uniformity of a sample. This property is key in the design of a class of QMC methods such as Sobol's low discrepancy sequence.
The S-ersatz therefore displays the following properties:
1. It is an unanchored discrepancy.
A sensitivity analysis setting
To explore the extent to which the concept of discrepancy is apt to distinguish influential from non-influential inputs, we benchmark the performance of Equations 6-10 and of the S-ersatz (Algorithm 1) in a sensitivity analysis setting. Specifically, we assess how well each discrepancy measure ranks the most influential parameters, i.e., those that convey the most uncertainty to the model output. Since only a small fraction of the model inputs tends to be responsible for variations in the output (a manifestation of the Pareto principle, where 80% of the effects are caused by 20% of the inputs [30,31]), most practitioners are content with the correct identification of top ranks only [32]. As a yardstick of quality, we compare the ranking produced by each discrepancy measure against the ranking produced by the Jansen estimator [33], one of the most accurate variance-based total-order estimators [7,15]. We consider the total-order index to be an appropriate standard because it captures first-order as well as interaction effects [17].
To minimize the influence of the benchmarking design on the results of the analysis, we randomize the main factors that condition the accuracy of sensitivity estimators: the sampling method τ , base sample size N s , model dimensionality d, form of the test function and distribution of model inputs φ [7,8]. We describe these factors with probability distributions selected to cover a wide range of sensitivity analysis settings, from low-dimensional, computationally inexpensive designs to complex, high-dimensional problems formed by inputs whose uncertainty is described by dissimilar mathematical functions (Fig. 3). Although not exhaustive, this approach permits us to go beyond classic benchmarking exercises, which tend to focus on a handful of test functions or just move one design factor at-a-time [34,35].
We first create a 2 9 × 5 sample matrix using quasi-random numbers [28,29], where the i-th row represents a random combination of τ, N s , d, . . . values and each column is a factor whose uncertainty is described with its selected probability distribution (Fig. 3). In the i-th row of this matrix we do the following:
1. We construct two sampling matrices with the sampling method as defined by τ (i) :
• A N (i) s (d (i) + 1) × d (i) sample matrix formed by an A matrix and d (i) A B k matrices,
where all columns come from A except the k-th, which comes from B. This sampling design is required to run a sensitivity analysis with the Jansen estimator [36,37]. We refer to this matrix as the "Jansen matrix".
• A N (i) s (d (i) +1
)×d (i) sample matrix formed by an A matrix only. Since discrepancy measures do not require a specific sampling design, we match the number of rows required by the Jansen estimator to ensure that the comparison between the latter and the discrepancy measures is done on the same total number of model runs. We refer to this matrix as the "Discrepancy matrix".
2. We define the distribution of each input in these matrices according to the value set by φ (i) (Fig. S1).
3. We run a metafunction rowwise through both the Jansen and the Discrepancy matrix and produce two vectors with the model output, which we refer to as y J and y D respectively. Our metafunction, whose functional form is defined by (i) , is based on the Becker metafunction [8] and randomizes over 13 univariate functions representing common responses in physical systems and in classic sensitivity analysis functions (from cubic, exponential or periodic to sinusoidal, see Fig. S2). A detailed explanation of the metafunction can be found in Becker [8] and in Puy et al. [7]. 4. We use y J to produce a vector with the total-order indices T , calculated with the Jansen estimator [33]. We also use y D to produce seven vectors with the discrepancy values D, one vector for each of the seven discrepancy measures tested.
5. We rank-transform T and D using savage scores, which emphasize and downplay top and low ranks respectively [38,39]. To check how well discrepancy measures match the ranks produced with the Jansen estimator, we calculate for each discrepancy measure the Pearson correlation between T and D, which we denote as r.
Results
Discrepancy measures for sensitivity analysis
We conduct our analysis in a non-orthogonal domain to make the total number of model runs used by each discrepancy measure match that of the Jansen estimator, defined as N s (d + 1) (Section 2.3). We observe that the symmetric, the centered, the wraparound and the S-ersatz present consistently high r values throughout most of the domain investigated, with lower values concentrating largely on the leftmost part of the domain (characterized by simulations with low sample sizes and increasing dimensionality). In
Factor
Sampling method: τ ∼ DU(1, 2) 1: Random numbers.
2: Quasi-random numbers [28,29]. Sample size: N s ∼ DU(10, 100) Model dimensionality: d ∼ DU (3,50). contrast, no discernible pattern is visible for the star, L2, centered or modified measures, which comparatively present a much larger number of low and negative r values (Fig. 4a).
Overall, the distribution of r values is left-skewed for all discrepancy measures. This suggests that they are able to properly approximate the savage-transformed ranks produced by the Jansen estimator in a non-negligible number of SA settings. According to median values, the discrepancy measure that better matches the Jansen savage-transformed ranks is the symmetric (r = 0.81), followed by the wrap-around (r = 0.78) and the centered (r = 0.74). The S-ersatz also displays a good performance (r = 0.72), and its spread is as small as that of the symmetric discrepancy measure (Fig. 4b).
To check whether these median r values come from different distributions, we conduct a pairwise Mood test on medians with corrections for multiple testing. We cannot reject the null hypothesis of a difference in medians between the wraparound and the symmetric, between the modified and the star, or between the ersatz, the L2 and the centered measure (95% CL, Fig. 4c). A hierarchical cluster analysis suggests that the difference in the distribution of r values is mainly between two groups: the group formed by the modified and the star discrepancy, and the group formed by all the rest. The star and the modified discrepancy present the most similar distributions, followed by the S-ersatz and the symmetric discrepancy (Fig. 4d).
The capacity of discrepancy measures in matching the savage-transformed ranks of the Jansen estimator seems to be mostly determined by high-order interactions between the benchmark factors selected in our analysis (Fig. 3). The model dimensionality (d) and the base sample size (N s ) are the only factors with a visible direct effect on the accuracy of discrepancy measures, especially on the S-ersatz: higher dimensionalities and larger sample sizes tend to respectively diminish and increase their performance (Fig. 5). Interestingly, the variability in the performance of discrepancy measures does not seem to be critically determined by the functional form of the model ( ), the underlying distribution (φ) or the sampling method used to design the sample matrix (τ ).
As displayed in Figs. 4-5, some simulations yield r < 0. To explore the reasons underlying this rank reversal, we plot all simulations where at least one discrepancy measure yielded r < 0 and cross-check the other r values. We observe that the production of negative r values is measure-specific: in the same simulation some discrepancy measures yielded r < 0 while others produced very high r values and hence accurately matched the rankings of the Jansen estimator (Figs. S4-S13). This indicates that certain discrepancy measures, especially the modified and the star, may be more volatile than the rest when used in an SA setting.
Computational complexity
The numerical efficiency of a sensitivity analysis method (how much time it requires to run its algorithmic implementation) is an important property to take into account when deciding which SA approach to use. If the model of interest is already computationally burdensome, the extra computational strain added by a demanding SA method may make the implementation of the latter unfeasible. To pinpoint the computational requirements of discrepancy measures, we calculate the time it takes to evaluate each expression using the R package microbenchmark [40], which uses sub-millisecond accurate timing functions. We use the implementations of Equations 6-10 in the R package sensitivity [41], which are written in C++. Our implementation of the S-ersatz algorithm uses base R language [42]. To gain robust insights into the time complexity of all these discrepancy measures, we explore their efficiency through a wide range of sample sizes N s (100-5,000) and dimensionalities d (3-100), which we treat as random factors following the approach described in Section 2.3.
The results, which are presented in Fig. 6
Discussion and conclusions
Sensitivity analysis (SA) is a key method to ensure the quality of model-based inferences and promote responsible modelling [3,4]. Here we contribute to the advancement of SA by proposing an approach that is as easy to understand as the visual inspection of input-output scatterplots. Our contribution is fourfold:
1. We turn the concept of discrepancy, usually applied to explore the input space of a mathematical simulation, into a tool to explore its output. 2. We make SA more accessible to the non-specialist by linking the sensitivity of a given input to this input's capacity to leave "holes" in the input-output scatterplot. 3. We provide numerical evidence that discrepancy is a good proxy of the total-order sensitivity index, a recommended best-practice in variance-based global SA [9,17]. 4. We introduce an ersatz discrepancy whose behavior in an SA setting approximates the best discrepancy measures while at a much reduced computational cost.
It has been argued that a desirable property of a good SA method is its capacity to be translated into comprehensible language [9]. In other words, a satisfactory SA method allows an easy explanation as to why a factor is identified as more influential than another. Variance-based methods have this property:
• The first-order index S i says that x 1 is more important than x 2 when fixing x 1 leads on average to a greater reduction in the output variance than fixing x 2 . • The total-order T i says that x 1 is more important than x 2 when fixing all factors but x 1 leaves on average a greater residual variance than doing the same on x 2 . • With Shapley coefficients, x 1 is more influential than x 2 if all combinations of factors including x 1 yields a higher outcome than the same conducted on x 2 [13,14].
The same clarity applies when using discrepancy measures:
• x 1 is more influential than x 2 when the scatterplot of x 1 against y displays a more discernible shape than the scatterplot of x 2 against y.
Our exploration of the performance of discrepancy measures as SA tools reveals the existence of two main groups: (1) the group formed by the wraparound, the centered, the symmetric, the L2 and the S-ersatz, and (2) the group formed by the modified and the star discrepancies. The second group matches the behavior of the Jansen estimator worse than the first group. This is because both the modified and the star discrepancy give the origin of the domain ([0] d ) a special meaning: points further away from the origin affect less the measures. As for the wraparound, the centered, the symmetric, the L2 and the S-ersatz, their results are similar because they also treat the origin and corner points in the same way. Their differences due to the dimensionality do not play an important role here given our focus on 2D sub-projections.
Note that the quality of the sampling is paramount. As we are constructing 2D subprojections between x k and y, it is important to have a uniform distribution on the sample axis. If this is not the case, the measure will be biased by a non-uniformity on the sample side. Randomized QMC methods such as Sobol' low-discrepancy sequences are appropriate as their properties guarantee the uniformity on sub-projections [28,43]. Essentially, it would only be necessary to ensure good 1D-subprojections, which means that simpler QMC methods such as Latin Hypercube sampling could alternatively be used [43]. This makes our method even more straightforward to implement in practice.
The use of discrepancy measures in an SA setting can be extended to higher dimensions to appraise high order interactions. Both the centered and the wrap-around discrepancy, however, are known to have shortcomings with regards to dimensionality: the centered suffers from the curse of dimensionality, whereas the wrap-around is not sensitive to a shift of dimension [21,44]. Our results may therefore change should we increase the sub-projections' dimensionality. Recently, some work has been done to combine the complementary benefits of both the centered and the wraparound discrepancy into a single measure: the mixture discrepancy M D 2 -see [44]. This method could also prove to be efficient here as it should give a more uniform importance to every part of the domain.
Finally, our work shows that ersatz methods can be a good alternative to classical discrepancy measures in an SA setting, and that there is potential to develop new ersatz discrepancies. The newly added Newcomb-Benford measure [45] may be a venue worth exploring further. x y Function
Discrepancy measures for sensitivity analysis
f 1 (x) = x 3 f 2 (x) = 1 if x > 1 2 , otherwise 0 f 3 (x) = (e x −1) e−1 f 4 (x) = (10 − 1 1.1 ) −1 (x + 0.1) −1 f 5 (x) = x f 6 (x) = 0 f 7 (x) = 4(x − 0.5) 2 f 8 (x) = 2 − 0.2 cos(7πx)
Fig. 1 :
1Scatterplots of xk, k = 1, 2 against y for three different two-dimensional functions. The red dots show the running mean across 100 simulations. The functions are F1, F2 and F3 in Azzini and Rosati
9: end for 10: O[M] ← 1 Use M to identify which elements in O should be 1. 11: S-ersatz ← {o ∈ O|o = 1}/N s Compute the proportion of grid cells with 1.
6 (second and third columns), the plane is partitioned into 16 and 64 cells respectively. The ratio of sampled cells to the total number of cells is 3/4 = 0.75, 11/16 = 0.68 and 40/64 = 0.62 in a), and 3/4 = 0.75, 12/16 = 0.75 and 59/64 = 0.92 in b)
Fig. 2 :
2The S-ersatz discrepancy. a) and b) show planes with sampling points produced using random and QMC (Sobol' sequence) respectively, for a total sample size of Ns = 2 b , b = 2, 4, 6. c) Evolution of the S-ersatz through different Ns.2. It is invariant under rotation and ordering.3. It has a geometrical meaning. 4. It is simple to approach and implement. 5. It is fast to compute.Furthermore, as opposed to classical discrepancy measures, both the upper and lower bounds are exactly known. The worst case would place all points in a single cell, while the ideal case would place a point per cell. This implies S-ersatz ∈ [1/N s , 1]. Other methods don't have simple expressions for their lower bounds which work for all combinations of number of samples and dimensions.
Fig. 3 :
3Tree diagram with the uncertain inputs, their distributions and their levels.
Fig. 4 :
4Results of the benchmarking. a) Distribution of the Pearson correlation r between the savage scores-transformed ranks yielded by each discrepancy measure and the savage scores-transformed ranks produced by the Jansen[33] estimator. Each dot is a simulation that randomizes the sample size Ns, the dimensionality d, the underlying probability distributions φ, the sampling method τ and the functional form of the metafunction . The total number of simulations is 2 9 . b) Density plots of r values. c) Tile plot showing the p-values of a pairwise Mood test on medians. d) Hierarchical clustering.
Fig. 5 :
5Scatterplot of x k against the model output r (the Pearson correlation) for each discrepancy measure. The red hexbins display simulations where r < 0. The number of simulations in each facet is Ns = 2 9 .
Fig. 6 :
6, match the expected numerical complexity of the different discrepancy measures [O(N 2 s d)] and of the S-ersatz [O(N s d)]. Time complexity of discrepancy measures. In the d facet Ns = 500, while in the Ns facet, d = 5.
Figure S1 :
S1Distributions used in the sensitivity analysis.
Figure S2 :Figure S3 :Figure S4 :
S2S3S4(x) = (−1) |4x| [0.125 − mod(x, 0.25)] + 0.125 f 11 (x) = (−1) |32x| [0.0325 − mod(x, 0.0325)] + 0.0325 f 12 (x) = x 2 f 13 (x) = cos(x) Univariate functions included in the meta-model, based on Becker's meta-function[1]. Fraction of simulations that yielded r < 0. The symmetric discrepancy measure is not plotted because all its simulations yielded r > 0. Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S5 :
S5Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S6 :Figure S7 :
S6S7Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label. Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S8 :
S8Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S9 :Figure S10 :
S9S10Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label. Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S11 :
S11Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Figure S12 :Figure S13 :
S12S13Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label. Simulations where at least one discrepancy measure yield negative r values. The simulation ID is shown in the facet label.
Supplementary Materials Arnald Puy * 1 , Pamphile T. Roy 2 , and Andrea Saltelli 3 1 School of Geography, Earth and Environmental Sciences, University of Birmingham, Birmingham B15 2TT, United Kingdom. E-mail: [email protected] 2 Quansight, Vienna, Austria. 3 Barcelona School of Management, Pompeu Fabra University, Carrer de Balmes 132, 08008 Barcelona, Spain.
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AcknowledgementsWe thank Bertrand Iooss and Art Owen for their insights into discrepancy measures. All mistakes are our own. AS has worked in this paper within the framework of the i4Driving project, funded by the European Union's Horizon Europe research and Innovation program (Grant Agreement ID 101076165). PR is partially supported by grant "SciPy: Fundamental Tools for Biomedical Research" (EOSS5-0000000176) from the Chan Zuckerberg Initiative DAF, an advised fund of the Silicon Valley Community Foundation.Code availabilityThe R code to replicate our results is available in Puy[46]and in https://github.com/ arnaldpuy/discrepancy. PR will also contribute the new methods to the Python library SALib[47].Author contributionsAS conceptualized the paper and proposed the ersatz measure. AP ran the simulations and lead the work. PR wrote the presentation of the classic discrepancy measures, their properties and analysed the results in terms of discrepancy. AP and AS wrote the rest of the paper. All authors revised the final version.
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|
[] |
[
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2",
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2",
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2",
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2",
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2",
"Finite-size effects on the ferroelectricity in rhombohedral HfO 2"
] |
[
"Francesco Delodovici \nUniversité Paris-Saclay, CentraleSupélec\nCNRS\nLaboratoire SPMS\n91190Gif-sur-YvetteFrance\n\nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n",
"Paolo Barone \nConsiglio Nazionale delle Ricerche CNR-SPIN\nArea della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly\n",
"Silvia Picozzi \nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n",
"Francesco Delodovici \nUniversité Paris-Saclay, CentraleSupélec\nCNRS\nLaboratoire SPMS\n91190Gif-sur-YvetteFrance\n\nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n",
"Paolo Barone \nConsiglio Nazionale delle Ricerche CNR-SPIN\nArea della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly\n",
"Silvia Picozzi \nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n",
"Francesco Delodovici \nUniversité Paris-Saclay, CentraleSupélec\nCNRS\nLaboratoire SPMS\n91190Gif-sur-YvetteFrance\n\nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n",
"Paolo Barone \nConsiglio Nazionale delle Ricerche CNR-SPIN\nArea della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly\n",
"Silvia Picozzi \nConsiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly\n"
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"Consiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly",
"Consiglio Nazionale delle Ricerche CNR-SPIN\nArea della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly",
"Consiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly",
"Université Paris-Saclay, CentraleSupélec\nCNRS\nLaboratoire SPMS\n91190Gif-sur-YvetteFrance",
"Consiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly",
"Consiglio Nazionale delle Ricerche CNR-SPIN\nArea della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly",
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"Université Paris-Saclay, CentraleSupélec\nCNRS\nLaboratoire SPMS\n91190Gif-sur-YvetteFrance",
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"Consiglio Nazionale delle Ricerche CNR-SPIN\nUniversità degli Studi \"G. D'Annunzio\"\nI-66100ChietiItaly"
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[] |
In this work we analyze the finite-size effects on the structural properties and on the polarization of the rhombohedral phase of HfO2 subjected to a biaxial compressive strain. We show how the presence of surface charges affects the polarization, leading to a strong reduction with respect to its bulk value. This reduction can be ascribed to two mechanisms: i) the coupling between compressive strain and the phase-transition order parameter; ii) the changes in the ferroelectric distortion. We give two alternative explanations of this phenomenon: from an atomistic point of view, analyzing the evolution of the bond lengths, and from a symmetry-analysis point of view, considering the changes in the amplitude of the symmetry-allowed distortions, when a slab configuration is considered. These results are independent on the slab-thickness in the considered range, suggesting the absence of a critical thickness for ferroelectricity in HfO2, in agreement with the proposed improper nature of hafnia ferroelectricity. arXiv:2204.05252v1 [cond-mat.mtrl-sci]
|
10.1103/physrevb.106.115438
|
[
"https://export.arxiv.org/pdf/2204.05252v1.pdf"
] | 248,085,283 |
2204.05252
|
dc5639ed30d7620b30de823cd97989883b4df79a
|
Finite-size effects on the ferroelectricity in rhombohedral HfO 2
Francesco Delodovici
Université Paris-Saclay, CentraleSupélec
CNRS
Laboratoire SPMS
91190Gif-sur-YvetteFrance
Consiglio Nazionale delle Ricerche CNR-SPIN
Università degli Studi "G. D'Annunzio"
I-66100ChietiItaly
Paolo Barone
Consiglio Nazionale delle Ricerche CNR-SPIN
Area della Ricerca di Tor Vergata, Via del Fosso del Cavaliere, 100I-00133RomeItaly
Silvia Picozzi
Consiglio Nazionale delle Ricerche CNR-SPIN
Università degli Studi "G. D'Annunzio"
I-66100ChietiItaly
Finite-size effects on the ferroelectricity in rhombohedral HfO 2
In this work we analyze the finite-size effects on the structural properties and on the polarization of the rhombohedral phase of HfO2 subjected to a biaxial compressive strain. We show how the presence of surface charges affects the polarization, leading to a strong reduction with respect to its bulk value. This reduction can be ascribed to two mechanisms: i) the coupling between compressive strain and the phase-transition order parameter; ii) the changes in the ferroelectric distortion. We give two alternative explanations of this phenomenon: from an atomistic point of view, analyzing the evolution of the bond lengths, and from a symmetry-analysis point of view, considering the changes in the amplitude of the symmetry-allowed distortions, when a slab configuration is considered. These results are independent on the slab-thickness in the considered range, suggesting the absence of a critical thickness for ferroelectricity in HfO2, in agreement with the proposed improper nature of hafnia ferroelectricity. arXiv:2204.05252v1 [cond-mat.mtrl-sci]
I. INTRODUCTION
Ferroelectricity in hafnium oxide has been at the center of continuous interests and deep analysis in the last decade. The stabilization of different crystalline phases represents a particularly delicate issue. In closer detail, the detected polar phases [1][2][3][4] of the pristine oxide [5] are only metastable in bulk configurations at standard thermodynamical conditions, its ground state rather being a non-polar monoclininc phase [6]. However, the polar phases become thermodynamically competitive with the monoclinic phase when hafnia is realized in the form of films [1]. In this respect, a variety of different factors play a role, such as mechanical strain induced by the substrate [7][8][9][10], doping [7,11], defects [12][13][14], grains size and films thickness [15]. Despite many open issues, it is anyway beyond question that its compatibility with Si-based semiconductors [16], along with the robustness of the polarization hysteresis at the nanoscale level [17,18] and its simple chemistry, make HfO 2 extremely appealing for ferroelectric-based technological applications, when compared to ordinary ferroelectric perovskite-like films [19].
From the theoretical modelling point of view, an extremely interesting topic is represented by the investigation of how the order parameter describing ferroelectricity couples to surfaces and interfaces [20,21]. In closer detail, it is partic-ularly interesting to determine whether a minimal thickness for the development of ferroelectric instabilities in the film exists or not. The presence of such a critical thickness is generally related to the screening of surfaces charges associated with the film spontaneous polarization via depolarizing fields [21][22][23]. In proper ferroelectrics, the occurrence of surfaces can have dramatic consequences on the stability of the polar phase for films thinner than the critical thickness, which in the case of prototypical BaTiO 3 is on the nanometer order under short-circuit condition [22,24]. Ferroelectrics where the onset of polarization emerges as a secondary effect, being mainly governed by the coupling to a primary non-polar distortion, may be expected to present robustness against depolarizing effects. This characteristic makes improper ferroelectrics particularly appealing for their potentiality in technological applications. Nonetheless, their behaviour in the ultra-thin limit remains unclear, as well as the existence of a critical thickness [25,26].
In this paper, we present a theoretical analysis, based on ab-initio simulations, of the effects of finite-size on the stability of polar R3m phase of hafnium oxide. We considered different slab configurations under open-boundary conditions with clean surfaces in the range of experimentally-reachable nanometric thicknesses. We stress that these electrostatic bound-ary conditions are highly unfavourable to proper ferroelectricity. Indeed, when polarization can be considered as the leading order parameter of the phase transition, these conditions correspond to vanishing electric displacement in the vacuum [21]. Under this configuration, below a critical thickness, the intense depolarizing field experienced by the slab arising from the unscreened surface charges is strong enough to completely suppress the polar phase. Further on, we analyze the evolution of polarization as a function of the depth in the slab (i.e. distance from the surfaces) by means of Born effective charges of relaxed bulk phases. Finally, we justify the observed effects in terms of microscopic modelling and in terms of suppression of the dominant symmetry-allowed distortions connecting the high and the low-symmetry phase.
II. SLAB CONFIGURATIONS
In this paragraph we report some structural information about the simulations and refer to appendices for further computational details. We orient the relaxed primitive cell according to the experimentally-detected growth direction [3]. Thus, the [111] direction of the rhombohedral setting is parallel to the normal of the film surface: we chose this direction to coincide with the z axis in our simulations. In this configuration, the polarization is also directed along z. Hereafter, we refer to calculations performed in the equivalent hexagonal setting. The relaxed primitive vectors are: a = (a, 0, 0), b = (−a/2, a √ 3/2, 0), c = (0, 0, c), where a= 7.134 A, c=8.738Å. With this choice, the unit cell consists of 12 formula units or, equivalently, of 36 atoms.
Previous computational modelling works [3,27] show that the polarization undergoes a rapid increase, when biaxial compressive strain is applied normally to the polarization direction. In closer detail, the polarization of the relaxed bulk configuration at equilibrium is approximately 0.06 µC/cm 2 , as computed with the Berry phase method [28], while higher values, above 25 µC/cm 2 , are obtained when compress-ing a above 5%. Given this significant difference in the P value, we expect finite size effects to be "masked" in equilibrium conditions by the small polarization magnitude; rather, we expect finite size effects to be more noticeable (and therefore easier to investigate), upon application of compressive strain. Therefore, we focus hereafter on the hexagonal cell subjected to a bi-axial 5% inplane compression. The bulk primitive cell under this strain condition is defined by a=6.78Å and c opt =9.66Å. The optimum c opt is obtained by minimizing the total energy of the primitive cell with fixed (compressed) in-plane lattice constant and relaxed internal degrees of freedom describing atomic positions. The polarization of this phase, computed with the Berry-phase method, is 22.8 µC/cm 2 , in good agreement with values reported in literature [27,29] In order to build the slab configurations, the compressed unit cell is repeated along its [001] hexagonal direction and "cut" at three different thicknesses. The resulting asymmetric slabs, before relaxation, are 12.88, 19.32, 38.64Å thick, corresponding to 4 layers, 6 layers and 12 layers, respectively. Hereafter, we define a "layer" as the stoichiometric combination of Hf and O occupying one third of the primitive hexagonal cell volume. As such, each layer consists of 4 Hf atoms and 8 O atoms. With this definition, the 6 (12) layers slab consists of a primitive cell repeated 2 (4) times along the c axis.
III. RESULTS
A. Atomic displacements
We observe the geometrical relaxation to produce two main effects in all the considered configurations: i) the contraction of the film thickness; ii) the relaxation of the atomic positions within the film. To give a quantitative description of the first effect, we consider, as a marker for the contraction experienced by the slabs, the thickness of the most central unit cell i.e. the c axis of an embedded hexagonal cell. In fact, the central layers can be reasonably assumed to be marginally affected by the surfaces. The comparison between the Hf-projected DOS (not shown) in these central layers with the bulk Hfprojected DOS indeed confirms such hypothesis. For the 4-layer-thick slab the marker is ill-defined; thus, in order to have a value comparable to those obtained for thicker slabs, we multiply the distance between the two central layers by 3. The 12, 6 and 4-layer slab experience respectively a contraction of ≈1.4%, ≈1.6%, ≈1.5%. This contraction can be traced back to the coupling of the strain acting along c with the different symmetry-allowed distortions (as will be detailed below, see section IV B).
To analyze the relaxations of atomic positions in the film, we compare the ferroelectric distortion in the bulk phases and in the relaxed slab. To this purpose, we consider the displacements of each atom with respect to a reference paraelectric (PE) phase: the R3m. In a bulk configuration, the polar displacements from a PE phase characterize the ferroelectric (FE) phase. From the same quantities, in a slab configuration, one can deduce how finite-size effects affect ferroelectricity. To decouple the displacements from the effects of the cell compression, we adjust the reference PE phase to have the same height of the central cell in the relaxed slab. Fig. 1 illustrates the vertical displacements δz = z slab − z PE of each atom in the HfO 2 slab. The top and bottom panel report respectively the displacement of the 6-layers and of the 12-layers slabs. The dot-dashed lines represent the displacements characterizing the bulk phases. The R3m bulk phase has two different Wyckoff positions for hafnium and four for oxygen, thus 6 lines appear in the panels (two oxygen lines at -0.2, in red, are almost degenerate, while a third one close to 0 overlaps with a Hf line in black) . Each of this high-symmetry position transforms in a unique way under geometric relaxation, therefore each layer presents 4 different oxygen displacements and 2 for hafnium. The surface layers experience the largest changes: the absolute displacements of hafnium at the bottom (first) layer change on average by 0.06Å , the oxygens closer to vacuum on average by 0.07Å whereas those toward the bulk change by 0.08Å . At the top layer, the hafnium displacements change by 0.16Å on average, while those of oxygens remain closer to the bulk value, changing on average by 0.03Å for O toward the bulk and of 0.04Å for O toward the vacuum. We stress that the four hafnium atoms at the top surface in the relaxed slab have the same z coordinate. This is analogous to what happens in the PE bulk phase. The changes in the ferroelectric distortion rapidly converge to constant values in the central layers, with the second layer acting as an intermediate buffer. In the bulk region the displacements of both hafnium and oxygen atoms do not match the displacements characterizing the bulk phases: the discrepancy for oxygen reaches 0.06Å and 0.03Å for hafnium. Based on these observations, it is reasonable to expect some changes in the electric polarization throughout the slab. Nonetheless, the complexity of the distortions makes these effects hard to deduce only from the evolution of the ferroelectric displacements. Therefore, we move to a direct analysis of the polarization throughout the slab.
B. Polarization
In order to investigate the evolution of the polarization in the slab as a function of thickness, we employed the Born effective charges, computed for the relaxed and for the compressed bulk phases. We do not employ the Berry-phase method, due to the presence of metallic states confined at the top surface and associated with oxygen p-orbitals, as revealed by the computed projected density of states. The polarization difference with respect to the centrosymmetric phase is by definition the contraction of the tensor Z * with the vector of the atomic displacements from the R3m phase, normalized to the bulk volume. For the layer polarization, the dipole moment is normalized to the volume of the layer, hereby defined as one third of the volume of the adjusted bulk. These simplified definitions allow us to investigate the dependence of the dipole-moment density on the "depth" (i.e. distance from the surfaces) within the slab.
We decouple the effects contributing to the change in polarization: the slab contraction and the local atomic distortions. Fig. 2 reports the evolution of the polarization as a function of the layer for three different slab thicknesses. The polarization of different bulk configurations obtained with the BP method is reported as a comparison: 0.07 and 22.8 µC/cm 2 respectively for the relaxed bulk and the bulk subjected to a biaxial (along a,b primitive vectors) compressive (-5%) strain. In this regard, it should be noticed that the use of effective charges to compute the polarization results in an underestimate with respect to the BP value: P computed with effective charges amounts to 0.06 and 20.03 µC/cm 2 , respectively, for the relaxed equilibrium bulk and for the bulk subjected to a biaxial compressive (-5%) strain. This discrepancy in the compressed case (i.e. almost 2.8 µC/cm 2 ) corresponds to 9% of the BP value, which is still a reasonable margin of error for our analysis.
We stress once again that we employed the effective charges of the bulk subjected to a biaxial -5% strain state to compute the effects of finite size on the polarization. This corresponds to assuming that the local distortions following the relaxation do not affect significantly the atomic responses to an external electric field. In order to analyze the effects of the slab contraction, we compute the change in P arising from the compression along c of the relaxed R3m bulk to the value of c characterising the central hexagonal cell embedded in the 6 layers slab. In this way, we observe that the slab contraction induces a decrease in P by about 33% (6.7 µC/cm 2 ) with respect to the c-relaxed bulk, as can be observed in Fig.2: the polarization of the compressed bulk, 13.33 µC/cm 2 , and the one of the c-relaxed bulk, 20.03 µC/cm 2 , are respectively represented by the blue and the continuous red line. Such a reduction of the polarization induced by the slab contraction can be phenomenologically traced back to the same strainpolarization coupling which is responsible for the remarkable increase of P following the 5% biaxial in-plane compression of the bulk configuration and consequent elongation of the cell along the out-of-plane direction. As the slab contraction causes a reduction of the strain deformation along the c axis with respect to the bulk phase, the strain-polarization coupling is responsible for a corresponding reduction of P. The introduction of finite-size causing the local rearrangement of atoms induces a further significant reduction of P. In fact, when moving from the surfaces where the polarization assumes opposite sign and different absolute values, the polarization settles to approximately 4.5 and to 3.3 µC/cm 2 , respectively, in the 12(6) and 4 layers slabs. The reduced value of P in the thinnest slab is connected with the lack of a proper bulk-like volume in the slab, as in this configuration the central layers are just below surfaces. In all the slabs, the polarization in the central layers is therefore reduced by more than 70% with respect to the corresponding value in the uncompressed bulk. Nonetheless, it remains finite. We note that the bulk Z * represent a poor approximation to the electric response at the surfaces. Therefore, the value of P at the extreme layers is likely biased and should not be commented.
It is interesting to notice that, despite the open-boundary conditions employed to simulate the slab, the depolarization field does not completely suppress the polarization, as it happens for proper-ferroelectrics slabs below their critical thickness under open-circuit conditions [21]. Since the polarization of the central layers remains finite as the thickness is lowered, we may infer that the critical thickness for the R3m phase of HfO 2 films is either smaller than the thickness of the 4-layer slab, or it completely vanishes. We note that the extreme case of a three layer configuration still presents a finite value of P in the central layer (≈ 2 µC/cm 2 ), but this can hardly be intended as a "proper bulk-like" layer. Therefore, the incomplete suppression of P may be an indirect sign of the improper character of ferroelectricity in this polar phase of HfO 2 . To explain the suppression of P in the central layers, we refer to the atomistic model of Zhang et al. reported in [27], where the effects of compressive strain on the R3m phase are analyzed. The authors show that the increase of P under certain strain states corresponds to the change in the contribution to P of two competing Hf-O bonds, hereafter named "primary" and "secondary", appearing in the unit cell with the ratio of 3 to 1. The primary bonds involve the Hf and O with high multiplicity (9b, corresponding to Hf 1 , O 1 and O 2 as reported in Fig. 1). Instead, the secondary bonds involve the Hf and O with low multiplicity (3a, corresponding to Hf 2 , O 3 and O 4 in Fig. 1). Fig. 3 reports the hexagonal unit cell with a primary bond highlighted by the green rectangle and the secondary bond by the blue one. For small strains, the contribution to the polarization of the two bonds compensate one another. However, above a certain strain threshold, the contribution from the 3 primary bonds grows and determines the increase of P. Fig. 4 reports the trend of the two bonds lengths across the 12-layers slab. The secondary and the primary bonds respectively increase and decrease with respect to their bulk value. Specifically, the primary ones approach the value of the bonds in the centro-symmetric phase. Since the two bonds compete in determining the sign of the polarization, the resulting effect of such a global distortion is to reduce the polarization in the central portion of the slab.
IV. INTERPRETATION OF POLARIZATION SUPPRESSION
Γ 4-U X 5- X 2-U Γ 4- X 2-U X 5- X 2-U Γ 4-U X 5- X 2-U Γ 4-U X 5-U X 3-
B. Suppression of the main distortion modes
An alternative approach relies on the analysis of the main symmetry-allowed distortion modes connecting the hafnia high-temperature cubic phase (F m − 3m) to the R3m phase. We employed standard crystallographic tools [30,31] to identify the symmetry-adapted distortions and found six of them, two of which have negligible amplitudes. Five symmetry modes are associated with the wave vector q X at the Brillouinzone boundary, and the last one to a polar distortion at Γ. Under -5% strain condition, the four dominant modes (in the hexagonal setting), ordered by decreasing amplitude, are: X 2− , X 5− ,Γ 4− and X 3− . The top panel in Fig. 5 reports the ab-initio total energy computed along the distortions. The only soft mode is X 2− corresponding to an instability of 8 meV/f.u. On the other hand, the high-symmetry phase is stable against all the remaining modes. This is a canonical behaviour for an improper ferroelectric, where the polarization emerges through the coupling to the leading order parameter, which in this case is a zone-boundary distortion. When examining the atomic distortions related to each mode, we note that the leading order mode distorts the angles formed by a Hf atom with the surrounding oxygen cage and changes the relative bond lengths. The X 5− mode moves apart, along the c direction, the Hf with different Wyckoff positions. These correspond to the Hf atoms forming the primary and the secondary bonds in Fig.3. The polar mode Γ 4− moves apart the oxygen and the hafnium atoms along the c direction. The X 3− mode displaces the Hf forming the primary bonds along a nontrivial direction and those forming the secondary bonds simply along c. Going back to Fig.5, the central panel reports the couplings between the different modes. The coupling of the polarization with X 2− does not significantly affect the soft character of the leading mode: the energy gain is 7 meV/f.u. . Instead, the X 5− significantly enhances the soft character of the leading mode, the energy gain when coupled to the order parameter amounting to 54 meV/f.u. . It is worth noticing that including the Γ 4− mode further stabilize this coupling, even though only by 9 meV/f.u. . In this sense, we may see the coupling of the X 5− mode with the order parameter as a condition favouring the excitation of the polar distortion. When all the four dominant modes are coupled, the largest gain in energy is obtained: 128 meV/f.u. below the high symmetry phase at λ = 0. The full relaxation, which includes the neglected structural distortions, leads to a further energy decrease of 20 meV/f.u., i.e. less than one sixth of the energy gain corresponding to the simultaneous activation of the dominant modes. The bottom panel in Fig.5 describes the dependence of the energy dispersion of X 2− and of the coupling X 2− U Γ 4− U X 5− on the applied strain. The compression to -5% weakens the character of the leading mode: its stability is reduced by almost 30 meV/f.u. . Instead, the three-modes-coupled distortion is strongly stabilized by the compression: when strain is applied, the energy gain is about 27 meV/f.u. . We stress that this energetically favourable pairing with the compressive strain is not shown by the X 2− U Γ 4− distortion. This is a further sign of the special role assumed by X 5− when compressive strain is applied. Based on this analysis, we can describe the suppression of the polarization in the central portion of the slab in terms of suppression of the dominant modes.
To this purpose, we define a structural "fingerprint" for each mode, which consists in geometrical quantities catching unambiguously the distortions of each single mode. From their trends throughout the slab, we can indeed deduce the variations of the modes amplitude. The finger-print describing the order parameter X 2− can be taken as the variation of the angle α (between the oxygen labelled O 2 and O 4 in Fig.6 with the vertex located at the Hf atom), with respect to the reference value of π (i.e. the α value in the PE phase at λ = 0). This angle is also affected by Γ 4− , but the change deriving from this mode is much smaller (only 26%) than the one from X 2− . The other angles in the cage are significantly affected also by the mode X 3− : the distortion coming from this mode is almost 80% of the one coming from X 2− . Therefore, they cannot be taken as fingerprints for the order parameter. The polar distortion can be easily described in terms of the distance between the oxygens and the Hf average z-coordinates. The effects of the other dominant modes on this quantity is negligible. The X 5− mode can be well represented by the finite distance between the z-coordinates of the Hf atoms having different Wyckoff positions. Specifically, we averaged the z-coordinate of the Hf with multiplicity 3. Fig.6 reports the trend of the three structural fingerprints for the different layers in the 6-layer slab configuration. The first panel reports the difference of the angle α−α para representing X 2− and its bulk value (≈ 12°) as a function of the layer. The reduction of about 30% with respect to the bulk value, occurring in the central layers, is a clear indication of the mode suppression. The central panel reports the trend for the fingerprint of the polar mode. Also in this case we observe a clear reduction of the fingerprint in the central layers. Given the detected improper character of ferroelectricity, we note that the suppression of the polarization should be interpreted as caused by the suppression of the X 2− mode. This dependence can also be inferred by the strong correlation between the trends of the two fingerprints: their suppression should not be intended as two independent phenomena. The third panel illustrates that the X 5− distortion is not suppressed as the depth increases. On the contrary, its amplitude oscillates around the bulk value. This is interesting to notice, given the considerable contribution to the stabilization of the polar phase coming from the coupling of this mode with the order parameter. This sug-gests that finite-size effects reduce the amplitude of the order parameter in such a way to lower the polarization in the central volume, while minimizing the reduction of energetic stability with respect to the pure bulk phase. Given the connection with the polar mode reported in Fig.5, this result also explains the incomplete suppression of the polarization. In fact, the finite size has no effect at all on the X 5− mode, which is responsible for "encouraging" the excitation of Γ 4− . Therefore, the role of X 5− when coupled with the order parameter is two-fold even in finite configuration: it contributes to the stabilization of the ferroelectric phase and it prevents the polar mode from completely vanishing.
V. CONCLUSIONS
In this paper we addressed the effects of surfaces on the R3m polar phase of hafnia. We have shown how the introduction of the finite size leads to two types of consequences. The first one consists in the contraction of the slab, resulting from the coupling between strain and the order parameter. This contraction leads to a reduction of polarization by about one third compared to the bulk value. This effect should diminish with increasing thickness, but the thicknesses range that we analyzed does not allow us to observe this trend clearly. The second consequence is the relaxation of the atomic positions towards a configuration that further decreases the electric dipole density in the central layers. The values achieved are finite, but much smaller than the corresponding bulk values. We explained these effects by means of two approaches: through an atomistic perspective, looking at bond lengths to identify polarization, and through a more fundamental approach based on symmetry-allowed distortion analysis. The trend in bond lengths throughout the slab clearly shows that the presence of the surfaces tends to balance the weight of different competing bonds that contribute with opposite sign to layer-polarization. This balance does not lead to full cancellation though, thus polarization is not completely suppressed. The analysis of the dis-tortions allowed us to identify the improper nature of ferroelectricity in hafnium oxide rhombohedral phase, the leading-order parameter being the mode at the boundary-zone X 2− instead of P. By assigning a structural fingerprint to each of the leading modes, we showed how the introduction of the surfaces leads to a suppression of the leading mode and, consequently, of the polarization in the central layers of the slab. However, even in this case the suppression is not complete and the amplitude of the distortions is not fully cancelled. Finally, we found that these results are independent on the slab-thickness in the considered range (1-3.5 nm). Therefore, we may infer the critical-thickness for the R3m phase of HfO 2 either to be lower than 1 nm or to completely vanish.
VI. ACKNOWLEDGMENTS
We acknowledge support from the Italian Ministry for Research and Education through the PRIN-2017 project "TWEET: Towards Ferroelectricity in Two dimensions" (IT-MIUR Grant No. 2017YCTB59) and computational support from CINECA through the ISCRA initiative with Grant N. HP10CEI2UQ (Fe-CoSMO project). We also acknowledge support from the computational resources at "Gabriele d'Annunzio" University of Chieti.
Appendix A: Computational details
We performed density functional theory simulations, as implemented in the Vienna Ab-initio Simulation Package (VASP) [32] at the level of the revised Perdew, Burke, and Ernzerhof functional for solids [33,34]. The bulk phases were relaxed until the difference in the total energy between two successive self-consistent steps is smaller than 10 −7 eV and forces are smaller than 10 −3 eV/Å. The same convergence criteria apply for the 4-layers and 6-layers slabs. Instead, the 12-layers slab was relaxed until all the forces are smaller than than 10 −2 eV/Å. The cutoff for the expansion on the plane waves basis is set to 600 eV. The k-point mesh is set to 5x5x3 for bulk calculations and to 4x4x1 for the slabs.
We relax the slabs with at least 30Å of vacuum including dipole corrections to prevent periodic-images spurious interaction. No external field is applied.
In the distortion modes analysis, we compute the energy of 10 intermediate structures along the distortion paths, with the same computational parameters reported above.
FIG. 1 .
1Top: the primitive cell of the rhombohedral phase defined by the vectors aR,bR,cR. Hf (O) atoms shown by a yellow (red) sphere. Center: panel a and b report a schematic representation of the ferroelectric distortion in bulk and slab (central layer) configurations, respectively. The atomic positions of the paraelectric phase are represented by the shaded spheres. Bottom: structural relaxation along the c primitive vector for the 6-layers slab (top panel) and the 12-layers slab (bottom panel). The black triangles and the red circles show, respectively, the displacements along the z direction of Hf and O symmetry-inequivalent types for each layer in the relaxed slab with respect to the relaxed paraelectric R-3m. The dotdashed lines represent the ferroelectric displacements in the bulk configurations. The atoms are labelled with different Wyckoff positions and multiplicity in each panel to facilitate the comparison between the panels.
polarization computed with different methods compared to the layer-projected value. Three slab thickness are reported; top: four-layers slab; center: sixlayers slab; bottom: twelve-layers slab. The dotted-dashed lines represent the P computed with the Berry-phase method. The continuous lines report the P computed using the Borneffective charges. The blue line corresponds to the P of the bulk phase with compressed c. The triangles report the value of P per each layer.
FIG. 3 .
3Section of the R3m hexagonal unit cell. The primary and the secondary (vertical) Hf-O bonds are highlighted respectively by green and blue rectangles. In the unit cell, these bonds appear with a ratio of 3 to 1. A shorter vertical bond corresponds to both: above for the primary and below for the secondary bonds, respectively.
FIG. 4 .
4The trend of the long-short primary (3x) and the long-short secondary (1x) bonds in the 12-layer slab. The dotted-dashed lines represents the corresponding bulk value. Each color has to be compared with the closest black line that represents the corresponding bulk bond lengths. The red line corresponds to the value of the bonds in the R − 3m paraelectric phase. The values for the two surfaces are not reported, since they reflect a different behaviour occurring at the surfaces.
FIG. 5 .
5Symmetry modes analysis connecting the high temperature cubic phase to the ferroelectric R3m. The top panel reports the ab-initio total energy for the dominant singlemodes as a function of the generalized displacement coordinate λ. The central panel reports the energy trend of the coupled-dominant modes. The bottom panel reports the dependence of the leading order parameter X2− and of the coupling X2− U Γ4− U X5− on the generalized coordinate, in the case of 0% and -5% compressive strain. In the bottom panel the configurations at λ = 0 are artificially set to zero to simplify the comparison: the -5% undistorted phase is more than 220 meV/f.u. higher in energy with respect to the noncompressed phase.
FIG. 6 .
6The evolution of the structural fingerprint for three dominant modes in the 6-layer slab. The first panel reports the fingerprint for the leading order parameter. The angle α is the one connecting the O2 and O4 with the vertex at Hf in the structure above the panels at left. The central panel reports the fingerprint for the polarization. The panel at bottom reports the fingerprint for X5− mode. It represents the zcomponent of the distance between Hf atoms with different Wyckoff positions which are reported in the picture above the panels at right.
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|
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[
"A Review of Smart Materials in Tactile Actuators for Information Delivery",
"A Review of Smart Materials in Tactile Actuators for Information Delivery"
] |
[
"Xin Xie [email protected]. \nDepartment of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA\n",
"Sanwei Liu \nDepartment of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA\n",
"Chenye Yang \nDepartment of Electrical Engineering and Computer Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n",
"Zhengyu Yang [email protected] \nDepartment of Electrical and Computer Engineering\nNortheastern University\n02115BostonMAUSA\n",
"Tian Liu \nDepartment of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA\n",
"Juncai Xu \nInstitute of Water Resources and Hydropower Research\nBeijing Post Code\nChina, China\n",
"Cheng Zhang \nMedtronic, Inc\n85286TempeAZUSA\n",
"Xianglin Zhai [email protected]. \nDepartment of Chemistry\nLouisiana State University\n70803Baton RougeLAUSA\n"
] |
[
"Department of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA",
"Department of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA",
"Department of Electrical Engineering and Computer Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Department of Electrical and Computer Engineering\nNortheastern University\n02115BostonMAUSA",
"Department of Mechanical and Industrial Engineering\nNortheastern University\n02115BostonMAUSA",
"Institute of Water Resources and Hydropower Research\nBeijing Post Code\nChina, China",
"Medtronic, Inc\n85286TempeAZUSA",
"Department of Chemistry\nLouisiana State University\n70803Baton RougeLAUSA"
] |
[] |
As the largest organ in the human body, the skin provides the important sensory channel for humans to receive external stimulations based on touch. By the information perceived through touch, people can feel and guess the properties of objects, like weight, temperature, textures, and motion, etc. In fact, those properties are nerve stimuli to our brain received by different kinds of receptors in the skin. Mechanical, electrical, and thermal stimuli can stimulate these receptors and cause different information to be conveyed through the nerves. Technologies for actuators to provide mechanical, electrical or thermal stimuli have been developed. These include static or vibrational actuation, electrostatic stimulation, focused ultrasound, and more. Smart materials, such as piezoelectric materials, carbon nanotubes, and shape memory alloys, play important roles in providing actuation for tactile sensation. This paper aims to review the background biological knowledge of human tactile sensing, to give an understanding of how we sense and interact with the world through the sense of touch, as well as the conventional and state-of-the-art technologies of tactile actuators for tactile feedback delivery.
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10.3390/c3040038
|
[
"https://export.arxiv.org/pdf/1708.07077v2.pdf"
] | 29,326,265 |
1708.07077
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b3e6687da07e8351d1c4c9800d984acc23630861
|
A Review of Smart Materials in Tactile Actuators for Information Delivery
Xin Xie [email protected].
Department of Mechanical and Industrial Engineering
Northeastern University
02115BostonMAUSA
Sanwei Liu
Department of Mechanical and Industrial Engineering
Northeastern University
02115BostonMAUSA
Chenye Yang
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
02139CambridgeMAUSA
Zhengyu Yang [email protected]
Department of Electrical and Computer Engineering
Northeastern University
02115BostonMAUSA
Tian Liu
Department of Mechanical and Industrial Engineering
Northeastern University
02115BostonMAUSA
Juncai Xu
Institute of Water Resources and Hydropower Research
Beijing Post Code
China, China
Cheng Zhang
Medtronic, Inc
85286TempeAZUSA
Xianglin Zhai [email protected].
Department of Chemistry
Louisiana State University
70803Baton RougeLAUSA
A Review of Smart Materials in Tactile Actuators for Information Delivery
Received: date; Accepted: date; Published: dateReviewsmart materialsactuatorstactile display 1 Mechanoreceptors
As the largest organ in the human body, the skin provides the important sensory channel for humans to receive external stimulations based on touch. By the information perceived through touch, people can feel and guess the properties of objects, like weight, temperature, textures, and motion, etc. In fact, those properties are nerve stimuli to our brain received by different kinds of receptors in the skin. Mechanical, electrical, and thermal stimuli can stimulate these receptors and cause different information to be conveyed through the nerves. Technologies for actuators to provide mechanical, electrical or thermal stimuli have been developed. These include static or vibrational actuation, electrostatic stimulation, focused ultrasound, and more. Smart materials, such as piezoelectric materials, carbon nanotubes, and shape memory alloys, play important roles in providing actuation for tactile sensation. This paper aims to review the background biological knowledge of human tactile sensing, to give an understanding of how we sense and interact with the world through the sense of touch, as well as the conventional and state-of-the-art technologies of tactile actuators for tactile feedback delivery.
Mechanoreceptors
Human skin contains a variety of different mechanoreceptors (touch receptors), each with its own structure, placement, frequency response, spatial resolution, adaptation speed, and necessary magnitude of skin indentation to produce a response. Mechanoreceptors are the sensing units that lie underneath the very outer skin surface for retrieving stimulations. The presence and spacing of mechanoreceptors varies between glabrous (naturally hairless) and hairy skin. There are four main types of mechanoreceptors that react to different kinds of stimuli information, such as vibration, shear, texture, and pressure. These four receptors are called Pacinian corpuscles, Meissner's corpuscles, Merkel's discs, and Ruffini endings [1]. Each receptor has a similar basic sensing element, except that their packaging and depth in the skin are adapted to their specific sensing purposes. Figure 1 shows a 3D diagram of tactile receptors in the skin. The four mechanoreceptors can be divided into two major classifications, fast adapting types and slow adapting types, based on their sensing capabilities. The fast adapting receptors include Meissner's corpuscles and Pacinian corpuscles, and are only sensitive to transitions like vibration and fluttering. The slow adapting receptors include Merkel's discs and Ruffini endings, and they are sensitive to steady or static stimulations, like pure indentation. The receptors can be further identified as type I and type II, based on their depth below the skin's surface. Type I receptors are referred to as Meissner's corpuscles and Merkel's discs; they are shallower receptors located at around 0.7 mm to 0.9 mm below the skin surface. Type II receptors are deeper ones like Pacinian corpuscles and Ruffini endings, which are often located at about 1-2 mm depth below the skin. The deeper receptors have larger receptive area [2]. Merkel's disks and Ruffini's corpuscles are classified as slowly adapting mechanoreceptors. Pressure, stretch, and static discrimination are sensed by these receptors. Higher force and larger displacement are needed to convey static stimuli sensations [5]. Receptors terminating in Merkel cells are found near the surface of the skin, and have an excellent spatial resolution, with an ability to resolve stimuli separated by as little as 0.5 mm in glabrous skin [6]. Merkel receptors are the primary receptors that are used in reading Braille. However, their best sensitivity to skin indentation is found in the range of 5 Hz to 15 Hz, at which frequency, a minimum skin indentation on the order of 50 µm is typically required to produce a response [7].
Meissner's corpuscles, which are located in the shallowest area below the skin's surface, are particularly suitable to sense low-frequency stroking and vibration. Meissner's corpuscles have a maximum sensitivity between about 20 Hz to 50 Hz, and have a minimum sensitivity to skin indentation of about 14 µm [8]. Meissner's corpuscles are located with a high density of about 150 receptors/cm 2 , but they have a relatively lower spatial resolution and respond rather uniformly across their entire 3-5 mm receptive field. Pacinian corpuscles differ from Meissner's corpuscles in their shape, depth, and response threshold. They are more sensitive to the vibrations with frequency ranging from a few tens of Hz to hundreds of Hz, with the highest sensitivity around 250 Hz [9]. The highest sensitivity may be found in Pacinian corpuscles, which have demonstrated sensitivity to less than 1 µm skin indentations around 250-300 Hz, and an effective frequency range of about 60-400 Hz [8]. Pacinian corpuscles have a large receptive field, and can sense larger vibrations from a distance of on the order of a centimeter away from the receptor. However, smaller vibrations near the 250 Hz frequency of optimal sensitivity produce a response that is localized directly over the Pacinian corpuscle [9], thereby enabling improved spatial localization with these highly sensitive receptors. The characteristics of mechanoreceptors have a promising degree of overlap with the known characteristics of microactuators and MEMS systems. Pacinian corpuscles also adapt faster than Meissner's corpuscles and have a lower response threshold [10]. For these reasons, Pacinian corpuscles are often considered as the better candidate for retrieving the information primarily about dynamic motion stimuli and high-frequency vibrations.
Stimulation Methods and Actuation Technologies
Ideally, the display's resolution should leverage the approximately one unit per mm 2 spacing of mechanoreceptors in human finger pads [11], and be extendable to full 2D. It should be refreshable in real time (i.e., it should refresh at least as quickly as human mechanoreceptors can react), allowing the information conveyed by the display to keep up with rapidly changing inputs. The display should also code information in a way that is easily detected and interpreted, so that it is intuitive and easy to learn to use. The psychophysics literature offers clear insights into humans' ease of sensing various types of tactile stimuli (static vs moving, vibrating vs quasistatic, and low vs high amplitude). In particular, humans are much more sensitive to motions and changing stimuli than they are to static patterns [12], whether those stimuli are visual, audible, or tactile. The display should therefore code information not only as static patterns, but also as simulated motion against the user's finger pads. Finally, its power consumption should be compatible with portable use, and it should be manufacturable by efficiently scalable means, to ensure that its cost is compatible with the resources of its intended user base. There are several ways to provide the stimuli for tactile displays. The stimulation methods can be divided into approximately three categories: thermal, electrical, and mechanical stimuli.
Thermal
Thermal stimulation, or thermal flow, is usually used for adding quality characteristics in the information delivery, such as mimicking the color rendering of a vision system. A 3 × 3 thermal tactile display device, with multiple heat sources that can display temperatures from 5 °C to 55 °C and produce different temperature field distributions, has been developed [13]. Thermal sensation modeling for the fingertip has been developed, showing that skin is more sensitive to rapid temperature change, which leads to a difficulty in presenting long time stimulation duration [14]. The combination of both vibrotactile and thermal stimuli is used in the generation of haptic sensation [15]. In spite of its capability of representing color identification and discrimination, thermal tactile displays alone are not well-suited to present rich information, because of their low spatial resolution and their lower transition-sensing time between on/off states [16].
Electrotactile
Electrotactile stimulation uses electrical current flow from electrodes embedded in the device to deliver stimuli to the sensing nerves of the skin, mimicking pressure or vibration without any actual mechanical actuators involved [17]. A 4 × 4 electrotactile matrix called SmartTouch is developed to selectively stimulate the Merkel's discs and Meissner corpuscles. The sensation is generated by electrodes that run electrical current pulses of 1-3 mA with a duration of 0.2 ms through the nerves of the skin [18]. A sensory substitution system that employs electrotactile and vibrotactile displays was developed [19]. There are also oral electrotactile displays. An array of 7 × 7 tactual actuators fabricated on polyimide-based flexible film was placed on the roof of the mouth, to deliver electrotactile stimulation with relatively low stimulation intensities [20]. Although electrotactile systems are structurally simple and easily controlled, challenges remain in spatial resolution, safety, comfort level, and power consumption.
Mechanical
Mechanical tactile stimulation is the most commonly used method to create a sensation, not only because the mechanoreceptors tend to respond to direct and physical mechanical stimuli easily, but also because of the finer spatial resolution as compared with thermal and electrotactile stimulation.
Static Indentation
Braille is the most well-known example of static tactile information acquisition. Braille can be printed on paper or produced on a refreshable Braille reader, in which the dots of each Braille cell are driven up and down by an array of stacked piezoelectric bending beam actuators. Braille is an excellent means of providing text information to those who are Braille-literate, as long as the information is available electronically as text, or can be scanned using optical character recognition. The most commonly-used technology for conveying tactile graphics is embossing of images on paper (e.g., thermoform or microcapsule paper) [21]. Refreshable Braille displays offer a quasi-static Braille display on a refreshed basis by employing extruded pins driven by piezoelectric bimorph actuators. This provides an opportunity for intensive information delivery using an electronic device that is more compact as compared with printed Braille books.
Vibration
Skin nerves tend to sense vibration easier than they sense static indentation [22]. Fast adapting Pacinian corpuscles are primarily responsible for vibrotactile perception in human skin. The lower thresholds of force and displacement for vibration, as compared with static indentation, make vibrotactile stimulation a widely and well-researched candidate for tactile displays. Thus, technologies have been developed to deliver vibrations for tactile display. As human beings, we can distinguish successive pulses with a time gap of 5 ms [23], which is even better than our vision system, in which the minimum time gap is 25 ms [24]. This means that vibration can be utilized to create rhythm or patterns for information with complex meaning. Further variation in vibro-rhythm can be realized by changing the amplitude and frequency.
Surface Acoustic Waves
Surface acoustic waves (SAW) generated by SAW transducers can stimulate the skin to provide a sensation of continuous roughness. Both passive and active sensation capture of SAW tactile devices have been reported [25][26][27]. Ultrasonic motors are used for generating vibration that can be directly sent to the user's finger, which is described as passive type SAW transducers. In contrast, active tactile transducers utilize standing waves of a SAW and friction shift to form vibration.
Electrorheological and Magnetorheological Fluids
Electrorheological (ER) and magnetorheological (MR) fluids are special classes of materials that can respond to the electrical field and magnetic field, respectively. They are both colloid suspensions with dielectric or ferromagnetic particles (1-100 µm) that are sensitive to electric or magnetic potential. Under normal conditions, when there is no electric or magnetic stimulus to the materials, the ER or MR fluids remain in liquid form. Upon application of the electric or magnetic field, the particles align themselves nearly parallel to the direction of the fields. This causes the viscosity to change, and the liquids become solid gels as the field applied increases. Such properties have been used to make tactile displays [28][29][30].
Vibrotactile Actuators for Tactile Feedback
Vibrotactile displays have been researched and developed during the past few decades, because mechanoreceptors sense vibration sensations more easily and rapidly than they sense quasistatic mechanical stimuli. The actuators used to generate vibration range from large-scale electric motors to MEMS-scale hydraulic pumps. Each has advantages and drawbacks for tactile displays.
Rotary Electromagnet Actuators
Rotary DC motors [31] are utilized to produce a vibrotactile sensation. The motors rotate when a DC current is applied. An off-centered mass affixed to the output shaft of the motor (often referred as an eccentric mass) offers the vibration. The feeling of vibration created by these motors varies linearly with the voltage or current applied. A small voltage creates a small and slow vibration, whereas a large applied voltage generates a strong and fast vibration. This type of actuator is the most commonly used vibrational actuator in toys, game controls, and virtual reality tactile devices (Axonvr). The benefits of rotary actuators include their cost-effectiveness, relatively strong vibration, and relatively lower requirements for electronics. Their drawbacks include their slow response time, which is usually in the range of tens of milliseconds. In addition, because the rotation it generates is non-directional, it is also not suitable for high-quality precision tactile feedback. Also, the size of the rotary motor is usually relatively large, which makes it a poor candidate for high-resolution tactile displays.
Linear Electromagnetic Actuators
Linear electromagnetic actuators (LEA) are another main way of using electromagnetic inputs to generate vibration. When a current passes through the conductive wires wrapped to form a coil, an electromagnetic field is generated. That field either pushes or drags a permanent magnet inside the coil, depending on its physical orientation and the direction in which the current flows into the coil; the motion of the magnet, in turn, causes a tactile vibration. LEA actuators are also commonly used in mobile phones, because of their low cost and the appropriateness of their compact size for the mobile phone scale. The actuator only works at the resonance frequency of the system. Although the LEA reacts faster than a rotary motor, it is still rather slow for a fast response tactile system. In addition, making the actuator can be complicated.
Electroactive Polymer
Electroactive polymers (EAP) are a group of polymers that change shape or size when an electric field is applied. When they change shape or size, a vibration is formed. Electroactive polymer actuators provide quasistatic millimeter-scale actuation, and are small enough to be arrayed with a pitch of a few millimeters, but they typically have actuation times on the order of tenths of seconds to tens of seconds [32,33]. The EAP is robust, but slow in refresh rate. In addition, a voltage higher than 300 V is normally needed to activate the actuator, which is not ideal in a portable tactile device.
Shape Memory Alloy
Shape memory alloy (SMA) actuators are metal alloys that remember their original shapes; their shapes change under the response to a temperature change, for example, via Joule heating. This effect happens because of the reversible phase changing inside the alloy. Though SMA tactile displays have been developed to provide large displacement and high force, their slow response times and large power consumptions make real-time vibrational graphical tactile display almost impossible [34][35][36].
Carbon Nanotube
As one of the promoted smart materials, carbon nanotube (CNT) plays an important role in providing actuation for tactile sensation. With the ultra-strong sp 2 carbon-carbon bond, carbon nanotubes (CNTs) have a great combination of high mechanical (5-fold stiffness vs steel, 10-fold strength vs carbon fiber) [37][38][39][40], electronic (current density is 1000 times higher than copper) [41], and thermal conducting properties [42]. This high aspect ratio, lightweight [43], and stable material can also be synthesized and fabricated into various sizes and shapes, as well as form composite networks with other functional materials [44,45]. All these advantages make CNTs an ideal candidate for making high performance mechanical, electrical, and even thermal tactile actuators [46][47][48][49].
Generally, the role CNTs play at actuation are utilizing the high electrical conductivity of CNTs to sense signals and transfer energy; realizing large stroke and fast response actuation by stretching, twisting, and bending the CNT based structures; or a combination of both. Baughman, et al. [50] have first demonstrated the electromechanical actuators based on sheets of single-walled CNTs, indicating CNT networks are a high potential durable media for various applications, such as low voltage-activated microcantilevers for medical catheters, or temperature insensitive material for jet engines. Lima, et al. [48] have demonstrated strong and fast hybrid CNT artificial muscles powered by electrically, chemically, and photonically activated guests.
A few approaches have been developed for tactile CNT actuators with different mechanisms. By adapting the high mechanical strength and high conductivity of MWCNT, Wang et al. [51] have developed a PDMS/MWCNT coplanar electrode layer to overcome crosstalk effects in the tactile array ( Figure 2). Other approaches, such as activating CNT composites via different, have also shown promising results for tactile applications. Camargo et al. [52] have demonstrated a tactile device powered by optically activating the liquid crystal-carbon nanotube (LC-CNT) composite Braille dots ( Figure 3). Pyo et al. [53] have demonstrated a CNT-PDMS composite tactile sensor, with the advantages of low-cost, batch production.
Due to a scaling degradation, large-scale CNT networks have much lower performance compared with ideal individual CNTs [46,47]; recent research has demonstrated that transfer of weak van der Waals CNT contacts into larger covalent junctions via appropriate electrical pulses is a promising method to overcome the challenge [54], which could further optimize the performance of CNT tactile actuators.
Piezoelectric Actuators
The piezoelectric effect was discovered in 1880 by two physicists [55]. It describes the phenomenon that in a certain group of materials, an electrical potential is generated when a mechanical load (pressing or squeezing) is applied on the material. In most crystals like metal, the unit cell that is the minimum repeating structure is symmetric. In contrast, in piezoelectric crystal structures, the unit cell is not symmetric. Piezoelectric crystals are electrically neutral in their initial state. They can have an electric polarization when no load is applied (being both ferroelectric and piezoelectric) or no electric polarization when no load is applied (being purely piezoelectric). When a mechanical load is applied, the positive and negative charges separate, generating an electrical potential across the piezoelectric material [56][57][58][59]. This process is also reversible. When a voltage is applied to the opposing faces of the piezoelectric material, the material needs to rebalance the electrical charges inside it, which causes a mechanical deformation. In order to create the piezoelectric effect in piezoceramics, in which the piezoelectric crystal grains are randomly oriented, the material needs to be heated to high-temperature under a strong electric field in a process called poling. The heat allows more free movement of the molecules, and the poling directions of each grain are forced into nearly the same direction under the strong electric field. After the poling process, the piezoelectric effect is obtained in the treated material. Frequently used piezoelectric materials include piezoelectric ceramics, such as lead zirconate titanate (PZT), and piezoelectric polymers, such as polyvinylidene fluoride (PVDF). The crystal structure of piezoelectric ceramics is close to a cube, in which, for example, eight metal ions with two negative charges take positions at the eight corners, six oxygen ions with two native charges take positions at the six faces, forming tetragons inside the cube, and a metal ion with four positive charges is located at the center of the cube and the tetragon. When the temperature of the material is above a certain value, called the Curie temperature (which depends on the material), each crystal has a symmetric structure with no electric dipole. When the temperature goes below the Curie temperature, the crystal cube turns dissymmetric, resulting in an electric dipole in a certain direction. If such dipoles randomly distribute inside a material, the material does not show net polarization. However, piezoelectric ceramics are ferroelectric below the Curie temperature, with permanent polarization and deformation after being poled by an electric field. In this way, such materials have the ability to generate an electric field in response to strain, or to undergo strain in response to external electric field. Piezoelectric polymers, such as PVDF, are also widely used as electrical generators. The structure of PVDF depends on the structure of the molecular chain, resulting in different polymer phases. Similarly, when poled, PVDF is a ferroelectric material that exhibits the piezoelectric effect.
The relationship between the mechanical deformation and the voltage applied to the piezoelectric materials is defined as a piezoelectric coefficient, which is mathematically defined as in Equation (1):
= (1)
Three axes, termed as 1, 2, and 3, are used to identify directions in piezoceramic material. The terms 1, 2, and 3 represent the axis of X, Y, and Z in the spatial 3D set of axes. The axis 3 is defined as the axis that is parallel to the polarization direction during the poling process. In practice, the piezoelectric coefficient is described as dij, in which the subscripts i and j represent the poling direction of the piezoelectric material, and the direction of the mechanical strain, respectively. A larger dij value means that the material has a greater mechanical deformation under a same electric field in the specific direction defined by i and j. The most commonly used dij are d33 and d31. The term d33 describes a positive strain in the direction of its length when an electric field is in the same direction. The term d31 describes a negative strain in the transverse direction as the electric field. The piezoelectric effect has been used to produce vibrotactile motion. Piezoelectric materials can be either solid ceramics or soft gel-like polymers, and they change shape when a voltage is applied. This process is reversible, so piezoelectric materials are also often used in sensors to detect mechanical deformation or the corresponding force applied on the material. Piezoelectric materials responds very quickly (microseconds) to electrical stimuli, and thus, are often used to produce high-frequency vibration.
Piezoelectric Bending Beam Actuators
The piezoelectric effect is found not only in piezoceramics, but also in polymers like PVDF (polyvinylidene fluoride). Piezoelectric polymer can provide a good pulling force, not a good pushing force, because of its mechanical flexibility. It also has a smaller piezoelectric coefficient compared to piezoelectric ceramic; thus, it is not suitable for applications in which high-frequency vibration with reasonable voltage is needed.
Piezoelectric actuators may be structured in various ways to produce extensional or bending actuation. When a single, long, thin plate or beam of a piezoceramic is poled in its thinnest dimension, it forms a piezoelectric unimorph actuator. When a voltage is subsequently applied across its thinnest dimension, the unimorph undergoes changes in length and thickness; in other words, it forms an extensional actuator. An alternative structure, a piezoelectric bimorph actuator, may be created by laminating two unimorphs together, so that the structure's thickness is doubled. The two unimorphs may be laminated, so that their polarizations are either parallel (called Y-poled) or antiparallel (called X-poled). When voltages are applied across the two layers of the bimorph, the bimorph can provide extensional actuation (like a unimorph actuator) or bending (when one layer is contracting, and the other is expanding under the influence of an applied voltage), depending on polarities of the applied voltages.
Piezoelectric bimorph bending-beam actuators may be used for vibrotactile applications. The vibration occurs when an alternating voltage is applied to the actuator. A device called the Optacon [60] was created in the 1970s to permit visually-impaired people to read text from a page without first translating the information into Braille. The system includes a camera that can be manually scanned across written text; an image of each character is then transferred into a pattern of vibrating piezoelectric beams in which the beam tips replicate the pattern of black and white of the character on the page. The piezoelectric beams are bimorph actuators that vibrate to deliver vibrotactile sensation. Although the Optacon is no longer in production, existing units are still in use by a group of people who find its capabilities indispensable. An alternative integrates piezoelectric bending beam actuators perpendicular to the tactile sensing plane, enabling large bending beam actuators to be tightly packed for fully 2D displays [61]. The large amount of piezoelectric material required potentially increases the cost of this type of architecture, as for other piezoelectric bending beam actuator systems.
Piezoelectric Extensional Actuators
Although most tactile displays that engage the piezoelectric effect utilize bending actuators, there are some tactile displays that use extensional piezoelectric actuators. Piezoelectric extensional actuators are not able to provide as large of a displacement as bending actuators, but with some motion amplification mechanisms, they can be compact and still efficient. Piezoelectric extension actuators and MEMS technology are also increasingly being leveraged to create tactile displays, as in [62][63][64][65][66][67][68][69][70]. Important challenges nonetheless remain, including spatial resolution, refresh rate, fabrication complexity, and cost.
Pneumatic Actuation
In pneumatic actuation, air pressure changes are used to provide direct or indirect vibration to the user's skin. The pressurized airflow directly causes a pressure or acts as a driver to move mechanical parts to interact with a finger [71,72]. Pneumatic actuator systems can be compact and light, but they require external air pumps to generate air pressure, and high-frequency output can be a problem for pneumatic tactile display.
Conclusions
Although refreshable 2D graphical interfaces have tremendous potential for conveying complex information, it is challenging to create actuators that are compact enough to be arrayed into an arbitrarily large number of rows and columns, while still being robust, easy to sense, and rapidly switchable. Each of the technologies presented in this chapter has its own advantages and disadvantages, certain types of actuator should be chosen in various applications in terms of the tradeoff between tactile feedback effectiveness, system complexity, power consumption, and cost, as shown in Table 2.
Research on high performance micro actuators is important, because they can be manufactured economically not only for assistive technology (e.g., interfaces through which people with blindness or low vision can perceive their environment), but also for enhancing the multi-sensory displays for private communication (e.g., silent, covert communication for military personnel), or advancing the tactile feedback in virtual reality, or providing motion and force in microrobotics. The main challenge in creating a small-sized actuator for high resolution tactile displays is that conventional macro actuators are large in size, making high resolution hard to implement, whereas MEMS actuators are small, making adequate force and displacement for human sensing difficult to attain. Suggested research work should investigate how actuators can bridge the gap between macroscale forces and microscale compactness to create effective tactile stimuli at the MEMS size scale.
Figure 1 .
1Tactile receptors in the skin[3].
Figure 2 .Figure 3 .
23Assembly of PDMS/MWCNT-based tactile sensor array with coplanar electrodes. Adapted from[51]. Schematic concept of carbon nanotube (CNT) composite film driven by the light source to form "blister" shape and flat shape for actuation. (a) Default state: the light source is OFF. The feature is a high-relief dot; (b) the feature flattens when it is light-actuated. This actuation is reversible when the light is turned off. Adapted from[52].
Figure 4 .
4Schematic diagrams showing the piezoelectric effect. The dashed outline represents the object's original shape; the solid outline represents the object's deformed shape.
Figure 5 .
5Schematic view of piezoelectric beams generating voltage in (a) 33 mode, and (b) 31 mode. The dashed outline represents the original shape; the solid outline represents the deformed shape.
Table 1 .
1Characteristics of the four mechanoreceptors[4].Receptors
Type (Depth)
Type (Adapting
Speed)
Receptive
Field
Frequency
(Hz)
Sensing Property
Merkel's discs
I
Slow
Small
5-15
Pressure, Texture
Ruffini endings
II
Slow
Large
15-400
Stretch
Meissner's corpuscles
I
Fast
Small
20-50
Stroke, Fluttering
Pacinian corpuscles
II
Fast
Large
60-400
Vibration
Table 2 .
2Comparison of common vibrotactile actuation types.Conflicts of Interest:The authors declare no conflict of interest.Feedback
Effectiveness
System
Complexity
Size
Power
Consumption
Response
Speed
Cost
Rotary Motor
Bad
Good
Bad
Marginal
Good
Good
LEA
Marginal
Good
Bad
Marginal
Good
Good
EAP
Good
Marginal
Good
Bad
Bad
Marginal
SMA
Good
Bad
Good
Bad
Bad
Marginal
Piezoelectric
Good
Marginal
Marginal
Good
Good
Bad
Pneumatic
Good
Bad
Bad
Bad
Marginal
Bad
CNT
Good
Marginal
Bad
Marginal
Bad
Bad
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Batch fabricated flexible tactile sensor based on carbon nanotube-polymer composites. S Pyo, J I Lee, M O Kim, T Chung, Y Oh, S C Lim, J Park, J Kim, Proceedings of the Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII). the Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII)Barcelona, SpainPyo, S.; Lee, J.I.; Kim, M.O.; Chung, T.; Oh, Y.; Lim, S.C.; Park, J.; Kim, J. Batch fabricated flexible tactile sensor based on carbon nanotube-polymer composites. In Proceedings of the Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII), Barcelona, Spain, 16-20 June 2013.
Sculpting carbon bonds for allotropic transformation through solid-state re-engineering of-sp 2 carbon. H Y Jung, P T Araujo, Y L Kim, S M Jung, X Jia, S Hong, C W Ahn, J Kong, M S Dresselhaus, S Kar, 10.1038/ncomms5941Nature Commun. 5Jung, H.Y.; Araujo, P.T.; Kim, Y.L.; Jung, S.M.; Jia, X.; Hong, S.; Ahn, C.W.; Kong, J.; Dresselhaus, M.S.; Kar, S.; et al. Sculpting carbon bonds for allotropic transformation through solid-state re-engineering of-sp 2 carbon. Nature Commun. 2014, 5, doi:10.1038/ncomms5941.
Passively-Switched Vibrational Energy Harvesters. T Liu, ProQuest Dissertations PublishingBoston, MA, USA; Ann Arbor, MI, USANortheastern UniversityPh.D. ThesisProQuest Number10276237Liu, T. Passively-Switched Vibrational Energy Harvesters. Ph.D. Thesis, Northeastern University, Boston, MA, USA, 2017; ProQuest Dissertations Publishing: Ann Arbor, MI, USA, 2017; ProQuest Number10276237.
Passively-switched energy harvester for increased operational range. T Liu, R S Pierre, C Livermore, 10.1088/0964-1726/23/9/09504523Smart Mater. Struct.Liu, T.; Pierre, R.S.; Livermore, C. Passively-switched energy harvester for increased operational range. Smart Mater. Struct. 2014, 23, doi:10.1088/0964-1726/23/9/095045.
A compact architecture for passively-switched energy harvesters. T Liu, C Livermore, 10.1088/1742-6596/660/1/012090J. Phys. Conf. Ser. 660Liu, T.; Livermore, C. A compact architecture for passively-switched energy harvesters. J. Phys. Conf. Ser. 2015, 660, doi:10.1088/1742-6596/660/1/012090.
Passively tuning harvesting beam length to achieve very high harvesting bandwidth in rotating applications. T Liu, C Livermore, Proceedings of the PowerMEMS. the PowerMEMSAtlanta, GA, USA, 2Liu, T.; Livermore, C. Passively tuning harvesting beam length to achieve very high harvesting bandwidth in rotating applications. In Proceedings of the PowerMEMS Atlanta, GA, USA, 2-5 December 2015;
Smart materials and structures for energy harvesters. T Liu, S Liu, X Xie, C Yang, Z Yang, X Zhai, arXiv:1709.00493Liu, T.; Liu, S, Xie, X.; Yang, C.; Yang, Z.; Zhai, X. Smart materials and structures for energy harvesters. arXiv 2017, arXiv:1709.00493.
Miniature Shape Memory Alloy Actuator for Tactile Binary Information Display. R Velazquez, E Pissaloux, J Szewczyk, M Hafez, 10.1109/robot.2005.1570302Proceedings of the 2005 IEEE International Conference on Robotics and Automation. the 2005 IEEE International Conference on Robotics and AutomationBarcelona, SpainVelazquez, R.; Pissaloux, E.; Szewczyk, J.; Hafez, M. Miniature Shape Memory Alloy Actuator for Tactile Binary Information Display. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 18-22 April 2005; doi:10.1109/robot.2005.1570302.
Evaluation of RU-Netra Tactile Feedback Navigation System for the Visually Impaired. C Shah, M Bouzit, M Youssef, L Vasquez, 10.1109/iwvr.2006.1707530Proceedings of the 2006 International Workshop on Virtual Rehabilitation. the 2006 International Workshop on Virtual RehabilitationNew York, NY, USAShah, C.; Bouzit, M.; Youssef, M.; Vasquez, L. Evaluation of RU-Netra Tactile Feedback Navigation System for the Visually Impaired. In Proceedings of the 2006 International Workshop on Virtual Rehabilitation, New York, NY, USA, 29-30 August 2006; doi:10.1109/iwvr.2006.1707530.
MEMS-enabled, vibrational tactile actuators for high resolution tactile displays. X Xie, Y Zaitsev, L F Velásquez-García, S J Teller, C Livermore, Scalable, 10.1088/0960-1317/24/12/125014J. Micromech. Microeng. 24Xie, X.; Zaitsev, Y.; Velásquez-García, L.F.; Teller, S.J.; Livermore, C. Scalable, MEMS-enabled, vibrational tactile actuators for high resolution tactile displays. J. Micromech. Microeng. 2014, 24, doi:10.1088/0960- 1317/24/12/125014.
Compact, planar, translational piezoelectric bimorph actuator with Archimedes' spiral actuating tethers. C Yang, S Liu, X Xie, C Livermore, 10.1088/0960-1317/26/12/124005J. Micromech. Microeng. 26Yang, C.; Liu, S.; Xie, X.; Livermore, C. Compact, planar, translational piezoelectric bimorph actuator with Archimedes' spiral actuating tethers. J. Micromech. Microeng. 2016, 26, doi:10.1088/0960-1317/26/12/124005.
Passively self-aligned assembly of compact barrel hinges for high-performance, outof-plane mems actuators. X Xie, C Livermore, Proceedings of the 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS). the 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS)Las Vegas, NV, USAXie, X.; Livermore, C. Passively self-aligned assembly of compact barrel hinges for high-performance, out- of-plane mems actuators. In Proceedings of the 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NV, USA, 22-26 January 2017; pp. 813-816.
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ultra-low leak micro valve using liquid surface tension sealing for vacuum applications. C Yang, X Xie, S Liu, C Livermore, Resealable, 10.1109/transducers.2017.7994481Proceedings of the 2017 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS). the 2017 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS)Kaohsiung, TaiwanYang, C.; Xie, X.; Liu, S.; Livermore, C. Resealable, ultra-low leak micro valve using liquid surface tension sealing for vacuum applications. In Proceedings of the 2017 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), Kaohsiung, Taiwan, 18-22 June 2017; doi:10.1109/transducers.2017.7994481.
Compact, scalable, high-resolution, MEMS-enabled tactile displays. X Xie, Y Zaitsev, L Velasquez-Garcia, S Teller, C Livermore, Proceedings of the Solid-State Sensors, Actuators, and Microsystems Workshop. the Solid-State Sensors, Actuators, and Microsystems WorkshopHilton Head Island, SC, USAXie, X.; Zaitsev, Y.; Velasquez-Garcia, L.; Teller, S.; Livermore, C. Compact, scalable, high-resolution, MEMS-enabled tactile displays. In Proceedings of the Solid-State Sensors, Actuators, and Microsystems Workshop, Hilton Head Island, SC, USA, 8-12 June 2014; pp. 127-130.
Test and Analysis of Hydraulic Fracture Characteristics of Rock Single Crack. J Xu, X Xie, C Yang, Z Shen, 10.4172/2476-2296.1000164Fluid Mech. Open Access. 4Xu, J.; Xie, X.; Yang, C.; Shen, Z. Test and Analysis of Hydraulic Fracture Characteristics of Rock Single Crack. Fluid Mech. Open Access 2017, 4, doi:10.4172/2476-2296.1000164.
High Performance Micro Actuators for Tactile Displays. X Xie, ProQuest Dissertations PublishingBoston, MA, USA; Ann Arbor, MI, USANortheastern UniversityPh.D. ThesisProQuest No. 10273384Xie, X. High Performance Micro Actuators for Tactile Displays. Ph.D. Thesis, Northeastern University, Boston, MA, USA, 2017; ProQuest Dissertations Publishing: Ann Arbor, MI, USA, 2017; ProQuest No. 10273384.
Position controlled pneumatic tactile display for tangential stimulation of a finger pad. J Yoo, S Yun, S Lim, J Park, K Yun, H Lee, 10.1016/j.sna.2015.03.023Sens. Actuators A Phys. 229Yoo, J.; Yun, S.; Lim, S.; Park, J.; Yun, K.; Lee, H. Position controlled pneumatic tactile display for tangential stimulation of a finger pad. Sens. Actuators A Phys. 2015, 229, 15-22, doi:10.1016/j.sna.2015.03.023.
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[
"The infrared spectrum of protonated buckminsterfullerene, C 60 H +",
"The infrared spectrum of protonated buckminsterfullerene, C 60 H +"
] |
[
"Julianna Palotás \nInstitute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands\n",
"Jonathan Martens \nInstitute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands\n",
"Giel Berden \nInstitute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands\n",
"Jos Oomens \nInstitute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands\n",
"\nHoff Institute for Molecular Sciences\nUniversity of Amsterdam\nScience Park 9081098XHAmsterdamThe Netherlands\n"
] |
[
"Institute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands",
"Institute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands",
"Institute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands",
"Institute for Molecules and Materials\nFELIX Laboratory\nRadboud University\nToernooiveld 76525EDNijmegenThe Netherlands",
"Hoff Institute for Molecular Sciences\nUniversity of Amsterdam\nScience Park 9081098XHAmsterdamThe Netherlands"
] |
[] |
AUTHOR CONTRIBUTIONSJP, JM and GB carried out the experiments, which were conceptualized by JO and GB. JP and JO wrote the manuscript with input from all other authors.
|
10.1038/s41550-019-0941-6
|
[
"https://export.arxiv.org/pdf/1910.04533v1.pdf"
] | 204,008,634 |
1910.04533
|
45d4ce5acb97ba4a7edcdc149586f6a239be48b9
|
The infrared spectrum of protonated buckminsterfullerene, C 60 H +
10 Oct 2019
Julianna Palotás
Institute for Molecules and Materials
FELIX Laboratory
Radboud University
Toernooiveld 76525EDNijmegenThe Netherlands
Jonathan Martens
Institute for Molecules and Materials
FELIX Laboratory
Radboud University
Toernooiveld 76525EDNijmegenThe Netherlands
Giel Berden
Institute for Molecules and Materials
FELIX Laboratory
Radboud University
Toernooiveld 76525EDNijmegenThe Netherlands
Jos Oomens
Institute for Molecules and Materials
FELIX Laboratory
Radboud University
Toernooiveld 76525EDNijmegenThe Netherlands
Hoff Institute for Molecular Sciences
University of Amsterdam
Science Park 9081098XHAmsterdamThe Netherlands
The infrared spectrum of protonated buckminsterfullerene, C 60 H +
10 Oct 2019
AUTHOR CONTRIBUTIONSJP, JM and GB carried out the experiments, which were conceptualized by JO and GB. JP and JO wrote the manuscript with input from all other authors.
Although fullerenes have long been hypothesized to occur in interstellar environments, their actual unambiguous spectroscopic identification is of more recent date. 1-4 C 60 , C 70 and C + 60 now constitute the largest molecular species individually identified in the interstellar medium (ISM). Fullerenes have significant proton affinities and it was suggested that C 60 H + is likely the most abundant interstellar analogue of C 60 . 5 We present here the first laboratory infrared (IR) spectrum of gaseous C 60 H + . Symmetry breaking relative to C 60 produces an IR spectrum that is much richer than that of C 60 . The experimental spectrum is used to benchmark theoretical spectra indicating that the B3LYP density functional with the 6-311+G(d,p) basis set accurately reproduces the spectrum. Comparison with IR emission spectra from two planetary nebulae, SMP LMC56 and SMC16, that have been associated with high C 60 abundances, indicate that C 60 H + is a plausible contributor to their IR emission.
Buckminsterfullerene C 60 is undoubtedly one of the most iconic molecules of our time. Since its discovery in 1985, 6 its physico-chemical properties have been extensively characterized, including its ion chemistry and spectroscopic properties. IR spectra have been reported in condensed and gas phases, 7-10 and spectra for ionized forms are available as well. 4,8,11,12 The high cosmic abundance of carbon combined with the high stability of fullerenes 13 initiated a quest for their detection in inter-and circumstellar environments. [14][15][16][17] This search culminated in the identifications of neutral C 60 and C 70 in a young planetary nebula (Tc1) 1 based on diagnostic IR features. Accurate gas-phase laboratory spectra in the near-IR range led to the identification of C + 60 as carrier of two of the diffuse interstellar bands near 9600Å. 4 The question of whether or not fullerenes can form in H-rich regions of the interstellar medium (ISM) has been under debate. 1,3 Hydrogenation produces stable fullerene derivatives and partially hydrogenated fullerenes (fulleranes) have been suggested to occur in circumstellar envelopes and in the ISM 3,18,19 . On the other hand, hydrogenation and the concomitant change in orbital hybridization from sp 2 to sp 3 reduces the stability of the fullerene cage, which under the conditions of the ISM would likely lead to dehydrogenation and restoration of the original fullerene 20 or to breakdown of the carbon cage. 5 However, in this latter paper, Kroto also noted that protonation does not compromise cage stability and hypothesized that "protonated C 60 is likely to be the most abundant fullerene analogue," analogous to high abundances of protonated carbon monoxide, HCO + .
Ion chemistry studies 21 have determined the proton affinity (PA) of C 60 at 860 kJ/mol. This relatively high value, just above the PA of ammonia, makes C 60 H + (Figure 1a) one of the most relevant stable fullerene derivatives and underpins Kroto's statement above. However, a) Corresponding author: [email protected] to our knowledge, no IR spectra have been reported for protonated fullerenes. Electronic spectra have been suggested to suffer from broad, unresolved features due to tunneling of H + between the identical C-atom sites. 5 IR spectra have been reported for deposited fullerene films exposed to atomic hydrogen 22 and for some stable C 60 H n hydrogenated fullerenes. 20 Here we present the first experimental IR spectrum of protonated C 60 , recorded in the gas phase via infrared multiple-photon dissociation (IRMPD) using the FELIX free-electron laser and an ion trap mass spectrometer. 23 This IR spectrum is compared with astronomical spectra of objects that were associated with high C 60 abundances as well as with theoretical IR spectra to evaluate the performance of different computational approaches. Contrasting the spectrum with that of C 60 yields a textbook example of the effects of symmetry breaking on vibrational selection rules.
C 60 H + generated by atmospheric-pressure chemical ionization (APCI) produces the mass spectrum in Figure 1b, showing the radical cation of C 60 at m/z 720, clearly resolved from the higher-intensity mass peak at m/z 721. At a natural carbon-13 abundance of 1.1%, monoisotopic 12 C 60 occurs at only 51%, while the 13 C 12 C 59 isotopomer has an abundance of 34%. Hence, the m/z 721 base peak is a superposition of two ions: the protonated ion C 60 H + and the 13 C 12 C +• 59 radical cation; the small mass difference of 0.0045 u cannot be resolved in our mass spectrometer. From the observed isotope pattern, the relative populations of C 60 H + and C +• 60 are derived as 65% and 35%, respectively. Figure 1c shows the mass spectrum after isolation of the m/z 721 ion in the trap (see Methods). Tuning the laser frequency to a vibrational resonance of C 60 H + leads to the absorption of multiple IR photons inducing dissociation forming the m/z 720 ion. Note that only C 60 H + can undergo fragmentation into this channel -by loss of an H-atom -while 13 C 12 C +• 59 cannot. Moreover, due to its high stability, C +• 60 does not undergo dissociation under our experimental conditions, as was tested by iso- b) Mass spectrum as generated by the APCI source without mass isolation. The predicted isotope pattern for a 35%/65% mixture of C +• 60 and C60H + (as represented by the sticks) closely reproduces the experimental pattern. c) Mass spectrum after isolation of the m/z 721 peak. With the IR laser frequency on-resonance with an IR absorption of C60H + at 1300 cm −1 , fragmentation by loss of an H-atom occurs leading to an increase of ions in the m/z 720 mass channel. d) Increasing the number of IR pulses leads to depletion of the C60H + ion population. Fragmentation of 13 C 12 C +· 59 is not observed under our experimental conditions, so that the m/z 721 peak intensity levels off to a constant value. Note that increasing the radiation power, saturation is reached sooner, but the limiting value is the same, namely the fraction of 13
C 12 C +• 59 ions.
lating the m/z 720 ion.
Increasing the number of laser pulses, the C 60 H + population undergoes dissociation until it is depleted. The remaining intensity at m/z 721 is then exclusively due to 13 C 12 C +• 59 . Figure 1d confirms its approximately 35% contribution to the ion population. By monitoring the fragmentation of m/z 721 into channel m/z 720 as function of wavelength, we selectively measure the IRMPD spectrum of C 60 H + , which is not contaminated by contributions from 13 C 12 C +• 59 . Buckminsterfullerene C 60 belongs to the icosahedral point group (I h ), 10,24 which leads to IR selection rules that leave the large majority of the 174 normal modes inactive. There are only four triply-degenerate modes belonging to the T 1u irreducible representation that are infrared active. As a consequence, C 60 has an extremely sparse IR spectrum, as reproduced in Figure 2. 25 Protonation of aromatic systems occurs through σbonding to one of the peripheral C-atoms. 26 Protonation of C 60 is therefore expected to occur on one of the 60 Catoms, and not centrally over one of the 5-or 6-membered rings. Then, since all C-atoms are symmetrically identical, there exists only one isomer of C 60 H + . The C-atoms do not coincide with one of the five-fold rotation axes of C 60 , so that the proton in C 60 H + is not on a C 5axis and thus breaks the five-fold rotation symmetry. In fact, protonation removes all symmetry of the original C 60 molecule, except for one mirror plane that includes the proton and one of the principal axes of C 60 . Protonation reduces the symmetry to C s , in which vibrational modes belong either to A or A irreps, both giving allowed IR transitions. In sharp contrast to C 60 , one thus expects a rich vibrational spectrum for C 60 H + .
The top panel of Figure 2 presents the IRMPD spectrum of C 60 H + compared with that of C 60 below it. The C 60 H + spectrum indeed features a significantly larger number of vibrational bands due to symmetry lowering to C s . Figure 3 compares the experimental spectrum with computed harmonic spectra at various levels of density functional theory (DFT), as further detailed in the Methods Section. Figure 3 and Supplementary Table 1 clearly testify that the B3LYP/6-311+G(d,p) method outcompetes all others in the prediction of the IR spectrum. The total integrated intensity of all fundamental vibrational modes at this level of theory is 790 km/mol, versus 215 km/mol for C 60 .
Our statement that there exists only a single isomer of C 60 H + ignored the -perhaps naive but intriguingpossibility of endohedral protonation, i.e. inside the cage. The optimized geometry is 193 kJ/mol higher in energy than the exohedrally protonated structure. Moreover, the predicted spectrum is in poorer agreement with the experimental spectrum (see Supplementary Figure 1) and will not be further considered.
Overall, the C 60 H + spectrum features strong bands in the 1150 -1570 cm −1 (6.4 -8.7 µm) range and weaker ones in the long-wavelength range. This general spectral shape resembles that of ionized PAHs, 27 but a detailed inspection reveals significant differences (see Supplementary Figure 2). For C 60 H + , the strong bands have mainly CC stretching character, with some of the vibrations near 1165 cm −1 having additional CH bending character. For PAHs, the CC-stretch bands extend to shorter wavelengths (6.2 µm). 28 The C 60 H + bands in the 7.7 -8.6 µm range overlap with strong PAH cation bands, but bands to the blue (6.5 -7.1 µm) fall in a relatively silent region of the PAH spectrum.
Towards longer wavelengths, main bands in the PAH spectrum are due to CH out-of-plane vibrations at 11. stronger bands at 17.7 and 19.1 µm (565 and 525 cm −1 ) are close to the two main bands of neutral C 60 (17.4 and 18.9 µm), with red-shifts relative to C 60 of 10 and 5 cm −1 , respectively. This is on the order of the bandwidth in the interstellar emission spectra. 1 C 60 H + possesses a single CH-stretch mode with a computed integrated intensity of 27 km/mol, which we were unable to detect, probably due to the lower laser power available in this frequency range. As detailed in the Methods section, we estimate this band to occur at 2850 cm −1 , which deviates from the 100 cm −1 broad CH stretch feature centered at 2910 cm −1 reported for solid hydrogenated fullerenes. 22 On the other hand, this position coincides roughly with aliphatic (not aromatic) CH stretches observed in gas-phase spectra of hydrogenated PAHs. 29 KBr pellet spectra of fulleranes (C 60 H x with x ≈ 30) show broad absorption features between 2800 and 2940 cm −1 . 20 Since the first spectroscopic identification of C 60 , 1 various astronomical objects have been particularly associated with high fullerene abundance. 2,3 Objects with low PAH emission are of particular interest to avoid confusion due to overlapping PAH bands. In Figure 4, the IR emission spectra of the planetary nebulae SMC16 and LMC56 30,31 are overlaid onto our laboratory spectrum of C 60 H + (see also Supplementary Figures 3 and 4). The two laboratory bands in the 17 -20 µm region fall within the bandwidth of the astronomical bands, although they are slightly shifted towards longer wavelengths with respect to the center position of the astronomical features, overlapping nearly exactly with the two bands in neutral C 60 . The asymmetric lineshape of the astronomical features, with a shoulder towards longer wavelengths, may then be speculated to be due to the combined emission from C 60 and C 60 H + . Perhaps most striking is the close correspondence in the 6 -9 µm range: the laboratory spectrum falls closely within the envelope in the observational spectrum, with some of the individual peaks coinciding remarkably well with partially resolved structures on the envelope. In contrast to the two bands at longer wavelength, the entire 6 -9 µm emission feature cannot be explained by C 60 alone (dashed lines), nor does it appear to be due to PAHs (Supplementary Figure 2). The data presented here suggest that a mixture of protonated and neutral C 60 and higher fullerenes may form a plausible set of carriers for these emission features. A contribution from radical cation fullerenes 8,12,32 is also possible, but requires further investigation. Weaker bands in the remainder of the C 60 H + spectrum are also consistent with the observational data. The 10.45 µm band appears visible in the LMC56 spectrum, and is consistent with the SMC16 spectrum, although overlapped by a broad emission plateau attributed to SiC. 30 Note that relative intensities in emission due to radiative cooling are amplified for bands at longer wavelengths with respect to bands at shorter wavelengths, while in photo-dissociation spectroscopy, this trend is opposite (see e.g. Figure 6 in Ref. 23).
The laboratory IR spectrum of C 60 H + overlaid onto emission spectra of objects hypothesized to harbour high abundances of fullerenes appears to be supportive of Kroto's hypothesis that C 60 H + is abundant. However, C 60 H + alone cannot entirely explain the broad but partially resolved emission feature between 6 and 9 µm in these sources, but neither can available IR spectra of C +• 60 . 8,12,32 We therefore speculate that a mixture of these species and their C 70 (and perhaps higher fullerene) analogues, is currently the most attractive explanation. New spectral data for C 70 H + (currently ongoing in our lab) and C +• 70 are anticipated to further resolve this question. Higher spectral and/or spatial resolution astronomical data from future instruments such as the James Web Space Telescope (JWST), may resolve individual contributions more clearly.
METHODS
Experiments were performed in a modified 3-D quadrupole ion trap mass spectrometer (Bruker Amazon
Speed ETD) coupled to the beamline of the Free-Electron Laser for Infrared eXperiments (FELIX). The modifications, including optical access to the trapped ions, have been described in detail in Ref. 33.
To produce protonated C 60 , an atmospheric pressure chemical ionization (APCI) source was employed, which is particularly efficient for less polar species, such as fullerenes 34,35 and polycyclic aromatic hydrocarbons 36,37 . Typically both protonated and radical cation species can be observed from APCI of these species.
C 60 (99+ %) was obtained from MER Corporation (USA). The powder is dissolved in toluene to a concentration of 1 mmol, and this stock solution was further diluted in a 50/50 % methanol-toluene mixture to provide a final C 60 concentration of about 10 µM. This solution is infused at a flow rate of 8 µl/min, nebulized with nitrogen at 3.5 bar and 220 • C, and vaporized at 450 • C . The fine droplets are sprayed towards the corona discharge needle. The potential difference between the end plate and capillary is 4500 V and the corona current is set to 6000 nA. Ions produced in the APCI source enter the vacuum of the mass spectrometer through a capillary, are guided by the ion-transfer optics and are then trapped in the radio-frequency quadrupole ion trap.
After mass isolation of the m/z 721 ions in the trap, the ions are irradiated with tunable IR radiation from FELIX in order to measure the IR multiple-photon dissociation (IRMPD) spectrum of the trapped ions. 23 The resonant multiple-photon absorption of IR photons by the ions leads to an increase of the internal energy and eventually to unimolecular dissociation. Energy randomization via intramolecular vibrational redistribution (IVR) enables excitation of the molecules to the dissociation threshold, with the excitation energy being dissipated in all other vibrational degrees of freedom of the molecule.
After irradiation of the trapped ion cloud, a mass spectrum is recorded and the normalized fragmentation yield (Y ) is derived, taking into account the 35% contribution of 13 C 12 C +· 59 to the signal at m/z 721:
Y = I m720 0.65[I m720 + I m721 ]
(1)
The fragment fluence S, which is proportional to the absorption cross section, is then obtained as
S = − ln(1 − Y )(2)
S is then plotted as a function of the IR wavelength to provide a surrogate for the ion's IR spectrum. For every wavelength point, 5 mass spectra were averaged. S was corrected linearly for wavelength-dependent variations in the laser pulse energy and for the irradiation time (number of applied laser pulses). From repeated scans of the spectrum, we estimate that dissociation yields are reproducible to within about 5% . The FEL wavelength is calibrated with a grating spectrometer with an accuracy of ±0.01 µm. Spectra have been recorded in the vibrational fingerprint region between 6 µm and 25 µm. 38 . The FELIX pulses have an energy of up to 120 mJ and are produced at a 10 Hz repetition rate. Each pulse has a duration of about 7 µs and consists of a series of micropulses spaced by 1 ns. The micropulses are Fourier-transform limited and have a bandwidth of 0.4% of the IR wavelength.
The high dissociation threshold of the protonated fullerene ions makes the application of photo-dissociation spectroscopy challenging, especially in the long wavelength range. In order to facilitate the on-resonance IR laser induced dissociation, the standard helium pressure settings of the ion trap, normally used to efficiently trap the ions, was reduced to its minimum value to minimize collisional deactivation of the ions during IR excitation 33 . The degree of on-resonance dissociation is controlled by varying the laser pulse energy and by irradiating with a variable number of laser pulses (typically between 2 and 10) to prevent excessive depletion of the C 60 H + precursor ion, which would manifest itself as saturation at strong absorption bands in the IR spectrum.
Computational investigations of the C 60 H + system are carried out at the density functional theory (DFT) level using different combinations of functionals and basis sets. The hybrid B3LYP functional, known for its reliable performance in predicting IR spectra, was used in combination with a series of basis sets of increasing size: 3-21G, 4-31G (which has long been the standard in characterization of IR spectra for large PAHs 32,39 ) and 6-311+G(d,p). In addition, the non-hybrid BP86 functional (with the 4-31G basis set) was tested, analogous to strategies used in the computation of PAH IR spectra, 40 as well as the more recent dispersion-corrected functional M06-2X.
All calculations employed the Gaussian16 software package as installed at the Cartesius supercomputer at SurfSARA, Amsterdam. Geometry optimizations were performed with the standard convergence criteria and vibrational spectra were computed within the harmonic oscillator approximation and harmonic frequencies were scaled uniformly as recommended by Ref. 41: 0.9679 for B3LYP/6-311+G(d,p), 0.9752 for B3LYP/4-31G, 0.965 for B3LYP/3-21G, 1.0033 for BP86/4-31G, 0.952 for M06-2X/6-311+G(d,p). The computational cost for the geometry optimization and the harmonic frequency calculation at the B3LYP/6-311+G(d,p) level of theory (2168 primitive gaussians) amounts to approximately 10 days on our 16-core computer cluster. For comparison with experiment, the computed stick spectra were convoluted with a Gaussian lineshape function with a full width at half maximum (FWHM) of 20 cm −1 , which is on the order of bandwidths typically observed in IRMPD spectra and which is believed to be due to a combination of laser bandwidth, rotational envelope and anharmonic broadening and shifting induced by the IRMPD process. 23 To estimate the frequency of the single CH-stretch band in C 60 H + , which we were unable to observe experimentally, we use the B3LYP/6-311+G(d,p) harmonic value and an appropriate frequency scaling factor. It is known that the scaling factor for CH stretch bands typically deviates from that for bands at longer wavelengths. 40 Therefore, we estimate this factor using B3LYP/6-311+G(d,p) computed spectra for pyrene and phenanthrene and comparing the CH-stretch frequencies with accurate gas-phase IR data. 42 A scale factor of 0.9610±0.0005 is established, in agreement with Ref. 40. This gives a value of 2850 cm −1 for the CH stretch mode of C 60 H + .
We note here that the astronomical spectra shown in this paper are emission spectra, resulting from radiative cooling of the species in the ISM after UV excitation and internal conversion. 43 Despite the fact that the IR emission process leaves its imprint on the appearance of the spectrum, as has been addressed in several modelling studies, 44,45 spectral comparisons are often performed with linear vibrational spectra obtained through laboratory transmission spectra or quantum-chemical calculations. A generic redshift of 5 -10 cm −1 or a frequency scaling is often applied to empirically correct for the small deviations. The laboratory spectrum of C 60 H + reported in this contribution was obtained through IRMPD, which leaves its own imprint on the spectrum, which has been modelled in several studies as well. 23,46 Again, deviations from linear vibrational spectra are small enough that spectral analyses are very commonly based on comparisons with linear spectra, as we do here in Figure 3.
The main challenge in correctly modelling IR emission and IRMPD spectra is that this requires parameters for line shifting and broadening as a consequence of anharmonicity, which are typically not available and are difficult, if not impossible, to determine accurately from quantum-chemical investigations. It is then interesting to note that although the processes of IR multiple-photon absorption and IR radiative cooling are different, they are dependent on the same molecular parameters, including these elusive mode-specific anharmonic parameters. To some extent, it may therefore be more appropriate to compare IR emission spectra with IRMPD spectra than with linear IR spectra. In any case, shifts are typically small, as is for instance evidenced by the good correspondence between our experimental IRMPD spectrum and the computed (linear) scaled harmonic spectrum of C 60 H + at the B3LYP/6-311+G(d,p) level.
From these considerations, we suggest that the band positions in our experimental spectrum can be used directly in astronomical searches for C 60 H + . FWHM bandwidths of about 20 cm −1 and slightly smaller at wavelengths near 20 µm are also expected to be close to those in an interstellar emission spectrum. Relative intensities may be somewhat less reliable as a consequence of nonlinearities resulting from the IRMPD method; moreover a correction for the bias towards longer wavelengths in an emission spectrum needs to be taken into account. 23
FIG. 1 .
1Structure of C60H + and experimental aspects of recording its IRMPD spectrum. a) Protonation of C60 occurs on one of the sixty equivalent C-atoms.
FIG. 2 .
23,12.0 and 12.7 µm. The C 60 H + spectrum features weak bands near 9.25, 10.45 and 13.15 µm (1090, 955 and 760 cm −1 ), roughly characterized as ring breathing (9.25 µm) and cage deformation (10.45 and 13.15 µm). Experimental IRMPD spectrum of C60H + (top) compared with a Fourier transform infrared absorption spectrum of a thin film of neutral C60 (bottom) taken from Ref.25. The C60 spectrum shows the four IR allowed modes as well as two very weak bands around 1500 cm −1 , which have been attributed to combination modes. The effects of symmetry breaking on the IR selection rules in C60H + are striking. Band positions in µm are indicated on the IRMPD spectrum. The stick spectra below the experimental data show the theoretical calculations for both molecules at the B3LYP/6-311+G(d,p) level of theory.
FIG. 4 .
4Comparison of the laboratory spectrum of C60H + (pink) with the Spitzer IRS emission spectra from the SMP SMC16 and LMC56 planetary nebulae. 30,31 Dashed lines indicate band positions for neutral C60 at 7.0, 8.5, 17.3 and 18.9 µm.
Cami, J., Bernard-Salas, J., Peeters, E. & Malek, S.E. Detection of C 60 and C 70 in a young planetary nebula. Science 329, 1180-
ACKNOWLEDGEMENTSWe gratefully acknowledge the expert support by the FELIX staff. This work is supported by the European MCSA ITN network "EUROPAH" (grant# 722346) and the Dutch Astrochemistry Network (DAN-II, grant# 648.000.030) of NWO. For the computational work, we acknowledge support by NWO under the "Rekentijd" program (grant# 17603) and the SurfSARA staff.COMPETING INTERESTSThe authors declare no competing interests.
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Wavenumber (cm -1 ). Wavenumber (cm -1 )
The experimental IR spectrum of C60H + compared with DFT computed spectra using different basis sets and functionals. The best match, marked in red, is obtained using the B3LYP functional with the 6-311+G(d,p) basis set. The RMS deviation between band centers in the experimental spectrum and in the scaled and convoluted theoretical spectra amounts to 1.8 cm −1 for this level of theory and is much smaller than for the other computational approaches. see Supplementary Table 1FIG. 3. The experimental IR spectrum of C60H + com- pared with DFT computed spectra using different ba- sis sets and functionals. The best match, marked in red, is obtained using the B3LYP functional with the 6-311+G(d,p) basis set. The RMS deviation between band centers in the experimental spectrum and in the scaled and convoluted theoretical spectra amounts to 1.8 cm −1 for this level of theory and is much smaller than for the other computational approaches (see Supplementary Table 1).
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[
"A NOTE ON THE NUCLEAR DIMENSION OF CUNTZ-PIMSNER C * -ALGEBRAS ASSOCIATED WITH MINIMAL SHIFT SPACES",
"A NOTE ON THE NUCLEAR DIMENSION OF CUNTZ-PIMSNER C * -ALGEBRAS ASSOCIATED WITH MINIMAL SHIFT SPACES"
] |
[
"Zhuofeng He \nSchool of Mathematics and Science\nResearch Center for Operator Algebras\nEast China Normal University\nShanghaiChina\n",
"Sihan Wei \nSchool of Mathematics and Science\nResearch Center for Operator Algebras\nEast China Normal University\nShanghaiChina\n"
] |
[
"School of Mathematics and Science\nResearch Center for Operator Algebras\nEast China Normal University\nShanghaiChina",
"School of Mathematics and Science\nResearch Center for Operator Algebras\nEast China Normal University\nShanghaiChina"
] |
[] |
For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner C * -algebra OX has nuclear dimension 1 when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.
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10.4153/s0008414x22000645
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[
"https://export.arxiv.org/pdf/2112.15519v4.pdf"
] | 245,634,524 |
2112.15519
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636d12d3f9885d15330fa6717afcdc9a3b011cd7
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A NOTE ON THE NUCLEAR DIMENSION OF CUNTZ-PIMSNER C * -ALGEBRAS ASSOCIATED WITH MINIMAL SHIFT SPACES
3 Mar 2023
Zhuofeng He
School of Mathematics and Science
Research Center for Operator Algebras
East China Normal University
ShanghaiChina
Sihan Wei
School of Mathematics and Science
Research Center for Operator Algebras
East China Normal University
ShanghaiChina
A NOTE ON THE NUCLEAR DIMENSION OF CUNTZ-PIMSNER C * -ALGEBRAS ASSOCIATED WITH MINIMAL SHIFT SPACES
3 Mar 2023Cuntz-Pimsner algebras · Nuclear dimension · Minimal shift spaces Mathematics Subject Classification (2010) Primary 46L05, 37A55
For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner C * -algebra OX has nuclear dimension 1 when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.
Introduction
The Cuntz-Pimsner C * -algebra O X is an invariant of conjugacy associated to any shift space X. This interplay between shift spaces and C * -algebras starts from the study of the C * -algebra O A of a two-sided shift of finite type represented by a {0, 1}-matrix A in a canonical way, see [11], in which the associated C * -algebra is originally called a Cuntz-Krieger algebra. In the next thirty years, the C * -algebra O X , to every shift space X, is constructed and studied in [1], [5], [7], [8], [9], [13], [14], [16], [17], [18] by several authors (for example, K. Matsumoto, S. Eilers, T. Carlsen, K. Brix and their collaborators, to name a few), but in different manners for their own uses. We additionally remark that the associated C * -algebra considered in the paper is first defined by T. Carlsen in [7] using a Cuntz-Pimsner construction, which is why we call it a Cuntz-Pimsner C * -algebra, as is also pointed out in [4].
Among these approaches, the cover (X, σ X ), of a one-sided shift space X, is a dynamical system constructed by T. Carlsen in [4], and used to define the O X as the full groupoid C * -algebra of GX . In particular, the reason why Carlsen considers the groupoid C *algebra of the cover but not the shift space X itself is that every such cover defines a dynamical system whose underlying map is a local homeomorphism, while this is not always the case for a one-sided shift. Actually, a one-sided shift on an infinite space is a local homeomorphism if and only if it is of finite type, as in [19].
In [6], it is shown that for every shift space X with the property (*), there is a surjective homomorphism ρ : O X → C(X) ⋊ σ Z, which sends the diagonal subalgebra D X onto the canonical commutative C * -subalgebra C(X), with X the corresponding two-sided shift space of X and σ the natural two-sided shift operation. Besides, if X has the property (**), then
kerρ ∼ = K n X ,
where n X is a positive integer related to the structure of the left special elements in X, namely, the number of right shift tail equivalence classes of X containing a left special element. Consequently, for every minimal shift space X, if it has the property (**), which is equivalent to X having finitely many left special elements, then its Cuntz-Pimsner C * -algebra O X is an extension of a unital simple AT-algebra by a finite direct sum of the compact operators. Also note that this extension makes O X falls into a class of C *algebras considered by H. Lin and H. Su in [15], called the direct limits of generalized Toeplitz algebras.
In [3], K. Brix considers the C * -algebra O α of a one-sided Sturmian shift X α for α an irrational number, by describing the cover of X α . In particular, he proves that the cover X α of X α is a union of the two-sided Sturmian shift X α and a dense orbit consisting of isolated points. The unique dense orbit corresponds to the unique point ω α in X α which has two preimages under the shift operation. This is the first concrete description of covers of non-sofic systems, whereas the cover of a sofic system is a specific class of shifts of finite type. We remark here that the uniqueness of ω α benefits from the well-known fact that X α has the smallest complexity growth for shift spaces with no ultimately periodic points: p X (n) = n + 1 for all n ≥ 1.
There are two corollaries from the concrete description of the cover of a Sturmian system in [3]: one for a reducing of the exact sequence in [6] to its simplest form, that is, O α is an extension of C(X α ) ⋊ σ Z by K; one for the precise value of dynamic asymptotic dimension of the associated groupoid. The latter together with the exact sequence make the O α be of nuclear dimension 1, where the nuclear dimension is a concept that plays a key role in the classification programs for C * -algebras.
In this note, we generalize this interesting approach and show that for every minimal one-sided shift X with finitely many left special elements, the Cuntz-Pimsner algebra O X has nuclear dimension 1. More specifically, with our concrete description, the cover of each such space will be a finite disjoint union: a copy of the corresponding minimal two-sided shift space X (induced from the projection limit of the original one-sided shift), and n X dense orbits, each consisting of isolated points. This also recovers the whole situation of the exact sequence in [6]. We also hope that with this description, more K-information can be read out from the groupoid for many other minimal shifts, such as non-periodic Toeplitz shifts X with lower complexity growth (which is to sufficiently make X have finitely many left special elements, or equivalently, have the property (**)).
Finally, we also want to emphatically point out that there is a large class of minimal shifts for which our results apply, such as those with bounded complexity growth (see Example 3.3.7 for the definition and Proposition 3.3.8 for details). This class of minimal shifts includes minimal Sturmian shifts considered by K. Brix, which are defined to be the minimal shifts associated with irrational rotations; minimal shifts associated with interval exchange transformations, whose complexity functions are known to satisfy p X (n + 1) − p X (n) ≤ d where d is the number of subintervals; minimal shifts constructed from (p, q)-Toeplitz words in [10], where p, q are natural numbers and p|q, whose complexity functions are shown to be linear; or also minimal shifts associated with a class of translations on 2-torus in [2], whose complexity functions satisfy p X (n) = 2n + 1, to name a few.
1.1. Outline of the paper. The paper is organized as follows. Section 2 will provide definitions, including basic notions of one-sided shift spaces, the corresponding two-sided shift spaces and C * -algebras. In Section 3, we recall definitions of past equivalence, right tail equivalence, covers and their properties. A couple of technical preparations will also be presented for the later use. Section 4 is devoted to the main body of the paper, in which we give a concrete description to the cover of a minimal shift with finitely many left special elements. We divide the description into three parts: (i) for isolated points in the cover, see Theorem 4.1.1; (ii) for the surjective factor π X , see Theorem 4.2.4 and Theorem 4.2.5; (iii) for non-isolated points in the cover, see Theorem 4.3.1. Finally, we conclude our main result for the nuclear dimension of O X in Section 5.
1.2. Acknowledgements. The authors were partially supported by Shanghai Key Laboratory of PMMP, Science and Technology Commission of Shanghai Municipality (STCSM), grant #13dz2260400 and a NNSF grant (11531003). The first named author was also supported by Project funded by China Postdoctoral Science Foundation under Grant 2020M681221. We would like to thank members of Research Center for Operator Algebras, for providing online discussions weekly during the difficult period of COVID-19. The second named author would also like to thank his advisor Huaxin Lin from whom he is learning a lot as a doctoral student. The authors thank the anonymous referee for many helpful comments and suggestions as well.
Preliminaries
Throughout the paper, we denote by N the set of nonnegative integers. For a finite set S, we will always use #S to denote its cardinality.
2.1. Shift spaces. Let A = {0, 1}. Endowed with the product topology, the spaces A Z and A N are homeomorphic to the Cantor space, i.e., the totally disconnected compact metric space with no isolated point. Note that A Z and A N can be given the following metrics:
d(x, y) = sup{1/2 N : x k = y k for all 0 ≤ |k| ≤ N − 1}, d(x, y) = sup{1/2 N : x k = y k for all 0 ≤ k ≤ N − 1}.
We use A * and A ∞ to denote the monoid of finite words and the set of infinite one-sided sequences with letters from A, that is,
A * = n≥1 A n ∪ {ǫ}, A ∞ = A N ,
where ǫ is the unique empty word in A * . For a word µ ∈ A * , we use |µ| to denote the length of µ and write |µ| = n if µ ∈ A n . For the empty word, we usually define |ǫ| = 0. Besides, the length of any element µ in A ∞ is defined to be ∞. Let µ ∈ A * and ν ∈ A * ⊔ A ∞ , we say µ occurs in ν, if there exists a ∈ A * and b ∈ A * ⊔ A ∞ such that ν = aµb.
If µ occurs in ν, we also say µ is a factor of ν.
A full-shift is a continuous map σ :
x → σ(x) from A N to A N (or A Z to A Z ) such that (σ(x)) n = x n+1 .
A one-sided (two-sided, respectively) shift space is a nonempty compact σ-invariant subspace X of A N (or A Z respectively) together with the restriction σ| X . Note that by σ-invariant, we mean σ(X) ⊂ X. Any two-sided shift is a homeomorphism, and any onesided shift σ : X → X is injective if and only if X is finite. Throughout the paper we will only consider one-sided shifts on infinite compact spaces. If X is a shift space, x ∈ X and −∞ < n ≤ m < ∞, we define x (n−1,m] = x [n,m+1) = x [n,m] = x n x n+1 · · · x m . We also use x (−∞,m] = x (−∞,m+1) or x [n,∞) = x (n−1,∞) to denote the natural infinite positive and negative parts of x respectively.
For any two-sided shift space X, we use X + to stand for the corresponding one-sided shift space, that is,
X + = {x [0,∞) : x ∈ X}.
If X is a one-sided shift space, then X is used in this paper, to denote the inverse limit of the projective system
X σ ← X σ ← · · · σ ← X σ ← · · · .
Note that X is a two-sided shift space under a canonical identification.
For a shift space X, its language L(X) will play a central role, whose elements are those finite words over A occurring in some x ∈ X. A language uniquely determines a shift space, or in other words, x ∈ X if and only if any factor µ of x is an element of L(X). This fact implies that for any two-sided shift space Y , σ(Y ) = Y , and therefore for any one-sided shift space X, σ(X) = X if and only if X = (X) + . Any topologically transitive one-sided shift (for the definition of topologically transitivity, see Proposition 3.3.2) is automatically surjective since its image is a dense compact subset. Definition 2.1.1. Let X be a one-sided shift space and x ∈ X. We define the forward and backward orbits of x to be Orb + σ (x) = {σ n (x) : n ≥ 0} and Orb − σ (x) = {y ∈ X : ∃n > 0(σ n (y) = x)} respectively, and the whole orbit of x to be Orb σ (x) = Orb + σ (x) ∪ Orb − σ (x). 2.2. C * -algebras and groupoids. Definition 2.2.1 (cf. [21], Definition 2.1). Let A and B be C * -algebras. A * -homomorphism π : A → B is said to have nuclear dimension at most n, denoted dim nuc (π) ≤ n, if for any finite set F ⊂ A and ε > 0, there is a finite-dimensional subalgebra F and completely positive maps ψ : A → F and ϕ : F → B such that ψ is contractive, ϕ is n-decomposable in the sense that we can write
F = F (0) ⊕ F (1) ⊕ · · · F (n)
satisfying ϕ| F (i) is c.p.c order zero for all i, and for every a ∈ F, π(a) − ϕψ(a) < ε.
The nuclear dimension of a C * -algebra A, denoted dim nuc (A), is defined as the nuclear dimension of the identity homomorphism id A .
We now recall the definitions of groupoid and its dynamic asymptotic dimension. Definition 2.2.2 (cf. [20], (3.1)). Let X be a local homeomorphism on a compact Hausdorff space X. We then obtain a dynamical system (X, T ). The corresponding Deaconu-Renault Groupoid is defined to be the set
G X = {(x, m − n, y) ∈ X × Z × X : T m (x) = T n (y), m, n ∈ N}, with the unit space G 0 X = {(x, 0, x) :
x ∈ X} identified with X, range and source maps r(x, n, y) = x and s(x, n, y) = y, and operations (x, n, y)(y, m, z) = (x, n + m, z) and (x, n, y) −1 = (y, −n, x).
By Lemma 2.3 in [4], Lemma 3.1 and Lemma 3.5 in [20], the groupoid G X considered in the paper will all be locally compact, Hausdorff, amenable andétale, where X is the cover of X in the sense of Definition 3.4.3. They are also principal since all such X have no periodic point, as is shown in Section 4.
The
Cuntz-Pimsner C * -algebra O X of a one-sided shift space X is defined to be the (full) groupoid C * -algebra C * (G X ). The diagonal-subalgebra D X is defined to be C( X) ⊂ O X .
Finally we recall the definition of dynamic asymptotic dimension forétale groupoids.
U 0 , U 1 , · · · , U d of G 0 that covers s(K) ∪ r(K) such that for each i, the set {g ∈ K : s(g), r(g) ∈ U i } is contained in a relatively compact subgroupoid of G.
It is known that for a minimal Z-action on a compact space, the associated groupoid has dynamic asymptotic dimension 1, see Theorem 3.1 in [12].
definitions and preparations
From now on, to avoid invalidity or triviality, we only consider infinite one-sided shift space X with σ(X) = X. We use X to denote the associated two-sided shift space.
3.1.
Left special elements and past equivalence.
Definition 3.1.1 (cf. [8], 2.2)
. Let X be a one-sided shift space and z ∈ X. We say that
z is left special if there exists z ′ ∈ X such that z −1 = z ′ −1 and z [0,∞) = z ′ [0,∞) . If z ∈ X is left special, we also say x = z [0,∞) is left special in X.
We use Sp l (X) and Sp l (X) to denote the collections of left special elements in X and X respectively.
We say x ∈ X has a unique past if #(σ k ) −1 ({x}) = 1 for all k ≥ 1. Moreover, we say x ∈ X has a totally unique past if σ n (x) has a unique past for all n ≥ 1.
It is clear from the definition that for any one-sided shift space X with σ(X) = X, a point x ∈ X is left special precisely when x has at least two preimages under σ, that is, #σ −1 ({x}) ≥ 2. Therefore for any such one-sided shift on an infinite space, left special elements always exist, or σ will be injective which implies that X is finite. It is also immediate that x has a totally unique past if and only if x / ∈ Orb σ (ω) for any ω ∈ Sp l (X).
Proposition 3.1.2. Suppose that Sp l (X) contains no periodic point of X. Then #Sp l (X) < ∞ if and only if #Sp l (X) < ∞.
Proof. The map π + : z → z [0,∞) induces a surjective map from Sp l (X) to Sp l (X). Therefore, if Sp l (X) is finite, so is Sp l (X). Now assume that Sp(X) is infinite. If Sp(X) is finite, then we can take x ∈ Sp l (X) with infinitely many preimages in Sp l (X) under π + . Denote this infinite preimage by F . Since A is finite, the Pigeonhole principle ensures the existence of an infinite subset
F 1 ⊂ F such that for every x = y ∈ F 1 , x [−1,∞) = y [−1,∞) . Then we choose n 1 ≤ −1 such that there exists x 1 , y 1 ∈ F 1 with (x 1 ) n 1 −1 = (y 1 ) n 1 −1 but x [n 1 ∞) = y [n 1 ,∞) for all x, y ∈ F 1 .
This means that there is some z 1 ∈ F 1 such that (z 1 ) [n 1 ,∞) ∈ Sp l (X). Similarly, choose an infinite subset F 2 ⊂ F 1 , an integer n 2 ≤ n 1 − 1 with the same property as the first step and a point z 2 ∈ F 2 such that (z 2 ) [n 2 ,∞) ∈ Sp l (X). Repeating this procedure, we have a strictly decreasing sequence of negative integers {n k } k≥1 and an infinite sequence {z k } k≥1 ⊂ Sp l (X) with the following property:
(z k ) [n k ,∞) ∈ Sp l (X) and (z k ) [n k ,∞) = (z k+1 ) [n k ,∞) (k = 1, 2, · · · ).
Note that it follows from the latter condition that (z k ) [n k ,∞) all lie on a single orbit in X. Since Sp l (X) is finite, it has to contain a periodic point, which is a contradiction.
Notation. Let S ⊂ X be a set and l ∈ N. We define S [0,l] to be the set whose elements are the prefixes of x ∈ S of length l + 1.
Definition 3.1.3 (cf. [8], 2.4). Let X be a one-sided shift space and l ≥ 1. For x ∈ X, set P l (x) = {µ ∈ L(X) : |µ| = l, µx ∈ X} = (σ l ) −1 ({x}) [0,l−1] .
For x, y ∈ X, we say x and y are l-past equivalent and write x ∼ l y, if P l (x) = P l (y). In particular, x and y are said to be past equivalent if x ∼ l y for some l ≥ 1.
We call x isolated in past equivalence if there exists l ≥ 1 such that x ∼ l y implies x = y.
If x ∼ l+1 y, then x ∼ k y for all 1 ≤ k ≤ l. Consequently, if x is isolated in l-past equivalent, then x is isolated in k-past equivalent for every k ≥ l.
Lemma 3.1.4. Suppose that x ∈ X has a unique past. Then for every l ≥ 1, there exists N ∈ N such that, whenever y ∈ X with y [0,N ] = x [0,N ] , #P l (y) = 1.
Proof. Assume that there exists l 0 ≥ 1 such that for every n ∈ N, we can always find some y n ∈ X with y n [0,n] = x [0,n] but #P l 0 (y n ) ≥ 2. We are then given a sequence {y n } n≥0 which is easily seen to converge to x as n → ∞.
Note that the alphabet A is finite, we now claim that there exists two distinct words µ, ν in L(X) of length l 0 such that two sequences of natural numbers {n k } k≥0 and {m k } k≥0 can be chosen, satisfying µy n k ∈ X and νy m k ∈ X.
In fact, from the Pigeonhole principle, there is at least one word µ with |µ| = l 0 such that µ can be a prefix of infinitely many y n , say, y n k for k ≥ 1. However, if µ is the unique word with such property, then all others in L(X) with length l 0 can only be prefixes of finitely many of y n , and which means that for some natural number N , y n will only have the unique prefix µ whenever n ≥ N . This is then a contradiction.
Finally, note that since y n → x as n → ∞, every finite word occurring in µx and νx is an element of L(X). This proves µx, νx ∈ X and hence x does not have a unique past.
3.2.
Right tail equivalence and j-maximal elements. Definition 3.2.1 (cf. [8], 2.2 (a slightly different version)). Let x, x ′ ∈ X. The notation x ∼ rte x ′ is used to mean x and x ′ are right tail equivalent, in the sense that there exist Then every j ∈ J X has a unique j-maximal element. In particular, an element ω ∈ Sp l (X) is j-maximal if and only if
M, M ′ ∈ N satisfying σ M (x) = σ M ′ (x ′ ). Set J X = Sp l (X)/ ∼ rte . Let j ∈ J X and ω ∈ j. We say ω is j-maximal if for any ω ′ ∈ j, there is an m ∈ N such that σ m (ω ′ ) = ω.ω ∈ j and σ m (ω) / ∈ Sp l (X) for all m ∈ N \ {0}.
Proof. Let η ∈ j be arbitrary. Since Sp l (X) is finite and contains no periodic point, we
can take K ∈ N such that σ K (η) ∈ Sp l (X) but σ k (η) / ∈ Sp l (X) for all k ≥ K + 1. Denote σ K (η) by ω. We prove the proposition by showing that ω is j-maximal. Let ω ′ ∈ j \ {ω}. Since ω ∼ rte ω ′ , there are M, M ′ ∈ N such that σ M (ω) = σ M ′ (ω ′ ). Take the minimal nonnegative integer M so that there is M ′ ∈ N with σ M (ω) = σ M ′ (ω ′ ). If M > 0, then ω M −1 = w ′ M ′ −1 , which means that ω [M,∞) = ω ′ [M ′ ,∞) is left special. Note that ω [M,∞) = σ k+M (η)
. However, this contradicts to the assumption that σ k (η) / ∈ Sp l (X) for all k ≥ K + 1. Consequently, M = 0, in other words, ω = σ M ′ −1 (ω ′ ). This proves the existence of j-maximal elements.
The uniqueness follows directly from the absence of periodic point in Sp l (X). Finally, the above argument verifies the second assertion at the same time.
Definition 3.2.3. Let X be a one-sided shift space with finitely many left special elements. From now on, for any j ∈ J X , we will always denote the unique j-maximal element by ω j . For every j ∈ J X , define
U j = {ω ∈ j : Orb − σ (ω) ∩ j = ∅}.
Note that for all ω ∈ U j , αω has a unique past whenever αω ∈ X for some α ∈ A.
Lemma 3.2.4. Suppose Sp l (X) is finite and contains no periodic point. For every ω ∈ Sp l (X), there is N ∈ N such that σ n (ω) is isolated in l-past equivalence for all l > n ≥ N .
Proof. Let ω ∈ Sp l (X). From Proposition 3.2.2, let m ∈ N be such that w j = σ m (ω) is j-maximal for some j ∈ J X . Since Sp l (X) is finite, there exists N ′ ∈ N with the following property:
for all y, y ′ ∈ X, y, y ′ ∈ Sp l (X) and y [0,
N ′ ] = y ′ [0,N ′ ] implies y = y ′ .
Let N = N ′ + m. Then σ N (ω) is isolated in N ′ + 1-past equivalence, and therefore for every l > n ≥ N , σ n (ω) is isolated in l-past equivalence as well.
3.3. Property (*) and (**). Definition 3.3.1 (cf. [8], Definition 3.1). A one-sided shift space X has property (*) if for every µ ∈ L(X), there exists x ∈ X such that P |µ| (x) = {µ}. We will also say X has property (*) if X does so.
Proposition 3.3.2. Let X be a one-sided shift space. Suppose that X is topologically transitive, namely, there is a point x 0 ∈ X such that its forward orbit is dense in X. If Sp l (X) is finite and contains no periodic point in X, then X has property (*).
Proof. Actually, the proof is basically the same as that of Example 3.6 in [8] for the minimal case, which goes like follows. Since X is transitive, take x 0 ∈ X with a dense forward orbit, which follows that every word in L(X) occurs in x 0 . Therefore it suffices to show that, for every word µ occurring in x 0 , there exists y 0 such that P |µ| (y 0 ) = {µ}. Now since x 0 is a transitive point, µ appears in x 0 infinitely many times. Consider the intersection Orb + σ (x 0 ) ∩ Sp l (X). Since Sp l (X) is finite and contains no periodic point, this intersection has to be finite, which means that there exists N ≥ 1 such that σ n (x 0 ) / ∈ Sp l (X) for all n ≥ N . This follows that for all n ≥ N , σ n (x 0 ) has only one preimage. Upon taking L > N + |µ| with (x 0 ) [L−|µ|+1,L] = µ, we conclude that σ L+1 (x 0 ) has only one preimage of length |µ|, and which is exactly µ. Example 3.3.5. Every non-regular Toeplitz shifts has property (*), as is shown in [8].
We now prove that this is the case for every non-periodic Toeplitz shift. The same notations as in [22] will be used in the following proposition. Proposition 3.3.6. Let η be a non-periodic Toeplitz sequence. Then X η has property (*).
Proof. Let µ ∈ L(X η ). Without loss of generality, assume η [0,m−1] = µ for m = |µ|. We show that
P |µ| (η [m,∞) ) = {µ}. Suppose µ ′ ∈ L(X η ) with µ ′ η [m,∞) ∈ X η . Then µ ′ η [m,∞)
can be approximated by a sequence σ n k (η). Write µ = µ 1 µ 2 · · · µ |µ| . We then note that η m−1 = µ |µ| .
Consider the p m−1 -skeleton of η, say,η ∈ (A ∪ {∞}) N . Thenη is a periodic sequence with period orbit {η, σ(η), · · · , σ p m−1 −1 (η)}. From the Pigeonhole principle, there is 0 ≤ l ≤ p m−1 − 1 such that there exists infinitely many n k j (j = 1, 2, · · · ) satisfying n k j − (m − 1) ≡ l mod p m−1 for some l ∈ {0, 1, · · · , p m−1 − 1}, which follows (σ n k j (η)) n = η m−1 for all n ∈ (l + m − 1) + p m−1 N, and therefore (µ ′ η) n = η m−1 for all n ∈ (l + m − 1) + p m−1 N. Due to the fact that the p-skeleton of a given Toeplitz sequence is the "maximal" periodic part with the given period,η plays the central role.
Hence, the assumption that µ ′ η [m,∞) and µη [m,∞) has a common right infinite section yields that l = 0. We then conclude that for all n ∈ m − 1 + p m−1 N,
(µ ′ η) n = η m−1 = µ m
and, in particular, µ ′ m = µ m . By repeatedly applying this procedure to m−1, m−2, · · · , 0, we therefore have µ ′ = µ. Example 3.3.7. Let X be a one-sided shift. The complexity function p X is defined on positive integers, which sends every n ≥ 1 to the number of finite words in L(X) of length n. Namely, p X (n) = #{µ ∈ L(X) : |µ| = n}. We say that X has a bounded complexity growth if there exists K > 0 such that p X (n + 1) − p X (n) ≤ K for all n ≥ 1. Then every minimal one-sided shift space with a bounded complexity growth has property (**), as is shown in Proposition 3.3.8.
Proposition 3.3.8. If X is a minimal one-sided shift space with a bounded complexity growth, then X has property (**).
Proof. It suffices to show that X has only finitely many left special elements. Let K ∈ N be a growth bound of X. We actually have #Sp l (X) ≤ K.
If not, then we take K + 1 distinct points {ω 1 , · · · , ω K+1 } ⊂ Sp l (X) and an integer N ∈ N such that the following K + 1 finite words
ω 1 [0,N ] , ω 2 [0,N ] , · · · , ω K+1 [0,N ]
are distinct. Note that these finite words are all of length N + 1 and each of which can be extended to the left in at least two different ways. This immediately follows that p X (N + 2) − p X (N + 1) ≥ K + 1, a contradiction. The proposition then follows.
Covers of one-sided shift spaces.
Definition 3.4.1. We use I to denote the set {(k, l) ∈ N × N : 1 ≤ k ≤ l} and D its diagonal {(k, k) ∈ I : k ≥ 1}. The partial order on I is defined by
(k 1 , l 1 ) (k 2 , l 2 ) ⇔ (k 1 ≤ k 2 ) ∧ (l 1 − k 1 ≤ l 2 − k 2 ).
For the later use, we prove a lemma first.
Lemma 3.4.2. Let F ⊂ I be an infinite set. Then F has an infinite subchain, or in other words, an infinite totally ordered subset of F.
Proof. Take (k 0 , l 0 ) ∈ F satisfying l 0 − k 0 = min{l − k : (k, l) ∈ F}.
Set F 0 = {(k, l) ∈ F : k ≤ k 0 }. Then F 0 is nonempty. If F \ F 0 = ∅, take then (k 1 , l 1 ) ∈ F \ F 0 such that l 1 − k 1 = min{l − k : (k, l) ∈ F \ F 0 } and set F 1 = {(k, l) ∈ F \ F 0 : k ≤ k 1 }. By repeating this step, we are given a sequence of sets {F n } n≥0 . If each of F n is finite, then every F n is nonempty, and this is when {(k n , l n )} becomes an infinite chain. Conversely, if one of F n is infinite, say, F N , then by a partition of F N into the following k N +1 − k N parts:
O N k = {(k ′ , l ′ ) ∈ F N : k ′ = k} (k N < k ≤ k N +1 )
, we see that there exists one of O N k being infinite, which is a chain as well.
As in [4], for every (k, l) ∈ I, we define an equivalence relation
k,l ∼ on X by x k,l ∼ x ′ if x [0,k) = x ′ [0,k) and P l (σ k (x)) = P l (σ k (x ′ )).
We write k [x] l for the k,l ∼ equivalence class of x and k X l the set of k,l ∼ equivalence classes. It is clear that k X l is finite. We then have a projective system
(k 1 ,l 1 ) Q (k 2 ,l 2 ) : k 2 X l 2 ∋ k 2 [x] l 2 → k 1 [x] l 1 ∈ k 1 X l 1 for all (k 1 , l 1 ) (k 2 , l 2 ).
Definition 3.4.3 (cf. [4], Definition 2.1). Let X be a one-sided shift space with σ(X) = X. By the cover X of X, we mean the projective limit lim
←− ( k X l , (k,l) Q (k ′ ,l ′ ) ). The shift operation σ X on X is defined so that k σ X (x) l = k [σ( k+1xl )] l where k+1xl is a representative of a k+1,l ∼ -equivalence relation class inx.
The following sets give a base for the topology of X: U (z, k, l) = {x ∈ X : z k,l ∼ kxl } for z ∈ X and (k, l) ∈ I. It is known that σ X is a surjective local homeomorphism, see [4] for details. Definition 3.4.4 (cf. [4], Definition 2.1). Let π X : X → X to be the map which sends eachx ∈ X to a point x = π(x) so that x [0,k) are determined uniquely by ( kxl ) [0,k) for every (k, l) ∈ I. Define ı X : X → X by k ı X (x) l = k [x] l for every (k, l) ∈ I.
In fact, π X is a continuous surjective factor map from ( X, σ X ) to (X, σ) and ı X is an injective map (not necessarily continuous) such that π X • ı X = id X .
Before the sequel, we recall the following lemmas. . Let X be a one-sided shift space. Any isolated point in the cover X is contained in the image of ı X and each fibre π −1 X ({x}) contains at most one isolated point. In particular, if x ∈ X is isolated in past equivalence, then ı X (x) is an isolated point in X.
Lemma 3.4.6 (cf. [3], Lemma 4.4). Let X be a one-sided shift space. Suppose that x ∈ X has a unique past, then anyx ∈ π −1 X ({x}) also has a unique past. We also note the following lemmas for the later use.
Lemma 3.4.7. Let X be a one-sided shift space with property (**). Suppose that ω, ω ′ ∈ X are left special elements, {(k m , l m )} m≥1 is an infinite sequence in I where {k m } m≥1 is an unbounded sequence with k m < k m+1 for all m ≥ 1, and {k m+1 − k m } m≥1 are distinct. Assume that to every m ≥ 1, an integer 0 ≤ n (km,lm) < l m is associated such that
P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) σ n (k m+1 ,l m+1 ) (ω ′ ))
for all m ≥ 1. Then the sequence {n (km,lm) } m≥1 is unbounded.
Proof. Assume that {n (km,lm) } m≥1 is bounded. Then there exists an infinite subsequence {(k m i , l m i )} i≥1 and an n ∈ N such that n (km i ,lm i ) = n for all i.
By passing to the subsequence {(k m i , l m i )} i≥1 and checking the equality of P lm i , we assume, without loss of generality, that n (km,lm) = n for some n and all m ≥ 1. Note that 0 ≤ n < l m . Now that we have
P lm (σ n (ω ′ )) = P lm (ω [km,k m+1 ) σ n (ω ′ ))
for all m ≥ 1. The condition k m < k m+1 together with the property (**) then infer that
σ n (ω ′ ) = ω [km,k m+1 ) σ n (ω ′ ) for all m ≥ 1. Also note that since {k m+1 −k m } m≥1 are distinct, ω [km,k m+1 ] are all distinct.
The condition 0 ≤ n < l m follows that #P lm (σ n (ω ′ )) ≥ 2, and therefore #P lm (ω [km,k m+1 ) σ n (ω ′ )) ≥ 2 for all m ≥ 1. This immediately tells us that every ω [km,k m+1 ) σ n (ω ′ ) lies on the forward orbit of some left special element. However, since w [km,k m+1 ] σ n (ω ′ ) are distinct points lying in the backward orbit of σ n (ω ′ ), we will then have infinitely many distinct special left elements, which contradicts to the assumption that X has property (**).
Lemma 3.4.8. Let X be a minimal one-sided shift with property (**) and x ∈ X. If x has a totally unique past, then ı X (x) ∈ X is not isolated. Consequently, π −1 X ({x}) contains no isolated point for any x having a totally unique past.
Proof. Let z ∈ X and (k, l) ∈ I be so that ı X (x) ∈ U (z, k, l). Then z
k,l ∼ x. Denote P l (σ k (x)) = {µx [0,k) } with |µ| = l − k. It suffices to find an elementx in X such that z k,l ∼ kxl but k ′x l ′ k ′ ,l ′ ≁ x for some (k ′ , l ′ ) ∈ I.
Let ω j be an arbitrary j-maximal element for some j ∈ J X . Since X is minimal, then µx [0,k) occurs infinitely many times in the forward orbit of ω j . Take L ∈ N sufficiently large so that
(σ L (ω j )) [0,k) = µx [0,k) . Setx = ı X (σ L+l−k (ω j )). Then (σ L+l−k (ω j )) [0,k) = x [0,k) and P l (σ L+l (ω j )) = {µx [0,k) }.
This verifies k x l k,l ∼ z. However, it is clear that k ′x l ′ k ′ ,l ′ ≁ x for some sufficiently large l ′ , since σ L (ω j ) sits in the forward orbit of a left special element.
The description of covers
In this section, X is always assumed to be a one-sided minimal shift space over the alphabet A = {0, 1}, having property (**). We will, as before, still use X to denote the corresponding two-sided shift space. Note that X is also minimal. We also remark that similar conclusions can be drawn for an arbitrary finite alphabet A, but we instead restrict in this paper to the binary shifts for the simplicity of formulations.
First we point out the isolated points in X.
Isolated points in cover.
Theorem 4.1.1. The set of isolated points in X is dense in X, which is exactly
ı j∈J X Orb σ (ω j ) ,
where ω j 's are the unique j-maximal elements.
Proof. Write I( X) for the set of isolated points in X. We know from Lemma 3.4.8 that, every isolated point of X has the form ı X (x) for some x ∈ X which does not have a totally unique past. This means x ∈ Orb σ (ω) for some ω ∈ Sp l (X). Assume now ω ∈ j 0 where j 0 is one of right tail equivalence classes. By the definition of j-maximal elements, we immediately see that x ∈ Orb σ (ω j 0 ). This implies the inclusion
I( X) ⊂ ı j∈J X Orb σ (ω j ) .
Conversely, according to the proof of Lemma 3.2.4, for every j ∈ J X , there is a point z ∈ Orb + σ (ω j ) isolated in past equivalence, which makes, from Lemma 3.4.5, ı X (z) an isolated point in X. On the other hand, recall that as a local homeomorphism, σ X preserves isolatedness and non-isolatedness, which follows that every point in ı(Orb σ (ω j )) is isolated in X. Since j is arbitrary,
ı j∈J X Orb σ (ω j ) ⊂ I( X).
This proves the second assertion. We now show that the set of isolated points in X is dense. Let z ∈ X and (k, l) ∈ I. To show the density, it suffices to take x ∈ Orb(ω j ) such that z k,l ∼ x for some j ∈ J X . We may assume z / ∈ Orb σ (ω j ) for all j ∈ J X . The argument of the existence of such x is then exactly the same as that of Lemma 3.4.8.
4.2.
The surjective factor π X . Recall that for every j ∈ J X , the set U j is defined to be U j = {ω ∈ j : Orb − σ (ω) ∩ j = ∅}. Remark 4.2.1. Elements in U j are called adjusted and j-maximal elements ω j are called cofinal in [8]. It is easy to see that U j is nonempty for every j ∈ J X . Lemma 4.2.2. For every j ∈ J X and ω ∈ U j , #π −1 X ({ω}) = 3. Proof. We first show that there are at least three distinct elements in π −1 X ({ω}). The construction below of these three preimages is similar to that of [3].
For every α ∈ {0, 1} and (k, l) ∈ I \ D, let µ α (k,l) ∈ L(X) with |µ α (k,l) | = l − k − 1 be such that µ α (k,l) αω ∈ X. Note that such finite word µ α (k,l) is unique because ω ∈ U j . Now since X has property (**), we can take x α (k,l) ∈ X satisfying
P l (x α (k,l) ) = {µ α (k,l) αω [0,k) } (α = 0, 1). Define k x α l = ω [0,k) x α (k,l) . Note that for α = 0, 1, we have ω k,l ≁ k x α l and k x 0 l k,l ≁ k x 1 l . We now define representatives on each (k, k) ∈ D. Take x α (k,k) ∈ X with P k (x α (k,k) ) = {ω [0,k) }. Let k x α k = ω [0,k) x α (k,k) . Now for α = 0, 1, setx α ∈ X satisfying k (x α ) l = [ k x α l ]
. It is clear that π(x α ) = ω and {ı(ω),x 0 ,x 1 } are three distinct elements. It is now enough to show thatx α are well-defined. We will only verify the case for α = 0, since the other one is exactly the same. For the simplicity of notations, we drop all the superscripts and abbreviatex 0 tox, for instance.
(i) Let (k 1 , l 1 ) (k 2 , l 2 ) be indices in I \ D. It is trivial that
( k 1 [x] l 1 ) [0,k 1 ) = ω [0,k 1 ) = ( k 2 [x] l 2 ) [0,k 1 ) , so it remains to show that {µ (k 1 ,l 1 ) 0ω [0,k 1 ) } = P l 1 (x (k 1 ,l 1 ) ) = P l 1 (ω [k 1 ,k 2 ) x (k 2 ,l 2 ) ).
For every ν ∈ P l 1 (ω [k 1 ,k 2 ) x (k 2 ,l 2 ) ), νω [k 1 ,k 2 ) ∈ P k 2 −k 1 +l 1 (x (k 2 ,l 2 ) ), and since l 1 + k 2 − k 1 ≤ l 2 , νω [k 1 ,k 2 ) is the suffix of an element in P l 2 (x (k 2 ,l 2 ) ), which follows ν = ν ′ 0ω [0,k 1 ) where ν ′ is the suffix of µ (k 2 ,l 2 ) with length l 1 − k 1 − 1. However, as 0ω has a unique past, ν ′ = µ (k 1 ,l 1 ) .
(ii) Let (k 1 , k 1 ) ∈ D and (k 2 , l 2 ) ∈ I \ D with k 1 ≤ k 2 . We shall confirm that l 2 ) ). The inequality k 2 ≤ l 2 infers that νω [k 1 ,k 2 ) is the suffix of some element in P l 2 (x (k 2 ,l 2 ) ) = {µ (k 2 ,l 2 ) 0ω [0,k 2 ) }, which has to be ω [0,k 2 ) . Therefore, ν = ω [0,k 1 ) .
{ω [0,k 1 ) } = P k 1 (ω [k 1 ,k 2 ) x (k 2 ,l 2 ) ). Since for every ν ∈ P k 1 (ω [k 1 ,k 2 ) x (k 2 ,l 2 ) , νω [k 1 ,k 2 ) ∈ P k 2 (x (k 2 ,
(iii) The case for (k 1 , k 1 ), (k 2 , k 2 ) ∈ D where k 1 ≤ k 2 is quite similar to the case (ii) and hence we omit the verification. Now that we have shown #π −1 X ({ω}) ≥ 3. We next prove that these are exactly the only three elements on this fibre.
Takex ∈ π −1 X (ω) and write kxl = k [ω [0,k) x (k,l) ] l for some x (k,l) ∈ X.
Claim. If there exists (k 0 , l 0 ) ∈ I such that #P l 0 (x (k 0 ,l 0 ) ) = 1, thenx ∈ {x 0 ,x 1 }. This is immediate. Suppose P l 0 (x (k 0 ,l 0 ) ) = {µ0ω [0,k 0 ) }, then every x (k ′ ,l ′ ) with (k ′ , l ′ ) (k 0 , l 0 ) are determined. Also note that for all (k ′ , l ′ ) with (k 0 , l 0 ) (k ′ , l ′ ) are also unique determined because 0ω has a unique past. Then x (k,l) are all determined, because I is directed in the sense that given any two points (k ′ , l ′ ), (k ′′ , l ′′ ) ∈ I, we can always find (k ′′′ , l ′′′ ) ∈ I with (k ′ , l ′ ) (k ′′′ , l ′′′ ) and (k ′′ , l ′′ ) (k ′′′ , l ′′′ ). Now assume that #P l (x (k,l) ) ≥ 2 for all (k, l) ∈ I. We then show thatx = ı X (ω), which will finish the proof. Fix any (k 0 , l 0 ) ∈ I. Note that this leads to the fact that, for every (k, l) ∈ I with (k 0 , l 0 ) (k, l), there exists ω (k,l) ∈ Sp l (X) and integers 0 ≤ n (k,l) ≤ l − 1 such that
x (k,l) = σ n (k,l) (ω (k,l) ). The finiteness of Sp l (X) implies that there is ω ′ ∈ Sp l (X) and infinitely many (k m , l m ) ∈ I satisfying (k 0 , l 0 ) (k m , l m ), k m < k m+1 for all m ≥ 1, lim m→∞ k m = ∞, and
x (km,lm) = σ n (km,lm) (ω ′ ), for all m ≥ 1.
Upon passing to a subsequence, we may assume, according to Lemma 3.4.2, that {(k m , l m )} m≥1 is a chain in the sense that (k m , l m ) (k m ′ , l m ′ ) whenever m ≥ m ′ . Besides, we may assume that {k m+1 − k m : m ≥ 1} are distinct. By the definition of cover, we then have
P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) σ n (k m+1 ,l m+1 ) (ω ′ )).
Now Lemma 3.4.7 applies, indicating that n (km,lm) is unbounded. Hence we may assume, without loss of generality, that n (km,lm) → ∞ as m → ∞.
On the other hand, from Lemma 3.2.4, we can take an N ∈ N such that σ n (ω ′ ) is isolated in l past equivalence whenever l > n ≥ N . Choose M ∈ N such that n (km,lm) > N whenever m > M . This follows that
x (km,lm) = σ n (km,lm) (ω ′ ) is l − isolated whenever m > M, l > n (km,lm) . In particular, x (km,lm) is l m -isolated because l m > n (km,lm) . Then, we know from P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) x (k m+1 ,l m+1 ) ) that x (km,lm) = σ n (km,lm) (ω ′ ) = ω [km,k m+1 ) x (k m+1 ,l m+1 ) for all m > M . Finally, since k 1 < k 2 < · · · < k m < k m+1 < · · · and lim m→∞ k m = ∞, we conclude that for all m > M ,
x (km,lm) = σ km (ω), and therefore kmxlm = km [ω] lm . Recall that (k m , l m ) (k 0 , l 0 ) for every m, we then have k 0x l 0 = k 0 [ω] l 0 . Finally, as the above discussion can be applied to every (k 0 , l 0 ) ∈ I, x = ı X (ω), the lemma follows.
Definition 4.2.3. Let x ∈ X and {z m } m≤0 be a sequence in X. We say {z m } m≤0 is a directed path terminating at x if z 0 = x and σ(z m−1 ) = z m for all m ≤ 0. It is not hard to see that for every one-sided shift space X with #Sp l (X) < ∞ and every x ∈ X, the number of directed path in X terminating at x is finite. We denote this number by d(x).
It immediately follows that for any fixed non-maximal element ω ∈ Sp l (X) and m 0 = min{m > 0 : σ m (ω) ∈ Sp l (X)}, we have that d(ω) = d(σ(ω)) = · · · = d(σ m 0 −1 (ω)) and d(ω) =
ω ′ ∈σ −1 ({ω}) d(ω ′ ).
Since π X is a factor, σ s (z) = σ s (ω). If z = ω, then there exists 1 ≤ j ≤ s < m 0 (ω) such that σ j (ω) is left special, but this contradicts to the minimality of m 0 . Therefore, z = ω and hencez ∈ {ı X (ω),x 1 ,x 2 , · · · ,x d(ω) }. This shows that there are no more elements in the fibre. Noting that π −1 X ({σ s (ω)}) = d(ω) + 1 = d(σ s (ω)) + 1, (ii) and (iii) follow as desired.
For the last part of the subsection, we consider those z ∈ X having totally unique past.
Theorem 4.2.5. Let z ∈ X \ j∈J X Orb σ (ω j ). Then #π −1 X (z) = 1.
Proof. Letz ∈ X with π X (z) = z. Let us show thatz = ı X (z).
Write kzl = k [z [0,k) z (k,l) ] l .
We turn to prove that z k,l ∼ z [0,k) z (k,l) for all (k, l) ∈ I. Obviously they have the same initial sections of length k. Therefore, it remains to verify that P l (z [k,∞) ) = P l (z (k,l) ). Write P l (z [k,∞) ) = {µz [0,k) } where µ is the unique prefix of length l − k. We turn to show the following claims to finish the proof.
Claim 1. µz [0,k) z (k,l) ∈ X: Since z has a unique past, so doesz. Take the uniquẽ z ′ ∈ X so that σ X (z ′ ) =z. Note that this implies
σ l−k π X (z ′ ) = π X σ l−k X (z ′ ) = π X (z) = z,
and hence π X (z ′ ) = µz. Denotez ′ = k [ kz
k σ l−k X (z ′ ) l = kzl = k [σ l−k ( lz ′ l )] l ,
which tells us z [0,k) z (k,l) = kzl k,l ∼ σ l−k ( lz ′ l ). Therefore, µz [0,k) ∈ P l (σ l ( lz ′ l )) = P l (z (k,l) ). Claim 2. #P l (z (k,l) ) = 1: Since z has a totally unique past, σ k (z) has a unique past. By Lemma 3.1.4, we can choose N 1 ∈ N with the following property: whenever y ∈ X with y [0,N 1 ] = σ k (z) [0,N 1 ] , #P l (y) = 1.
Set N = N 1 + k + 1. Since (N, l + N − k) (k, l), we have Nzl+N −k k,l ∼ kzl ,
which follows that P l (σ k ( Nzl+N −k )) = P l (z (k,l) ).
However, since σ k ( Nzl+N −k ) = z k z k+1 · · · z N −1z(N,l+N −k) , it has a prefix of length N −k = N 1 + 1, equal to σ k (z) [0,N 1 ] . Therefore, by how we choose N 1 , we conclude that #P l (z (k,l) ) = #P l (σ k ( Nzl+N −k )) = 1.
This completes the proof.
4.3.
Non-isolated points in the cover.
Theorem 4.3.1. LetΛ X = X \ j∈J X Orb σ X (ı X (ω j ))
be the non-isolated points in the cover. ThenΛ X ∼ = X, i.e., there is a canonical conjugacy from (Λ X , σ X ) to (X, σ), where X is the two-sided shift associated with X.
Proof. Note that since the set of isolated points is open,Λ X is closed and invariant. We first show that every element ofΛ X has a unique past. For this, by Lemma 3.4.6, we only need to verify that, for any fixed k > 0, ω ∈ Sp l (X) andz ∈ π −1 X (σ k (ω)) \ {ı X (σ k (ω))},z has a unique past.
Denote z = σ k (ω). Then π X (z) = z. Define
m z = min{m > 0 : ∃ ω ′ ∈ Sp l (X) (σ m (ω ′ ) = z)}.
Note that because ω is left special and z = σ k (ω), m z is well-defined. Then we claim that the sets
E i = {ỹ ∈X : σ iX (ỹ) =z}
are singletons for i = 1, 2, · · · , m z . In fact, for i = 1, if there areỹ 1 ,ỹ 2 ∈ E 1 , then
σ • π X (ỹ 1 ) = π X • σX(ỹ 1 ) = π X • σX(ỹ 2 ) = σ • π X (ỹ 2 ),
and therefore π X (ỹ 1 ) = π X (ỹ 2 ) ∈ σ −1 ({z}). This meansỹ 1 ,ỹ 2 ∈ π −1 X (σ −1 ({z})) with σX(ỹ 1 ) = σX(ỹ 2 ). However, according to the final paragraph of Theorem 4.2.4 and the minimality of m z , the restriction of σX from π −1 X (σ −1 ({z})) to π −1 X ({z}) is injective and onto, which means thatỹ 1 =ỹ 2 , and therefore E 1 is a singleton. Note that by the minimality of m z , we can clearly apply the same argument to the case i = 2, 3, · · · , m z .
For i = m z + 1, from the construction in Lemma 4.2.2, there is a unique element corresponding to the prefix 0 or 1. Therefore E mz +1 is a singleton as well. Repeating this procedure and noting that there exists K > 0 such that x has a unique past whenever k ≥ K and x ∈ σ −k (z), we conclude thatz has a unique past.
On the other hand, it is quite clear that σX is a surjective map restricted onΛ X , and from which we can then conclude that σX is a homeomorphism fromΛ X ontoΛ X . Now we construct a map from X toΛ X . This is a natural construction which is similar to that of the Sturmian case. Specifically, (i) If x ∈ X such that σ k (x [0,∞) ) has a unique past for all k ≥ 0, we set
Φ(x) =x = ı X (x [0,∞) ),
where ı X (x [0,∞) ) is the unique element in π −1 X (x [0,∞) ) by Theorem 4.2.5. Explicitly, for every x = (x n ) n∈Z ∈ X, since X is an inverse limit, we regard x as a sequence of right infinite words in X:
x = {x [−n,∞) } n≥1 .
Then we have σX
• Φ(x) = σX • ı X (x [0,∞) ) = ı X (x [1,∞) ) = Φ • σ(x).
(ii) If x ∈ X such that there is some k ≥ 0 making σ k (x [0,∞) ) don't have a unique past, since X has property (**), we can choose K ≥ 0 such that every element in Orb + σ (σ K (x))
is not left special anymore. Therefore, it is enough to determine Φ(σ K (x)). By abuse of notation, we denote σ K (x) by x. Let k be the smallest natural number such that
x [−k,∞) ∈ Sp l (X).
Then there is a unique element in π −1 X (x [−k,∞) ) corresponding to the prefix x −k−1 ∈ {0, 1}. Now by applying this argument to x [−(k+1),∞) , together with the assumption that X only has finitely many of special elements, we get a unique element Φ(x) inΛ X . Similar to the case (i), It is straightforward to verify that σX • Φ(x) = Φ • σ(x) holds naturally.
Finally, to see that Φ is a homeomorphism, we first notice that since the topology on X andΛ X are both generated by the cylinder sets and that Φ doesn't change any finite prefix of any right infinite word in the sequence x = {x [−n,∞) }, Φ is clearly continuous and injective. For the surjectivity of Φ, letz ∈Λ X . Sincez has a unique past, σ −ñ X (z) is well-defined for all n ≥ 1. Then define a sequence z in X by
z = {π X • σ −ñ X (z)} n≥1 Since σ • π X • σ −n−1 X (z) = π X • σX • σ −n−1 X (z) = π X • σ −ñ X (z)
, we see that z corresponds to an element in the projective system X σ ← X and defines a point in X. From the construction above, we immediately have Φ(z) =z. This verifies the surjectivity of Φ. Finally, since X is compact andΛ X is Hausdorff, Φ is a homeomorphism.
We now close Sect. 4 by summarizing in the following theorem the main results in the section. Theorem 4.3.2. Let (X, σ) be a one-sided minimal shift over {0, 1} on an infinite space X with finitely many left special elements. Let X be its cover. Then we have the following.
(1) The set I( X) of isolated points in X is a disjoint union:
I( X) = j∈J X ı(Orb σ (ω j ))
which forms a dense open subset of X;
(2) The subsystem ( X \ I( X), σ X | X\I( X) ) on the set of non-isolated points is invertible and conjugate to the canonical two-sided shift space X of X;
(3) For every x ∈ X \ j∈J X Orb σ (ω j ), #π −1 X (x) = 1. Moreover, for every x ∈ j∈J X Orb σ (ω j ),
#π −1 X (x) = d(x) + 1, where d(x)
is the number of directed path in X terminating at x. Remark 4.3.3. Last but not least, since we only consider systems with alphabet A = {0, 1} in order to simplify our proofs, we would also like to mention how our results depend on the number of symbols. In fact, all but (3) in Theorem 4.3.2 hold for systems over any finite alphabet A. In fact, Lemma 4.2.2 may fail even for A = {0, 1, 2}. This is because for a left special element, say ω, we don't know exactly what is the preimage of ω, for it could be any of {0ω, 1ω}, {1ω, 2ω}, {0ω, 2ω} or {0ω, 1ω, 2ω}. On the other hand, we see that the proof of (1) and (2) have nothing to do with the number of symbols.
A commutative diagram and the nuclear dimension
We conclude our main result in this short section, concerning the nuclear dimension of the Cuntz-Pimsner C * -algebra O X associated to every minimal one-sided shift over an infinite space X with finitely many special elements.
U i } is a finite set for i = 0, 1. This implies that the groupoid generated by {g ∈ K : s(g), r(g) ∈ U i } is a relatively compact subgroupoid for i = 0, 1. This shows G X has dynamic asymptotic dimension 1. Now from Theorem 8.6 of [12], dim nuc (O X ) ≤ 1.
However, by the exact sequence and Proposition 2.9 of [21],
1 = max{dim nuc (K n X ), dim nuc (C(X) ⋊ σ Z)} ≤ dim nuc (O X )
≤ dim nuc (K n X ) + dim nuc (C(X) ⋊ σ Z) + 1 = 2.
We then conclude that dim nuc (O X ) = 1. This finishes the proof.
Remark 5.0.2. An alternative argument for the last part of Theorem 5.0.1 would just be that, as a C * -algebra of a groupoid associated with a minimal system over an infinite compact metric space, O X is not AF. This follows immediately that dim nuc (O X ) ≥ 1.
Suppose that Sp l (X) is finite and contains no periodic point of X.
Definition 3.3.3 (cf.[8], Definition 3.2). Let X be a one-sided shift space with property (*). If, in addition, Sp l (X) is finite and contains no periodic point in X, then we say X has property (**)The Proposition 3.3.2 together with Proposition 3.1.2 implies the following corollary.
Corollary 3.3.4. A transitive one-sided shift space X has property (**) if and only ifSp l (X) is finite and contains no periodic point. In particular, if X is minimal, then X has property (**) exactly when Sp l (X) is finite.
Lemma 3.4. 5
5(cf. [3], Lemma 4.2)
.
There are precisely n X distinct discrete orbits in X each of which forms an open invariant subspace X, where n X = #J X is the number of right tail equivalence classes in Sp l (X). The union of these isolated orbits forms an open dense subset in X.
′ l ] l . We then have ( lz′ l ) [0,l) = µz [0,k) . On the other hand,
Theorem 4.2.4. For every x ∈ j∈J X Orb σ (ω j ), #π −1 X ({x}) = d(x) + 1. Proof. First, we verify the situation for which x has a unique past, that is, d(x) = 1. This could happen for example when x lies in the backward orbit of some ω ∈ U j . Then it is clear that either 0x ∈ X or 1x ∈ X. In any case, the procedure of Lemma 4.2.2 defines a non-isolated point in π −1 X ({x}) and an exactly same argument as in Lemma 4.2.2 shows that #π −1 X ({x}) = 2 = d(x) + 1. For the case when x ∈ Sp l (X), according to the definition of the integer-valued function d, there are at most d(x) finite prefixeswith |µ i (k,l) | = l − k for sufficiently large (k, l) ∈ I such that µ i (k,l) x ∈ X and moreover, for each pair of µ i (k,l) x and µ j (k,l) x (i = j) and every n ∈ N, µ i (k,l) x = σ n (µ j (k,l) x) and µ j (k,l) x = σ n (µ i (k,l) x). Since X has property ( * * ) as assumed, we can takeis not one of the points we constructed above, thenx = ı X (x). This proves that for any left special element x in the whole orbit of any maximal left special element ω j ,Finally, let us consider those elements x ∈ j∈J X Orb σ (ω j ) which are not left special. This divides into the following three cases:(i) x lies in the backward orbit of some ω ∈ Sp l (X) having a unique past;(ii) x lies in the forward orbit of some maximal element ω j for j ∈ J X ; (iii) there are distinct left special elements ω, ω ′ such that ω lies in the backward orbit of x and ω ′ lies in the forward orbit of x.Note that the case (i) has already been included in the the first paragraph above. For (ii) and (iii), let ω be a left special element and m 0 (ω) = min{m > 0 : σ m (ω) ∈ Sp l (X)}.Without loss of generality, we may say m 0 (ω) = ∞ if ω is a maximal element. We now reach (ii) and (iii) by showing thatFor this, write π −1 X ({ω}) = {ı X (ω),x 1 ,x 2 , · · · ,x d(ω) }, wherex 1 ,x 2 , · · · ,x d(ω) are the elements in π −1 X ({ω}) constructed above. We then have π −1 X ({σ s (ω)}) = {σ s X (ı(ω)), σ s X (x 1 ), σ s X (x 2 ), · · · , σ s X (x d(ω) )}, It is clear that these d(ω) + 1 elements are distinct and in the preimage of σ s (ω). Therefore it suffices to show that there are no more elements in the fibre. Suppose thatỹ ∈ π −1 X ({σ s (ω)}). Since σ X is surjective, there existsz such that σ s X (z) =ỹ. Take z = π X (z). the abelian C * -algebra of the space of n X discrete orbits and the commutativity of the diagram is induced by π X . From the description of the cover X, the unit space of its groupoid G X decomposes into two parts:In particular, the groupoid restricted toΛ X is isomorphic to X ⋊ σ Z by Theorem 4.3.1, whose C * -algebra is * -isomorphic to the crossed product C(X) ⋊ σ Z, and the groupoid restricted to the open subset j∈J X ı X (Orb σ (ω j )) is the sum of full equivalence relations restricted on each discrete orbit ı X (Orb σ (ω j )) (j ∈ J X ), whose C * -algebra is * -isomorphic to the direct sum K n X . Then the exactness of the second row follows from Proposition 4.3.2 in[20]. For the nuclear dimension of O X , we first claim thatClaim. G X has dynamic asymptotic dimension 1.To see this, let K be an open relative compact subset of G X . Denote the groupoid restricted onΛ X = X by GΛ. It has already been verified that GΛ is a minimal reversible groupoid, or in other words, a groupoid of an invertible minimal action on an infinite compact space, which follows that it has asymptotic dimension 1.Then there are open subsetsŨ 0 ,Ũ 1 of its unit space G 0 Λ that cover s(K ∩ GΛ) ∪ r(K ∩ GΛ), and the set {g ∈ K ∩ GΛ : s(g), r(g) ∈Ũ i } is contained in a relatively compact subgroupoid of GΛ for i = 0, 1. LetIt is clear that U i are open and cover s(K)∪r(K). On the other hand, since the right most one is an discrete open set and K is relatively compact, the set {g ∈ K \ GΛ : s(g), r(g) ∈
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|
[] |
[
"A generic physics-informed neural network-based framework for reliability assessment of multi-state systems",
"A generic physics-informed neural network-based framework for reliability assessment of multi-state systems"
] |
[
"Taotao Zhou \nCenter for Risk and Reliability\nUniversity of Maryland\nCollege ParkMDUSA\n",
"Xiaoge Zhang [email protected] \nDepartment of Industrial and Systems Engineering\nThe Hong Kong Polytechnic University\nKowloon, Hong Kong\n",
"Enrique Lopez Droguett \nDepartment of Civil and Environmental Engineering\nUniversity of California\nLos AngelesCAUSA\n\nGarrick Institute for the Risk Sciences\nUniversity of California\nLos AngelesCAUSA\n",
"Ali Mosleh \nGarrick Institute for the Risk Sciences\nUniversity of California\nLos AngelesCAUSA\n"
] |
[
"Center for Risk and Reliability\nUniversity of Maryland\nCollege ParkMDUSA",
"Department of Industrial and Systems Engineering\nThe Hong Kong Polytechnic University\nKowloon, Hong Kong",
"Department of Civil and Environmental Engineering\nUniversity of California\nLos AngelesCAUSA",
"Garrick Institute for the Risk Sciences\nUniversity of California\nLos AngelesCAUSA",
"Garrick Institute for the Risk Sciences\nUniversity of California\nLos AngelesCAUSA"
] |
[] |
In this paper, we leverage the recent advances in physics-informed neural network (PINN) and develop a generic PINN-based framework to assess the reliability of multi-state systems (MSSs). The proposed methodology consists of two major steps. In the first step, we recast the reliability assessment of MSS as a machine learning problem using the framework of PINN. A feedforward neural network with two individual loss groups are constructed to encode the initial condition and state transitions governed by ordinary differential equations (ODEs) in MSS. Next, we tackle the problem of high imbalance in the magnitude of the back-propagated gradients in PINN from a multi-task learning perspective. Particularly, we treat each element in the loss function as an individual task, and adopt a gradient surgery approach named projecting conflicting gradients (PCGrad), where a task's gradient is projected onto the norm plane of any other task that has a conflicting gradient. The gradient projection operation significantly mitigates the detrimental effects caused by the gradient interference when training PINN, thus accelerating the convergence speed of PINN to high-precision solutions to MSS reliability assessment. With the proposed PINN-based framework, we investigate its applications for MSS reliability assessment in several different contexts in terms of time-independent or dependent state transitions and system scales varying from small to medium. The results demonstrate that the proposed PINN-based framework shows generic and remarkable performance in MSS reliability assessment, and the incorporation of PCGrad in PINN leads to substantial improvement in solution quality and convergence speed.
|
10.1016/j.ress.2022.108835
|
[
"https://arxiv.org/pdf/2112.00220v1.pdf"
] | 244,773,339 |
2112.00220
|
ac956a2b791b5da1b64e7e339a6b9e082d8193c3
|
A generic physics-informed neural network-based framework for reliability assessment of multi-state systems
1 Dec 2021
Taotao Zhou
Center for Risk and Reliability
University of Maryland
College ParkMDUSA
Xiaoge Zhang [email protected]
Department of Industrial and Systems Engineering
The Hong Kong Polytechnic University
Kowloon, Hong Kong
Enrique Lopez Droguett
Department of Civil and Environmental Engineering
University of California
Los AngelesCAUSA
Garrick Institute for the Risk Sciences
University of California
Los AngelesCAUSA
Ali Mosleh
Garrick Institute for the Risk Sciences
University of California
Los AngelesCAUSA
A generic physics-informed neural network-based framework for reliability assessment of multi-state systems
1 Dec 2021Preprint submitted to Reliability Engineering & System Safety December 2, 2021(Xiaoge Zhang) The source codes of this paper will be made publicly accessible at GitHub: https://github.com/zxgcqupt/PINNs4SRA upon the publication of this paper.multi-state systemsreliability assessmentphysics-informed neural networkgradient projectionMarkov process * Corresponding author
In this paper, we leverage the recent advances in physics-informed neural network (PINN) and develop a generic PINN-based framework to assess the reliability of multi-state systems (MSSs). The proposed methodology consists of two major steps. In the first step, we recast the reliability assessment of MSS as a machine learning problem using the framework of PINN. A feedforward neural network with two individual loss groups are constructed to encode the initial condition and state transitions governed by ordinary differential equations (ODEs) in MSS. Next, we tackle the problem of high imbalance in the magnitude of the back-propagated gradients in PINN from a multi-task learning perspective. Particularly, we treat each element in the loss function as an individual task, and adopt a gradient surgery approach named projecting conflicting gradients (PCGrad), where a task's gradient is projected onto the norm plane of any other task that has a conflicting gradient. The gradient projection operation significantly mitigates the detrimental effects caused by the gradient interference when training PINN, thus accelerating the convergence speed of PINN to high-precision solutions to MSS reliability assessment. With the proposed PINN-based framework, we investigate its applications for MSS reliability assessment in several different contexts in terms of time-independent or dependent state transitions and system scales varying from small to medium. The results demonstrate that the proposed PINN-based framework shows generic and remarkable performance in MSS reliability assessment, and the incorporation of PCGrad in PINN leads to substantial improvement in solution quality and convergence speed.
Introduction
Reliability assessment and optimization of engineered systems have received growing attention in a broad range of sectors, such as power grid [1,2], transportation systems [3,4,5], computing systems [6,7], electrical and mechanical systems [8,9,10,11]. Over the last few years, the increasing occurrences of extreme events have posed more than ever pressing needs for highly reliable infrastructure systems so that they can still operate at a desirable performance under extreme natural conditions. The malfunction and failure of these critical systems can lead to catastrophic consequences in terms of economic loss and human fatality. Take the 2021 Texas power crisis as an example, the inadequately winterizing of power equipment significantly compromised the reliability of power transmission system and resulted in the partial failure of the power grid, which eventually led to a massive power outage and left 4.5 million homes and businesses without power for several days [12].
In the context of reliability assessment and optimization of engineered systems, one of the popular means in characterizing engineered systems is to model it as a multi-state system (MSS) [13,14,15,16].
Differing from the binary-state reliability models, which assumes that a system and its components only have two states (i.e., perfectly operational or complete failure), MSS introduces a finite number of intermediate states for each system component to indicate a wide range of performance levels that lie between the perfectly functioning state and the completely failed state [14]. The rich intermediate states in MSS models enable the representation of the deterioration behavior of engineered systems in a finer granularity than that of traditional binary-state system models. As a consequence, MSS models have become an appealing tool for modelling and assessing system reliability in a broad array of industrial applications.
For example, Qiu and Ming [17] modeled a power distribution system as a MSS and developed a universal generating function (UGF)-based approach to quantify its reliability, where power transmission loss was taken into consideration. Liu et al. [18] modeled the stochastic dependency among state transitions of a MSS or component via copula functions and studied the reliability assessment of MSS with state transition dependency. Iscioglu and Erem [19] evaluated the reliability of a MSS consisting of n identical independent units with two different types of dependency among components. Mi et al. [20] developed an evidential network-based method to analyze the reliability of complex MSS with epistemic uncertainty.
In general, the methods used for reliability assessment of MSS can be roughly categorized into five classes: stochastic process particularly Markov process method [21,22], extensions of conventional binary reliability model such as multi-state fault tree method [23], Monte Carlo simulation (MCS) method [24,25], universal generating function (UGF) method [26], and Bayesian network [27]. Among them, MCS is one of the most popularly used approaches for system reliability assessment owing to its easiness to implement, advantages in uncertainty representation and propagation as well as flexibility in characterizing com-plex system behavior and interactions among system components. For example, Zio et al. [28] exploited the flexibility of MCS and developed quantitative measures to estimate the importance of components in a multi-state series-parallel system. Echard et al. [29] developed an efficient active learning method that combined Kriging with Monte Carlo Simulation to perform reliability assessment in structural systems. Schneider et al. [30] treated social network as a multi-state commodity and applied reliability measures commonly used in MSS to quantify the influence of a given actor in the social network.
In principle, the reliability of MSS can be accurately estimated using the standard MCS method as long as sufficient MCS samples are generated. However, despite the popularity of MCS, its computational effort grows exponentially in accordance with the number of components and component-wise states in MSS. The number of MCS samples needed to estimate the reliability of MSS in large-scale systems at a high precision easily gets computationally unaffordable. Such a flaw in MCS makes it inapplicable in time-sensitive applications that require real-time decision-making support. One alternative approach is to build data-driven surrogate models for the concerned MSS by taking advantage of the recent advances in deep learning. Unfortunately, deep learning faces similar issues when dealing with MSS in the highdimensional space. To be specific, a considerable amount of data needs to be collected to represent MSS in a wide range of scenarios (e.g., different degradation conditions, deterioration trajectories) in order to train a deep learning model. The collection of such a representative training dataset for MSS can take a long time and might incur unaffordable costs in some cases.
A promising direction to address the aforementioned issues is to encode physical laws (or empirical laws) in the development of machine learning models, which is referred to as Physics-Informed Machine Learning (PIML) in the literature [31,32,33]. A representative example along this front is the family of Physics-Informed Neural Networks (PINNs) [31]. The physical laws (i.e., conservation laws) governing system behaviors in the form of partial differential equation (PDE) or ordinary differential equation (ODE) are usually rigorously derived from first principles. In PINN, physical laws are typically incorporated as a soft loss term in the objective function of deep learning models. The incorporation of physical laws in the deep learning models substantially prunes the parameter search space as solutions violating the physical laws are discarded immediately. As a result, encoding physical laws in machine learning models essentially reduces the number of training points that are required to tune a deep learning model. The benefits of exploiting physical laws in building efficient deep learning models have been showcased in several recent studies [34,35,36,37,38,39]. It is worthwhile noting that the loss functions of PINNs is complicated and involve multiple terms, which would compete with each other during the training process [32]. Hence, since PINNs is a highly non-convex optimization problem, it is essential to assure the stability and robustness in the training of PINNs, which remains an active research topic yet [40].
The application of PINN in MSS reliability assessment has been rarely studied in the literature even though several features of MSS make it a natural fit to be formulated and solved as a PINN-type problem.
Specifically, the stochastic behavior of component state transitions in MSS is commonly characterized as Markov process [41,42,43,44], which are usually difficult to derive analytical solutions. Numerical methods are typically adopted, such as differential equation solver and Monte Carlo simulation. These numerical methods are prohibitively computational expensive, and they get computationally unaffordable easily when extensive uncertainty and sensitivity analysis are needed. The ODE governing the Markov processes [45] makes PINN a viable solution for MSS reliability assessment. Secondly, existing approaches often discretize the life span of MSS into multiple equally-sized time intervals. The side effect of doing this is that the reliability of MSS can only be performed in the pre-specified discrete time instants. In contrast, the development of PINN for MSS reliability assessment frees us from the MSS life span discretization, and it allows to estimate the reliability of MSS at any time instant in a mesh-free fashion.
To address the above issues, in this paper, we are motivated to develop a generic framework casting reliability assessment of MSS as a machine learning problem by exploiting the power of PINN. Towards this goal, one common pain point in adopting PINN is that the original formulation by Raissi et al. [31] often struggles to approximate the exact solution of PDEs in high precision due to the extremely imbalanced gradients during the training of PINN via back-propagation [32,40,46]. To address the imbalanced gradients among loss terms in PINN, we treat each loss term as an individual task and tackle this problem from a multi-task learning perspective following the approach proposed by Yu et al. [47].
The key idea is to project a task's gradient onto the normal plane of the gradient of any other task that has a conflicting gradient. Compared to previous studies, we make the following contributions:
1. Formulation of a generic physics-informed neural network-based framework to tackle the system reliability assessment problem in MSS. The developed PINN-based framework provides a novel and effective paradigm for assessing the reliability of complex MSS.
2. To address the issue of the extremely imbalanced loss function in PINNs, we integrate the gradient surgery method with PINN to deconflict gradients during the training of PINN via back-propagation.
The incorporation of the gradient surgery approach in PINN significantly accelerates the convergence speed of PINN and substantially improves the solution quality in MSS reliability assessment.
3. We investigate the applications of the PINN-based framework in several different scenarios in terms of system state transition rates (e.g., homogeneous continuous-time Markov chain and non-homogeneous continuous-time Markov chain) and system scales (e.g., small-scale MSS, medium-scale MSS). Besides, we examine the quality of the solutions from PINN by comparing them with that of Matlab solver.
The rest of the paper is structured as follows. Section 2 provides a brief introduction to the multi-state systems (MSS) and describes the technical background for PINN. Section 3 develops the proposed methodology to build PINN for MSS reliability assessment. Section 4 shows the applications of PINN in MSS reliability assessment and compares its performance with two other alternatives. Section 5 ends this paper with concluding remarks and discusses future research directions.
Background
In this section, we briefly introduce the technical background of reliability modeling of multi-state system and the mathematical formulation of physics-informed neural network.
MSS Reliability Model
Traditional binary reliability models only allow two operational states: perfectly functioning or complete failure. Whereas, MSS reliability assessment associates the system and its components with multiple intermediate states as indicated by either performance capacity or damage severity during performance degradation. Suppose the performance of a MSS is characterized by M + 1 discrete ordered states [13], represented by the following set:
S = {0, 1, · · · , M }(1)
where 0 denotes the worst state, and M denotes the best state. The others are intermediate states between
the worst and the best states.
Suppose the probabilities associated with the M + 1 states in the MSS at time t is denoted by the following vector:
p (t) = [p 0 (t) , p 1 (t) , · · · , p M (t)](2)
As the probability vector p (t) constitutes the exhaustive set of all the MSS states, it needs to satisfy the following constraint:
M i=0 p i (t) = 1, ∀t ∈ [1, 2, · · · , T ](3)
where T denotes the MSS operation period.
In general, the system dynamics in a MSS at each time instant t is characterized by a state-transition diagram as shown in Fig. 1. Each node in Fig. 1 represents the probability associated with the state i,
Q (t) = λ 0,0 (t) λ 0,1 (t) · · · λ 0,M (t) λ 1,0 (t) λ 1,1 (t) · · · λ 1,M (t) . . . . . . . . . . . . λ M,0 (t) λ M,1 (t) · · · λ M,M (t) (4) where λ i,i (t) = − M j=0,j =i λ i,j (t) , ∀i ∈ S.
Obviously, the sum of all the elements in each row is zero.
With the properly defined state transition matrix Q (t), then the evolution of the states in MSS over time can be described using the Kolmogorov forward equation:
p (t) = p (t) Q (t) p (t = 0) = s 0(5)
where p (t) refers to the first-order derivative of p (t) at the time instant t and p (t = 0) denotes the initial system state at the time instant 0. Given the initial system state s 0 , MSS evolves over time following the state transition matrix Q (t).
The MSS reliability can be derived by aggregating the state probability associated with system states that perform its desired function during the mission time. Mathematically, it is formulated as below:
R = M i=0 δ i p i (t)(6)
where δ i is a binary variable indicating whether state i satisfies the desired property at the system level.
If δ i = 1, then state i meets the intended function, otherwise, state i does not meet the requirement; p i (t) denotes the probability of state i at the time instant t.
Physics-Informed Neural Networks
In this section, we explain the underlying architecture of physics-informed neural networks (PINNs) and describes the mathematical formulation of PINNs.
In several industrial applications, the behavior of dynamical systems are described by general nonlinear partial differential equation (PDE) [32,48]. Consider a PDE represented in a general form formulated in Eq. (7):
u (x, t) + N x [u] = 0, x ∈ Ω, t ∈ [0, T ](7)
where u (x, t) denotes the latent solution, N x [•] is a nonlinear differential operator, x denotes a vector of space coordinates, and t denotes the time. The domain Ω of the PDE is bounded based on the prior knowledge of the dynamic systems, and [0, T ] is the time interval within which the system evolves.
It is well-known that neural networks are universal function approximators to learn the unknown relationship between any inputs and outputs. As a result, neural networks can be used to approximate the solution to the PDE function shown in Eq. (7). Suppose we denote the left-hand-side of Eq. (7) as
f (t, x): f : u (x, t) + N x [u](8)
f (t, x) now acts as a constraint modelling the physical law described by the PDE in Eq. (7). The first term u (x, t) in Eq. (7) can be approximated by a neural network, where x and t are inputs to the neural network. The neural network for approximating u (x, t) together with Eq. (8) (here, Eq. (8) acts as an equality constraint) result in a physics-informed neural network. Regarding the nonliner differentiator
N x [u]
, its value can be derived using the same neural network that is used to approximate u (x, t), where automatic differentiation can be applied to differentiate compositions of functions following the chain rule [49].
The neural network approximating u (x, t) has the same parameters as the network representing
f (t, x).
The weights of the neural network can be optimized by minimizing the following function:
L M SE = L u M SE + W * L f M SE (9)
where W is a factor denoting the weight associated with the loss term L f M SE , and
L u M SE = 1 N u Nu i=1 u N N t i u , x i u ; θ − u i 2(10)
and
L f M SE = 1 N f N f i=1 f t i f , x i f 2(11)
where {t i u , x i u , u i } denotes the initial and boundary data points on u (t, x), u N N (t i u , x i u ; θ) denotes the prediction of the neural network on the inputs (t i u , x i u ), θ refers to the weights in the neural network, t i f , x i f represents the collocation points for f (t, x), N u and N f represent the number of points generated for u (x, t) and f (t, x), respectively.
In Eq. (9), L u M SE measures the loss of the neural network when approximating the function u (x, t), while L f M SE enforces the physical law imposed by Eq. (7) into the neural network via a series of collocation points t i f , x i f . With the training data consisting of boundary and collocation points, we seek to minimize the loss function formulated in Eq. (9) through optimizing the weights in the neural network via gradient descent algorithms. As illustrated in Eq. (9), PINNs provide a rigorous way to seamlessly integrate the information from both the measurement data and physical laws, where physical laws are encoded into the loss function of the neural network via automatic differentiation. Consideration of underlying physical laws prunes the feasible solution space to the neural network parameters, and thus significantly reduces the number of training points as well as the size of the neural network (e.g., number of layers, number of hidden node in each layer etc.) during model training.
Proposed Framework
In this section, we introduce the proposed PINN-based framework for MSS reliability assessment in details. The proposed framework consists of two major steps. In the first step, we recast MSS reliability assessment as a machine learning problem in the framework of PINN. Next, we outline the gradient surgery approach to minimize gradient conflicts among multiple tasks during the training of PINN.
PINNs for MSS Reliability Assessment
As introduced in Section 2.1, there are two key components in the reliability assessment of a MSS that need to be appropriately characterized in the framework of PINN. The first key component is the initial state s 0 denoting the states of the MSS at the time instant 0. The second core component is the state transition, which is described by the Kolmogorov forward equations as shown in Eq. (5).
For the sake of demonstration, Fig. 2 illustrates the configuration of a PINN composed of two hidden layers with each layer having m hidden units for reliability assessment in MSS. In practice, a PINN can consist of as many hidden layers and hidden units as needed. Suppose we discretize the operation horizon [1, 2, · · · , T ] of a MSS into T time steps, which is typically referred to as collocation points. Each time we feed a specific time step t into the PINN, we obtain the probability associated with each state in p (t). With automatic differentiation, we derive the first order derivative of p (t) corresponding to each state with respect to the time instant t. Next, we compute the loss function between the first order derivative of p (t) and the exact solution p (t) Q (t) as illustrated in Eq. (5).
In addition to incorporating the differential equations characterizing state transitions, another constraint that needs to be modeled is the initial state. Basically, at the time instant 0, the system starts at a specific condition characterized by the probability associated with each state in MSS. Suppose s i 0 denotes the probability corresponding to state i at the time instant 0, then, in conjunction with the loss function for state transition modeling, we have the loss function for the MSS reliability assessment as below: Combining the two loss terms together, the goal is to minimize the loss function L (θ) through optimizing the parameter θ. With a set of training data, we can reduce the loss of the neural network iteratively via backpropgation using gradient descent algorithms, such as Adam [50].
L (θ) = M i=0 u i N N (t = 0; θ) − s i 0 2 Lu:Loss on initial condition +W 1 T T t=1 u N N (t; θ) Q (t) − du N N (t; θ) dt
After the PINN is properly trained, then it can be used to estimate the probability corresponding to any state at any given time instant t. Once the probability associated with each state is accurately predicted, the reliability of MSS at system level can be inferred following Eq. (6). Different from existing methods, a significant advantage of PINNs is that they allow to estimate the reliability of MSS in a continuous manner. Most of existing approaches can only estimate the probability associated with each state at predetermined time instants, while PINN is mesh-free and it allows to tackle MSS reliability assessment in a comprehensive fashion.
Conflicting Gradients Projection for Physics-Informed Neural Networks
As reported by Wang et al. [46], PINN faces a fundamental mode of failure that is closely related to the stiffness of the back-propagated gradient flows because the loss terms L u and L f are highly imbalanced in magnitude. In particular, the loss term L f characterizing the PDE residual dominates the loss function and, consequently, the optimization algorithm is heavily biased towards minimizing the loss term L f . As a result, PINN performs poorly in fitting the initial conditions, and leads to quite unstable and erroneous predictions [51,52].
In this study, we aim to tackle this problem from a multi-task learning perspective because it shares several features with PINN in common. In the multi-task learning, the ultimate goal is to train a network on all tasks jointly. Towards this goal, multi-task learning faces the same problem arising from unmatched gradients. More specifically, the gradient might be dominated by the value from one task at the cost of degrading the performance of the other task. In addition to imbalanced gradients, the gradients corresponding to different tasks (or loss terms in PINN) might be conflicting along the direction of descent with one another in a way that is detrimental to the progress of the optimization. These factors combined together result in the fact that the optimizer struggles to make progress in optimizing the weights of the network because the reduction in the loss value specific to one task eventually leads to the oscillation of losses in other tasks (see the demonstration in Section 4).
To resolve this issue, Yu et al. [47] proposed a projecting conflicting gradients (PCGrad) approach to minimize the gradient interference, which consists in projecting the gradient of a task onto the norm plane of any other task that has a conflicting gradient. In this paper, we adopt the PCGrad method to deconflict gradients during the training of PINN, where we treat each loss term as an individual task in the learning process. Specifically, consider two gradients ∆ i and ∆ j corresponding to the i-th and the j-th loss term in PINN. PCGrad first checks whether there are conflicts between ∆ i and ∆ j using the cosine similarity defined in Eq. (13).
ω (∆ i , ∆ j ) = ∆ i • ∆ j ∆ i ∆ j(13)
where • denotes the norm of the corresponding vector.
The cosine similarity results in a value within the range [−1, 1], where -1 denotes exactly the opposite direction, 1 means exactly the same, and 0 indicates orthogonality or decorrelation. If the cosine similarity between ∆ i and ∆ j is negative, then PCGrad projects ∆ i to the norm plane of ∆ j or the other way around. If the cosine similarity between ∆ i and ∆ j is non-negative, then the original gradients ∆ i and ∆ j remain the same. Suppose we project ∆ i to the norm plane of ∆ j , then we have the gradient of ∆ i after the projection as:
∆ P C i = ∆ i − ∆ i • ∆ j ∆ j 2 ∆ j(14)
where ∆ P C i denotes the gradient after the projection. Fig. 3 demonstrates the core idea in PCGrad. As it can be observed in Fig. 3(a), there is a high degree of conflict between the two gradients ∆ i and ∆ j . PCGrad either projects the gradient ∆ i onto the norm vector of the gradient ∆ j as illustrated in Fig. 3(b), or projects the gradient ∆ j onto the norm vector of the gradient ∆ i as shown in Fig. 3(c). Such operation amounts to removing the conflicting component from the gradient task, thus mitigating the destructive gradient interference among different tasks. In a similar way, PCGrad repeats the same procedures for all the other tasks following a randomly sampled order. Algorithm 1 summarizes the steps in PCGrad for projecting conflicting gradients in PINNs.
As the gradient projection operation accounts for the gradient information for all the tasks in a holistic manner, it significantly mitigates the conflicts among different tasks and results in a set of gradients with Algorithm 1: Conflicting gradients projection using PCGrad Data: neural network weights θ, number of tasks K 1 ∆ i ← ∇ θ L i (θ) , ∀i = 1, 2, · · · , K /* L i denotes the i-th loss term in PINN */
2 ∆ P C i ← ∆ i ∀i 3 for ( i = 1; i <= K; i = i + 1 ) { 4 for ( j uniformly ∼ [1, 2, · · · , m, · · · , K] , where m = i ) { 5 if ω (∆ P C i , ∆ j ) < 0 then 6 Set ∆ P C i = ∆ P C i − ∆ P C i •∆j ∆j 2 ∆ j /* Subtract the projection of ∆ P C i onto ∆ j */ 7 end 8 } 9 } 10 return update ∆θ = K i=1 ∆ P C i minimal gradient interference.
Numerical Examples
In this section, we demonstrate the proposed framework for MSS reliability assessment using a smallscale MSS of a single propulsion module in a railway system under either time-independent state transitions or time-dependent state transitions. We also illustrate its performance in assessing the reliability of a medium-scale MSS regarding a flow transmission system. The performance of the proposed framework is examined in comparison with the solution derived by the differential equation solver implemented in Matlab. is delivered when all the series components, and one converter out of the two works normally.
3. Failed state: State 2 means the state of complete failure with no power delivered.
The transition probabilities across the three states are described by the following matrix:
Q = − (2γ + λ) 2γ λ 0 − (γ + λ) (γ + λ) 0 0 0 (15)
where γ denotes the sum of failure rates of the series components γ = 2.8 × 10 −5 hour −1 , and λ denotes the failure rate associated with each converter λ = 7.26 × 10 −5 hour −1 .
As it can be seen from Eq. (15), the transition rate is time-independent, thus leading to a homogeneous continuous-time Markov chain (CTMC). The propulsion module starts at full capacity and denote its initial state as below:
s 0 = [1, 0, 0](16)
In MSS, we are interested in estimating the probabilities associated with the three states in the operation time horizon t ∈ (0, T ]. Following the methodology described in Section 3, together with Eq.
(5), we derive four loss terms that belong to two individual groups in the PINN as formulated below:
L u (θ) = [u N N (t = 0; θ) − s 0 ] 2 L f (θ) = du 0 N N (t;θ) dt − u 0 N N (t; θ) × [− (2γ + λ)] 2 du 1 N N (t;θ) dt − [u 0 N N (t; θ) × 2γ − u 1 N N (t; θ) × (γ + λ)] 2 du 2 N N (t;θ) dt − [u 0 N N (t; θ) × λ + u 1 N N (t; θ) × (γ + λ)] 2(17)
where Table 1 summarizes the performance comparison quantitatively. Clearly, PINN with PCGrad achieves a RMSE that is lower than that of PINN without PCGrad by at least one order of magnitude across the three time ranges.
Example 2
In this example, we extend the Example 4.1 by imposing time-inhomogeneity in the component failures.
Differing from Example 4.1, the transition rate is time-dependent in this example, thus leading to a nonhomogeneous CTMC. In particular, we assume that the transition rates follow a Weibull distribution and the corresponding transition rates are defined as below:
λ (t) = λ 0 αt α−1 γ (t) = γ 0 βt β−1(18)
where λ 0 and γ 0 denote the initial failure rates associated with each state, and their values are the same as the Example in Section 4.1. More specifically, λ 0 has a value of 7.26 × 10 −5 hour −1 and γ 0 has a value of 2.8 × 10 −5 hour −1 .
For the sake of illustration, we set the parameters in the Weibull distribution as α = 2.0 and β = 2.0.
The corresponding transition matrix of this problem is shown in Eq. (19). In the non-homogeneous CTMC, one difference worthy of mention is that each loss term in the loss group L f now involves the input time t when constructing the ODE residual. Following similar steps, the problem can be formulated using the proposed PINN-based framework. To train the PINN, we generate 300 collocation points within the time range [0, 300] with equal intervals. The initial state is the same as Example 4. The architecture of PINN is exactly the same as in Example 4. As the gradient during back-propagation is complex in the non-homogeneous CTMC case, we impose a monotonically decreasing learning rate that follows a polynomial decay schedule with an initial learning rate of 1 * 10 −3 and a final learning rate of 8 * 10 −5 . Fig. 9 compares the convergence performance of the two loss groups during the 150000 iterations. Clearly, PINN with PCGrad converges to lower loss values in both loss groups L u and L f than that of the PINN without PCGrad.
Q (t) = − (2γ (t) + λ (t)) 2γ (t) λ (t) 0 − (γ (t) + λ (t)) (γ (t) + λ (t)) 0 0 0 (19) ,WHUDWLRQ /RVV u LQ3,11ZLWKRXW3&*UDG u LQ3,11ZLWK3&*UDG ,WHUDWLRQ /RVV f LQ3,11ZLWKRXW3&*UDG f LQ3,11ZLWK3&*UDG
To test the performance of the trained PINN model, we generate another 301 points within the time range [0, 300] with equal intervals. Fig. 10 illustrates the histogram of the mean absolute error between PINN's predictions with Matlab's solutions with respect to the three system states. As it can be observed, PINN with PCGrad has similar performance when estimating the probability corresponding to state 0, while it achieves slightly better performance than the case of no PCGrad when predicting the probabilities associated with state 1 and state 2. derived by the Matlab solver, as shown in Fig. 11. As it can be seen, PINN accurately captures the changing trend of probability associated with each system state over time.
In this application, we are interested in the system-level performance, which is measured by the maximum flow that can be transmitted from point C to point E. At the system level, there are 12 states (2 × 2 × 3) in total. The state transition diagram at the system level is shown in Fig. 14, where the corresponding system performance is presented in the lower parts of the circle, and the label along each arc denotes the transition probability from one state to another state. For the details in deriving the system-level state transition diagram, refer to page 83 in chapter 2 of the Ref. [54]. 6 g 6 =1. 8 The differential equations governing the transitions among different system performance rates p i (t) are shown in Eq. (21).
dp1(t) dt = − λ (1) 2,1 + λ (2) 2,1 + λ (3) 3,2 p 1 (t) + µ (1) 1,2 p 2 (t) + µ (2) 1,2 p 3 (t) + µ (3) 2,3 p 4 (t) , dp2(t) dt = λ (1) 2,1 p 1 (t) − µ (1) 1,2 + λ (2) 2,1 + λ (3) 3,2 p 2 (t) + µ (2) 1,2 p 5 (t) + µ(3)
2,3 p 6 (t) ,
dp3(t) dt = λ (2) 2,1 p 1 (t) − µ (2) 1,2 + λ (1) 2,1 + λ (3) 3,2 p 3 (t) + µ (1) 1,2 p 5 (t) + µ (3) 2,3 p 7 (t) , dp4(t) dt = λ (3) 3,2 p 1 (t) − µ (3) 2,3 + λ (1) 2,1 + λ (2) 2,1 + λ (3) 2,1 p 4 (t) + µ(1)
1,2 p 6 (t) + µ
1,2 p 7 (t) + µ
(3) 1,2 p 8 (t) , dp5(t) dt = λ (2) 2,1 p 2 (t) + λ (1) 2,1 p 3 (t) − µ (2) 1,2 + µ (1) 1,2 + µ(3)
3,2 p 5 (t) + µ
2,3 p 9 (t) ,
dp6(t) dt = λ (3) 3,2 p 2 (t) + λ (1) 2,1 p 4 (t) − µ (3) 2,3 + µ (1) 1,2 + µ (2) 2,1 + µ(3)
2,1 p 6 (t) + µ
1,2 p 9 (t) + µ
1,2 p 10 (t) ,
dp7(t) dt = λ (3) 3,2 p 3 (t) + λ (2) 2,1 p 4 (t) − µ (3) 2,3 + µ (2) 1,2 + µ (1) 2,1 + µ (3) 2,1 p 7 (t) + µ (1) 1,2 p 9 (t) + µ (3) 2,3 p 11 (t) , dp8(t) dt = λ (3) 2,1 p 4 (t) − µ (3) 1,2 + λ (1) 2,1 + λ (2) 2,1 p 8 (t) + µ(1)
1,2 p 10 (t) + µ
1,2 p 11 (t) ,
dp9(t) dt = λ(3)
3,2 p 5 (t) + λ
2,1 p 6 (t) + λ
(1) 2,1 p 7 (t) − µ (3) 2,3 + µ (2) 1,2 + µ (1) 1,2 + λ(3)
2,1 p 9 (t) + µ
1,2 p 12 (t) ,
dp10(t) dt = λ(3)
2,1 p 6 (t) + λ
(1) 2,1 p 8 (t) − µ (3) 1,2 + µ (1) 1,2 + λ(2)
1,2 p 10 (t) + µ
1,2 p 12 (t) ,
dp11(t) dt = λ (3) 2,1 p 7 (t) + λ (2) 2,1 p 8 (t) − µ (3) 1,2 + µ (2) 1,2 + λ (1) 2,1 p 11 (t) + µ (1) 1,2 p 12 (t) , dp12(t) dt = λ (3) 2,1 p 9 (t) + λ (2) 2,1 p 10 (t) + λ (1) 2,1 p 11 (t) − µ (3) 1,2 + µ (2) 1,2 + µ (1) 1,2 p 12 (t) .(21)
From the state transition diagram in Fig. 14, we observe that there are five unique performance rates at the system level, namely: state 1: g 1 = 3.5; state 2: g 2 = 2.0; states 4 and 6: g 4 = g 6 =1.8; states 3 and 7: g 3 = g 7 =1.5; states 5, 8, 9, 10, 11 and 12: g 5 = g 8 = g 9 = g 10 = g 11 = g 12 = 0. Hence, the reliability of the system-level performance rate is formulated as follows:
P (G = 3.5) = p 1 (t) ; P (G = 2.0) = p 2 (t) ; P (G = 1.8) = p 4 (t) + p 6 (t) ; P (G = 1.5) = p 3 (t) + p 7 (t) ; P (G = 0) = p 5 (t) + p 8 (t) + p 9 (t) + p 10 (t) + p 11 (t) + p 12 (t) .
Following the framework proposed in Section 3, we reformulate this problem in the context of PINN. PCGrad achieves much better performance than the case of no PCGrad. In particular, when t is less than 0.05, PINN without PCGrad fails to capture the changing trend of the probability associated with each system performance rate while PINN with PCGrad matches with the Matlab solver consistently when estimating the probability corresponding to each system performance rate.
Next, we compute the RMSE of the differences in the solutions derived by PINN and the Matlab solver. Table 3 summarizes the RMSE between PINN and the Matlab solver in terms of the five system performance rates. Clearly, the RMSE of PINN with PCGrad is less than the RMSE of no PCGrad by nearly two orders of magnitude. In other words, PCGrad substantially improves the solution quality when
Summary
As demonstrated in the previous three numerical examples, the benefits of deconflicting gradients using PCGrad in PINN are multi-fold. First of all, it substantially reduces the number of iterations and the amount of training data needed to tune the PINN, thus facilitating to achieve data and computationefficient PINN. It also alleviates the oscillation of loss values during the training of PINN, and allows PINN to converge to a better solution with a much lower RMSE in relation to the solutions derived using the Matlab solver than that of PINN without PCGrad. Last but not least, the introduction of gradient projection frees us from tuning the weight parameter W as shown in Eq. (12) because all the tasks are treated independently in the PINN with PCGrad and the weight parameter W is not needed any more.
Conclusion
Reliability assessment of multi-state systems is of significant concerns in a broad range of areas. In this paper, we exploit the power of physics-informed neural network and formulate a generic PINN-based framework for MSS reliability assessment. The developed framework tackles the problem of MSS reliability assessment from a machine learning perspective, and provides a viable paradigm for effective reliability modeling. The proposed methodology follows a two-step procedure. In the first step, MSS reliability assessment is reformulated as a machine learning problem in the framework of PINN, where loss functions are constructed to characterize the constraints associated with the initial condition and state transitions in MSS. Afterwards, to mitigate the high imbalance in the magnitude of gradients during the training of PINN, we leverage the projecting conflicting gradients (PCGrad) method to map the gradient of a task onto the norm plane of the other task that has a conflicting gradient. The embedding of PCGrad into the optimization algorithms significantly speeds up the convergence of the PINN to high-quality solutions.
The proposed PINN-based framework demonstrates promising performance in evaluating the reliability of MSS in a variety of scenarios.
Future work can be carried out along the following directions. First of all, we investigate PINN's applications in the MSS, where state transitions are characterized by either homogeneous or non-homogeneous CTMC. It is worth exploring how to adopt PINN to analyze the reliability of semi-Markov MSS. Another direction worthy of investigation is to explore more effective ways to incorporate the equations governing state transitions in MSS into the neural network. In this paper, the ODE residuals are embedded into the neural network in a soft manner by appropriately penalizing the loss function. The drawback of this approach is that PINN might still violate the state transition equations in some scenarios. Thus, it is meaningful to investigate other alternative approaches so as to guarantee that PINN is strictly in compliance with the equations governing the underlying state transitions in MSS. Last but not least, we add the training points representing the ODE for state transitions at one shot, it is essential to develop more effective methods to add the ODE residual points in an adaptive manner, for example, adaptively adding training points in the locations with the largest expected reduction in the ODE loss in batch mode.
Figure 1 :
1and each branch labeled λ i,j (t) denotes the corresponding one-step transition probability from state i to state j at the time instant t. Mathematically, the state transition probabilities among all the states at Graphical representation of state transitions in a multi-state system the time instant t can be represented by the following matrix:
Figure 2 :
2Configuration of the proposed PINN-based framework for multi-state system reliability assessment
2 L
2f :Loss on approximating state transition equation (12) where M refers to the number of states in the MSS, T is the number of time steps, u N N (θ) denotes the neural network with its weights represented by the parameter θ, u i N N (t = 0; θ) indicates the probability corresponding to state i at the time instant 0 as predicted by the neural network, W is a weighting factor, and du i N N (t;θ) dt reveals the derivative with respect to the time t that is estimated by the neural network u N N (θ) via automatic differentiation. As shown in Eq. (12), there are two key components in the loss function L (θ). The first component L u uses a mean squared error metric to evaluate the loss corresponding to the initial states of the MSS, while the second component L f enforces the structure modeling of the state transition in MSS and estimates the residual when approximating the governing equation characterizing the state transitions in MSS.
Figure 3 :
3Gradient deconflict process in PCGrad. (a) Two conflicting gradients ∆i and ∆j. (b) PCGrad projects the gradient ∆i onto the norm vector of the gradient ∆j. (c) PCGrad projects the gradient ∆j onto the norm vector of the gradient ∆i
Figure 4 :
4A three-state transition diagram representing the performance deterioration of a propulsion module. The label along each arc denotes the corresponding transition probability from one state to another state.Consider a single propulsion module in a multi-voltage propulsion system designed for the Italian high-speed railway system. The propulsion module consists of a series of four components (transformer, filter, inverter and motor) and two parallel converters[53]. The propulsion module can be represented by a three-state Markov model as illustrated inFig. 4. The three states in the propulsion module correspond to three different levels of power delivery as described below:1. Full operation: State 0 denotes a fully operational state. In this state, the propulsion delivers the maximum power (2200 kW) when all components are working.2. Degraded state: State 1 indicates a degraded state. In this state, only half of the power (1100 kW)
Figure 5 :
5u i N N (t; θ) denotes the prediction of neural network on the i-th state in the three-state MSS at the time instant t. A snapshot of the convergence of the two loss groups Lu and L f Ideally, the value of each component in the loss group L f should be strictly zero. In PINN, we approximate these equations by embedding a soft loss term in the objective function. In this example, the PINN consists of two hidden layers with each hidden layer having 50 units and the Tanh activation function. The last layer of the PINN is a fully-connected layer, and it has three outputs with each output corresponding to one state in the MSS, and softmax is employed as the activation function in the fullyconnected layer to ensure that the range of output value is within the range [0, 1] 1 . To train the neural network, we generate 5000 collocation points for t ∈ [0, 60, 000] with equal intervals representing Eq. (5).Next, we use the Adam algorithm with a learning rate of 0.001 to optimize the weights of the neural network. The two PINNs trained using Adam with and without PCGrad have the same architecture and initial weights. In the PINN with PCGrad, the two loss groups L u and L f are treated as two individual tasks in the PINN. Whereas, in the PINN without PCGrad, the two loss groups are combined together via equal weights 2 . The paradigm of adopting the proposed PINN-based framework to model MSS reliability assessment apparently differs from existing methods for MSS reliability analysis. The proposed PINNbased framework provides an insightful point of view to take advantage of the power in neural network to tackle this challenging problem.
Fig. 5
5illustrates a snapshot demonstrating the convergence of the two loss groups L u and L f during PINN training. Obviously, without PCGrad, the decrease in the loss group L f (note loss L f dominates in the loss function in the snapshot) leads to severe oscillation of the other loss group L u , which eventually translates into the slow convergence of the PINN. In contrast, the situation is completely different after PCGrad is applied. The values of both loss groups L u and L f drop down in a steady trend, and PCGrad converges to a solution with a loss value that is much lower than that of PINN without PCGrad by one order of magnitude.
Fig. 6 Figure 6 :
66compares the convergence performance of the two loss groups after the PINN is trained for 80,000 iterations. It can be noted that both L u and L f converge to a much lower values in the PINN with PCGrad than that of PINN without PCGrad. In particular, PINN with PCGrad achieves much better performance in approximating the transition equations L f than the case of no PCGrad as partially reflected by the significant gap in the early stage of iterations as indicated inFig. 6. To examine the solution quality of PINN trained with PCGrad, we derive solutions to the PDE using the Runge-Kutta method for the 5000 time instants within the time range [0, 80, 000] with a step size of 16 in Matlab. Next, we use the root squared mean error (RMSE) in relation to the results from Runge-Kutta using Matlab solver that is averaged over the three states as the performance metric to compare the probabilities associated with the three states generated by PINN with that of the Runge-Kutta method. Note that Convergence performance comparison of two loss groups in the MSS for the homogeneous propulsion module PINN is trained with the data in the time range [0, 60000]. In other words, we examine the performance of PINN in both interpolating and extrapolating the state probabilities.
Figure 7 :
7Performance comparison of PINN's predictions on the probabilities associated with each state over time in the MSS
Figure 8 :
8Histogram of the mean absolute error between PINN and the Matlab solver with respect to each system state in the homogeneous propulsion system that PCGrad outperforms the optimization without PCGrad when estimating the probabilities of state 1 and state 2. At the same time, PCGrad overestimates the probability associated with state 0. Another interesting observation is that both optimization methods with and without PCGrad perform well in extrapolation for the time range t ∈ (60, 000, 80, 000] due to the incorporation of PDE equations.Fig. 8displays the histogram of the mean absolute error between PINN's predictions and the Matlab solver with respect to the three system states. Not surprisingly, PINN with PCGrad achieves substantially lower mean absolute error in all three system states than that of PINN without PCGrad. In other words, PCGrad significantly improves the quality of the state estimation in MSS.
Figure 9 :
9Convergence performance comparison of PINN for reliability assessment of the non-homogeneous propulsion module
Figure 10 :Figure 11 :
1011Next, we evaluate the quality of the solutions of PINN quantitatively by comparing them with solutions Histogram of the mean absolute error between PINN and the Matlab solver with respect to each system state in the non-Performance comparison of PINN's predictions on the probabilities associated with each state over time in the non-homogeneous propulsion system
Figure 12 :
12MSS-based representation of the flow transmission systemConsider a flow transmission system consisting of three pipes.Fig. 12shows the MSS structure of the flow transmission system[54], where the oil flows from point C to point E in the flow transmission system. The performance of the flow transmission system is measured by its capacity in the unit of tons per minute. Both element 1 and element 2 have two states: operational and failed state. In the operational state, element 1 and element 2 have a capacity of 1.5 and 2 tons per minute, respectively.Whereas, their capacity degrades to zero if they are in a state of total failure. Differing from element 1 and element 2, element 3 has three states: a state of total failure corresponding to a capacity of 0, a state of partial failure corresponding to a capacity of 1.8 tons per minute, and a fully operational state with a capacity of 4 tons per minute.
Figure 13 :
13Component-wise state transitions in the flow transmission system
Fig
.13 illustrates the state transitions of each component in the flow transmission systemthe failure rate and repair rate associated with the k-th element when the element transitions between state i and state j in the flow transmission system. The specific values of the failure rates and repair rates are shown as below:
Figure 15 :
15Specifically,we generate 500 collocation points within the time range [0, 0.2] with equal intervals, and initialize the state of the system at the time instant 0 as follows: of the architecture of the neural network, the PINN used for reliability assessment in this numerical example has the same configuration as in Example 4.1, such as network architecture, learning rate, activation function and number of training steps. The PINN is trained for 40000 iterations using the Adam algorithm with a learning rate of 1 * 10 −3 . To compare the model performance, we generate 501 points within the time range [0, 0.2] with an equal step size of 0.0004.Fig. 15shows the histogram of the mean absolute error between PINN's predictions and Matlab's solutions regarding the five systemlevel performance rates. PINN with PCGrad outperforms the case of no PCGrad when system-level performance rates are 3.5, 1.8, and 0. When system-level performance rates G are at 2.0 and 1.5, PCGrad maintains almost the same level of performance as the case of no PCGrad. Histogram of the mean absolute error between PINN and the Matlab solver with respect to each system state in the flow transmission system
Fig
.16 visualizes the results of PINN when estimating the probabilities associated with each system performance rate in comparison with the solutions derived from the Matlab solver. Again, PINN with
Figure 16 :
16Instantaneous probabilities associated with different system performance rates in the flow transmission system performing reliability assessment in MSS.
Table 1 :
1Performance comparison of PINN with and without PCGrad in the reliability assessment of the homogeneous flow transmission systemMethods
Time Range
t ∈ (0, 80000] t ∈ (0, 60000] t ∈ (60000, 80000]
RMSE of PINN without PCGrad
0.111139
0.1461463
0.0062300
RMSE of PINN with PCGrad
0.008277
0.0108075
0.0006953
Fig. 7 compares the predictions of the two PINN models with the Matlab solver. It can be observed
Table 2
2summarizes the
Table 2 :
2Performance comparisons of PINN with and without PCGrad in the reliability assessment of the non-homogeneous flow transmission systemMethods
Time Range
t ∈ [0, 300]
RMSE of PINN without PCGrad
0.00048846
RMSE of PINN with PCGrad
0.00044518
Table 3 :
3Performance comparisons of PINN with and without PCGrad against Matlab solver in the reliability assessment of the flow transmission systemMethods
Time Range
t ∈ [0, 0.2]
RMSE of PINN without PCGrad
0.013388
RMSE of PINN with PCGrad
0.000454
In this paper, PINN has the same architecture across the three numerical examples. The only difference is the number of outputs in the fully-connected layer.2 In the case of no PCGrad, the losses are combined following the same way in the subsequent two examples.
AcknowledgementThe work described in this paper is partially supported by a grant from the Research Committee of The Hong Kong Polytechnic University under project code 1-BE6V.
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In coded bi-directional cooperation, two nodes wish to exchange messages over a shared half-duplex channel with the help of a relay. In this paper, we derive performance bounds for this problem for each of three decode-and-forward protocols.The first protocol is a two phase protocol where both users simultaneously transmit during the first phase and the relay alone transmits during the second. In this protocol, our bounds are tight.The second protocol considers sequential transmissions from the two users followed by a transmission from the relay while the third protocol is a hybrid of the first two protocols and has four phases. In the latter two protocols the bounds are not identical. Numerical evaluation shows that in some cases of interest our bounds do not differ significantly.Finally, in the Gaussian case with path loss, we derive achievable rates and compare the relative merits of each protocol. This case is of interest in cellular systems. Surprisingly, we find that in some cases, the achievable rate region of the four phase protocol contains points that are outside the outer bounds of the other two protocols.
|
10.1109/tit.2008.929913
|
[
"https://arxiv.org/pdf/cs/0703017v4.pdf"
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cs/0703017
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5e49a40f2260fdfb420d0d90207937ace8c21174
|
Performance Bounds for Bi-Directional Coded Cooperation Protocols
5 Jun 2008 June 5, 2008
Sang Joon Kim
School of Engineering and Applied Sciences
Department of Electrical and Computer Engineering
Harvard University
02138CambridgeMA
Patrick Mitran [email protected].
School of Engineering and Applied Sciences
Department of Electrical and Computer Engineering
Harvard University
02138CambridgeMA
Vahid Tarokh
School of Engineering and Applied Sciences
Department of Electrical and Computer Engineering
Harvard University
02138CambridgeMA
Sang Joon Kim
University of Waterloo
WaterlooCanada
Vahid [email protected]
University of Waterloo
WaterlooCanada
Performance Bounds for Bi-Directional Coded Cooperation Protocols
5 Jun 2008 June 5, 2008arXiv:cs/0703017v4 [cs.IT] 1 DRAFT 2Index Terms Cooperationcapacity boundsperformance boundsbi-directional communicationnetwork coding
In coded bi-directional cooperation, two nodes wish to exchange messages over a shared half-duplex channel with the help of a relay. In this paper, we derive performance bounds for this problem for each of three decode-and-forward protocols.The first protocol is a two phase protocol where both users simultaneously transmit during the first phase and the relay alone transmits during the second. In this protocol, our bounds are tight.The second protocol considers sequential transmissions from the two users followed by a transmission from the relay while the third protocol is a hybrid of the first two protocols and has four phases. In the latter two protocols the bounds are not identical. Numerical evaluation shows that in some cases of interest our bounds do not differ significantly.Finally, in the Gaussian case with path loss, we derive achievable rates and compare the relative merits of each protocol. This case is of interest in cellular systems. Surprisingly, we find that in some cases, the achievable rate region of the four phase protocol contains points that are outside the outer bounds of the other two protocols.
I. INTRODUCTION
Consider two users, denoted by a and b, who wish to share independent messages over a shared channel. Traditionally, this problem is known as the two-way channel [2], [10].
In many realistic broadcast environments, such as wireless communications, it is not unreasonable to assume the presence of a third node which may aid in the exchange of a and b's messages. In particular, if a is a mobile user and b is a base station, then we may suppose the presence of a relay station r to assist in the bi-directional communication.
Traditionally, without the presence of the relay station, communication between nodes a and b is performed in two steps: first a transmits its message to b followed by similar transmission from b to a (illustrated in Fig. 1.i). In the presence of relay node r, one might initially assume that four phases are needed (see Fig. 1.ii). However, by taking advantage of the shared wireless medium, it is known that the third and fourth transmissions may be combined ( Fig. 1.iii) into a single transmission using, for example, ideas from network coding [1], [13]. In particular, if the messages of a and b are w a and w b respectively and belong to a group, then it is sufficient for the relay node to successfully transmit w a ⊕ w b simultaneously to a and b. In [4], [5], such a three phase coded bi-directional protocol is considered when the group is Z k 2 , the binary operator is component-wise modulo 2 addition (i.e., exclusive or) and encoding is performed linearly to produce parity bits. As each user transmits sequentially, each user is amenable to receive "side-information" from the opposite user during one of the first two phases.
The works of [7] and [8] not only consider the three phase protocol, but combine the first two phases into a single joint transmission by nodes a and b followed by a single transmission by the relay which forwards its received signal ( Fig. 1.iv). Coded bi-directional cooperation may also be extended for the case of multiple relaying nodes [11], [12]. In [9], achievable rate regions are derived assuming full duplex capabilities at all nodes.
In this paper, we are interested in determining fundamental bounds on the performance of coded bidirectional communications assuming various decode-and-forward protocols for half-duplex channels. In the case of a two phase protocol where both users transmit simultaneously in the first phase followed by a transmission from the relay, we derive the exact performance 1 . In the case of three or more phase protocols, we take into account any side information that a node may acquire when it is not transmitting and derive inner and outer bounds on the capacity regions. We find that a four phase hybrid protocol is sometimes strictly better than the outerbounds of two or three phase decode-and-forward protocols previously introduced in the literature. This paper is structured as follows. In Section II, we define our notation and the protocols that we consider. In Section III, we derive performance bounds for the protocols while in Section IV, we numerically compute these bounds for fading Gaussian channels.
II. PRELIMINARIES
A. Notation and Definitions
We first start with a somewhat more general formulation of the problem. We consider an m node set, denoted as M := {1, 2, . . . , m} (where := means defined as) for now, where node i has message W i,j that it wishes to send to node j. Each node i has channel input alphabet X * i = X i ∪ {∅} and channel output alphabet
Y * i = Y i ∪ {∅},
where ∅ is a special symbol distinct of those in X i and Y i and which denotes either no input or no output. In this paper, we assume that a node may not simultaneously transmit and receive at the same time. In particular, if node i selects X i = ∅, then it receives Y i ∈ Y i and if X i ∈ X i , then necessarily Y i = ∅, i.e., X i = ∅ iff Y i = ∅ 2 . Otherwise, the effect of one node remaining silent on the received variable at another node may be arbitrary at this point. The channel is assumed discrete memoryless. In Section IV, we will be interested in the case
X * i = Y * i = C ∪ {∅}, ∀i ∈ M.
The objective of this paper is to determine achievable data rates and outer bounds on these for some particular cases. We use R i,j for the transmitted data rate of node i to node j, i.e., W i,j ∈ {0, . . . , ⌊2 nRi,j ⌋ − 1} := S i,j . 1 Similar results were independently derived in [6]. 2 Thus, FDM cannot be allowed as it violates the half-duplex constraint. June 5, 2008 DRAFT For a given protocol P, we denote by ∆ ℓ ≥ 0 the relative time duration of the ℓ th phase. Clearly, ℓ ∆ ℓ = 1. It is also convenient to denote the transmission at time k, 1 ≤ k ≤ n at node i by X k i , where the total duration of the protocol is n and X (ℓ) i denotes the random variable with alphabet X * i and input distribution p (ℓ) (x i ) during phase ℓ. Also, X k i corresponds to a transmission in the first phase if k ≤ ∆ 1 n, etc. We also define X k S := {X k i |i ∈ S}, the set of transmissions by all nodes in the set S at time k and similarly X
coders byŴ j,i (Y 1 i , . . . , Y n i , W i,1 , . . . , W i,m )
. Given a block size n, a set of encoders and decoders has associated error events E i,j := {W i,j =Ŵ i,j (·)}, for decoding the message W i,j at node j at the end of the block, and the corresponding encoders/decoders result in relative phase durations {∆ ℓ,n }, where the subscript n indicates that the phase duration depends on the choice of block size (as they must be multiples of 1/n).
A set of rates {R i,j } is said to be achievable for a protocol with phase durations {∆ ℓ }, if there exist encoders/decoders of block length n = 1, 2, . . . with P [E i,j ] → 0 and ∆ ℓ,n → ∆ ℓ as n → ∞ ∀ℓ. An achievable rate region (resp. capacity region) is the closure of a set of (resp. all) achievable rate tuples for fixed {∆ ℓ }.
B. Basic Results
In the next section, we will use a variation of the cut-set bound. We assume that all messages from different sources are independent, i.e., ∀i = j, W i,k and W j,l are independent ∀k, l ∈ M. In contrast to [2], we relax the independent assumption from one source to different nodes, i.e., in our case W i,j and W i,k may not be independent. Given subsets S, T ⊆ M, we define W S,T := {W i,j |i ∈ S, j ∈ T } and R S,T = lim n→∞ 1 n H(W S,T ). Lemma 1: If in some network the information rates {R i,j } are achievable for a protocol P with relative durations {∆ ℓ }, then for every ǫ > 0 and all S ⊂ {1, 2, · · · , m} = M
R S,S c ≤ ℓ ∆ ℓ I(X (ℓ) S ; Y (ℓ) S c |X (ℓ) S c , Q) + ǫ,(1)
for a family of conditional distributions p (ℓ) (x 1 , x 2 , . . . , x m |q) and a discrete time-sharing random variable Q with distribution p(q). Furthermore, each p (ℓ) (x 1 , x 2 , . . . , x m |q)p(q) must satisfy the constraints of phase ℓ of protocol P.
Proof: Replacing W (T ) by W S,S c and W (T c ) by W S c ,M in (15.323) -(15.332) in [2], then all the steps in [2] still hold and we have
H(W S,S c ) = H(W S,S c |W S c ,M ) ≤ n k=1 I(X k S ; Y k S c |X k S c ) + nǫ n , where ǫ n → 0 since i∈S,j∈S c P [E i,j ] → 0 and the distributions p(x k 1 , . . . , x k m , y k 1 , . . . , y k m ) are those induced by encoders for which P [E i,j ] → 0 as n → ∞.
Defining Q 1 , Q 2 , . . . to be discrete random variables uniform over {1, . . . , n·∆ 1,n }, {n·∆ 1,n +1, . . . , n· ∆ 1,n + n · ∆ 2,n }, . . ., we thus have
H(W S,S c ) ≤ ℓ n · ∆ ℓ,n I(X Qℓ S ; Y Qℓ S c |X Qℓ S c , Q ℓ ) + nǫ n ,(2)
Defining the discrete random variable Q := (Q 1 , Q 2 , . . .), then
1 n H(W S,S c ) ≤ ℓ ∆ ℓ,n I(X (ℓ) S ; Y (ℓ) S c |X (ℓ) S c , Q) + ǫ n ,(3)
where X (ℓ)
S := X Qℓ S . Finally, since the distributions p (ℓ) (x 1 , x 2 , . . . , x m |q)p(q) are those induced by encoders for which P [E i,j ] → 0, if
there is a constraint on the encoders (such as a power constraint), this constraint is also valid for the distributions p (ℓ) (x 1 , x 2 , . . . , x m |q)p(q).
C. Protocols
In bi-directional cooperation, two terminal nodes denoted a and b exchange their messages. The messages to be transmitted are W a := W a,b , W b := W b,a and the corresponding rates are R a := R a,b and The simplest protocol for the bi-directional channel, is that of Direct Transmission (DT) (Fig. 2.i).
R b := R b,
Here, since the channel is memoryless and ǫ > 0 is arbitrary, the capacity region from Lemma 1 is :
R a ≤ sup p (1) (xa) ∆ 1 I(X (1) a ; Y (1) b |X (1) b = ∅) R b ≤ sup p (2) (xb) ∆ 2 I(X (2) b ; Y (2) a |X (2) a = ∅),
where the distributions are over the alphabets X a and X b respectively.
With a relay node r, we suggest three different decode-and-forward protocols, which we denote as Multiple Access Broadcast (MABC) protocol, Time Division Broadcast (TDBC) and Hybrid Broadcast (HBC). Then, the message from a (resp. b) to r is W a,r = W a (resp. W b,r = W b ) and the corresponding rate is R a,r = R a (resp. R b,r = R b ). Also, in our protocols, all phases are contiguous, i.e., they are performed consecutively and are not interleaved or re-ordered. 3 In the MABC protocol ( Fig. 2.ii), terminal nodes a and b transmit information simultaneously during phase 1 and the relay r transmits some function of the received signals during phase 2. With this scheme, we only divide the total time period into two regimes and neither node a nor node b is able to receive any meaningful side-information during the first phase due to the half-duplex constraint.
In the TDBC protocol ( Fig. 2.iii), only node a transmits during the first phase and only node b transmits during the second phase. In phase 3, relay r performs a transmission based on the received data from the first two phases. Here, node a attempts to recover the message W b based on both the transmissions from node b in the second phase and node r in the third phase. We denote the received signal at node a in the second phase as second phase side information. Likewise, node b may also recover W a based on first phase side information and the received signal at node b during the third phase.
Finally, we consider a Hybrid Broadcast (HBC) protocol ( Fig. 2.iv) which is an amalgam of the MABC and TDBC protocols. In this scheme, there are 4 distinct transmissions, two of which result in side-information at a and b. 3 If we relax the contiguous assumption, the achievable region could increase by cooperation between interleaving phases.
III. PERFORMANCE BOUNDS
A. MABC Protocol
Theorem 2: The capacity region of the half-duplex bi-directional relay channel with the MABC protocol is the closure of the set of all points (R a , R b ) satisfying
R a < min ∆ 1 I(X (1) a ; Y (1) r |X (1) b , X (1) r = ∅, Q), ∆ 2 I(X (2) r ; Y (2) b |X (2) a = X (2) b = ∅, Q) R b < min ∆ 1 I(X (1) b ; Y (1) r |X (1) a , X (1) r = ∅, Q), ∆ 2 I(X (2) r ; Y (2) a |X (2) a = X (2) b = ∅, Q) R a + R b < ∆ 1 I(X (1) a , X (1) b ; Y (1) r |X (1) r = ∅, Q) over all joint distributions p(q)p (1) (x a |q)p (1) (x b |q)p (2) (x r |q) with |Q| ≤ 5 over the alphabet X a ×X b ×X r .
Remark: If the relay is not required to decode both messages, then the region above is still achievable, and removing the constraint on the sum-rate R a + R b yields an outer bound.
Proof: Achievability: Random code generation: For simplicity of exposition only, we take |Q| = 1 and therefore consider distributions p (1) (x a ), p (1) (x b ) and p (2) (x r ). First we generate random (n · ∆ 1,n )-
length sequences x (1) a (w a ) with w a ∈ S a and x (1) b (w b ) with w b ∈ S b , and (n · ∆ 2,n )-length sequences x (2) r (w r ) with w r ∈ Z L where L = max(⌊2 nRa ⌋, ⌊2 nRb ⌋), according to p (1) (x a ), p (1) (x b ) and p (2) (x r )
respectively.
Encoding: During phase 1, encoders of node a and b send the codewords x Decoding: a and b estimatew b andw a after phase 2 using jointly typical decoding. Since w r = w a ⊕w b and a knows w a , node a can reduce the number of possible w r to ⌊2 nRb ⌋ and likewise at node b, the cardinality is ⌊2 nRa ⌋.
Error analysis: For convenience of analysis, first define E (ℓ) i,j as the error event at node j that node j attempts to decode w i at the end of phase ℓ using jointly typical decoding. Let A
P [E a,b ] ≤ P [E (1) a,r ∪ E (1) b,r ∪ E (2) r,b ](4)≤ P [E (1) a,r ∪ E (1) b,r ] + P [E (2) r,b |Ē (1) a,r ∩Ē (1) b,r ](5)
June 5, 2008 DRAFT Following the well-known MAC error analysis from (15.72) in [2]:
P [E (1) a,r ∪ E (1) b,r ] ≤P [D (1) {a,b},{r} (w a , w b )] + 2 nRa 2 −n·∆1,n(I(X (1) a ;Y (1) r |X (1) b ,X (1) r =∅)−3ǫ) + 2 nRb 2 −n·∆1,n(I(X (1) b ;Y (1) r |X (1) a ,X (1) r =∅)−3ǫ) + 2 n(Ra+Rb) 2 −n·∆1,n(I(X (1) a ,X (1) b ;Y (1) r |X (1) r =∅)−4ǫ)(6)
Also,
P [E (2) r,b |Ē (1) a,r ∩Ē (1) b,r ] ≤ P [D (2) {r},{b} (w a ⊕ w b )] + P [∪w a =wa D (2) {r},{b} (w a ⊕ w b )] ≤ P [D (2) {r},{b} (w a ⊕ w b )] + 2 nRa 2 −n·∆2,n(I(X (2) r ;Y (2) b |X (2) a =X (2) b =∅)−3ǫ)(7)
Since ǫ > 0 is arbitrary, with the conditions of Theorem 2 and the AEP property, we can make the right hand sides of (6) - (7)
R a ≤ ∆ 1 I(X (1) a ; Y (1) r , Y (1) b |X (1) r , X (1) b , Q) + ∆ 2 I(X (2) a ; Y (2) r , Y (2) b |X (2) r , X (2) b , Q) + ǫ (8) = ∆ 1 I(X (1) a ; Y (1) r |X (1) b , X (1) r = ∅, Q) + ǫ,(9)
where (9) follows since in the MABC protocol, we must have
Y (1) a = Y (1) b = X (1) r = ∅ (10) X (2) a = X (2) b = Y (2) r = ∅.(11)
We find the outer bounds of the other cut-sets in the same manner:
S 2 : R b ≤ ∆ 1 I(X (1) b ; Y (1) r |X (1) a , X (1) r = ∅, Q) + ǫ.(12)S 3 : N/A(13)S 4 : R a + R b ≤ ∆ 1 I(X (1) a , X (1) b ; Y (1) r |X (1) r = ∅, Q) + ǫ,(14)S 5 : R a ≤ ∆ 2 I(X (2) r ; Y (2) b |X (2) a = X (2) b = ∅, Q) + ǫ,(15)S 6 : R b ≤ ∆ 2 I(X (2) r ; Y (2) a |X (2) a = X (2) b = ∅, Q) + ǫ.(16)
Since ǫ > 0 is arbitrary, together, (9), (12) -(16) and the fact that the half-duplex nature of the channel constrains X
a to be conditionally independent of X
b given Q yields the converse. By Fenchel-Bunt's theorem in [3], it is sufficient to restrict |Q| ≤ 5.
R a < min ∆ 1 I(X (1) a ; Y (1) r |X (1) b = X (1) r = ∅, Q),∆ 1 I(X (1) a ; Y (1) b |X (1) b = X (1) r = ∅, Q) + ∆ 3 I(X (3) r ; Y (3) b |X (3) a = X (3) b = ∅, Q) R b < min ∆ 2 I(X (2) b ; Y (2) r |X (2) a = X (2) r = ∅, Q),∆ 2 I(X (2) b ; Y (2) a |X (2) a = X (2) r = ∅, Q) + ∆ 3 I(X (3) r ; Y (3) a |X (3) a = X (3) b = ∅, Q) over all joint distributions p(q)p (1) (x a |q)p (2) (x b |q)p (3) (x r |q) with |Q| ≤ 4 over the alphabet X a ×X b ×X r .
Proof: Random code generation: First, we generate a partition of S a randomly by independently assigning every index w a ∈ S a to a set S a,i , with a uniform distribution over the indices i ∈ {0, . . . , ⌊2 nRa0 ⌋− 1}. We denote by s a (w a ) the index i of S a,i to which w a belongs and likewise, a partition for w b ∈ S b is similarly constructed. For simplicity of exposition, we take |Q| = 1. For any ǫ > 0 and distributions
p (1) (x a ) , p (2) (x b ) and p (3) (x r ), we generate random (n · ∆ 1,n )-length sequences x (1) a (w a ) with w a ∈ S a , (n·∆ 2,n )-length sequences x (2) b (w b ) with w b ∈ S b and (n·∆ 3,n )-length sequences x (3) r (w r ) with w r ∈ Z L , L = ⌊2 n·max{Ra0,Rb0} ⌋.
Encoding: During phase 1 (resp. phase 2), the encoder at node a (resp. node b) sends the codeword
x (1) a (w a ) (resp x (2) b (w b ))
. Relay r estimatesŵ a andŵ b after phases 1 and 2 respectively. The relay then constructs w r = s a (ŵ a ) ⊕ s b (ŵ b ) in Z L , and sends x i,j as the error events from node i to node j assuming node j attempts to decode w i at the end of phase ℓ using jointly typical decoding ands a ors b if available. Also we use the same definitions of A
P [E a,b ] ≤P [E (1) a,r ∪ E (2) b,r ∪ E (3) r,b ∪ E (3) a,b ] (17) ≤P [E (1) a,r ] + P [E (2) b,r ] + P [E (3) r,b |Ē (1) a,r ∩Ē (2) b,r ] + P [E (3) a,b |Ē (1) a,r ∩Ē (2) b,r ∩Ē (3) r,b ].(18)a ;Y (1) r |X (1) b =X (1) r =∅)−3ǫ) (19) P [E (2) b,r ] ≤ P [D (2) {b},{r} (w b )] + 2 nRb 2 −n·∆2,n(I(X (2) b ;Y (2) r |X (2) a =X (2) r =∅)−3ǫ) (20) P [E (3) r,b |Ē (1) a,r ∩Ē (2) b,r ] ≤ P [D (3) {r},{b} (s a (w a ) ⊕ s b (w b ))] + P [∪s a =sa(wa) D (3) {r},{b} (s a ⊕ s b (w b ))] ≤ P [D (3) {r},{b} (s a (w a ) ⊕ s b (w b ))] + 2 nRa0 2 −n·∆3,n(I(X (3) r ;Y (3) b |X (3) a =X (3) b =∅)−3ǫ) (21) P [E (3) a,b |Ē (1) a,r ∩Ē (2) b,r ∩Ē (3) r,b ] ≤ P [D (1) {a},{b} (w a )] + P [∪w a =wa D (1) {a},{b} (w a ), s a (w a ) = s a (w a )] ≤ P [D (1) {a},{b} (w a )] + 2 n(Ra−∆1,nI(X (1) a ;Y (1) b |X (1) b =X (1) r =∅)−Ra0+3ǫ)(22)
Since ǫ > 0 is arbitrary, with the proper choice of R a0 , the conditions of Theorem 3 and the AEP property, we can make the right hand sides of (19) -(22) vanish as n → ∞. Similarly, P [E b,a ] → 0 as n → ∞. By Fenchel-Bunt's theorem in [3], it is sufficient to restrict |Q| ≤ 4.
Theorem 4:
The capacity region of the bi-directional relay channel with the TDBC protocol is outer bounded by the union of
R a ≤ min{∆ 1 I(X (1) a ; Y (1) r , Y (1) b |X (1) b = X (1) r = ∅, Q),∆ 1 I(X (1) a ; Y (1) b |X (1) b = X (1) r = ∅, Q) + ∆ 3 I(X (3) r ; Y (3) b |X (3) a = X (3) b = ∅, Q)} R b ≤ min{∆ 2 I(X (2) b ; Y (2) r , Y(2)
a |X
(2) a = X
(2) r = ∅, Q),
∆ 2 I(X (2) b ; Y (2) a |X (2) a = X (2) r = ∅, Q) + ∆ 3 I(X (3) r ; Y (3) a |X (3) a = X (3) b = ∅, Q)} R a + R b ≤∆ 1 I(X (1) a ; Y (1) r |X (1) b = X (1) r = ∅, Q) + ∆ 2 I(X (2) b ; Y (2) r |X (2) a = X (2) r = ∅, Q)
over all joint distributions p(q)p (1) (x a |q)p (2) (x b |q) p (3) (x r |q) with |Q| ≤ 5 over the alphabet X a ×X b ×X r .
Remark: If the relay is not required to decode both messages, removing the constraint on the sum-rate R a + R b yields an outer bound.
Proof outline: The proof of Theorem 4 follows the same argument as in the proof of the converse part of Theorem 2.
C. HBC Protocol
Theorem 5: An achievable region of the half-duplex bi-directional relay channel with the HBC protocol is the closure of the set of all points (R a , R b ) satisfying
R a < min ∆ 1 I(X (1) a ; Y (1) r |X (1) b = X (1) r = ∅, Q) + ∆ 3 I(X (3) a ; Y (3) r |X (3) b , X (3) r = ∅, Q), ∆ 1 I(X (1) a ; Y (1) b |X (1) b = X (1) r = ∅, Q) + ∆ 4 I(X (4) r ; Y (4) b |X (4) a = X (4) b = ∅, Q) R b < min ∆ 2 I(X (2) b ; Y (2) r |X (2) a = X (2) r = ∅, Q) + ∆ 3 I(X (3) b ; Y (3) r |X (3) a , X (3) r = ∅, Q),∆ 2 I(X (2) b ; Y (2) a |X (2) a = X (2) r = ∅, Q) + ∆ 4 I(X (4) r ; Y (4) a |X (4) a = X (4) b = ∅, Q) R a + R b <∆ 1 I(X (1) a ; Y (1) r |X (1) b = X (1) r = ∅, Q) + ∆ 2 I(X (2) b ; Y (2) r |X (2) a = X (2) r = ∅, Q)+ ∆ 3 I(X (3) a , X (3) b ; Y (3) r |X (3) r = ∅, Q) over the joint distribution p(q)p (1) (x a |q)p (2) (x b |q)p (3) (x a |q)p (3) (x b |q) p (4) (x r |q) over the alphabet X 2 a × X 2 b × X r with |Q| ≤ 5. Proof outline: Generate random codewords x (1) a (w a ), x (2) b (w b ), x (3) a (w a ), x (3) b (w b ).
Relay r receives data from terminal nodes during phases 1 -3, which is decoded by the relay using a MAC protocol to recover w a , w b . Theorem 5 then follows the same argument as the proof of Theorem 3.
Theorem 6:
The capacity region of the bi-directional relay channel with the HBC protocol is outer bounded by the union of
R a ≤ min ∆ 1 I(X (1) a ; Y (1) r , Y (1) b |X (1) b = X (1) r = ∅, Q) + ∆ 3 I(X (3) a ; Y (3) r |X (3) b , X (3) r = ∅, Q), ∆ 1 I(X (1) a ; Y (1) b |X (1) b = X (1) r = ∅, Q) + ∆ 4 I(X (4) r ; Y (4) b |X (4) a = X (4) b = ∅, Q) R b ≤ min ∆ 2 I(X (2) b ; Y (2) r , Y (2) a |X (2) a = X (2) r = ∅, Q) + ∆ 3 I(X (3) b ; Y (3) r |X (3) a , X (3) r = ∅, Q), ∆ 2 I(X (2) b ; Y (2) a |X (2) a = X (2) r = ∅, Q) + ∆ 4 I(X (4) r ; Y (4) a |X (4) a = X (4) b = ∅, Q) R a + R b ≤∆ 1 I(X (1) a ; Y (1) r |X (1) b = X (1) r = ∅, Q) + ∆ 2 I(X (2) b ; Y (2) r |X (2) a = X (2) r = ∅, Q)+ ∆ 3 I(X (3) a , X (3) b ; Y (3) r |X (3) r = ∅, Q) over all joint distributions p(q)p (1) (x a |q)p (2) (x b |q)p (3) (x a , x b |q) p (4) (x r |q) with |Q| ≤ 5 over the alphabet X 2 a × X 2 b × X r . Remark:
If the relay is not required to decode both messages, then removing the constraint on the sum-rate R a + R b in the region above yields an outer bound.
Proof outline: The proof of Theorem 6 follows the same argument as the proof of the converse part of Theorem 2.
X r [k] = ∅, then Y a [k] = g ra X r [k] + Z a [k] and Y b [k] = g rb X r [k] + Z b [k]
and similar expressions hold if other pairs of nodes are silent, where the effective complex channel gain g ij between nodes i and j combines both quasi-static fading and path loss and the channels are reciprocal, i.e., g ij = g ji . For convenience, we define G ij := |g ij | 2 , i.e. G ij incorporates path loss and fading effects on received power.
Furthermore, we suppose the interesting case that G ab ≤ G ar ≤ G br . Finally, we assume full Channel State Information (CSI) at all nodes (i.e. each node is fully aware of g ab , g br and g ar ) and that each node has the same transmit power P for each phase, employs a complex Gaussian codebook and the noise is of unit power, additive, white Gaussian, complex and circularly symmetric. For convenience of analysis, we also define the function C(x) := log 2 (1 + x).
For a fading AWGN channel, we can optimize the ∆ i 's for given channel mutual informations in order to maximize the achievable sum rate (R a + R b ). First, we optimize the time periods in each protocol and compare the achievable sum rates obtained to determine an optimal transmission strategy in terms of sum-rate in a given channel. For example, applying Theorem 3 to the fading AWGN channel, the optimization constraints for the TDBC protocol are 4 :
R a ≤ min {∆ 1 C(P G ar ), ∆ 1 C(P G ab ) + ∆ 3 C(P G br )} (23) R b ≤ min {∆ 2 C(P G br ), ∆ 2 C(P G ab ) + ∆ 3 C(P G ar )}
We have taken |Q| = 1 in the derivation of (23) and (24), since a Gaussian distribution simultaneously maximizes each mutual information term individually as each node is assumed to transmit with at most power P during each phase. Linear programming may then be used to find optimal time durations. The optimal sum rate corresponding to the inner bounds of the protocols is plotted in Fig. 3. As expected, the optimal sum rate of the HBC protocol is always greater than or equal to those of the other protocols since the MABC and TDBC protocols are special cases of the HBC protocol. Notably, the sum rate of the HBC protocol is strictly greater than the other cases in some regimes. This implies that the HBC protocol does not reduce to either of the MABC or TDBC protocols in general.
In the MABC protocol, the performance region is known. However, in the other cases, there exists a gap between the expressions. An achievable region of the 4 protocols and an outer bound for the TDBC protocol is plotted in Fig. 4 (in the low and the high SNR regime). As expected, in the low SNR regime, the MABC protocol dominates the TDBC protocol, while the latter is better in the high SNR regime.
It is difficult to compute the outer bound of the HBC protocol numerically since, as opposed to the TDBC case, it is not clear that jointly Gaussian distributions are optimal due to the joint distribution p (3) (x a , x b |q) as well as the conditional mutual information terms in Theorem 6. For this reason, we do not numerically evaluate the outer bound. Notably, some achievable HBC rate pairs are outside the outer bounds of the MABC and TDBC protocols.
Fig. 1 .
1(i) Traditional approach, (ii) Naive four phase bi-directional cooperation, (iii) coded broadcast three phase protocol, (iv) two phase protocol.
∈ S}, a set of random variables with channel input distribution p (ℓ) (x S ) for phase ℓ, where x S := {x i |i ∈ S}. Lower case letters x i denote instances of the upper case X i which lie in the calligraphic alphabets X * i . Boldface x i represents a vector indexed by time at node i. Finally, it is convenient to denote by x S := {x i |i ∈ S}, a set of vectors indexed by time. Encoders are then given by functions X k i (W i,1 , . . . , W i,m , Y 1 i , . . . , Y k−1 i ), for k = 1, . . . , n and de-
a . The two distinct messages W a and W b are taken to be independent and uniformly distributed in the set of {0, . . . , ⌊2 nRa ⌋ − 1} := S a and {0, . . . , ⌊2 nRb ⌋ − 1} := S b , respectively. Then W a and W b are both members of the additive group Z L , where L = max(⌊2 nRa ⌋, ⌊2 nRb ⌋).
Fig. 2 .
2Proposed protocol diagrams. Shaded areas denote transmission by the respective nodes. It is assumed that all nodes listen when not transmitting.
a
(w a ) and x
b
(w b ) respectively. Relay r estimatesŵ a andŵ b after phase 1 using jointly typical decoding, then constructs w r =ŵ a ⊕ŵ b in Z L and sends x
r
(w r ) during phase 2.
T
) sequences of length n · ∆ ℓ,n according to the input distributions employed in phase ℓ. Also define the set of codewords x
i
(w i )|i ∈ S} and the events D
T }, where S and T are disjoint subsets of nodes.
tend to 0 as n → ∞. Similarly, P [E b,a ] → 0 as n → ∞. Converse: We use Lemma 1 to prove the converse part of Theorem 2. As we have 3 nodes, there are 6 cut-sets, S 1 = {a}, S 2 = {b}, S 3 = {r}, S 4 = {a, b}, S 5 = {a, r} and S 6 = {b, r}, as well as two rates R a and R b . The outer bound corresponding to S 1 is then
:
An achievable region of the half-duplex bi-directional relay channel with the TDBC protocol is the closure of the set of all points (R a , R b ) satisfying
r
(w r ) during phase 3. Decoding: Terminal nodes a and b estimate the indicess b (w b ) ands a (w a ) after phase 3 from x (3) r and then decodew b andw a if there exists a uniquew b ∈ S b,sb ∩ A (2) {b},{a} andw a ∈ S a,sa ∩ A (1) {a},{b} . Error analysis: Define E (ℓ)
T (w S ) as in the proof of Theorem 2. Then :
Fig. 3 .
3Achievable sum rates of the protocols (P = 15 dB, Gar = 0 dB)
Fig. 4 .
4Achievable rate regions and outer bounds with P = 0 dB (top) and P = 10 dB (bottom) (Gar = 0 dB, G br = 5 dB,G ab = −7 dB)
June 5, 2008 DRAFT IV. THE GAUSSIAN CASE In the following section, we apply the performance bounds derived in the previous section to the AWGN channel with pass loss. Definitions of codes, rate, and achievability in the memoryless Gaussian channels are analogous to those of the discrete memoryless channels. If X a [k] = ∅, X b [k] = ∅, X r [k] = ∅, then the mathematical channel model is Y r [k] = g ar X a [k] + g br X b [k] + Z r [k] and Y a [k] and Y b [k] are given by similar expression in terms of g ar , g br and g ab if only one node is silent. If X a [k] = X b [k] = ∅ and
The power constraint is satisfied almost surely as n → ∞ in the random coding argument for Gaussian input distributions with E[X 2 ] < P .June 5, 2008 DRAFT
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Information exchange in wireless networks with network coding and physical-layer broadcast. Proc. Conf. on Inf. Sci. and Sys. Conf. on Inf. Sci. and SysBaltimore, MD--, "Information exchange in wireless networks with network coding and physical-layer broadcast," in Proc. Conf. on Inf. Sci. and Sys., Baltimore, MD, Mar. 2005.
Network coding and random binning for multi-user channels. L L Xie, the 10th Canadian Workshop on Information Theory. Edmonton, Alberta, CanadaL. L. Xie, "Network coding and random binning for multi-user channels," in the 10th Canadian Workshop on Information Theory, Edmonton, Alberta, Canada, Jun. 2007, pp. 85 -88.
|
[] |
[
"Projection-Free Stochastic Bi-level Optimization",
"Projection-Free Stochastic Bi-level Optimization",
"Projection-Free Stochastic Bi-level Optimization",
"Projection-Free Stochastic Bi-level Optimization"
] |
[
"AmritZeeshan Akhtar ",
"SrujanSingh Bedi ",
"Teja Thomdapu ",
"Ketan Rajawat ",
"AmritZeeshan Akhtar ",
"SrujanSingh Bedi ",
"Teja Thomdapu ",
"Ketan Rajawat "
] |
[] |
[] |
Bi-level optimization, where the objective function depends on the solution of an inner optimization problem, provides a flexible framework for solving a rich class of problems such as hyper-parameter optimization and modelagnostic meta learning. This work puts forth the first Stochastic Bi-level Frank-Wolfe (SBFW) algorithm to solve the stochastic bi-level optimization problems in a projection-free manner. We utilize a momentum-based gradient tracker that results in a sample complexity of O( −3 ) for convex objectives and O( −4 ) for non-convex objectives. The stochastic compositional optimization problems, which are a special case of bi-level optimization problems entailing the minimization of a composition of two expected-value functions, are also considered within the same rubric.The proposed Stochastic Compositional Frank-Wolfe (SCFW) algorithm is shown to achieve a sample complexity of O( −2 ) for convex objectives and O( −3 ) for non-convex objectives, at par with the state-of-the-art rates of projection-free algorithms for single-level problems. The usefulness and flexibility of SBFW and SCFW algorithms is demonstrated via extensive numerical tests. We show that SBFW outperforms the state-of-the-art methods for the problem of matrix completion with denoising, and achieve improvements of up to 82% in terms of the wall-clock time required to achieve a particular level of accuracy. Furthermore, we demonstrate the improved performance of SCFW over competing projection-free variants on the policy evaluation problem in reinforcement learning.
|
10.1109/tsp.2023.3234462
|
[
"https://export.arxiv.org/pdf/2110.11721v2.pdf"
] | 256,149,421 |
2110.11721
|
f22f4fc9d3b4a2f0af13cb9419826dc7ea7e99f5
|
Projection-Free Stochastic Bi-level Optimization
AmritZeeshan Akhtar
SrujanSingh Bedi
Teja Thomdapu
Ketan Rajawat
Projection-Free Stochastic Bi-level Optimization
1
Bi-level optimization, where the objective function depends on the solution of an inner optimization problem, provides a flexible framework for solving a rich class of problems such as hyper-parameter optimization and modelagnostic meta learning. This work puts forth the first Stochastic Bi-level Frank-Wolfe (SBFW) algorithm to solve the stochastic bi-level optimization problems in a projection-free manner. We utilize a momentum-based gradient tracker that results in a sample complexity of O( −3 ) for convex objectives and O( −4 ) for non-convex objectives. The stochastic compositional optimization problems, which are a special case of bi-level optimization problems entailing the minimization of a composition of two expected-value functions, are also considered within the same rubric.The proposed Stochastic Compositional Frank-Wolfe (SCFW) algorithm is shown to achieve a sample complexity of O( −2 ) for convex objectives and O( −3 ) for non-convex objectives, at par with the state-of-the-art rates of projection-free algorithms for single-level problems. The usefulness and flexibility of SBFW and SCFW algorithms is demonstrated via extensive numerical tests. We show that SBFW outperforms the state-of-the-art methods for the problem of matrix completion with denoising, and achieve improvements of up to 82% in terms of the wall-clock time required to achieve a particular level of accuracy. Furthermore, we demonstrate the improved performance of SCFW over competing projection-free variants on the policy evaluation problem in reinforcement learning.
I. INTRODUCTION
We consider the two-level hierarchical optimization problem (P 1 ) min x∈X ⊂R m Q(x) := F (x, y (x)) (outer) (1) y * (x) ∈ arg min y G(y, x) (inner).
Here, the outer problem involves minimizing the objective function Q(x) with respect to x over the convex compact constraint set X ⊂ R m . The objective function is of the form Q(x) := F (x, y (x)), where y (x) is a solution of the inner optimization problem, which for a given x, entails minimizing the strongly convex function G(y, x)
with respect to optimization variable y. Observe that for bilevel problems of type P 1 , the inner and outer problems are inter-dependent and cannot be solved in isolation. Yet, these problems arise in a number of areas, such as meta-learning [1], continual learning [2], reinforcement learning [3], and hyper-parameter optimization [4], [5].
impractical in online settings, where a random subset of the entries of the matrix are observed at every iteration.
The bilevel optimization framework provides a way out, allowing the incorporation of denoisining step withing the inner-level subproblem. Mathematically, the bi-level matrix completion with denoising problem can be written as
min X * ≤α 1 |Ω 1 | (i,j)∈Ω1 (X i,j − Y i,j ) 2 ,(3)s. t. Y ∈ arg min V 1 |Ω 2 | (i,j)∈Ω2 (V i,j − M i,j ) 2 + λ 1 V 1 + λ 2 X − V 2 F ,
where M ∈ R n×m is the given incomplete noisy matrix, V 1 := i,j |V i,j | is the sum-absolute-value ( 1 ) norm, and λ 1 and λ 2 are regularization parameters. Note that the regularization over the discrepancy between X and denoised matrix Y results in bilevel formulation (3). A similar technique in deterministic settings is utilized in various other applications in machine learning and signal processing problems [37], [38]. Observe that (3) is in the form of bilevel formulation (1); however, when the entries are revealed in the form of randomly selected subsets
B. Related Work
We review some of the related work in the context of bi-level optimization, compositional optimization, and projection-free algorithms.
Bi-level optimization has had a long history, with the earliest applications in economic game theory [39]. Bi-level optimization has recently received great attention from the machine learning community due to the number of applications in the area [40]. A series of works that proposed to solve the problem of the form (1) has appeared recently [7]- [9], [19], [20], [26], [41]. Of these, the seminal works in [7], [8] proposed a class of double-loop approximation algorithms to iteratively approximate the stochastic gradient of the outer objective and incurred a sample complexity of O( −2 ) in order to achieve the -stationary point. The double loop structure of these approaches made them impractical for large-scale problems; [7] required solving an inner optimization problem to a predefined accuracy, while [8] required a large batch size of O( −1 ) at each iteration. To address this issue, various single-loop methods, involving simultaneous update of inner and outer optimization variables, have been developed [8], [9], [19], [20]. A single-loop two-time scale stochastic algorithm proposed in [19] incurred a suboptimal sample complexity of O( −2.5 ). This is further improved recently in [8], [9], [20], in which the authors have utilized the momentum-based variance reduction technique from [42] to obtain optimal convergence rates.
While all of the above-mentioned works seek to solve (1), they are projection-based and require a projection on to X at every iteration. In this work, we are interested in developing projection-free stochastic optimization algorithms for bi-level problems, which is still an open problem and the subject of the work in this paper.
Compositional optimization problems have been recently studied and various algorithms have been proposed in [13]- [16], [43]- [45]. The seminal work in [13] proposed a quasi-gradient method called stochastic compositional gradient descent (CGD) to solve the problem via a two time-scale approach. In [14], the authors proposed an accelerated SCGD method that achieved an improved sample complexity of O( −2 ) for convex objectives and O( −2.25 ) for non-convex objectives. Further, different variance-reduced SCGD variants have been proposed, such as SCVR [46], VRSC-PG [47], SARAH-Compositional [43], [48], and STORM-Compositional [44]. In the literature, we can also find some single time-scale algorithms to solve compositional problems [15], [16]. Work in [15] presented a nested averaged stochastic approximation (NASA) and proved a sample complexity of O( −2 ).
Recently, another single time-scale algorithm called the stochastically corrected stochastic compositional gradient method (SCSC) is proposed in [16] that converges at the same rate as the SGD methods for non-compositional stochastic optimization. It further adopted the Adam-type adaptive gradient approach and achieved the optimal sample complexity of O( −2 ). Again, all the above-mentioned algorithms either solve an unconstrained problem or use projection operation at each iteration to deal with the constraints. In this work, we developed a projection-free algorithm for compositional problems as well. Note that even-though compositional problems are a special case of bi-level problems, we have studied them separately in this work and proposed a novel projection-free algorithm specifically for compositional problems to achieve the optimal sample complexity.
Projection-free algorithms have been extensively studied to solve the single-level optimization problems of the form (1) in the literature [49]- [51]. A number of first-order projection-free algorithms have been developed for DRAFT stochastic optimization problems as well [10], [11], [52]- [54]. The stochastic FW method proposed in [52] achieves a sample complexity of O( −3 ) but requires an the batch size b = O(t), where t is the iteration index. The need for increasing batch sizes was dropped in [10], which worked with a standard mini-batch but still achieved the same sample complexity. Finally, an improved stochastic recursive gradient estimator-based algorithm called ORGFW was proposed in [11] and achieved a sample complexity of O( −2 ). For non-convex problems, [53] proposed an approach where the batch-size depends on the total number of iterations, resulting in a sample complexity of O( −4 ).
Later, work in [12] came up with a two-sample strategy and achieved O( −3 ) sample complexity. We remark that the idea of projection-free algorithms is limited only to single-level optimization problems in the existing literature.
Therefore, there are no corresponding oracle complexity bounds for projection-free algorithms in bilevel settings, and our work fills this crucial gap. We present our main contributions as follows.
C. Contributions
• Firstly, we propose a novel projection-free SBFW algorithm, utilizing the idea of momentum-based gradient tracking [42] in order to track the gradient of the outer objective function. The combination of this idea along with FW updates allows us to achieves the sample complexity of O( −3 ) and O( −4 ) for convex and non-convex cases, respectively (cf, Sec. IV).
• Secondly, in contrast to the existing literature, we consider the compositional problems (which is a special case of bi-level problems) separately in this paper and propose a novel Stochastic Compositional Frank Wolfe
(SCFW) algorithm. The SCFW algorithm is able to achieve a better convergence rate than bi-level problems under fewer assumptions on the inner objective functions. SCFW achieves the sample complexity of O( −2 ) and O( −3 ) for convex and non-convex cases, respectively (cf, Sec. IV). These results are interesting because they match with the existing best possible convergence rates of projection-based methods (which are computationally expensive) for compositional problems [14], [16], and projection-free methods for non-compositional (singlelevel) cases [11], [55].
• Finally, we test the proposed algorithm on matrix completion and the problem of policy evaluation in reinforcement learning and establish the efficacy of the proposed techniques as compared to state-of-the-art algorithms (cf. Sec. V-A). We achieve an improvement of up to 82% in the computation time for the proposed algorithm as compared to state-of-the-art methods.
A comprehensive list of all existing related works is provided in Table I. For bi-level problems, the second column of Table I represents the objective type (convex/non-convex) of outer function. The inner objective is strongly convex for all the methods in Table I.
Notation: First, we defined the compact notations we utilize in the convergence proofs. We denote column vectors with lowercase boldface x, its transpose as x , and its Euclidean norm by x . We use E t := E[·|F t ] to denote the conditional expectation with respect to given sigma field F t which contains all algorithm history (randomness) till step t. [3], [5], [19], the outer objective F (x, y (x)) := E θ [f (y (x); θ)] and the inner objective G(y, x) := E ξ [g(y, x; ξ)] are expected values of continuous and proper closed functions f : R m → R, and the function g is defined as g(·, ·, ξ) : R n × R m → R with respect to the independent random variables θ and ξ, respectively. Hence, the equivalent bi-level stochastic optimization problem is given by
x * := arg min x∈X ⊂R m Q(x) = E θ [f (x, y (x); θ)],(4a)s.t. y (x) ∈ arg min y∈R n E ξ [g(y, x; ξ)].(4b)
Besides the general form in (4), we are also interested in the special cases where it is possible to solve the inner optimization problem at a given x and obtain a smooth closed form expression for the optimal solution y (x). For instance, if the inner objective g is quadratic in y, i.e. g(x, y, ξ) = y − h(x, ξ) 2 , where h is a smooth function over x, then we can write y * (x) = h(x, ξ). Hence, the problem in (4) boils down to a stochastic compositional optimization problem given by
min x∈X ⊂R m C(x) := E θ [f (E ξ [h(x, ξ)], θ)] ,(5)
which involves the nested expectations in the objective. The problem in (5) has been independently considered in the literature and solved using two-time scale approaches [13], [56]. A single-time scale approach for the problem in (5) is also proposed in [15], [16] but all the existing approaches are projection-based. In this work, we are interested in developing first-order methods to solve the problem in (4) in a projection-free manner. We remark here that while the algorithms developed for problem in (4) could be readily applied to solve the problems of the form in (5), there is a scope to further propose faster algorithms for compositional problems in (5). Therefore, we will consider the compositional problems separately from bi-level problems and derive better convergence rates.
III. ALGORITHM DEVELOPMENT
In this section, we develop the proposed algorithm to solve the problem in (4). We note that solving the bi-level optimization problem in (4) is NP hard in general but we restrict our focus to problems where the inner objective is continuously twice differentiable in (x, y) and also strongly convex w.r.t y with parameter µ g > 0. Such an assumption is common in the related works [7]- [9] and ensures that for any x ∈ X , y (x) is unique. Applying any first order method to solve (4) requires the evaluation of the gradient of outer objective ∇Q(x) with respect to x. Let us denote the iteration index by t ∈ {1, 2, · · · , T }, and then we can write the standard projected gradient descent update as
x t+1 = P X [x t − α∇Q(x t )] ,(6)
Algorithm 1: Stochastic Bi-level Frank Wolfe Input: x 1 ∈ X , y 1 ∈ R m , η t , δ t , ρ t , β t , and d 1 = h 1 (θ 1 ; ξ 1 ) using (9)
1 for t = 2 to T do 2
Update approximate inner optimization solution
y t = y t−1 − δ t ∇ y g(x t−1 , y t−1 , ξ t ) 3
Gradient tracking evaluate h(x t , y t ; θ t , ξ t ) and h(x t−1 , y t−1 ; θ t , ξ t ) using (9) and compute
d t = (1 − ρ t )(d t−1 − h(x t−1 , y t−1 ; θ t , ξ t ) + h(x t , y t ; θ t , ξ t ) 4
Evaluate feasible direction s t = arg min s∈X s, d t
5
Update
solution x t+1 = (1 − η t )x t + η t s t Output: x T +1 orx selected uniformly from {x i } T i=1
where P X [·] denotes the projection onto the constraint set X and α is the constant step size. Note that the gradient calculation in (6) is achieved by application of implicit function theorem to the optimality condition for inner optimization problem ∇ y G(y, x) = 0, calculating total derivative followed by chain rule to obtain the expression
∇Q(x t ) =∇ x F (x t , y (x t )) − ∇ 2 xy G(y (x t ), x t ) × [∇ 2 yy G(y (x t ), x t )] −1 ∇ y F (x t , y (x t )).(7)
From (7), note that ∇Q(x t ) requires the information about y (x t ) which is not available in general, unless the second level problem (2) has a closed form solution. Hence, it is not possible to utilize gradient based algorithms to solve the problem in (4). This challenge is addressed in the literature via the utilization of approximate gradients [7]. Following a similar approach, we use a surrogate gradient defined as ∇S(x t , y t ) in place of original gradient ∇Q(x t ). The surrogate gradient is obtained by replacing y (x t ) in (7) with some y t ∈ R n (we will define the explicit value of y t later) to write
∇S(x t , y t ) = ∇ x F (x t , y t ) − ∇ 2 xy G(y t , x t ) × [∇ 2 yy G(y t , x t )] −1 ∇ y F (x t , y t ).(8)
Even after replacing y (x t ) with some y t ∈ R n , there is an additional challenge of evaluating the individual terms in (8) for the stochastic bi-level problems mentioned in (4). For instance, the term ∇
x F (x t , y t ) = E θ [∇ x f (y t ; θ)]
in (8) involves the evaluation of the expectation operator, which is not possible in practice due to the unknown data distribution. One standard approach is to use unbiased stochastic gradient instead of the original gradient, but an unbiased estimate of ∇S(x t , y) would still require the computation of Hessian inverse. To avoid such complicated matrix computations, we follow the approach presented in [19,Sec. E.4] and compute a mini-batch approximation of Hessian inverse using the samples returned by the sampling oracle. We assume the availability of sampling oracle such that for a given x ∈ X and y ∈ R n , it returns unbiased samples ∇ x f (x, y, θ), ∇ y f (x, y, θ), ∇ y g(x, y, ξ),
Algorithm 2: Stochastic Compositional Frank Wolfe Input: x 0 ∈ X , η t , δ t , ρ t , y 0 = h(x 0 , ξ 0 ), d 0 = ∇F (x 0 , y 0 , θ 0 , ξ 0 ) 1 for t = 1 to T do 2
Update inner function tracking
y t = (1 − δ t )(y t−1 − h(x t−1 , ξ t )) + h(x t , ξ t ) 3
Gradient tracking
d t = (1 − ρ t )(d t−1 − ∇F (x t−1 , y t−1 , θ t , ξ t )) + ∇F (x t , y t , θ t , ξ t ) 4
Evaluate feasible direction s t = arg min s∈X s, d t
5
Update
solution x t+1 = (1 − η t )x t + η t s t Output: x T +1 orx selected uniformly from {x i } T i=1
∇ 2 xy g(x, y, ξ), and ∇ 2 yy g(x, y, ξ) realized at random variables ξ and θ. Having access to such oracle, we can write the biased estimate (denoted by h(x t , y t ; θ t , ξ t )) of surrogate gradient ∇S(x t , y t ) in (8) as
h(x t , y t ; θ t , ξ t ) = ∇ x f (x t , y t ; θ t ) − M (x t , y t ;ξ t ) · ∇ y f (x t , y t ; θ t ),(9)
where
∇ x f (x t , y t ; θ t ) is an unbiased estimate of ∇ x F (x t , y t ), ∇ y f (x t , y t ; θ t ) is an unbiased estimate of ∇ y F (x t , y t ),
and M (x t , y t ;ξ t ) is a biased estimate of product ∇ 2 xy G(y t , x t ) · [∇ 2 yy G(y t , x t )] −1 . In the term M (x t , y t ;ξ t ),ξ t is defined asξ t := {ξ t,i : i ∈ {0, 1, · · · , k}} which represents a collection of (k + 1) i.i.d. samples ∇ 2 xy g(y t , x t ; ξ t,i ).
The explicit form of M (x t , y t ;ξ t ) is M (x t , y t ;ξ t ) =∇ 2 xy g(y t , x t ; ξ t,0 ) × k L g Π l i=1 I − 1 L g ∇ 2 yy g(y t , x t ; ξ t,i ) ,(10)
where l is selected uniformly from {0, 1, · · · , k − 1}. Further, for l=0, we use the convention
Π l i=1 I − 1 L g ∇ 2 yy g(x t , y t ; ξ t,i ) = I.
Hence, a stochastic version of the update in (6) would be given by
x t+1 = P X [x t − αh(x t , y t ; θ t , ξ t )] .(11)
Similar to the update in (11), different variants are proposed in the literature [7]- [9], [20]. But a significant challenge that remains un-addressed to date in the literature for the bi-level problems is associated with the projection operator in (11). Projection-based algorithms require to perform a computationally expensive projection step at each iteration t. One projection step is called projection oracle call. For instance, to achieve suboptimality, the projected subgradient method [57] entails O( −2 ) projection oracle calls. The projection is easy to evaluate when the constraint set is a simple convex set (onto which projection operation is computationally cheap such as probability simplex) or has a closed-form solution (set of unit-ball). However, the projection step is often computationally costly (e.g., nuclear norm constraint), and its complexity could be comparable to the problem at hand [23]. In the next subsection, we obviate the issue related to projections by proposing projection-free algorithms for both the bi-level and compositional stochastic optimization problems, which is the key novel aspect of work in this paper.
A. Stochastic Projection-Free Bi-level Optimization Algorithm
Before proceeding towards the design of a projection-free algorithm, a discussion regarding the particular choice of y t in (11) is due. A popular choice (see [9], [15], [17], [19]) for y t is the stochastic gradient descent update for the inner optimization problem given by
y t = y t−1 − δ t ∇ y g(y t−1 , x t−1 ; ξ t ),(12)
where ∇ y g(y t−1 , x t−1 ; ξ t ) is the unbiased estimate of the gradient E ξ [∇ y g(y t−1 , x t−1 ; ξ)], and δ t denotes the step size. Now we are ready to propose the first projection-free algorithm for the bi-level stochastic optimization problems. We propose to use a conditional gradient method (CGM) based approach instead of calculating the projection in (11). That is, we solve a linear minimization problem to find a feasible direction s t ∈ X for a given stochastic gradient direction h(x t , y t ; θ t , ξ t ), given by, s t := arg min s∈X s, h(x t , y t ; θ t , ξ t ) . This reduces the optimization problem of evaluating the projection operator in (11) to solving a linear program which is easier to solve in practice. Hence, the iterate in (11) gets modified to
s t := arg min x∈X x, h(x t , y t ; θ t , ξ t )(13)x t+1 =(1 − η t+1 )x t + η t+1 s t ,(14)
where η t is the step size. To this end, we would like to emphasize that naive use of h(x t , y t ; θ t , ξ t ) in (13) for the evaluation of s t which is then used in (14) can result in the iterate divergence due to the non-vanishing variance of the gradient estimate h. The standard approach to deal with this issue is to use a biased gradient estimate with low variance instead of an unbiased one. For example, a mini-batch approximation is proposed in [52], [53] with linearly increasing batch size with iteration index. Such an approach runs into memory issues when utilized in practice.
Another line of work suggests the use of gradient tracking given by
d t = (1 − ρ t )(d t−1 ) + ρ t h(x t , y t ; θ t , ξ t ),
where ρ t being the tracking parameter, which does not suffer from the problem of increasing batch size. But such gradient tracking schemes are shown to be suboptimal even for single-level optimization problems [10], so no scope for much harder bi-level problems are considered in this work. To address the issue of memory and iterate divergence, we took motivation from the momentum-based approach in [42] and propose to use the following gradient tracking scheme given by
d t =(1 − ρ t )(d t−1 − h(x t−1 , y t−1 ; θ t , ξ t )) + h(x t , y t ; θ t , ξ t ).(15)
We remark that such a tracking technique is recently utilized in [20] for projection-based bi-level optimization problems. However, in this work, our focus lies in developing projection-free algorithms, and hence analysis is significantly different from [20]. We will show in Lemma 2 ( supplementary material) that momentum-based tracking technique such as in (15) results in a reduced variance for the gradient estimate, hence eventually resulting April 5, 2022 DRAFT in improving the convergence rate of the algorithm to achieve the optimality guarantees. Now we proceed to modify the updates in (13)- (14) in order to propose the main projection-free updates as
s t := arg min x∈X x, d t (16) x t+1 =(1 − η t+1 )x t + η t+1 s t .(17)
The proposed algorithm is summarized in Algorithm 1. The details about the specific choice of step sizes δ t , ρ t ,
and η t are discussed in Sec. IV.
B. Stochastic Projection-Free Compositional Optimization Algorithm
In this section, we propose a separate novel algorithm for problems of the form in (5) in a projection-free manner.
To solve compositional problems in (5), the classical SGD methods requires the unbiased samples of the actual
gradient ∇C(x t ) given by ∇C(x t , θ t , ξ t ) := ∇h(x t , ξ t ) ∇f (E ξ [h(x t , ξ)], θ t ).
That is, for a given vector x t and random sample ξ t and θ t , this gradient computation requires E ξ [h(x t , ξ)] which is not available on a single query to sampling oracle. The existing projection-based stochastic compositional methods adopt some kind of tracking approach to approximate E ξ [h(x t , ξ)] using the samples returned by the oracle. For instance, the seminal work SCGD in [13] approximated h(x t ) at each iteration as
y t = (1 − δ t )(y t−1 ) + δ t h(x t , ξ t ) where δ t is a diminishing step size but achieved suboptimal convergence rate of O(1/ 4 ).
Recently, work in [16] proposed Adam-SCSC that introduces a stochastic correction to the original SCGD [13] using Adam-type adaptive gradient approach and achieved the convergence rate of O(1/ 2 ). However, all these existing algorithms require projection onto the set X at each iteration. In the proposed algorithm SCFW, slightly different from SBFW, we proposed to use momentum-based tracking technique to estimate y t as
y t = (1 − δ t )(y t−1 − h(x t−1 , ξ t )) + h(x t , ξ t ).(18)
After utilizing y t , we denote the stochastic gradient estimate of compositional objective as ∇C(x t , y t , θ t , ξ t ) =
∇h(x t , ξ t ) ∇f (y t , θ t ), and then similar to SBFW, we proposed to use the momentum based method to track the compositional gradient as
d t = (1 − ρ t )(d t−1 − ∇C(x t−1 , y t−1 , θ t , ξ t )) + ∇C(x t , y t , θ t , ξ t ).(19)
We will establish that using such momentum based tracking technique for both the inner function (18) as well as on the gradient of the objective (19) not only reduces the variance of the approximation noises with each iteration but also helps in establishing optimal convergence results. The rest of the steps essentially remains the same as SBFW; that is, we solve a linear minimization problem s t = arg min s∈X s, d t and then update the iterate as
x t+1 = (1 − η t+1 )x t + η t+1 s t .
The proposed SCFW algorithm is summarized in Algorithm 2. The details about the specific choice of step sizes δ t , ρ t and η t are provided in Sec. IV.
IV. CONVERGENCE ANALYSIS
This section presents the convergence rate analysis for the proposed Algorithm 1 and Algorithm 2. We provide the convergence rate results for both convex and non-convex objectives. Note that the inner objective is considered April 5, 2022 DRAFT to be strongly convex for both cases. Before proceeding towards the analysis, we first discuss the convergence criteria we utilize to evaluate the performance of the proposed algorithms.
Convergence Criteria: For the convex objective Q(x), we use the expected suboptimality
E[Q(x T ) − Q(x )]
after T number of iterations where x * is as defined in (4). The expectation here is with respect to the randomness in the objective as well as iterate x T . However, for non-convex Q(x), we use the expected Frank-
Wolfe gap E[G(x)]
as the performance metric where G(x) is defined as
G(x) := max v∈X v − x, −∇Q(x) .(20)
The Frank-Wolfe gap is a standard performance metric for the constrained non-convex settings as mentioned in [12], [53], [58]. From the definition in (20), we note that G(x) ≥ 0 for all x ∈ X and if ∃x ∈ X such that G(x ) = 0, then x is a first-order stationary point. Further, for comparison with other methods, we will use the stochastic first-order oracle (SFO) complexity which is a commonly used metric to compare stochastic first-order methods. SFO defines the total number of times an algorithm is required to call a first-order oracle (which provides stochastic gradients) to reach -approximate (or stationary) solution. Another metric that is often used to evaluate stochastic projection-free algorithms is the linear minimization oracle (LMO) complexity, which is the total number of times an algorithm needs to solve the linear minimization problem to reach -approximate (or stationary) solution.
However, since both the proposed SBFW and SCFW algorithms require O(1) LMO calls at each iteration, the LMO complexity is similar to the SFO complexity. We will discuss the SFO complexity of each algorithm after the main theorems.
Next, we state the assumptions required to perform the analysis in this work that is similar to the assumptions considered in the existing literature [18], [19].
Assumption 1 For the bi-level stochastic optimization problem in (4), we need the following statements to hold true for the analysis.
(i) The stochastic gradient estimates satisfies
E[ ∇ x f (x, y) − ∇ x f (x, y, θ) 2 ] ≤ σ 2 x , E[ ∇ y f (x, y) − ∇ y f (x, y, θ) 2 ] ≤ σ 2 y , E[ ∇ 2 xy g(x, y) − ∇ 2 xy g(x, y, θ) 2 ] ≤ σ 2 xy , E[ ∇ y g(x, y) − ∇ y g(x, y, ξ) 2 ] ≤ σ 2 g , for some σ 2 x > 0, σ 2 y > 0, σ 2 xy > 0, and σ 2 g > 0. (ii) For any given x ∈ X , the terms ∇ x f (x, y), ∇ y f (x, y), ∇ y g(x, y), ∇ 2
xy g(x, y) and ∇ 2 yy g(x, y) are Lipschitz continuous with respect to y with Lipschitz parameter L fx , L fy , L g , L gxy and L gyy , respectively.
(iii) For any given y ∈ R n , the terms ∇ y f (x, y), ∇ 2 xy g(x, y) and ∇ 2 yy g(x, y) are Lipschitz continuous with respect to x with positive constants L fy , L gxy and L gyy , respectively. Note that for the sake of simplicity, here we slightly abused the notation and used the same constants as in Assumption 1 (ii).
April 5, 2022 DRAFT
(iv) For all x ∈ X and y ∈ R n , it holds that E[ ∇ y f (x, y) ] ≤ C y and E[ ∇ 2 xy g(x, y) ] ≤ C xy for some for constants C y > 0 and C xy > 0.
(v) The inner function g(x, y) is µ g -strongly convex in y for any x ∈ X .
For SCFW, the basic assumptions are essentially the same as SBFW; however, for the sake of clarity, we state them here explicitly. It must be noted that for compositional problems, the inner function h(·) is not necessarily required to be strongly convex.
Assumption 2 For the compositional stochastic optimization problem in (4), we need the following statements to hold true for the analysis.
(i) (Sampling oracle) For a given x ∈ X and y ∈ R n , the oracle returns samples h(x, ξ), ∇h(x, ξ) and ∇f (y, θ) for some random sample ξ and θ. These samples are unbiased that is
E ξ [h(x, ξ)] = h(x) and E {ξ,θ} [∇h(x, ξ) ∇f (y, θ)] = ∇h(x) ∇f (y). (ii) (Bounded Variance) The inner function h(·) has bounded variance, i.e., E[ h(x, ξ) − h(x) ] ≤ σ 2 h .(21)
(iii) (Lipschitz continuous) The functions f (·) and h(·) are smooth, i.e., for any y 1 ,
y 2 ∈ R n ∇f (y 1 , θ) − ∇f (y 2 , θ) ≤ L f y 1 − y 2 ,(22)
and for any x 1 , x 2 ∈ X , it holds that
∇h(x 1 , ξ) − ∇h(x 2 , ξ) ≤ L h x 1 − x 2 .(23)
(iv) (Bounded second moments) The stochastic gradients of functions f (·) and h(·) have bounded second order moments, i.e., for any x ∈ X , and y ∈ R n
E[ ∇f (y, θ) 2 ] ≤ M f ; E[ ∇h(x, ξ) 2 ] ≤ M h .(24)
Note that for SCFW, we have assumed both the inner and outer functions to be smooth; the resulting composition function C(x) will also be smooth [59]. Its smoothness parameter L F can be easily obtained using Assumption 2 Lemma 1 Consider the proposed Algorithm 1 and x t be the iterates generated by it, then for the algorithm
(iii-iv) as L F = M 2 h L f + M f L h .parameter δ t ≤ min{ 2 3µg , µg 2(1+σ 2 g )L 2
g } and step size η t , the optimality gap of the lower level problem satisfies
E t [ y t − y (x t ) 2 ] ≤ 1 − δ t µ g 2 E t [ y t−1 − y (x t−1 ) 2 ] + 2η 2 t−1 δ t µ g C xy µ g 2 D 2 + 4δ 2 t σ 2 g .(25)
The proof Lemma 1 is provided in section C-A of the supplementary material. Lemma 1 quantifies how close y t is from the optimal solution of inner problem at x t and establishes the progress of the inner-level update.
April 5, 2022 DRAFT Lemma 2 Consider the proposed Algorithm 1 and x t be the iterates generated by it, then for the algorithm parameter δ t , ρ t and η t , we have
E t [ d t − ∇S(x t , y t ) − B t 2 ≤ (1 − ρ t ) 2 E t [ (d t−1 − ∇S(x t−1 , y t−1 ) − B t−1 ) 2 ] + 4L k δ 2 t L 2 g + 4L k η 2 t−1 D 2 + 2ρ 2 t σ 2 f .(26)
The proof of Lemma 2 is provided in section C-B of the supplementary material.
Lemma 2 describe the tracking error in the gradient approximation ∇S(x, y) at point x t and y t . The presence of (1 − ρ t ) 2 term in RHS of (26) shows that the variance of the tracking error reduces with iteration.
Next, we utilize Lemma 1-(2) to establish a bound on the gradient estimation error for ∇Q presented in the form of following Corollary 1.
Corollary 1 For the proposed Algorithm 1, with δ t = 2a0 t q , where a 0 = min{ 1 3µg , µg 2(1+σ 2 g )L 2 g }, ρ t = 2 t q , β t ≤ CxyCy µg(t+1) q and η t ≤ 2 (t+1) 3q/2 for 0 < q ≤ 1, the gradient approximation error E ∇Q(x t ) − d t 2 converges to zero at the following rate E ∇Q(x t ) − d t 2 ≤ C 1 (t + 1) q ,(27)
where
C 1 = 3(max{2 q y 1 − y (x 1 ) 2 , (2(C xy /µ g ) 2 D 2 +16a 2 0 σ 2 g )/(2a 0 −1)}+ CxyCy µg +8(2L k L 2 g +L k D 2 +σ 2 f )).
The proof of Corollary 1 is provided in section C-C of the supplementary material. The result in Corollary 1 is presented in general form and indicates that for properly chosen parameters q, the gradient approximation error in expectation decreases at each iteration and approaches zero asymptotically. We will use this upper bound to prove the convergence of the proposed algorithm SBFW for a different types of objective functions in the following theorem. Note that in the analysis of Corollary 1 we have set β t ≤ CxyCy µg(t+1) q . To satisfy this condition, the number of samples k at iteration t needed to approximate the Hessian inverse in (10)
is k = O(log((1 + t) q )).
Now we are ready to present the first main result of this work as Theorem 1.
Theorem 1 (Convergence Rate of SBFW) Consider the proposed Algorithm 1 and suppose Assumption 1 is satisfied. Then,
(i) (Convex Bi-Level): If Q is convex on X and we set δ t = a0 t 2/3 , where a 0 = min{ 2 3µg , µg 2(1+σ 2 g )L 2 g }, ρ t = 2 t 2/3 , η t = 2 t+1 and k = 2Lg 3µg (log(1 + t))
, then the output is feasible x T +1 ∈ X and satisfies
E[Q(x T +1 ) − Q(x )] ≤ 12D √ C 1 5(T + 1) 1 3 + 2L Q D 2 (T + 1) .(28)
(ii) (Non-Convex Bi-level): If Q is non-convex and we set δ t = a0 1 + t)) , then the output is feasiblex ∈ X and satisfies
t 1/2 , where a 0 = min{ 2 3µg , µg 2(1+σ 2 g )L 2 g }, ρ t = 2 t 1/2 , η t =E[G(x)] ≤ Q(x 1 ) − Q(x ) (T + 1) 1/4 + 16D √ C 1 3(T + 1) 1/4 + L Q D 2 (T + 1) 3/4 , here L Q = (L fy +L)Cxy µg + L fx + C y Lg xy Cy µg + Lg yy Cxy µ 2 g and C 1 = 3(max{2 y 1 − y (x 1 ) 2 , (2(C xy /µ g ) 2 D 2 + 16a 2 0 σ 2 g )/(2a 0 − 1)} + CxyCy µg + 8(2L k L 2 g + L k D 2 + σ 2 f )).
The proof of Theorem 1 is provided in Appendix A in the supplementary material. Theorem 1 shows that the optimality gap for SBFW decays as O(T −1/3 ) for general convex objectives and for non-convex case it establishes an upper bound on the expected Frank-Wolfe gap for the iterates generated by SBFW that converges to zero at least at the rate of O(T −1/4 ), where T is the total number of iterations. It must be noted that for at each iteration, SBFW requires 2k + 1 gradient samples to obtain gradient estimate: 2k samples for outer gradient estimate (15) and one sample for inner variable update (12). Further, we have set k ≈ O(log(t)). Hence, the SFO complexity of SBFW is comparable to the projection-based state-of-the-art methods [7], [9], [17], [19], [20], however, it shows slightly worse performance in terms of the outer level complexity. This is not surprising as we are tackling the outer level in a projection-free manner.
Next in order to study the convergence rate of the proposed algorithm SCFW, we state some results on the error of the inner function approximation and gradient approximation in the form of following lemma.
Lemma 3 Consider the proposed Algorithm 2 and x t be the iterates generated by it, then the sequence y t − h(x t ) 2 converges to zero at the following rate
E y t − h(x t ) 2 ≤ (1 − δ t ) 2 E (y t−1 − h(x t−1 )) 2 + 2δ 2 t σ 2 h + 2(1 − δ t ) 2 η 2 t−1 M h D 2 .(29)
The proof of Lemma 3 is provided in Appendix A-A. It shows that the distance between y t and h(x t ) decreases with iteration in expectation. Intuitively, this means that our tracking variable y t will converge to the unknown h(x t ). This result will be used to obtain the convergence rates of the proposed SCFW algorithm for different kinds of objective functions, which we provide as Theorem 2.
Note that as instead of directly using the gradient samples, we are approximating it using tracking technique, we provide an upper bound on the gradient approximation error as follows.
Corollary 2 For the proposed Algorithm 2, with δ t = 2 t p , ρ t = 2 t p and η t ≤ 2 (t+1) p for 0 < p ≤ 1, the gradient approximation error E ∇C(x t ) − d t 2 converges to zero at the following rate
E ∇C(x t ) − d t 2 ≤ A 1 (t + 1) p(30)
where
A 1 := 32[M h (M f + 28L f σ 2 h ) + (M f L h + 28M 2 h L f )D 2 ].
The proof of Corollary 2 in section D-A of the supplementary material. The result in Corollary 2 indicates that for properly chosen parameters p, the gradient approximation error in expectation decreases at each iteration and approaches zero asymptotically. We will use this upper bound to prove the convergence of the proposed algorithm SCFW, which we will discuss next.
Theorem 2 (Convergence Rate of SCFW) Consider the proposed Algorithm 2 and suppose that Assumption 2 is satisfied. Then, (i) (Convex Compositional): If C is convex on X and we set ρ t = 2 t , δ t = 2 t and η t = 2 t+1 , then the output is feasible x T +1 ∈ X and satisfies
E[C(x T +1 ) − C(x )] ≤ 8D √ A 1 3(T + 1) 1 2 + 2L F D 2 (T + 1) ,(31)
(ii) (Non-Convex Compositional): If C is non-convex and we set ρ t = 2 t 2/3 , δ t = 2 t 2/3 and η t = 2 (T +1) 2/3 , then the output is feasiblex ∈ X and satisfies
E[G(x)] ≤ C(x 1 ) − C(x ) (T + 1) 1/3 + 6D √ A 1 (T + 1) 1/3 + L F D 2 (T + 1) 2/3 , where L F = M 2 h L f + M f L h and A 1 := 32[M h (M f + 28L f σ 2 h ) + (M f L h + 28M 2 h L f )D 2 ].
The proof of Theorem 2 is provided in section D-B of the supplementary material. Interestingly, the optimality gap decays as O(T −1/2 ), which is the optimal rate even for projected stochastic compositional optimization problems with general convex objectives [14]. For the non-convex case, it establishes an upper bound on the expected Frank-Wolfe gap for the iterates generated by SCFW and shows that it converges to zero at least at the rate of
O(T −1/3 ),
where T is the total number of iterations. It must be noted that at each iteration, SCFW requires only two gradient samples to obtain gradient estimate. Hence, it has SFO complexity is O( −2 ) and O( −3 ) for convex and nonconvex objectives, respectively. Interestingly, these results match with the state-of-the-art methods [11], [12] for projection-free single level (or non-compositional) stochastic optimization.
Remark: Note that SCFW analysis is not a straightforward extension to SBFW, this is because we specifically re-derived inequality to exploit the special case of SCFW where we have an additional tracking step for inner function (h(x, ξ) of eq. (5)), hence requiring a separate analysis. In fact, note that we are able to achieve better
convergence rate for SCFW (O( −2 ), O( −3 )) as compared to SBFW (O( −3 ), O( −4 )).
V. NUMERICAL EXPERIMENTS
In this section, first, we consider the problem of low-rank matrix completion formulated in (3) to illustrate the performance of our proposed SBFW algorithm. Then we perform another experiment on sparse policy value evaluation to highlight the importance of the proposed SCFW algorithm. All the experiments are performed in MATLAB R2018a with Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz.
A. Importance of SBFW (cf. Algorithm 1)
We start with emphasizing that there are two challenges (C1 and C2) involved in solving the problem in (3); C1: the problem is stochastic bi-level in nature, and C2: the nuclear norm constraints are expensive to project onto. to the standard single level problem (solved via SFW [10]). We note that SBFW achieves a lower normalized error as compared to SWF (Fig. 1a). This advantage of SBFW is further confirmed in Fig. 1b, which plots the distance from the optimal value ē −ē0 with respect to the noise factor in the matrix completion problem. The proposed algorithm is able to better handle the noise in the observed matrices by a significant margin.
We address both the challenges by solving the problem in (3) via the proposed SBFW algorithm. Note that (3) is of the form (4) with f (·) = g(·) = · 2 F , and (θ t , ξ t ) being the independent random subset of entries revealed at every iteration t.
Synthetic Dataset Results:
We start with performing experiments to highlight the importance of denoising in matrix completion which lead to the bi-level formulation of the problem in (3). We follow the experimental setting of [10] and start with forming an observation matrix as M =X+E. Here,X = WW T with W ∈ R n×r containing normally distributed independent entries, and the noise matrix E =n(L + L T ) where L ∈ R n×n contains normally distributed independent entries andn ∈ (0, 1), is the noise factor. For the simulations, we set n = 250, r = 10, and α = X * . Further, we define the set of observed entries Ω by sampling M uniformly at random with probability 0.8.
We start with settingn = 0.5 and solve the problem in (3) using the proposed SBFW algorithm and compare it with the state-of-the-art conditional gradient method called SFW [10]. For SBFW, we set the step sizes as dictated in theory and set λ 1 = λ 2 = 0.05, while for SFW, the step sizes are set as defined in [10]. We use a batch size of b = 250 for both the algorithms and run them for 10 4 iterations. The performance is analyzed in terms of normalized error, defined asē
= (i,j)∈Ω (X i,j −X i,j ) 2 (i,j)∈Ω (X i,j ) 2(32)
where X is the output generated by the algorithm.
The evolution of the normalized error is shown in Fig. 1a. Since SFW is a projection-free algorithm, it addresses the challenge C2 and converges fast but to a suboptimal point, while SBFW converges to better accuracy but slowly.
Hence, the proposed algorithm solves the bi-level matrix completion problem efficiently (addressed challenge C1).
This also justifies the claim that additional denoising step improves the quality of matrix completion. Then, to further investigate the effectiveness of the proposed algorithm in addressing C1, we solve the problem for different 100k dataset with the other state-of-the-art algorithms such as TTSA [19], MSTSA [18], and SUSTAIN [20].We note that SBFW takes the least amount of time (shown in the inner bar plot) to attain same level of accuracy.
noise factorn ∈ (0, 1) and compare the normalized error in the solution at eachn with the normalized error obtained for zero noise case (i.e.n = 0) denoted asē 0 (shown in Fig. 1b). We note that the growth in the error difference is much slower for SBFW than SFW, which shows the effectiveness of the proposed algorithm in dealing with noise.
Real Dataset Results:
To test the scalability of our proposed projection-free algorithm, we run experiment over large size matrices of MovieLens 1 datasets, which contains user ratings of movies ranging from 0 to 5. Since, SBFW is a single loop algorithm, we compare its performance of SBFW with other state-of-the-art single loop projection-based bilevel algorithms such as SUSTAIN [20], TTSA [19], and MSTSA [18]. We start with Movielens 100k dataset of 10 5 ratings from 1000 users for 1700 movies. This dataset is denoted by observation matrix M of size 1000 × 1700. For the simulations, we define the set of observed entries Ω by sampling the matrix uniformly at random from M with a batch size of b = 5000. Fig. 2 plots the evolution of normalized error for 2500 iterations for all the algorithms. We note that the proposed algorithm is not the best in terms of the convergence rate when compared to projection-based schemes, which is expected from the slower theoretical convergence rates. However, when compared in terms of amount of clock time required to achieve the same level of normalized error (Fig. 2), he proposed scheme outperforms the other state-of-the-art methods. This gain is coming from the projection-free nature of the proposed algorithm, and we no longer required to perform a complicated projection at each iteration. To further highlight the importance of projection-free bilevel algorithm in practice, we perform additional experiments on larger dataset (of MovieLens 1M) which contains 1 million ratings from 6000 users and for 4000 movies. We plot the evolution of normalized error with time in Fig. 3, where we only compare SBFW against SUSTAIN which is the state-of-the art projection based bilevel algorithm. It is interesting to note that even though SUSTAIN has a 1 https://grouplens.org/datasets/movielens/ better theoretical convergence rate, it shows inferior performance in actual computation time (due to the projection operation) compared to SBFW as evident from Fig. 3. In Table II We have analyzed the performance of SBFW on different datasets but a remark is due on the importance of proposed SCFW and studying its convergence rate results separately in this work. This is detailed next.
B. Importance of SCFW (cf. Algorithm 2)
In this subsection, we consider the problem of policy evaluation which lies at the heart of reinforcement learning as discussed in [14]. We perform this experiments to justify the claim that for compositional stochastic problems, the proposed SCFW (cf. Algorithm 2) exhibits a faster convergence rate as compared to SBFW (cf. Algorithm 1).
Consider a Markov Decision Process (MDP) with finite state space S and action space A. For a fixed policy π which maps the current state s to action a ∈ A, the value function V π (s) for state s is given by
V π (s) = E {r s,ŝ + γV π (ŝ)|s, π} ∀s,ŝ ∈ S,(33)
where r s,ŝ is the reward for transitioning from s toŝ, γ ∈ (0, 1) is the discount factor, and the expectation is taken over all the possible future statesŝ conditioned on current state s and policy π. Looking at the bellman equation in (33), it is clear that evaluating the value function at all the states is impractical when |S| is moderately large.
Hence, we consider a linear function approximation for the value function such that V π (s) = φ T s w , for some w ∈ R m . Here, φ s ∈ R m denotes the m dimensional state features for state s. The goal is to learn an optimal w to obtain a suitable linear function approximation for the value function. The problem of finding w can be formulated as
min w 1≤α S s=1 φ T s w − q π,ŝ (w) 2 ,(34)
where q π,ŝ (w) = E r s,ŝ + γφ T s w|s, π = ŝ P π sŝ r s,ŝ + γφ T s w . For the purposeful behavior, we follow the same approach as in [60], where the agent favors a single action at each state. Out of 3, one action is randomly selected and is assigned with the probability 0.9, and the others are evenly assigned with probabilities. The feature vector of each state has dimension m = 100. However, the additional 1 constraint ensures the sparsity in optimization variable w.
In our experiment, we set α = 10 −1 and run both the proposed SCFW and SBFW algorithm for 1000 iterations.
The results of the experiment is reported in terms of objective convergence w t − w 2 in Fig. 4. Here, w t is the algorithmic solution at iteration t. The optimal solution w is obtained by running the algorithm for 10 5 iterations,
given that the complete information of transition probabilities is known. Observe that under the same setting, SCFW performs better as compared to SBFW. This performance gain is actually obtained due to the additional tracking of the inner function (see (18)) in SCFW algorithm. We further provide comparison with the proximal ASC-PG algorithm [14], where λ w 1 is added as a regularizer to the objective function. The parameter λ is tuned so that w 1 ≤ α holds. As expected, all the algorithms have managed to reduce the error with iteration. However, note that ASC-PG is solving a relaxed version of the constraint problem by including the 1 norm constraint as regularizer, that requires tuning of parameter λ. In fact, it might be possible to attain a better convergence plot for ASC-PG by tuning λ further, but at an additional cost of constraint violation. In contrast, the proposed algorithm is converging while strictly satisfying the constraint. Note that this experiment is just a representative example for SCFW and comparing with the gamut of RL algorithms is beyond the scope of the paper and hence we provide comparison only with the state-of-the-art proximal compositional algorithm [14].
VI. CONCLUSION
This paper presents the first projection-free algorithm for stochastic bi-level optimization problems with a strongly convex inner objective function. We utilize the concept of momentum-based tracking to track the stochastic gradient estimate and establish the oracle complexities of the proposed SBFW algorithm for the convex and non-convex outer objective functions. We also develop the first projection-free algorithm called SCFW for stochastic compositional problems. We show that tracking both the inner function and the gradient of the objective function with momentum technique reduces approximation noise which eventually helps obtain the optimal convergence rates. Numerical results show that the proposed projection-free variants have a significantly reduced wall-clock times as compared to their projection-based counterparts.
APPENDIX A PROOF OF THEOREM 1
From the initialization of variable x, we have x 1 ∈ X . Also since we obtain s t solving a linear minimization problem over the set X , we have s t ∈ X . Thus, x t+1 which is a convex combination of x t and s t , i.e. x t+1 = (1 − η t+1 )x t + η t+1 s t will also lie in the set X . Hence x T +1 ∈ X andx ∈ X . Now, starting with definition of Q(·), we have Q(x) = E θ [f (x, y (x); θ)]. Also note that we have set k = qLg µg (log(1 + t)), this ensures that the condition β t ≤ CxyCy µg(t+1) q required in the analysis of Corollary (1) is satisfied. Hence, we can use results from Corollary (1) with q = 2/3 for convex case and q = 1/2 for non-convex case.
1) Proof of Statement (i) (Convex case):
Using the smoothness assumption of Q we can write
Q(x t+1 )−Q(x t ) ≤ ∇Q(x t ), x t+1 − x t + L Q 2 x t+1 −x t 2 = η t ∇Q(x t ), s t − x t + L Q η 2 t 2 s t − x t 2 ,(35)
where L Q = (L fy +L)Cxy µg
+ L fx + C y
Lg xy Cy µg + Lg yy Cxy µ 2 g (see Lemma 4). Here, in the last expression we have
replace term x t+1 − x t = η t (s t − x t )
. Now adding and subtracting η t d t , s t − x t in (35) we get
Q(x t+1 ) ≤ Q(x t ) + η t ∇Q(x t ) − d t , s t − x t + η t d t , x − x t + L Q η 2 t D 2 2 ,(36)
here in last the inequality is obtained using optimality of s t . Now introducing η t ∇Q(x t ), x − x t in RHS of (36) and regrouping the terms we obtain
Q(x t+1 ) − L Q η 2 t D 2 2 ≤ Q(x t ) + η t ∇Q(x t ) − d t , s t − x + η t ∇Q(x t ), x − x t ≤ Q(x t ) + η t D ∇Q(x t ) − d t + η t ∇Q(x t ), x − x t ≤ Q(x t ) + η t D ∇Q(x t ) − d t − η t (Q(x t ) − Q(x )),(37)
here in the second inequality we use bound
η t ∇Q(x t )−d t , s t −x ≤ η t ∇Q(x t ) − d t s t − x ≤ η t D ∇Q(x t ) − d t
and in last inequality we used the bound ∇Q(x t ), x −x t ≤ Q(x )−Q(x t ). Subtracting Q(x ), taking expectation and using E X ≤ E X 2 we get
E[Q(x t+1 ) − Q(x )] ≤ (1 − η t )E[Q(x t ) − Q(x )] + η t D E ∇Q(x t ) − d t 2 + L Q η 2 t D 2 2 .(38)
Further, setting q = 2/3 hence, η t = 2 t+1 and using Corollary 1, we can bound the second term of (38)
η t D E ∇Q(x t ) − d t 2 ≤ 2D √ C1 (t+1) 4/3 . which gives E[Q(x t+1 ) − Q(x )] ≤ 1 − 2 t + 1 E[Q(x t ) − Q(x )] + 2D √ C 1 (t + 1) 4 3 + 2L Q D 2 (t + 1) 2 .(39)
Multiplying both side by t(t + 1) we can write
t(t + 1)E[Q(x t+1 ) − Q(x )] ≤ t(t − 1)E[Q(x t ) − Q(x )] + 2tD √ C 1 (t + 1) 1 3 + 2tL Q D 2 t + 1 ≤ t(t − 1)E[Q(x t ) − Q(x )] + 2D C 1 (t + 1) 2 3 + 2L Q D 2 ,(40)
Summing for t = 1, 2, · · · , T and rearranging we get
E[Q(x T +1 ) − Q(x )] ≤ 1 T (T + 1) 6 5 D C 1 (T + 1) 5 3 + 2L Q D 2 T ≤ 12D √ C 1 5(T + 1) 1 3 + 2L Q D 2 (T + 1) ,(41)
here we use the fact that T t=1 (t + 1) 2/3 ≤ 3 5 (T + 1) 5/3 .
2) Proof of Statement (ii) (Non-convex case):
Again starting with the smoothness assumption of Q we can write
Q(x t+1 ) − Q(x t ) ≤ ∇Q(x t ), x t+1 −x t + L Q 2 x t+1 −x t 2 = η t ∇Q(x t ), s t − x t + L Q η 2 t 2 s t − x t 2 (42) ≤ η t ∇Q(x t ), s t − x t + L Q η 2 t D 2 2 = η t ∇Q(x t ) − d t , s t − x t + η t d t , s t − x t + L Q η 2 t D 2 2 ,
where in the second expression we have replace term x t+1 − x t = η t (s t − x t ) and in third inequality we used compactness assumption of set while in the last expression we introduced η t d t , s t −x t . Next, introducing the following quantityυ t = arg max υ∈X υ − x t , −∇Q(x t ) , and using the optimality of s t that is s t = arg min s∈X d t , s , we write (42) as
Q(x t+1 ) − Q(x t ) ≤ η t ∇Q(x t ) − d t , s t − x t + η t d t ,υ t − x t + L Q η 2 t D 2 2 = −η t G(x t ) + Q(x t ) + η t ∇Q(x t ) − d t , s t − x t + η t d t − ∇Q(x t ),υ t − x t + L Q η 2 t D 2 2 ≤ −η t G(x t ) + Q(x t ) + 2η t D d t − ∇Q(x t ) + L Q η 2 t D 2 2 ,(43)
where the second expression is obtained by adding and subtracting the term η t υ t − x t , −∇Q(x t ) while the last inequality is obtained using compactness assumption.
Rearranging (43), summing for t = 1, 2, · · · , T and taking expectation, we get
T t=1 η t E[G(x t )] − L Q D 2 2 T t=1 η 2 t (44) ≤ Q(x 1 ) − E[Q(x T +1 )] + 2D T t=1 η t E d t − ∇Q(x t ) ≤ Q(x 1 ) − Q(x ) + 2D T t=1 η t E d t − ∇Q(x t ) ,(45)
here the last inequality follows from optimality of x . Using bound from Corollary 1 for q = 1/2, and Jensen's
equality we can write E d t − ∇Q(x t ) 2 ≤ √ C1
(t+1) 1 4 . Now, setting η t = 2 (T +1) 3/4 in (44) and rearranging we can write
E[G(x)] ≤ 1 T T t=1 E[G(x t )] ≤ Q(x 1 ) − Q(x ) 2T (T + 1) −3/4 + T t=1 2D √ C 1 T (t + 1) 1/4 + T t=1 L Q D 2 T (T + 1) 3/4 ≤ Q(x 1 ) − Q(x ) 2T (T + 1) −3/4 + 8D √ C 1 3T (T + 1) −3/4 + L Q D 2 (T + 1) 3/4 ≤ Q(x 1 ) − Q(x ) (T + 1) 1/4 + 16D √ C 1 3(T + 1) 1/4 + L Q D 2 (T + 1) 3/4 := O((T + 1) −1/4 ),(46)
here in the second inequality we use the fact that T t=1 (t + 1) −1/4 ≤ 4 3 (T + 1) 3/4 .
A. Proof of Lemma 3
Setting Ψ(·) = h(·) in Lemma (7), we can write
E t [ y t − h(x t ) 2 ] ≤ (1 − δ t ) 2 (y t−1 − h(x t−1 )) 2 + 2(1 − δ t ) 2 E t [ h ξt (x t ) − h ξt (x t−1 ) 2 ] + 2δ 2 t E t [ h ξt (x t ) − h(x t ) 2 ].(47)
Now, taking total expectation of (47) we get
E y t − h(x t ) 2 − (1 − δ t ) 2 E (y t−1 − h(x t−1 )) 2 ≤ 2δ 2 t σ 2 h + 2(1 − δ t ) 2 E h ξt (x t ) − h ξt (x t−1 ) 2 ≤ 2δ 2 t σ 2 h + 2(1 − δ t ) 2 M h E x t − x t−1 2 = 2δ 2 t σ 2 h + 2(1 − δ t ) 2 M h η 2 t−1 E s t−1 − x t−1 2 ≤ 2δ 2 t σ 2 h + 2(1 − δ t ) 2 η 2 t−1 M h D 2 ,(48)
here in the first inequality we have used Assumption 2(ii). In the second inequality we have used Assumption 2(iv)
while the next expression follows from the update step x t+1 = (1 − η t )x t + η t s t . The last inequality obtained using compactness assumption of set X .
SUPPLEMENTARY MATERIAL APPENDIX B PRELIMINARIES
A. Existing Results
We start the discussion by mentioning some of the existing results in Lemma 4 and 5 which are useful for the analysis in this paper. (a) For any x ∈ X and y ∈ R n ,
∇S(x, y) − ∇Q(x) ≤ L y (x) − y ,(49)
where L := L fx + L fy Cxy µg
+ C y Lg xy µg + Lg yy Cxy µ 2 g
and all the constants are as defined in Assumption1.
(b) The inner optimal solution y (x) is
Cxy µg -Lipschitz continuous in x, which implies that for any x 1 , x 2 ∈ X , it holds that y (x 1 ) − y (x 2 ) ≤ Cxy µg
x 1 − x 2 .
(c) The gradient of outer objective ∇Q is L Q -Lipschitz continuous in x, which implies that for any with k number of samples using (9), then (a) for any x ∈ X and y 1 , y 2 ∈ R n , we have
x 1 , x 2 ∈ X , it holds that Q(x 1 ) − Q(x 2 ) ≤ L Q x 2 − x 1 where L Q := (L fy +L)Cxy µg + L fx + CE t h(x, y 1 ;θ t ,ξ t )−h(x, y 2 ;θ t ,ξ t ) 2 ≤ L k E t y 1 −y 2 2(50)
(b) for any y ∈ R n and x 1 , x 2 ∈ X , we have
E t h(x 1 , y;θ t ,ξ t )−h(x 2 , y;θ t ,ξ t ) 2 ≤ L k E t x 1 −x 2 2 (51) where L k = 2L 2 fx + 6k[(Lg−µg) 2 (C 2 gxy L 2 fy +C 2 fy L 2 gxy )+k 2 C 2 gxy C 2 fy L 2 gyy ] µg(2Lg−µg)
.
B. General Inequalities
Before proceeding towards the main analysis of this work, we first present and establish a general mathematical inequality in the form of Lemma 6 which will be useful to the analysis in this work. Further, we will present a general upper bound in Lemma 7 on the expected estimation error when the momentum-based method is employed to track the function or gradient.
Lemma 6 Let ψ t be a sequence of real numbers which satisfy
ψ t+1 = 1 − c 1 (t + t 0 ) r1 ψ t + c 2 (t + t 0 ) r2(52)
for some r 1 ∈ (0, 1] such that r 1 ≤ r 2 ≤ 2r 1 , c 1 > 1, and c 2 ≥ 0. Then, ψ t+1 would converge to zero at the following rate
ψ t+1 ≤ c (t + t 0 + 1) r2−r1 ,(53)
where c = max{ψ 1 (t 0 + 1) r2−r1 , c2 c1−1 }.
Proof: We prove Lemma 6 by induction. The base step of induction holds (for t = 0) from the definition of c which implies that c ≥ ψ 1 (t 0 + 1) r2−r1 . Next, we assume that (53) holds for t = k, which means
ψ k+1 ≤ c (k + t 0 + 1) r2−r1 .(54)
Now it remains to show that (53) also holds for t = k + 1. To proceed, we set t = k + 1 in (52) to obtain
ψ k+2 = 1 − c1 (k+t0+1) r 1 ψ k+1 + c2 (k+t0+1) r 2 .(55)
From the definition of c, it holds that c 2 ≤ c(c 1 − 1) and we utilize the upper bound on ψ k+1 from (54) into (55) to obtain
ψ k+2 ≤ 1 − c1 (k+t0+1) r 1 c (k+t0+1) r 2 −r 1 + c(c1−1) (k+t0+1) r 2 .(56)
Simplify and rearrange the terms in (56) to write
ψ k+2 ≤ c 1 (k + t 0 + 1) r2−r1 − 1 (k + t 0 + 1) r2 .(57)
Next, note that for general p, q, it holds that 1
x p−q − 1 x p ≤ 1 (x+1) p−q , which allows us to write ψ k+2 ≤ c (k + t 0 + 2) r2−r1 .(58)
Thus, it holds that the statement in (52) holds for all t ≥ 0.
Lemma 7 Let us estimate function Ψ(x) = E ξ [Ψ(x, ξ)] by y t using step size δ t as follows
y t = (1 − δ t )(y t−1 − Ψ(x t−1 , ξ t )) + Ψ(x t , ξ t ).(59)
Then the expected tracking error
E t [ y t − Ψ(x t ) 2 ] satisfies E t [ y t − Ψ(x t ) 2 ] ≤ (1 − δ t ) 2 (y t−1 − Ψ(x t−1 )) 2 + 2(1 − δ t ) 2 E t [ Ψ(x t , ξ t ) − Ψ(x t−1 , ξ t ) 2 ] + 2δ 2 t E t [ Ψ(x t , ξ t ) − Ψ(x t ) 2 ].(60)
Proof: Consider the update equation in (59), add/subtract the term (1 − δ t )Ψ(x t−1 ) in the right hand side of (59) to obtain
y t = (1 − δ t )(y t−1 − Ψ(x t−1 , ξ)) + Ψ(x t , ξ t ) + (1 − δ t )Ψ(x t−1 ) − (1 − δ t )Ψ(x t−1 ).(61)
Subtract Ψ(x t ) from both sides in (61) and take norm square:
y t − Ψ(x t ) 2 = (1 − δ t )(y t−1 − Ψ(x t−1 )) (62) − (1 − δ t )(Ψ(x t−1 , ξ t ) − Ψ(x t−1 )) + Ψ(x t , ξ t ) − Ψ(x t ) 2 .
April 5, 2022 DRAFT Now, expand the square and calculate conditional expectation
E t = E[(·)|F t ] to obtain E t [ y t − Ψ(x t ) 2 ] = (1 − δ t ) 2 (y t−1 − Ψ(x t−1 )) 2 (63) − 2 (1 − δ t )(y t−1 − Ψ(x t−1 )), (1 − δ t )(E t [Ψ(x t−1 ) − Ψ(x t−1 , ξ t )]) + E t [Ψ(x t ) − Ψ(x t , ξ t )] +E t (1−δ t )(Ψ(x t−1 , ξ t )−Ψ(x t−1 ))+Ψ(x t )−Ψ(x t , ξ t ) 2 . Note that E t [Ψ(x t−1 ) − Ψ(x t−1 , ξ t )] = 0 and E t [Ψ(x t ) − Ψ(x t , ξ t )] = 0, which implies that E t [ y t − Ψ(x t ) 2 ] = (1 − δ t ) 2 (y t−1 − Ψ(x t−1 )) 2 +E t (1−δ t )(Ψ(x t−1 ,ξ t )−Ψ(x t−1 ))+Ψ(x t )−Ψ(x t , ξ t ) 2 ≤ (1 − δ t ) 2 (y t−1 − Ψ(x t−1 )) 2 + 2(1 − δ t ) 2 E t [ Ψ(x t , ξ t ) − Ψ(x t−1 , ξ t ) 2 ] + 2δ 2 t E t [ Ψ(x t , ξ t ) − Ψ(x t ) 2 ](64)
where the last inequality holds due to the fact that E X − E[X] + Y 2 ≤ 2E X 2 + 2E Y 2 for any two random variables X and Y .
APPENDIX C PROOFS FOR SBFW
Before proceeding towards the analysis, we provide a Lemma regarding the property of the bias induced by the gradient estimate.
Lemma 8 Under Assumption 1, consider the estimator defined in (9), then (i) define bias B t :=E[h(x t , y t ; θ t , ξ t )]−∇S(x t , y t ), it holds that we have,
B t ≤ (C xy C y /µ g ) (1 − (µ g /L g )) k ,(65)E h(x t , y t ; θ t , ξ t )−∇S(x t , y t )−B t 2 ≤ σ 2 f ,(66)
where σ 2 f =σ 2 x + 3 µ 2 g (σ 2 y +C 2 y )(σ 2 xy +2C 2 xy )+σ 2 y C 2 xy . (ii) For t ≥ 0, it is possible select k (required to approximate the Hessian inverse in (10)) such that B t ≤ β t where β t ≤ ct a for some constant c and a > 0.
Proof: For proof of Lemma (8)(i) see [ Lemma 11,[19]]. The proof Lemma (8)(ii) is straight forward. From (65) we have β t = O(1 − µ g /L g ) k . Now on setting k = O(log(t)) we can get the required condition as β t ≤ ct a . It shows that with proper selection of k, we can make the bias to decay polynomially to zero.
A. Proof of Lemma 1
Let us consider the term E t [ y t − y (x t−1 ) 2 ] and from the update step 2 of Algorithm 1, we can write
E t [ y t − y (x t−1 ) 2 ] = E t [ y t−1 − δ t ∇ y g(x t−1 , y t−1 , ξ t ) − y (x t−1 ) 2 ].(67)
By expanding the square and taking conditional expectation term inside the inner product terms, we obtain
E t [ y t − y (x t−1 ) 2 ] = E t [ y t−1 − δ t ∇ y g(x t−1 , y t−1 , ξ t ) − y (x t−1 ) 2 ] = E t [ y t−1 − y (x t−1 ) 2 ] + δ 2 t E t [ ∇ y g(x t−1 , y t−1 , ξ t ) 2 ] − 2δ t E t [ y t−1 −y (x t−1 ), ∇ y g(x t−1 , y t−1 , ξ t ) ](68)≤ E t [ y t−1 − y (x t−1 ) 2 ] + δ 2 t E t [ ∇ y g(x t−1 , y t−1 , ξ t ) 2 ] − 2δ t µ g E t [ y t−1 − y (x t−1 ) 2 ] (69) = (1 − 2δ t µ g )E t [ y t−1 − y (x t−1 ) 2 ] + δ 2 t E t [ ∇ y g(x t−1 , y t−1 , ξ t ) 2 ](70)
here (68) comes from the fact that E t [∇ y g(x t−1 , y t−1 , ξ t )] = ∇ y g(x t−1 , y t−1 ), while (69) comes from using the strong convexity property of function g. Now let us consider the last term E t ∇ y g(x t−1 , y t−1 , ξ t ) 2 in the right hand side of (70):
E t [ ∇ y g(x t−1 , y t−1 , ξ t ) 2 ] (71) = E t ∇ y g(x t−1 ,y t−1 ,ξ t )+∇ y g(x t−1 ,y t−1 ) −∇ y g(x t−1 ,y t−1 ) 2 ≤ 2E t [ ∇ y g(x t−1 , y t−1 , ξ t ) − ∇ y g(x t−1 , y t−1 ) 2 ] + 2 ∇ y g(x t−1 , y t−1 ) 2 ,(72)
where we use the inequality a + b 2 ≤ 2 a 2 + 2 b 2 . From Assumption 1(i), we can further upper bound (71) as
E t [ ∇ y g(x t−1 , y t−1 , ξ t ) 2 ] ≤ 2σ 2 g (1 + ∇ y g(x t−1 , y t−1 ) 2 ) + 2 ∇ y g(x t−1 , y t−1 ) 2 = 2σ 2 g + 2(1 + σ 2 g ) ∇ y g(x t−1 , y t−1 ) 2 (73) ≤ 2σ 2 g +2(1+σ 2 g ) ∇ y g(x t−1 , y t−1 )−∇ y g(x t−1 ,y (x t−1 )) 2 ≤ 2σ 2 g + 2(1 + σ 2 g )L 2 g y t−1 − y (x t−1 ) 2(74)
where we used the fact that ∇ y g(x t−1 , y (x t−1 )) = 0. Substituting the upper bound in (73) in (70) we obtain
E t [ y t − y (x t−1 ) 2 ] ≤ [(1 − 2δ t µ g ) + 2δ 2 t (1 + σ 2 g )L 2 g ]E t [ y t−1 − y (x t−1 ) 2 ] + 2δ 2 t σ 2 g ≤ (1 − δ t µ g )E t [ y t−1 − y (x t−1 ) 2 ] + 2δ 2 t σ 2 g .(75)
The last inequality in (75) is obtained by selecting δ t such that 2δ t (1 + σ 2 g )L 2 g ≤ µ g . To proceed next, we use Young's inequality to bound the term E t [ y t − y (x t ) 2 ] in (75) as
E t [ y t − y (x t ) 2 ] ≤ 1 + 1 α E t [ y t − y (x t−1 ) 2 ] + (1 + α)E t [ y (x t ) − y (x t−1 ) 2 ] ≤ 1 + 1 α E t [ y t − y (x t−1 ) 2 ] + (1 + α) Cxy µg 2 E t x t − x t−1 2 ] ≤ 1 + 1 α E t [ y t −y (x t−1 ) 2 ]+(1+α) Cxy µg 2 η 2 t−1 D 2(76)
where the second inequality comes from Lemma 4(b), and the last inequality comes from the update equation in (17) and the compactness of the domain X . Utilizing (75) into (76), we get
E t [ y t − y (x t ) 2 ] ≤ 1 + 1 α (1 − δ t µ g )E t [ y t−1 − y (x t−1 ) 2 ] + 1 + 1 α 2δ 2 t σ 2 g + (1 + α) C xy µ g 2 η 2 t−1 D 2 .(77)
To proceed next, we substitute α = 2(1−δtµg) δtµg which also implies that 1
+ 1 α (1 − δ t µ g ) = 1 − µgδt 2 : E t [ y t − y (x t ) 2 ] ≤ 1 − δ t µ g 2 E t [ y t−1 − y (x t−1 ) 2 ] + 2 − δ t µ g δ t µ g C xy µ g 2 η 2 t−1 D 2 + 1 + 1 α 2δ 2 t σ 2 g ≤ 1 − δtµg 2 E t y t−1 − y (x t−1 ) 2 + 2η 2 t−1 δtµg Cxy µg 2 D 2 +4δ 2 t σ 2 g ,(78)
where the second inequality comes from the fact that 2−δtµg δtµg < 2 δtµg while in the last inequality, we have assumed that δ t is chosen such that δ t ≤ 2 3µg giving 1 + 1 α ≤ 2. In Corollary 1 we will see that our choice of step sizes satisfies these conditions.
B. Proof of Lemma 2
Starting with update equation (15) and employing Lemma 7 we can write
E t [ d t − ∇S(x t , y t ) − B t 2 ≤ (1 − ρ t ) 2 E t [ (d t−1 − ∇S(x t−1 , y t−1 ) − B t−1 ) ] + 2(1 − ρ t ) 2 E t [ h(x t , y t ; θ t , ξ t ) − h(x t−1 , y t−1 ; θ t , ξ t ) 2 ] + 2ρ 2 t E t [ h(x t , y t ; θ t , ξ t ) − ∇S(x t , y t ) − B t ) 2 ] ≤ (1 − ρ t ) 2 E t [ (d t−1 − ∇S(x t−1 , y t−1 ) − B t−1 ) ] + 2E t [ h(x t , y t ; θ t , ξ t ) − h(x t−1 , y t−1 ; θ t , ξ t ) 2 + 2ρ 2 t σ 2 f ,(79)
here the last inequality is obtained using (66) and the fact that (1 − ρ 2 t ) ≤ 1. Now we introduce h(x t , y t−1 ; θ t , ξ t ) and bound the second term of RHS of (79) as
E t [ h(x t , y t ; θ t , ξ t ) − h(x t−1 , y t−1 ; θ t , ξ t ) 2 ] = E t h(x t , y t ; θ t , ξ t ) − h(x t−1 , y t−1 ; θ t , ξ t ) + h(x t , y t−1 ; θ t , ξ t ) − h(x t , y t−1 ; θ t , ξ t ) 2 (a) ≤ 2E t [ h(x t , y t ; θ t , ξ t ) − h(x t , y t−1 ; θ t , ξ t ) 2 + 2E t [ h(x t−1 , y t−1 ; θ t , ξ t ) − h(x t , y t−1 ; θ t , ξ t ) 2 (b) ≤ 2L k E t y t − y t−1 2 + 2L k E t x t−1 − x t 2 (c) ≤ 2L k δ 2 t E t ∇ y g(x t−1 , y t−1 , ξ t ) 2 + 2L k η 2 t−1 D 2 (d) ≤ 2L k δ 2 t L 2 g + 2L k η 2 t−1 D 2 ,(80)
here (a) comes from simple norm property, (b) comes from Lemma 5, (c) comes from update equation (12) For the simplicity of analysis we start with writing Lemma 1 for t = t + 1 and set δ t = 2a0 t q where a 0 = min{ 1 3µg ,
µg 2(1+σ 2 g )L 2 g } and η t = 2 (t+1) 3q 2
, which gives
E t y t+1 − y (x t+1 ) 2 ≤ 1 − 2a0 (t+1) q E t y t − y (x t ) 2 + 2 (t + 1) 3q−q C xy µ g 2 D 2 + 16a 2 0 (t + 1) 2q σ 2 g (81) = 1 − 2a0 (t+1) q E t [ y t − y (x t ) 2 ] + 2(Cxy/µg) 2 D 2 +16a 2 0 σ 2 g (t+1) 2q
.
Note such selection of δ t ensures that the conditions 2δ t (1 + σ 2 g )L 2 g ≤ µ g and δ t ≤ 2 3µg required in Lemma 1 are satisfied. Now taking full expectation and using Lemma 6 we get
E[ y t − y (x t ) ] ≤ b 1 (t + 1) q ,(82)where b 1 = max{2 q y 1 − y (x 1 ) 2 , (2(C xy /µ g ) 2 D 2 + 16a 2 0 σ 2 g )/(2a 0 − 1)}. Similarly, in Lemma 2 setting δ t = 2a0 (t) q , η t = 2 (t+1) 3q 2 and ρ t = 2 (t) q , we can write E t [ d t+1 − ∇S(x t+1 , y t+1 ) − B t+1 2 ] ≤ 1 − 2 (t + 1) q E t [ (d t − ∇S(x t , y t ) − B t ) 2 ] + 16L k L 2 g (t + 1) 2q + 16L k D 2 (t + 1) 3p + 8σ 2 f (t + 1) 2q ≤ 1 − 2 (t + 1) q E t [ (d t − ∇S(x t , y t ) − B t ) 2 ] + 16L k L 2 g + 16L k D 2 + 8σ 2 f (t + 1) 2q ,(83)
here the last inequality is obtained using the fact 1/(t + 1) 3q ≤ 1/(t + 1) 2p . Application of Lemma 6 gives
E t [ d t − ∇S(x t , y t ) − B t 2 ] ≤ b 2 (t + 2) q ,(84)
where b 2 = max{2 q d 1 − ∇S(x 1 , y 1 ) − B 1 2 , 8(2L k L 2 g + L k D 2 + σ 2 f )} = 8(2L k L 2 g + L k D 2 + σ 2 f ). As, we have initialize d 1 = h(x 1 , y 1 ; θ 1 , ξ 1 ) we can use the bound
d 1 − ∇S(x 1 , y 1 ) − B 1 2 = h(x 1 , y 1 ; θ 1 , ξ 1 ) − ∇S(x 1 , y 1 ) − B 1 2 ≤ σ 2 f
. Next, we can bound the term E ∇Q(x t ) − d t 2 as follows
E ∇Q(x t ) − d t 2 (85) = E ∇Q(x t ) − d t + B t + ∇S(x t , y t ) − B t − ∇S(x t , y t ) 2 ≤ 3E ∇Q(x t ) − ∇S(x t , y t ) 2 + 3 B t 2 + 3E ∇S(x t , y t ) + B t − d t 2
≤ 3E y (x t ) − y t 2 + 3β 2 t + 3b 2 (t + 1) q ≤ 3b 1 (t + 1) q + 3b 3 (t + 1) q + 3b 2 (t + 1) q :=
C 1 (t + 1) q ,(86)
here second inequality comes from simple norm property, while third inequality is obtained using Lemma (4)(a) on the first term, Lemma 8 on the second term and (84) on the third term. The last inequality comes from (82) and
using β t ≤ CxyCy µg(t+1) q := b3 (t+1) q and the constant C 1 = 3(b 1 + b 2 + b 3 ) is defined as C 1 = 3(max{2 q y 1 − y (x 1 ) 2 , (2(C xy /µ g ) 2 D 2 + 16a 2 0 σ 2 g )/(2a 0 − 1)} + 8(2L k L 2 g + L k D 2 + σ 2 f ) + CxyCy µg ).
APPENDIX D PROOFS FOR SCFW
Defining∇C(x t , y t ) = E[∇C(x t , y t , θ t , ξ t )], we provide an upper bound on E d t −∇C(x t , y t )
Application of Lemma 6 in (95) gives
E y t − h(x t ) 2 ≤ a 1 (t + 1) p ,(96)
where a 1 = max{2 p E y 1 − h(x 1 ) 2 , 8(σ 2 h + M h D 2 )}. As we have initialized y 1 = g ξ1 (x 1 ) we can bound
E y 1 − h(x 1 ) 2 = E g ξ1 (x 1 ) − h(x 1 ) 2 ≤ σ 2 h .
Further, using the fact that 0 < p ≤ 1, we can simplify a 1 as a 1 = 8(σ 2 h + M h D 2 ). Similarly, from Lemma 9 using the fact (1 − ρ t ) 2 ≤ 1, we can write
E d t+1 −∇C(x t+1 , y t+1 ) 2 (97) ≤ (1 − ρ t+1 ) 2 E d t −∇C(x t , y t ) 2 + 4ρ 2 t+1 M f M h + 4 (M f L h + 3M 2 h L f )η 2 t D 2 + 3M h L f δ 2 t+1 E t y t − h(x t ) 2 + 3δ 2 t+1 M h L f σ 2 h .(98)
Substituting δ t = 2 t p , ρ t = 2 t p and η t ≤ 2 (t+1) p , and using bound (96) we get
E d t+1 −∇C(x t+1 , y t+1 ) 2 (99) ≤ 1 − 2 (t + 1) p E d t −∇C(x t , y t ) 2 + 16 M f M h (t + 1) 2p + (M f L h + 3M 2 h L f )D 2 (t + 1) 2p + 3M h L f (t + 1) 2p a 1 (t + 1) p + 3M h L f σ 2 h (t + 1) 2p .(100)
Using Lemma 6 and the fact that
1 (t+1) 3p ≤ 1 (t+1) 2p gives E d t −∇C(x t , y t ) 2 ≤ a 2 (t + 1) p ,(101)
where a 2 := max{2 p E d 1 −∇C(x 1 , y 1 ) 2 , 16(M f M h + (M f L h + 3M 2 h L f )D 2 + 3M h L f a 1 + 3M h L f σ 2 h )}. As we have initialized d 1 = ∇C φ1 (x 1 , y 1 ), we can bound E d 1 −∇C(x 1 , y 1 ) 2 = E ∇C φ1 (x 1 , y 1 ) −∇C(x 1 , y 1 ) 2 ≤
2M h M f (see (88)). Also as 0 < p ≤ 1, we can further simplify a 2 to get a 2 = 16
(M f M h +(M f L h +3M 2 h L f )D 2 + 3M h L f a 1 + 3M h L f σ 2 h ). Now, introducing∇C(x t , y t ) can write E ∇C(x t ) − d t 2 = E ∇C(x t ) − d t −∇C(x t , y t ) +∇C(x t , y t ) 2 ≤ 2E d t −∇C(x t , y t ) 2 + 2E ∇ C(x t , y t ) − ∇C(x t ) 2 = 2E d t −∇C(x t , y t ) 2 + 2E ∇h(x t ) ∇f (y t ) − ∇h(x t ) ∇f (h(x t )) 2 ≤ 2E d t −∇C(x t , y t ) 2 + 2M h L f E y t − h(x t ) 2 ≤ 2a 2 (t + 1) p + 2M h L f a 1 (t + 1) p := A 1 (t + 1) p ,(102)
here the first inequality comes from simple norm property while in the next expression to it we simply substitute the values of gradients. In the second inequality, we used the Assumption 2 (iii-iv) while the last inequality is
E[C(x t+1 ) − C(x )] ≤ (1 − η t )E[C(x t ) − C(x )] + η t D E ∇C(x t ) − d t 2 + L F η 2 t D 2 2 .(103)
Setting η t = 2 t+1 and using Corollary 2 with p = 1, we can bound the second term of (103) as
η t D E ∇C(x t ) − d t 2 ≤ 2D √ A 1 (t + 1) 3/2 .(104)
Now using bound from (104) in (103) we obtain
E[C(x t+1 ) − C(x )] ≤ 1 − 2 t + 1 E[C(x t ) − C(x )] + 2D √ A 1 (t + 1) 3 2 + 2L F D 2 (t + 1) 2 .(105)
Multiplying both side by t(t + 1) we can write
t(t + 1)E[C(x t+1 ) − C(x )] (106) ≤ t(t − 1)E[C(x t ) − C(x )] + 2tD √ A 1 (t + 1) 1 2 + 2tL F D 2 t + 1 ≤ t(t − 1)E[C(x t ) − C(x )] + 2D A 1 (t + 1) 1 2 + 2L F D 2 ,
here in last the inequality we used the fact that t < t + 1. Summing for t = 1, 2, · · · , T and rearranging we get
E[C(x T +1 ) − C(x )] ≤ 1 T (T + 1) 2D A 1 T t=1 (t + 1) 1 2 + 2L F D 2 T ≤ 1 T (T + 1) 4 3 D A 1 (T + 1) 3 2 + 2L F D 2 T ≤ 8D √ A 1 3(T + 1) 1 2 + 2L F D 2 (T + 1) ,(107)
here in the second inequality we use the fact that T t=1 (t + 1) 1/2 ≤ 2 3 (T + 1) 3/2 and the last inequality is obtained using (T + 1) = T (1 + 1 T ) ≤ 2T . 2) Proof of Statement (ii) (Non-convex case): Again starting with the smoothness assumption of F and proceeding the same way as (42)-(44), we can write
T t=1 η t E[G(x t )] ≤ C(x 1 ) − C(x ) + 2D T t=1 η t E d t − ∇C(x t ) + L F D 2 2 T t=1 η 2 t .(108)
Using bound from Corollary 2 for p = 2/3, and Jensen's equality we can write
E d t − ∇C(x t ) ≤ E d t − ∇C(x t ) 2 ≤ √ A1 (t+1) 1 3 .(109)
April 5, 2022 DRAFT Now, using (109), setting η t = 2 (T +1) 2/3 in (108) and rearranging we can write
E[G(x)] ≤ 1 T T t=1 E[G(x t )] ≤ C(x 1 ) − C(x ) 2T (T + 1) −2/3 + T t=1 2D √ A 1 T (t + 1) 1/3 + T t=1 L F D 2 T (T + 1) 2/3 ≤ C(x 1 ) − C(x ) 2T (T + 1) −2/3 + 3D √ A 1 T (T + 1) −2/3 + L F D 2 (T + 1) 2/3 = (T +1)(C(x 1 )−C(x )) 2T (T + 1) 1/3 + 3D √ A 1 (T +1) T (T + 1) 1/3 + L F D 2 (T + 1) 2/3 ≤ C(x 1 ) − C(x ) (T + 1) 1/3 + 6D √ A 1 (T + 1) 1/3 + L F D 2 (T + 1) 2/3 := O((T + 1) −1/3 ),(110)
here in the second inequality we use the fact that T t=1 (t+1) −1/3 ≤ 3 2 (T +1) 2/3 and the last inequality is obtained using (T + 1) = T (1 + 1 T ) ≤ 2T .
We start the analysis by presenting intermediate Lemmas 1-2 and Corollary 2 which eventually leads to the main result of this section presented in Theorem 1.
SBFW for outer objective isO(log( −1 ) −3 ) ≈ O( −3 ) and O(log( −1 ) −4 ) ≈ O( −4 ) for convex and non-convex objective, respectively. Similarly, observe that E[ y t − y (x t ) ] ≤ O((t + 1) −q ) (see (82) in the supplementary material), where q = 2/3 (as δ t = O(t −2/3 ))for convex objective and q = 1/2 (as δ t = O(t −1/2 )) for non-convex objective. Hence, the SFO complexity of inner objective for SBFW is O( −1.5 ) and O( −2 ) for convex and nonconvex function, respectively. It can be seen that complexity for inner level objective of the proposed algorithm
Error variation vs noise factor.
Fig. 1 :
1This figure shows the benefit of bilevel formulation (solved via SBFW) for the matrix completion problem as compared
Fig. 2 :
2This figure compares the performance of the proposed SBFW algorithm for matrix completion problem on MovieLens
Fig. 3 :
3This figure compares the normalized error with respect to computation time required for SBFW and SUSTAIN [18] on MovieLens 1M dataset. We note that eve thought SUSTAIN has a better theoretical convergence guarantees, it performs poorly in terms of computation time required to achieve a particular level of normalized error. For instance, to achieve normalized error of 0.3, SUSTAIN requires 31 minutes while SBFGW requires only 5 minutes. So there is a percentage improvement of 15% by the proposed algorithm.
The constraint w 1
1≤ α in(34) for some sparsity tuning parameter α > 0 is useful in practice, where the sparsity needs to be ensured. For instance, when the number of states in S is large, the features of each state would become large, thus making the dimension of the optimization variable large. We propose to solve (34) in a projectionfree manner using the proposed SCFW algorithm. In the current context, the complete knowledge of transition probabilities {P π sŝ } is not known but revealed sequentially. For the experiments, we consider the number of states |S| = 100 with 3 actions available at every state. Given a pair of state and action, the agent can move any one of the next possible states. The transition probabilities and rewards for each transition are uniformly sampled in [0, 1].
Fig. 4 :
4Convergence results for ASCPG, SCFW and SBFW algorithms for sparse policy value evaluation (Sec.V-B).
obtained using bounds from 96 and 101 and definedA 1 := 32[M h (M f + 28L f σ 2 h ) + (M f L h + 28M 2 h L f )D 2 ].B. Proof of Theorem 2 1) Proof of Statement (i) (convex case): Starting with the smoothness of C with parameter L F = M 2 h L f +M f L h and proceeding the same way as (35)-(38) we can write
Let us consider the optimization problem P 1 . For most of the applications in practice [1]-April 5, 2022
DRAFT
II. PROBLEM FORMULATION
, we provide computation time comparisons (to complete 10 3 iteration) of both the algorithms (under same settings) over different real datasets. Observe that for large data set, SBFW is approximately 10× faster than the SUSTAIN and exhibit an improvement upto 82% in the computation time. This performance gain in terms of computation time comes from the fact that other methods require to perform projections over nuclear norm at each iteration which is computationally expensive due to the computation of full singular value decomposition. In contrast, SBFW solves only a single linear program at each iteration, which only requires the computation of singular vectors corresponding to the highest singular value.Dataset
#users
#movies
#ratings
Time
SUSTAIN
SBFW
%imp.
Movielens 100k
1000
1700
10 5
554 sec.
236 sec.
57%
Movielens latest
600
9000
10 5
66.6 mins.
12.9 mins.
81%
Movielens 1M
6000
4000
10 6
10.16 hrs.
1.82 hrs.
82%
TABLE II :
IIComparison of computation time of the proposed algorithm SBFW and the state-of-the-art projection based algorithm SUSTAIN over different size of real data sets.
while (d) comes from Lipschitz continuous Assumption 1(iii) and from the compactness of the set. Using (80) in (79), we get the desired expression.April 5, 2022
DRAFT
C. Proof Corollary 1
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|
[] |
[
"Cycle Consistent Adversarial Denoising Network for Multiphase Coronary CT",
"Cycle Consistent Adversarial Denoising Network for Multiphase Coronary CT"
] |
[
"Angiography Eunhee Kang \nDept. of Bio and Brain Engineering\nBio Imaging and Signal Processing Lab\nKAIST\nDaejeonRepublic of Korea\n",
"Hyun Jung Koo \nDept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea\n",
"Dong Hyun Yang \nDept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea\n",
"Joon Bum Seo \nDept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea\n",
"Jong Chul Ye \nDept. of Bio and Brain Engineering\nBio Imaging and Signal Processing Lab\nKAIST\nDaejeonRepublic of Korea\n"
] |
[
"Dept. of Bio and Brain Engineering\nBio Imaging and Signal Processing Lab\nKAIST\nDaejeonRepublic of Korea",
"Dept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea",
"Dept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea",
"Dept. of Radiology\nAsan Medical Center\nUniversity of Ulsan College of Medicine\nSeoulRepublic of Korea",
"Dept. of Bio and Brain Engineering\nBio Imaging and Signal Processing Lab\nKAIST\nDaejeonRepublic of Korea"
] |
[] |
Purpose: In multiphase coronary CT angiography (CTA), a series of CT images are taken at different levels of radiation dose during the examination. Although this reduces the total radiation dose, the image quality during the low-dose phases is significantly degraded. Recently, deep neural network approaches based on supervised learning technique have demonstrated impressive performance improvement over conventional model-based iterative methods for low-dose CT. However, matched low-and routine-dose CT image pairs are difficult to obtain in multiphase CT. To address this problem, we aim at developing a new deep learning framework.Method:We propose an unsupervised learning technique that can remove the noise of the CT images in the low-dose phases by learning from the CT images in the routine dose phases. Although a supervised learning approach is not applicable due to the differences in the underlying heart structure in two phases, the images are closely related in two phases, so we propose a cycle-consistent adversarial denoising network to learn the mapping between the low and high dose cardiac phases.Results: Experimental results showed that the proposed method effectively reduces the noise in the low-dose CT image while preserving detailed texture and edge information. Moreover, thanks to the cyclic consistency and identity loss, the proposed network does not create any artificial features that are not present in the input images. Visual grading and quality evaluation also confirm that the proposed method provides significant improvement in diagnostic quality.Conclusions:The proposed network can learn the image distributions from the routine-dose cardiac phases, which is a big advantages over the existing supervised learning networks that need exactly matched low-and routine-dose CT images.Considering the effectiveness and practicability of the proposed method, we believe that the proposed can be applied for many other CT acquisition protocols.2
|
10.1002/mp.13284
|
[
"https://arxiv.org/pdf/1806.09748v3.pdf"
] | 49,434,516 |
1806.09748
|
6627feb69fbc101a5addaeddf6c2642be7f0a364
|
Cycle Consistent Adversarial Denoising Network for Multiphase Coronary CT
7 Nov 2018
Angiography Eunhee Kang
Dept. of Bio and Brain Engineering
Bio Imaging and Signal Processing Lab
KAIST
DaejeonRepublic of Korea
Hyun Jung Koo
Dept. of Radiology
Asan Medical Center
University of Ulsan College of Medicine
SeoulRepublic of Korea
Dong Hyun Yang
Dept. of Radiology
Asan Medical Center
University of Ulsan College of Medicine
SeoulRepublic of Korea
Joon Bum Seo
Dept. of Radiology
Asan Medical Center
University of Ulsan College of Medicine
SeoulRepublic of Korea
Jong Chul Ye
Dept. of Bio and Brain Engineering
Bio Imaging and Signal Processing Lab
KAIST
DaejeonRepublic of Korea
Cycle Consistent Adversarial Denoising Network for Multiphase Coronary CT
7 Nov 20181
Purpose: In multiphase coronary CT angiography (CTA), a series of CT images are taken at different levels of radiation dose during the examination. Although this reduces the total radiation dose, the image quality during the low-dose phases is significantly degraded. Recently, deep neural network approaches based on supervised learning technique have demonstrated impressive performance improvement over conventional model-based iterative methods for low-dose CT. However, matched low-and routine-dose CT image pairs are difficult to obtain in multiphase CT. To address this problem, we aim at developing a new deep learning framework.Method:We propose an unsupervised learning technique that can remove the noise of the CT images in the low-dose phases by learning from the CT images in the routine dose phases. Although a supervised learning approach is not applicable due to the differences in the underlying heart structure in two phases, the images are closely related in two phases, so we propose a cycle-consistent adversarial denoising network to learn the mapping between the low and high dose cardiac phases.Results: Experimental results showed that the proposed method effectively reduces the noise in the low-dose CT image while preserving detailed texture and edge information. Moreover, thanks to the cyclic consistency and identity loss, the proposed network does not create any artificial features that are not present in the input images. Visual grading and quality evaluation also confirm that the proposed method provides significant improvement in diagnostic quality.Conclusions:The proposed network can learn the image distributions from the routine-dose cardiac phases, which is a big advantages over the existing supervised learning networks that need exactly matched low-and routine-dose CT images.Considering the effectiveness and practicability of the proposed method, we believe that the proposed can be applied for many other CT acquisition protocols.2
output data are paired in a meaningful way, i.e. there can exist various inputs (resp. outputs) that matches to one output (resp. input) despite them generating the same distribution.
In coronary CTA, even though the images at the low-dose and high-dose phases do not match each other exactly due to the cardiac motion, they are from the same cardiac volume so that they have important correspondence. Therefore, one can conjecture that the correctly denoised low-dose phase should follow the routine dose phase image distribution more closely and learning between two phase cardiac images is more effective than learning from totally different images. One of the most important contributions of this work is to show that we can indeed improve the CT images at the low-dose phase by learning the distribution of the images at the high-dose phases using the cyclic consistency by Zhu et al.
(cycle GAN) 22 or by Kim el al. (DiscoGAN) 23 . Specifically, we train two networks between two different domains (low dose and routine dose). Then, the training goal is that the two networks should be inverse of each other. Thanks to the existence of inverse path that favors the one to one correspondence between the input and output, the training of the GAN is less affected by the mode collapse. Furthermore, unlike the classic GAN which generates samples from random noise inputs, our network creates samples from the noisy input that are closely related. This also reduces the likelihood of mode collapse. Another important aspect of the algorithm is the identity loss 22 . The main idea of the identity loss is that a generator G : A → B should work as an identity for the target domain image y ∈ B such that G(y) y. This constraint works as a fixed-point constraint of the output domain so that as soon as the output signal is generated to match the target distribution, the network no longer changes the signal. Experimental results show that the proposed method is robust to the cardiac motion and contrast changes and does not create artificial features.
II. THEORY
The overall framework of the proposed network architecture is illustrated in Fig. 2.
We denote the low-dose CT domain by (A) and routine-dose CT domain by (B), and the probability distribution for each domain is referred to as P A and P B , respectively. The generator G AB denotes the mapping from (A) to (B), and G BA are similarly defined as the mapping from (B) to (A). As for the generator, we employ the optimized network for a noise reduction in low-dose CT images in our prior work 10 . In addition, there are two adversarial discriminators D A and D B which distinguish between measured input images and synthesized images from the generators. Then, we train the generators and discriminators simultaneously. Specifically, we aim to solve the following optimization problem:
min G AB ,G BA max D A ,D B L(G AB , G BA , D A , D B ).(1)
where the overall loss is defined by:
L(G AB , G BA , D A , D B ) =L GAN (G AB , D B , A, B) + L GAN (G BA , D A , B, A) +λL cyclic (G AB , G BA ) + γL identity (G AB , G BA ),(2)
where λ and γ control the importance of the losses, and L GAN , L cyclic and L identity denote the adversarial loss, cyclic loss, and identity loss. More detailed description of each loss follows.
II.A. Loss formulation
II.A.1. Adversarial loss
We employ adversarial losses using GAN as proposed in Zhu et al 22 . According to the original GAN 24 , the generator G AB and discriminator D B can be trained by solving the following min-max problem:
min G AB max D B L GAN (G AB , D B , A, B) = E x B ∼P B [log D B (x B )] + E x A ∼P A [log(1 − D B (G AB (x A )))],(3)
where G AB is trained to reduce a noise in the low-dose CT image x A to make it similar to the routine-dose CT image x B , while D B is trained to discriminate between the denoised CT image G AB (x A ) and the routine-dose CT image x B . However, we found that the original adversarial loss (3) is unstable during training process; thus, we changed the log-likelihood function to a least square loss as in the least squares GAN (LSGAN) 25 . Then, the min-max problem can be changed to the two minimization problems as follows:
min G AB E x A ∼P A [(D B (G AB (x A )) − 1) 2 ],(4)min D B 1 2 E x B ∼P B [(D B (x B ) − 1) 2 ] + 1 2 E x A ∼P A [D B (G AB (x A )) 2 ].(5)
The adversarial loss causes the generator to generate the denoised images that may deceive the discriminator to classify them as the real images at routine doses. At the same time, the adversarial loss will guide the discriminator to well distinguish the denoised image and the routine dose image. Similar adversarial loss is added to the generator G BA , which generates noisy images.
II.A.2. Cyclic loss
With the adversarial losses, we could train the generator G AB and G BA to produce the realistic denoised images and noisy CT images, respectively; but this does not guarantee that they have an inverse relation described in Fig. 2. To enable one to one correspondence between the noisy and denoised image, the cycle which consists of two generators should be imposed to bring the input x A to the original image. More specifically, the cyclic loss is defined by
L cyclic (G AB , G BA ) = E x A ∼P A [ G BA (G AB (x A )) − x A 1 ] + E x B ∼P B [ G AB (G BA (x B )) − x B 1 ],(6)
where · 1 denotes the l 1 -norm. Then, the cyclic loss enforces the constraint that G AB and G BA should be inverse of each other, i.e. it encourages G BA (G AB (x A )) ≈ x A and
G AB (G BA (x B )) ≈ x B .
II.A.3. Identity loss
In multiphase CTA, there are often cases where the heart phase and dose modulation are not perfectly aligned as originally planned. For example, in the multiphase CTA acquisition in Fig. 1, it is assumed that the systolic phase images should be obtained using low dose modulation, but due to the mismatch with the cardiac cycle from arrhythmia, the systolic phase image noise level may vary and even be in full dose. In this case, the input to the generator G AB can be at full dose, so it is important to train the generator so that it does not alter such clean images. Similarly, the generator G BA should not change the input images acquired at the low-dose level. To enforce the two generator G AB and G BA to satisfy these conditions, the following identity loss should be minimized:
L identity (G AB , G BA ) = E x B ∼P B [ G AB (x B ) − x B 1 ] + E x A ∼P A [ G BA (x A ) − x A 1 ].(7)
In other word, the generators should work as identity mappings for the input images at the target domain:
G AB (x B ) x B , G BA (x A ) x A(8)
Note that this identity loss is similar to the identity loss for the photo generation from paintings in order to maintain the color composition between input and output domains 22 .
The constraints in (8) ensure that the correctly generated output images no longer vary when used as inputs to the same network, i.e. the target domain should be the fixed points of the generator. As will be shown later in experiments, this constraint is important to avoid creating artificial features.
II.B. Network architecture
The network architecture of two generators G AB and G BA is illustrated in Fig. 3. This architecture is optimized for low-dose CT image denoising in Kang et al 10 . To reduce network complexity, images are used directly as inputs to the network instead of the wavelet transform coefficients as in our prior work 10 . The first convolution layer uses 128 set of 3 × 3 convolution kernels to produce 128 channel feature maps. We have 6 set of module composed of 3 sets of convolution, batch normalization, and ReLU layers, and one bypass connection with a ReLU layer. Convolution layers in the modules use 128 set of 3 × 3 × 128 convolution kernels. In addition, the proposed network has a concatenation layer that concatenates the inputs of each module and the output of the last module, which is followed by the convolution layer with 128 set of 3 × 3 × 896 convolution kernels. This concatenation layer has a signal boosting effect using multiple signal representation 10 and provides various paths for gradient backpropagation. The last convolution layer uses 15 sets of 3 × 3 × 128 convolution kernels.
Finally, we add an end-to-end bypass connection to estimate the noise-free image while exploiting the advantages of bypass connection in He et al 27 .
The network architecture of discriminators D A and D B is illustrated in Fig. 4. This is from PatchGAN 26 , which has 70 × 70 receptive field and classifies image patches whether they are real or synthesized. Specifically, it consists of 5 convolution layers including the last fully-connected layer. The first convolution layer uses 64 sets of 4 × 4 convolution kernels, and the number of convolution kernels in the following layers is twice that of the previous layer except the last fully connected layer. After the last fully connected layer, 5 × 5 feature maps are obtained, and we calculate the l 2 -loss. Arbitrary sized images can be applied to this discriminator network by summing up the l 2 -loss from each 56 × 56 patch, after which the final decision is made.
III. METHODS
III.A. Data: Cardiac CT scans
The study cohort comprised 50 CT scans of mitral valve prolapse patients and 50 CT scans of coronary artery disease patients, and the CT scan protocols are described in previous reports 28, 29 . The mean age of the population was 58 ± 13.2 years, and the mean body weight was 66.2 ± 12.6 kg. Using a second generation dual source CT scanner (Somatom Definition Flash, Siemens, Erlangen, Germany), electrocardiography (ECG)-gated cardiac CT scanning was performed. Retrospective ECG-gated spiral scan with ECG-based tube current modulation was applied to multiphase of 0-90% of the R-R interval which comprises with a full dose pulsing window of 30-80% of the R-R interval. The tube current was reduced to 20% of the maximum outside the ECG pulsing window 30 (Fig. 1)
III.B. Training details
Training was performed by minimizing the loss function (2) with λ = 10 and γ = 5. We used the ADAM optimization method to train all networks with β 1 = 0.5 and β 2 = 0.999.
The number of epochs was 160, which was divided into two phases to control the learning rate during the training. In the first 100 epochs, we set the learning rate to 0.0002, and linearly decreased it to zero over the next epochs. We performed early stopping at 160 epochs, since the early stopping was shown to work as a regularization 31 . The size of patch was 56 × 56 and the size of mini-batch was 10. Kernels were initialized randomly from a Gaussian distribution. We have updated the generator and the discriminator at each iteration. We normalized the intensity of the input low-dose CT images and the target routine-dose CT image using the maximum intensity value of the input images, and subtract 0. 5
III.D. Statistical analysis
VGA scores obtained from original CT images and denoising images were compared using chi-square test. Image noise and SNR were compared using paired t-test. P -values of < 0.05 indicated statistical significance. Statistical analyses were performed using commercial software (SPSS, Chicago, IL, United States).
IV. RESULTS
IV.A. Qualitative evaluation
To verify the performance of the proposed method, we tested 50 cases from the dataset of mitral valve prolapse patients which were not used in the training session. Also, we tested 50 cases from the dataset coronary artery disease patients which were not used to training the network. The results are described in Fig. 5 and 6, respectively. Each row indicates showed that the proposed method is good at reducing the noise in the input CT images while the texture information and edges are still intact. The difference images showed that the proposed method did not change the detailed information and only removes noise from the input CT images. The proposed method is robust to the type of heart disease as confirmed in another disease cases in Fig. 6. Results showed that the network does not create any artificial features that can disturb the diagnosis while maintaining the crucial information.
We also observed that the proposed method is automatically adapted to the noise levels of the input CT images. Specifically, there are some data which have similar noise level between phase 1 and phase 8 as shown in Fig. 7. If the input CT images have a noise level similar to the CT target images, we have found that the proposed generator G AB does not show any noticeable change, as shown in in Fig. 7. These results confirms the proposed generator G AB acts as the identity for the images in the target domain, as shown in (8).
To compared the performance with the state-of-the-art model based iterative method (MBIR), we compared our algorithm with the Siemens ADMIRE (Advanced Modeled Iterative Reconstruction) algorithm 33 . ADMIRE is the latest MBIR method from Siemens, which has been improved from SAFIRE (Sinogram Affirmed. Iterative Reconstruction) algorithm. ADMIRE incorporates statistical modeling, both in the raw projection data and in the image domains, such that a different statistical weighting is applied according to the quality of the projection 33 , so ADMIRE is only available for latest scanner (Siemens Flash system). Thus, we cannot provide ADMIRE images for all patients in our retrospective studies, so we obtained multiphase CTA images from a new patient case.
As shown in Fig. 9, both ADMIRE and the proposed method successfully reduced noise in low-dose CT images. However, the difference images between input and results showed that, in the case of ADMIRE, the edge information was somewhat lost and over-smoothing
IV.B. Visual grading score and SNR analysis results
All visual scores are significantly higher in denoising CT, representing that the image quality of denoising CT is better (P < 0.001) (Table III). Quantitatively, image noise was decreased, and SNR was significantly increased on denoising CT (P < 0.05) (Table IV, Fig. 8), except no statistically significant SNR changes detected in left ventricular cavity where contrast enhanced blood pool measured by the largest region of interest (P = 0.96).
IV.C. Application to AAPM Data Set
We have performed additional experiments with AAPM low-dose CT grand challenge dataset which consists of abdominal CT images from ten patients. We used the 8 patient data for training and validation, and the remaining 2 patient data for the test. In contrast to the existing supervised learning approaches for low dose CT denosing 9 , here, the training was conducted in an unsupervised manner using the proposed network, with the input and target images randomly selected from the entire data set. Fig. 10(b) showed that the proposed unsupervised learning method provided even better images than the supervised learning, while there are some remaining artifacts in Fig. 10(a)(c). In general, the denoising results by the proposed approach has the competitive denoising performance compared to the supervised learning approach 9 .
IV.D. Ablation study
To analyze the roles of each building block in the proposed network architecture, we performed ablation studies by excluding the identity loss and/or cyclic loss and using the same training procedures. The results with respect to two different noise levels are illustrated in Fig. 11 and Fig. 12, respectively. Recall that the input low-dose CT image in the sub- illustrate reconstruction images as well as the difference images between the input and the reconstruction results. We also indicate the artificial features that were not present in the input images by red arrows.
All the reconstruction result images in Fig. 11 show that the noise level is reduced and the edge information is well maintained. In contrast to the proposed method that does not generate any artificial features, the other methods generated some structures which are not using the sigmoid cross entropy loss function is often unstable during training. To address this, we used LSGAN 25 and the cycle-loss 22 . Convergence plots in Fig. 13 shows that the proposed networks converged stably. Here, L G AB and L D B denoted the loss of generator Eq. 4 and the loss of discriminator Eq. 5. If network reaches the optimal equilibrium state, L G AB and L D B should be reached at 0.25, which was also shown in Fig. 13. The cyclic loss also decreased steadily during training process and converged. This confirms that the network training was well done.
Another critical issue with GAN is the problem of mode collapse. The GAN mode collapse occurs when the generator network generates limited outputs or even the same output, regardless of the input. Unlike the classic GAN, which generates samples from random noise inputs, our network creates samples from the noisy input that are closely related. In addition, the presence of an inverse path reduces the likelihood of mode collapse, and the identity loss prevents the creation of artificial features. Thanks to the synergistic combination of these components of network architectures, the likelihood of mode collapse was significantly reduced, and we have not observed any case where the generated outputs from distinct inputs are the same.
However, there are some limitations of the present studies. The current method mainly focused on multiphase CTA, and the performance of the proposed method is confirmed in this specific application. Also, our training, validation, and test data are generated using the same reconstruction kernel (B26f: cardiac filter). Thus, it is not clear whether our approach can be generalized to different kernels, organs, etc. Even though we provided preliminary results using the AAPM data set, more extensive study is required to validate the generalizability of the proposed method. These issues are very important for clinical uses, which need to be investigated in separate works.
We agree that once a well-trained network from supervised learning tasks is available, one can use low-dose acquisition for all cardiac phases. However, extensive clinical evaluation is required to have such drastic protocol changes, which is unlikely to happen in the near future. On the other hand, the proposed approach still uses the current acquisition protocols, but provide enhanced images as additional information for radiologists, which can be easily accepted in the current clinical setting. Moreover, in contrast to supervised learning approaches for low-dose CT, the unsupervised learning approaches, such as the proposed one, do not require vendor-supported simulated low-dose data or additional matched full/low-dose acquisition. Therefore, we believe that the potential for the proposed method in terms of science and product development could be significant.
VI. CONCLUSION
In this paper, we proposed a cycle consistent adversarial denoising network for multiphase coronary CT angiography. Unlike the existing supervised deep learning approaches for lowdose CT, our network does not require exactly matched low-and routine-dose images.
Instead, our network was designed to learn the image distributions from the high-dose cardiac phases. Furthermore, in contrast to the other state-of-the-art deep neural networks with GAN loss that are prone to generate artificial features, our network was designed to prevent from generating artificial features that are not present in the input image by exploiting the cyclic consistency and identity loss. Experimental results confirmed that the proposed method is good at reducing the noise in the input low-dose CT images while maintaining the texture and edge information. Moreover, when the routine dose images were used as input, the proposed network did not change the images, confirming that the algorithm correctly learn the noise. Radiological evaluation using visual grading analysis scores also confirmed that the proposed denoising method significantly increases the diagnostic quality of the images. Considering the effectiveness and practicability of the proposed method, our method can be widely applied for other CT acquisition protocols with dynamic tube current modulation.
ACKNOWLEDGEMENT
The authors would like to thanks Dr. Cynthia MaCollough, the Mayo Clinic, the Amer-
Fig. 2
2Overview of the proposed framework for low-dose CT image denoising. There are two generator networks G AB and G BA and two discriminator networks D A and D B . A denotes the low-dose CT image domain and B denotes the routine-dose CT image domain. The network employes three losses such as adversarial loss (adv), cyclic loss, and additionally identity loss.
Fig. 3 A
3generator architecture optimized for the low-dose CT image denoising 10 .
Fig. 4 A
4network architecture of discriminator 26 .
Fig. 5
5Restoration results from the dataset of mitral valve prolapse patients. Intensity range of the CT image is (-1024, 976)[HU] and the difference image between the input and result is (-150, 150)[HU]. Yellow arrow indicates the distinctly different region between input image from phase 1 and target image from phase 8.
Fig. 6
6Restoration results from the dataset of coronary artery disease patients. Intensity range of the CT image is (-924, 576)[HU] and the difference image between the input and result is (-200, 200)[HU]. Yellow arrow indicates the distinctly different region between input image from phase 1 and target image from phase 8. the different patient case, and the restoration results from the first column are shown in the second column. The input low-dose CT images are from phase 1 and the target routinedose images are from phase 8. Due to the cardiac motion during CT scanning, the shape of the heart and image intensity from the contrast agent are different at the two phases. Distinct differences are indicated by the yellow arrows in the images. The denoised results
Fig. 7
7Restoration results from the dataset whose input CT images have similar noise level with the target CT images. Intensity range of the CT image is (-924, 576)[HU] and the difference image between the input and result is (-15, 15)[HU]. Yellow arrow indicates the distinct different region between input image from phase 1 and target image from phase 8.
Fig. 8
8Standard deviation and signal-to-noise ratio between original CT (red) and denoising CT (purple) images measured from selected structures. (LV, left ventricle; pRCA, proximal right coronary artery)
Fig. 9
9Restoration results from the ADMIRE algorithm and the proposed method. Intensity range of the CT image is (-800, 800)[HU] and the difference image between the input and result is (-100, 100)[HU]. Yellow arrows indicate the streaking noise and red arrows indicate the details in the lung.
Fig. 10
10Restoration results from the AAPM challenge dataset using the proposed method and the supervised learning method 9 . Images of (a) the liver, (b) various organs includes the intestine and kidney, etc, (c) the bones. Intensity range of the CT image is (-300, 300)[HU].was observed in the lung region, indicated by red arrows. On the other hand, no structural loss was observed in the proposed method. Moreover, in the left two columns ofFig. 9, we can clearly see the remaining streaking artifacts in the ADMIRE images, while no such artifacts are observed in the proposed method. A similar, consistent improvement by the proposed method was observed in all volume slices.
figure (a) and target routine-dose image in the sub-figure (e) have different shape of heart due to the cardiac motion. The results of the proposed method are illustrated in the second column, the results of the excluding the identity loss are in the third column, and the results of the excluding the identity loss and cyclic loss are illustrated in the fourth column. We
Fig. 11 (
11a) Input CT image, (b) proposed method, (c) proposed method without identity loss, (d) with only GAN loss, (e) target CT image, (f-h) difference images between input image and result images (b-d), respectively. Intensity range of the CT image is (-820, 1430)[HU] and the difference image between the input and result is (-200, 200)[HU]. Red arrow indicates the artificial features that were not present in the input image. present in the input images. The result of the excluding the identity loss (third column) are better than the network trained only with GAN loss without including cycle consistency and identity loss (fourth column), but both methods deformed the shape of the heart and removed some structures. Similar observations can be found in Fig. 12 where input CT image has a similar noise level with target CT image. While the proposed method does not change the original image, the other methods deformed the shape and created the features that were not present in the input image. Considering that artificial features can confuse radiologists in diagnosing the patient's disease, the result confirmed the critical importance of cyclic loss and the identity loss as proposed by our algorithm. V. DISCUSSION Unsupervised learning with GAN has become popular in computer vision literatuires, which has demonstrated impressive performance for various tasks, but the classical GAN 24 with only GAN loss, (e) target CT image, (f-h) difference images between input image and result images (b-d), respectively. Intensity range of the CT image is (-924, 576)[HU] and the difference image between the input and result is (-100, 100)[HU]. Red arrow indicates the artificial features that were not present in the input image.
Fig. 13
13Convergence plots according to the epochs during the training process.
ican Association of Physicists in Medicine (AAPM), and grant EB01705 and EB01785 from the National Institute of Biomedical Imaging and Bioengineering for providing the Low-Dose CT Grand Challenge data set. This work is supported by Industrial Strategic technology development program (10072064, Development of Novel Artificial Intelligence Technologies To Assist Imaging Diagnosis of Pulmonary, Hepatic, and Cardiac Disease and Their Integration into Commercial Clinical PACS Platforms) funded by the Ministry of Trade Industry and Energy (MI, Korea). This work is also supported by the R&D Convergence Program of NST (National Research Council of Science & Technology) of Republic of Korea (Grant CAP-13-3-KERI).
. A bolus of 70-90 mL of contrast material (Iomeprol, Iomeron 400; Bracco, Milan, Italy) was administered by a power injector (Stellant D; Medrad, Indianola, PA, USA) at a rate of 4.0 mL/s and followed by 40 mL saline. The bolus tracking method (region of interest, the ascending aorta; attenuation threshold level, 100 HU; scan delay, 8 s) was applied to determine scan time. In all CT scans, tube voltage and the tube current-exposure time product were adjusted according to the patients body size, and the scan parameters were as follows: tube voltage, 80-120 kV; tube current-exposure time product, 185-380 mAs; collimation, 128 × 0.6 mm; and gantry rotation time, 280 s. Mean effective radiation dose of CCTA was 11.4 ± 6.2 mSv. Standard cardiac filter (B26f) was used for imaging reconstruction.
and multiply two to make the input image intensity range as [−1, 1]. For training, we used 50 cases from the dataset of mitral valve prolapse patients. The proposed method was implemented in Python with the PyTorch 32 and NVIDIA GeForce GTX 1080 Ti GPU was used to train and test the network.III.C. EvaluationIII.C.1. Visual grading analysisImage quality was assessed using relative visual grading analysis (VGA). This VGA method is planned to be related to the clinical task to evaluate any structural abnormality that may present at specific anatomical structures in a CT images. Two expert cardiac radiologists established a set of anatomical structures to evaluate image quality.Table Idemonstrates the 13 anatomical structures used in this study. The VGA scoring scale are shown inTable II. All CT images including denoising CT images were uploaded on picturearchiving and communication system (PACS) for visual grading. Of all, randomly selected
25 CT scans from mitral valve prolapse patients and 25 CT scans from coronary artery
disease patients were included for VGA. Total 1300 CT images (50 selected CT scans × 13
structures × original and denoising CT) were scored. Two radiologists performed VGA in
consensus, and all CT scans are scored independently, without side-by-side comparison.
Table I
IStructures selected as diagnostic requirements to assess the diagnostic quality of cardiac images. The structures were evaluated to be sharp with clear visualization. (LCA, left coronary artery; LV, left ventricle; RCA, right coronary artery; RV, right ventricle)Organ
Structure
Left/right coronary artery LCA ostium
LCA distal 1.5 cm
LCA distal
RCA ostium
RCA 1.5 cm
RCA distal
Cardiac wall
LV septum
RV free wall margin
Cardiac cavity
LV trabeculation
Left arterial appendage
Aorta
Aortic root
Valve
Aortic valve
Mitral valve
Table II
IIVisual grading analysis scores used to evaluate the structure visibilityScore
Visibility of the structures in relation to
the reference images
1 Poor image quality
2 Lower image quality
3 Mild noise, but acceptable
4 Average
5 Good
6 Excellent
III.C.2. Quantitative analysis
The image noise and signal-to-noise (SNR) of all images were obtained at four anatomical
structures: ascending aorta, left ventricular cavity, left ventricular septum, and proximal
right coronary artery. The size of region of interest to evaluate SNR were varied to fit each
anatomic structure; however, it was confined into each structure without overlapping other
structures.
Table III
IIIComparison of visual scores between original image and denoising CT image using Chi
Table IV
IVComparison of standard deviation and signal-to-noise ratio between original CT anddenoising CT images measured from selected structures (LV, left ventricle; pRCA, proximal right
coronary artery;)
Image noise
P-value SNR
P-value
(standard deviation)
Original image Denoising
Original image Denoising
Ascending aorta 48.0 ± 26.8
39.0 ± 17.0 0.003 11.1 ± 6.5
12.3 ± 6.2 0.001
LV cavity
69.4 ± 37.9
48.5 ± 19.0 < 0.001 8.9 ± 12.7
9.0 ± 3.5 0.96
LV septum
63.2 ± 40.8
40.9 ± 14.0 < 0.001 2.9 ± 1.7
3.4 ± 1.3 0.015
pRCA
70.5 ± 50.7
62.1 ± 48.5 0.036 6.4 ± 3.9
7.6 ± 4.4 0.034
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|
[] |
[
"The Brown-Colbourn Conjecture on Zeros of Reliability Polynomials is False",
"The Brown-Colbourn Conjecture on Zeros of Reliability Polynomials is False"
] |
[
"Gordon Royle [email protected] ",
"Alan D Sokal [email protected] ",
"\nDepartment of Computer Science & Software Engineering\nUniversity of Western\nAustralia\n",
"\nDepartment of Physics\nStirling Highway Crawley\n6009WAAUSTRALIA\n",
"\nNew York University\n4 Washington Place10003New YorkNYUSA\n"
] |
[
"Department of Computer Science & Software Engineering\nUniversity of Western\nAustralia",
"Department of Physics\nStirling Highway Crawley\n6009WAAUSTRALIA",
"New York University\n4 Washington Place10003New YorkNYUSA"
] |
[] |
We give counterexamples to the Brown-Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown-Colbourn conjecture is false already for the complete graph K 4 . The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K 4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel. , 82B20, 90B15, 90B18, 90B25, 94C15 (Secondary).Running Head: Brown-Colbourn Conjecture is FalseUnivariate Brown-Colbourn conjecture. For any graph G, the zeros of the univariate reliability polynomial R G (p) all lie in the closed disc |p − 1| ≤ 1. In other words, if |p − 1| > 1, then R G (p) = 0.Subsequently, one of us[16]proposed a multivariate extension of the Brown-Colbourn conjecture:Multivariate Brown-Colbourn conjecture. For any graph G, if |p e − 1| > 1 for all edges e, then R G (p) = 0.Not long ago, Wagner [18] proved, using an ingenious and complicated construction, that the univariate Brown-Colbourn conjecture holds for all series-parallel graphs. 2 Subsequently, 1 Henceforth we omit the prefix "multi". In this paper a "graph" is allowed to have loops and/or multiple edges unless explicitly stated otherwise.2 Unfortunately, there seems to be no completely standard definition of "series-parallel graph"; a plethora of slightly different definitions can be found in the literature[9,7,13,14,4]. So let us be completely precise about our own usage: we shall call a loopless graph series-parallel if it can be obtained from a forest by a finite sequence of series and parallel extensions of edges (i.e. replacing an edge by two edges in series or two edges in parallel). We shall call a general graph (allowing loops) series-parallel if its underlying loopless graph is series-parallel. Some authors write "obtained from a tree", "obtained from K 2 " or "obtained from C 2 " in place of "obtained from a forest"; in our terminology these definitions yield, respectively, all connected series-parallel graphs, all connected series-parallel graphs whose blocks form a path, or all 2-connected seriesparallel graphs. See [4, Section 11.2] for a more extensive bibliography.
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10.1016/j.jctb.2004.03.008
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[
"https://export.arxiv.org/pdf/math/0301199v2.pdf"
] | 12,271,491 |
math/0301199
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71eed1dd723186995cc97690d3059a5bead387ac
|
The Brown-Colbourn Conjecture on Zeros of Reliability Polynomials is False
2 Jun 2004 January 18, 2003 revised April 2, 2004
Gordon Royle [email protected]
Alan D Sokal [email protected]
Department of Computer Science & Software Engineering
University of Western
Australia
Department of Physics
Stirling Highway Crawley
6009WAAUSTRALIA
New York University
4 Washington Place10003New YorkNYUSA
The Brown-Colbourn Conjecture on Zeros of Reliability Polynomials is False
2 Jun 2004 January 18, 2003 revised April 2, 2004Running Head: Brown-Colbourn Conjecture is FalseReliability polynomialall-terminal reliabilityBrown-Colbourn conjectureTutte polynomialPotts model Mathematics Subject Classification (MSC 2000) codes: 05C99 (Primary)05C40, 68M10, 68M15, 68R10, 82B20, 90B15, 90B18, 90B25, 94C15 (Secondary)
We give counterexamples to the Brown-Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown-Colbourn conjecture is false already for the complete graph K 4 . The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K 4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel. , 82B20, 90B15, 90B18, 90B25, 94C15 (Secondary).Running Head: Brown-Colbourn Conjecture is FalseUnivariate Brown-Colbourn conjecture. For any graph G, the zeros of the univariate reliability polynomial R G (p) all lie in the closed disc |p − 1| ≤ 1. In other words, if |p − 1| > 1, then R G (p) = 0.Subsequently, one of us[16]proposed a multivariate extension of the Brown-Colbourn conjecture:Multivariate Brown-Colbourn conjecture. For any graph G, if |p e − 1| > 1 for all edges e, then R G (p) = 0.Not long ago, Wagner [18] proved, using an ingenious and complicated construction, that the univariate Brown-Colbourn conjecture holds for all series-parallel graphs. 2 Subsequently, 1 Henceforth we omit the prefix "multi". In this paper a "graph" is allowed to have loops and/or multiple edges unless explicitly stated otherwise.2 Unfortunately, there seems to be no completely standard definition of "series-parallel graph"; a plethora of slightly different definitions can be found in the literature[9,7,13,14,4]. So let us be completely precise about our own usage: we shall call a loopless graph series-parallel if it can be obtained from a forest by a finite sequence of series and parallel extensions of edges (i.e. replacing an edge by two edges in series or two edges in parallel). We shall call a general graph (allowing loops) series-parallel if its underlying loopless graph is series-parallel. Some authors write "obtained from a tree", "obtained from K 2 " or "obtained from C 2 " in place of "obtained from a forest"; in our terminology these definitions yield, respectively, all connected series-parallel graphs, all connected series-parallel graphs whose blocks form a path, or all 2-connected seriesparallel graphs. See [4, Section 11.2] for a more extensive bibliography.
Introduction
Let us consider a connected (multi)graph 1 G = (V, E) as a communications network with unreliable communication channels, in which edge e is operational with probability p e and failed with probability 1 − p e , independently for each edge. Let R G (p) be the probability that every node is capable of communicating with every other node (this is the so-called all-terminal reliability). Clearly we have
R G (p) = A ⊆ E (V, A) connected e∈A p e e∈E\A (1 − p e ) ,(1.1)
where the sum runs over all connected spanning subgraphs of G, and we have written p = {p e } e∈E . We call R G (p) the (multivariate) reliability polynomial [7] for the graph G; it is a multiaffine polynomial, i.e. of degree at most 1 in each variable separately. If the edge probabilities p e are all set to the same value p, we write the corresponding univariate polynomial as R G (p), and call it the univariate reliability polynomial. We are interested in studying the zeros of these polynomials when the variables p e (or p) are taken to be complex numbers. Brown and Colbourn [5] studied a number of examples and made the following conjecture:
one of us [16,Remark 3 in Section 4.1] showed, by a two-line induction, that the multivariate Brown-Colbourn conjecture holds for all series-parallel graphs. 3 Both the univariate and multivariate conjectures remained open for general graphs, but most workers in the field suspected that they would be true. (At least the present authors did.) In this short note we would like to report that both the univariate and multivariate Brown-Colbourn conjectures are false! The multivariate conjecture is false already for the simplest non-series-parallel graph, namely the complete graph K 4 . As a corollary we will deduce that the univariate conjecture is false for a 4-vertex, 16-edge planar graph that can be obtained from K 4 by adding parallel edges, and for a 1512-vertex, 3016-edge simple planar graph that can be obtained from K 4 by adding parallel edges and then subdividing edges. So the Brown-Colbourn conjecture is not true even for simple planar graphs.
Furthermore, for the multivariate property we are able to obtain a complete characterization: a graph has the multivariate Brown-Colbourn property if and only if it is seriesparallel.
It is convenient to restate the Brown-Colbourn conjectures in terms of the generating polynomial for connected spanning subgraphs,
C G (v) = A ⊆ E (V, A) connected e∈A v e ,(1.2)
where we have written v = {v e } e∈E . This is clearly related to the reliability polynomial by
R G (p) = e∈E (1 − p e ) C G p 1 − p (1.3) C G (v) = e∈E (1 + v e ) R G v 1 + v (1.4)
where 1 denotes the vector with all entries 1, and division of vectors is understood componentwise. The multivariate Brown-Colbourn conjecture then states that if G is a loopless graph and |1 + v e | < 1 for all edges e, then C G (v) = 0. Loops must be excluded because a loop e multiplies C G by a factor 1 + v e but leaves R G unaffected. Some workers also prefer to use the failure probabilities q e = 1 − p e as the variables. The plan of this paper is as follows: In Section 2 we show that the multivariate Brown-Colbourn conjecture fails for the complete graph K 4 . In Section 3 we review the series and parallel reduction formulae for the reliability polynomial. In Section 4 we show that the univariate Brown-Colbourn conjecture fails for certain graphs that are obtained from K 4 by adding parallel edges and then optionally subdividing edges. In Section 5 we complete these results by showing that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel.
2 The multivariate Brown-Colbourn conjecture is false for K 4
For the complete graph K 4 , the univariate polynomial C G (v) is
C K 4 (v) = 16v 3 + 15v 4 + 6v 5 + v 6 . (2.1)
The roots of this polynomial all lie outside the disc |1 + v| < 1, so the univariate Brown-Colbourn conjecture is true for K 4 . Let us now consider the bivariate situation, in which the six edges receive two different weights a and b. There are five cases:
(a) One edge receives weight a and the other five receive weight b:
C K 4 (a, b) = (8b 3 + 5b 4 + b 5 ) + (8b 2 + 10b 3 + 5b 4 + b 5 )a (2.2) (b)
A pair of nonintersecting edges receive weight a and the other four edges receive weight b:
C K 4 (a, b) = (4b 3 + b 4 ) + (8b 2 + 8b 3 + 2b 4 )a + (4b + 6b 2 + 4b 3 + b 4 )a 2 (2.3)
(c) A pair of intersecting edges receive weight a and the other four edges receive weight b:
C K 4 (a, b) = (3b 3 + b 4 ) + (10b 2 + 8b 3 + 2b 4 )a + (3b + 6b 2 + 4b 3 + b 4 )a 2 (2.4)
(d) A 3-star receives weight a and the complementary triangle receives weight b:
C K 4 (a, b) = (9b 2 + 3b 3 )a + (6b + 9b 2 + 3b 3 )a 2 + (1 + 3b + 3b 2 + b 3 )a 3 (2.5)
(e) A three-edge path receives weight a and the complementary three-edge path receives weight b:
C K 4 (a, b) = b 3 + (7b 2 + 3b 3 )a + (7b + 9b 2 + 3b 3 )a 2 + (1 + 3b + 3b 2 + b 3 )a 3 (2.6)
We have plotted the roots a when b traces out the circle |1 + b| = 1, and vice versa. In cases (b) and (d) it turns out that the roots can enter the "forbidden discs" |1 + a| < 1 and |1 + b| < 1. This is shown in Figure 1 for case (b); blow-ups of the crucial regions are shown in Figure 2 both for case (b) and for case (d). As a result, counterexamples to the multivariate Brown-Colbourn conjecture can be obtained in these two cases: indeed, for any a lying in the region A + (resp. A − ), there exists b ∈ B − (resp. B + ) such that C K 4 (a, b) = 0, and conversely. Let us note for future reference that the endpoint of the region A ± (resp. B ± ) lies at a = −1 + e ±2πiα (resp. b = −1 + e ±2πiβ ), where α ≈ 0.120692 and β ≈ 0.164868 in case (b), and α ≈ 0.110198 and β ≈ 0.030469 in case (d).
We can understand this behavior analytically as follows: For each of the five cases, let us solve the equation C K 4 (a, b) = 0 for a in terms of b, expanding in power series for b near 0. We obtain:
b 2 + O(b 3 ) (b) a = −b ± 1 2 b 3/2 + O(b 2 ) (c) a = − 1 3 b + 1 8 b 2 + O(b 3 ) and a = −3b + 31 8 b 2 + O(b 3 ) (d) a = −3b ± i √ 3b 3/2 + O(b 2 ) and a = 0 (e) a = −b + 3 4 b 2 + O(b 3 ) and a = (−3 ± 2 √ 2)b + 9 16 (10 ∓ 7 √ 2)b 2 + O(b 3 )
The behavior is thus different in cases (a,c,e) on the one hand and cases (b,d) on the other:
Cases (a,c,e): Here the solution is of the form
a = γ 1 b + γ 2 b 2 + O(b 3 ) (2.7)
with γ 1 , γ 2 real. Therefore, if we set b = −1 + e iθ and expand in powers of θ, we obtain
|1 + a| 2 = 1 + (γ 2 1 − γ 1 − 2γ 2 )θ 2 + O(θ 4 ) . (2.8)
Provided that γ 2 1 − γ 1 − 2γ 2 > 0 -as indeed holds for all the roots in cases (a,c,e) -we have |1 + a| ≥ 1 for small θ, so no counterexample is found (at least for small θ).
Cases (b,d):
Here, by contrast, the solution is of the form
a = δ 1 b + δ 2 b 3/2 + O(b 2 ) (2.9)
with δ 1 < 0 and δ 2 = 0. Therefore, if we set b = −1 + e iθ and expand as before, we obtain
a = iδ 1 θ + e ±3πi/4 δ 2 θ 3/2 + O(θ 2 ) . (2.10)
Since Re(e ±3πi/4 δ 2 ) < 0 for at least one of the roots, we have Re a ∝ −| Im a| 3/2 for small θ; in particular, we have |1 + a| < 1 for small θ = 0. In fact, more can be said: suppose that we fix any λ > 0 and set b = λ(−1 + e iθ ). Then we have
a = iδ 1 λθ + e ±3πi/4 δ 2 λ 3/2 θ 3/2 + O(θ 2 ) ,(2.11)
so that once again Re a ∝ −| Im a| 3/2 for small θ. In particular, we will have |λ + a| < λ for small θ = 0, irrespective of how small λ was chosen. This observation will play a crucial role in Section 5 (see Proposition 5.5). 7
Series and parallel reduction formulae
Suppose that G contains edges e 1 , . . . , e n (with corresponding weights v 1 , . . . , v n ) in parallel between the same pair of vertices x, y. Then it is easy to see that the edges e 1 , . . . , e n can be replaced by a single edge of weight
v 1 v 2 · · · v n ≡ n i=1 (1 + v i ) − 1 (3.1) without changing the value of C G (v). [Reason:
x is connected to y via this "super-edge" if and only if x is connected to y by at least one of the edges e 1 , . . . , e n .] Suppose next that G contains edges e 1 , . . . , e n (with corresponding weights v 1 , . . . , v n ) in series between the pair of vertices x, y: this means that the edges e 1 , . . . , e n form a path in which all the vertices except possibly the endvertices x and y have degree 2 in G. Let G ′ be the graph in which the edges e 1 , . . . , e n are replaced by a single edge e * from x to y. Then it is not hard to see that
C G (v) = n j=1 i =j v i C G ′ (v ′ ) (3.2)
where the edge e * is given weight
v ′ * = v 1 ⊲⊳ v 2 ⊲⊳ · · · ⊲⊳ v n ≡ 1 n i=1 1/v i (3.3)
and all edges other than e 1 , . . . , e n , e * are given weight v ′ e = v e . [Reason: A connected spanning subgraph of G can omit at most one of the edges e 1 , . . . , e n , for otherwise at least one of the internal vertices of the path would be disconnected from both x and y. Moreover, x is connected to y via the "super-edge" e * if and only if none of the edges e 1 , . . . , e n are omitted. The relative weight of the cases with and without x connected to y via e * is thus
( n i=1 v i )/( n j=1 i =j v i ) = v * ;
and there is an overall normalization factor n j=1 i =j v i . See also [7, p. 35] for an equivalent formula.]
The formula for series reduction can be applied immediately to handle arbitrary subdivisions of a graph G. Given a finite graph G = (V, E) and a family of integers s = {s e } e∈E ≥ 1, we define G ⊲⊳s to be the graph in which each edge e of G is subdivided into s e edges in series. If s ≥ 1 is an integer, we define G ⊲⊳s to be the graph in which each edge of G is subdivided into s edges in series. All the edges in G ⊲⊳s or G ⊲⊳s obtained by subdividing the edge e ∈ E are assigned the same weight v e as was assigned to e in the original graph G. It follows immediately from (3.2)/(3.3) that
C G ⊲⊳s (v) = e∈E s e v se−1 e C G (v/s) (3.4) where (v/s) e ≡ v e /s e .
Remarks. 1. Series and parallel reduction formulae can be derived in the more general context of the q-state Potts model (also known as the multivariate Tutte polynomial): see e.g. [17,Section 2]. Parallel reduction is always given by (3.1), independently of the value of the parameter q. Series reduction is given by
v 1 ⊲⊳ v 2 ⊲⊳ · · · ⊲⊳ v n = q n i=1 (1 + q/v i ) − 1 . (3.5)
Please note that (3.5) reduces to (3.3) when q → 0, which is precisely the limit in which the multivariate Tutte polynomial Z G (q, v) tends (after division by q) to C G (v).
2. If one takes in C G (v) the further limit of v infinitesimal, one obtains the generating polynomial of minimal connected spanning subgraphs, i.e. spanning trees. Now, spanning trees are intimately related to linear electrical circuits, as was noticed by Kirchhoff in 1847 [10,12]. For v infinitesimal, the parallel reduction formula (3.1) becomes for the cases (c), (d) and (e) discussed in Section 2. We saw in Section 2 that in cases (b) and (d) one can obtain a counterexample to the multivariate Brown-Colbourn conjecture by choosing the weight a to lie anywhere in the region A + ; this leads to a root b lying in the region B − (see Figures 1 and 2). Note now that the pth power of the region 1 + A + will overlap the region 1 + B − whenever p > (1 − β)/α [just choose any point b ∈ B − close enough to the endpoint −1 + e −2πiβ = −1 + e 2πi(1−β) ; then one of the pth roots of 1 + b will lie in the region 1 + A + ]. And (3.1) tells us that p edges in parallel, each with weight v, are equivalent to a single edge with weight v eff satisfying 1 + v eff = (1 + v) p . This reasoning suggests that counterexamples to the univariate Brown-Colbourn conjecture might be found for the graphs K have, for all p > (1 − α)/β, a root v ∈ B + . These guesses are in fact correct, and we find the following counterexamples to the univariate Brown-Colbourn conjecture: Counterexamples are also obtained for each larger p; some typical numbers are shown in Table 1. Please note that all these counterexample graphs are planar.
v 1 v 2 · · · v n ≡ v 1 + v 2 + . . . + v n ,(3.• G = K (b,1,7) 4 , 30 edges: v ≈ −0.269253 ± 0.682304i, |1 + v| ≈ 0.999765 • G = K
The graphs
G = K (b/d,p 1 ,p 2 ) 4
are, of course, non-simple (except when p 1 = p 2 = 1); so one might cling to the hope that the univariate Brown-Colbourn conjecture is true at least for simple graphs (or, weaker yet, for simple planar graphs). But these hopes too are false. To see why this is the case, consider the following procedure: 1) Choose p 1 , p 2 so that the graph K (b,p 1 ,p 2 ) 4 has a root v 1 satisfying |1 + v 1 | < 1.
2) Choose any integer s ≥ 2.
3) Find an integer k large enough so that v k ≡ −1 + (1 + v 1 ) 1/k -defined using the root with | arg[(1 + v 1 ) 1/k ]| ≤ π/k -lies in the disc |1/s + v k | < 1/s. [It is always possible to find such a k, because the points v k lie on a logarithmic spiral that approaches the point v = 0 making a nonzero angle with the imaginary axis, while all the circles |1/s + v| = 1/s pass through v = 0 tangent to the imaginary axis.]
Then v k is a root for the graph K (b,kp 1 ,kp 2 ) 4
, by the rules for parallel reduction; and sv k is a root for the graph (K (b,kp 1 ,kp 2 ) 4 ) ⊲⊳s , by the rules for series reduction. And by construction we have |1 + sv k | < 1. Therefore, the graph (K (b,kp 1 ,kp 2 ) 4 ) ⊲⊳s , which is simple and planar, is the desired counterexample.
For example, if we take (p 1 , p 2 ) = (11, 1) and s = 2, counterexamples can be obtained for k ≥ 58: ) ⊲⊳(1,s) can probably be found by direct search. But the foregoing construction has the advantage that there is no need to compute the roots of extremely-high-degree polynomials; it suffices to compute the roots for the base case K (b/d,p 1 ,p 2 ) 4 (for which the polynomials are large but not huge) and then make simple manipulations on them.
Methodological remark. In this work we needed to compute accurately the roots of polynomials of fairly high degree (up to 93) with very large integer coefficients (up to about 10 27 ). To do this we used the package MPSolve 2.0 developed by Dario Bini and Giuseppe Fiorentino [2,3]. MPSolve is much faster than Mathematica's NSolve for high-degree polynomials (this is reported in [3], and we confirm it); it gives guaranteed error bounds for the roots, based on rigorous theorems [3]; its algorithms are publicly documented [3]; and its source code is freely available [2].
Let us mention, finally, that counterexamples with smaller values of |1 + v| can be found. Consider, for example, the complete graph K 6 in which a pair of vertex-disjoint triangles receives weight a and the remaining nine edges receive weight b. We have C K 6 (a, b) = (81b 5 + 78b 6 + 36b 7 + 9b 8 + b 9 ) + (324b 4 + 594b 5 + 480b 6 + 216b 7 + 54b 8 + 6b 9 )a + (486b 3 + 1314b 4 + 1665b 5 + 1224b 6 + 540b 7 + 135b 8 + 15b 9 )a 2 + (324b 2 + 1188b 3 + 2160b 4 + 2376b 5 + 1656b 6 + 720b 7 + 180b 8 + 20b 9 )a 3 + (81b + 432b 2 + 1134b 3 + 1800b 4 + 1854b 5 + 1254b 6 + 540b 7 + 135b 8 + 15b 9 )a 4 + (54b + 216b 2 + 504b 3 + 756b 4 + 756b 5 + 504b 6 + 216b 7 + 54b 8 + 6b 9 )a 5 + (9b + 36b 2 + 84b 3 + 126b 4 + 126b 5 + 84b 6 + 36b 7 + 9b 8 + b 9 )a 6 (4.1)
If we then substitute a = (1+v) p 1 −1 and b = (1+v) p 2 −1, counterexamples to the univariate Brown-Colbourn conjecture can be found for many pairs (p 1 , p 2 ). For example, for (p 1 , p 2 ) = (1, 6) we obtain a 60-edge non-planar graph whose roots include v ≈ −0.357514 ± 0.713815 i, yielding |1 + v| ≈ 0.960375. It would be interesting to know whether examples can be found in which |1 + v| is arbitrarily small. More generally, one can ask:
Question 4.1
What is the closure of the set of all roots of the polynomials C G (v) as G ranges over all graphs? Over all planar graphs? Over all simple planar graphs?
Brown and Colbourn [5] pointed out that the graphs G = C (p) n (the n-cycle with each edge replaced by p parallel edges) have roots that, taken together, are dense in the region |1 + v| ≥ 1. We have shown here that roots can also enter the region |1 + v| < 1. But how far into this latter region can they penetrate? Might the roots actually be dense in the whole complex plane? If this is indeed the case, it would mean that the univariate Brown-Colbourn conjecture is as false as it can possibly be. , as the strip length tends to infinity, penetrates into the "forbidden region" |1 + v| < 1. Some of these families consist of planar graphs.
Series-parallel is necessary and sufficient
In this section we shall prove that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel.
Let us begin by defining a weakened version of the Brown-Colbourn property:
Definition 5.1 Let G be a graph, and let λ > 0. We say that G
• has the univariate property BC λ if C G (v) = 0 whenever |λ + v| < λ;
• has the multivariate property BC λ if C G (v) = 0 whenever |λ + v e | < λ for all edges e.
Properties BC 1 are, of course, the original univariate and multivariate Brown-Colbourn properties; the properties BC λ become increasingly weaker as λ is decreased. The properties BC λ are intimately related to subdivisions: Indeed, Lemmas 5.2 and 5.3 are an immediate consequence of the formula (3.4) for subdivisions -which states that subdivision by s moves the nonzero roots from v to sv -together with the fact that |sλ + sv| < sλ is equivalent to |λ + v| < λ.
In the preceding section we have shown that not all graphs have the univariate property BC 1 . It is nevertheless true -and virtually trivial -that every connected graph has the univariate property BC λ for some λ > 0. (Since a non-identically-vanishing univariate polynomial has finitely many roots, it suffices to choose λ small enough so that none of the roots of C G (v) lie in the disc |λ + v| < λ.) By Lemma 5.2, an equivalent assertion is that G ⊲⊳s has the univariate property BC 1 for all sufficiently large integers s. 5 The situation is very different, however, when we consider the multivariate property BC λ . We begin with a simple but important lemma: Lemma 5.4 Let λ > 0, and suppose that the connected graph G has the multivariate property BC λ . Then every connected subgraph H ⊆ G also has the multivariate property BC λ .
Proof. Consider first the case in which H is a connected spanning subgraph (i.e. its vertex set is the same as that of G). Let us write
v = (v ′ , v ′′ ) where v ′ = {v e } e∈E(H) and v ′′ = {v e } e∈E(H)\E(G) . Then C H (v ′ ) = C G (v ′ , 0) = lim v ′′ →0 C G (v ′ , v ′′ ) . (5.1)
By hypothesis, C G (v ′ , v ′′ ) = 0 whenever |λ + v e | < λ for all e ∈ E(G). Now take v ′′ → 0 from within this product of discs (0 lies on its boundary). By Hurwitz's theorem 6 , either C H (v ′ ) is nonvanishing whenever |λ + v e | < λ for all e ∈ E(H), or else C H is identically zero. But the latter is impossible since H is connected. Now let H be an arbitrary connected subgraph of G (spanning or not). Construct a connected spanning subgraph H of G by hanging trees off some or all of the vertices of H without creating any new circuits. 7
Let us write v = {v e } e∈E( H) = (v ′ , v ′′ ) where v ′ = {v e } e∈E(H) and v ′′ = {v e } e∈E( H)\E(H) . Then C H (v) = C H (v ′ ) e∈E( H)\E(H) v e . (5.2)
Since H has multivariate property BC λ , so does H.
The following is the fundamental fact from which all else flows:
Proposition 5.5 The complete graph K 4 does not have the multivariate property BC λ for any λ > 0.
Proof. This is an almost immediate consequence of the observations made at the end of Section 2. In cases (b) and (d), for any λ > 0 there exists b with |λ + b| = λ for which at least one of the solutions to C K 4 (a, b) = 0 satisfies |λ + a| < λ. By slightly perturbing this pair, we can find a pair (a, b) with C K 4 (a, b) = 0 satisfying |λ + a| < λ and |λ + b| < λ. So K 4 does not even have the bivariate property BC λ .
We can deduce from Lemma 5.4 and Proposition 5.5 a necessary and sufficient condition for G to have various forms of the multivariate Brown-Colbourn property:
Theorem 5.6 Let G be a loopless connected graph. Then the following are equivalent:
(a) G has the multivariate property BC 1 . 6 Hurwitz's theorem states that if D is a domain in C n and (f k ) are nonvanishing analytic functions on D that converge to f uniformly on compact subsets of D, then f is either nonvanishing or else identically zero. Hurwitz's theorem for n = 1 is proved in most standard texts on the theory of analytic functions of a single complex variable (see e.g. [1, p. 176]). Surprisingly, we have been unable to find Hurwitz's theorem proven for general n in any standard text on several complex variables (but see [11, p. 306] and [15, p. 337]). So here, for completeness, is the sketch of a proof: Suppose that f (c) = 0 for some c = (c 1 , . . . , c n ) ∈ D, and let D ′ ⊂ D be a small polydisc centered at c. Applying the single-variable Hurwitz theorem, we conclude that f (z 1 , c 2 , . . . , c n ) = 0 for all z 1 such that (z 1 , c 2 , . . . , c n ) ∈ D ′ . Applying the same argument repeatedly in the variables z 2 , . . . , z n , we conclude that f is identically vanishing on D ′ and hence, by analytic continuation, also on D.
(b) G has the multivariate property BC λ for some λ > 0.
(c) G is series-parallel.
Proof. (a) =⇒ (b) is trivial.
(b) =⇒ (c): Let G be a loopless connected graph that is not series-parallel. Then G contains a subgraph H that is a subdivision of K 4 . 8 Suppose that H = (K 4 ) ⊲⊳s with s = (s 1 , . . . , s 6 ), and define s = max(s 1 , . . . , s 6 ). Now fix any λ > 0; then, by Proposition 5.5 we can find a vector v = (v 1 , . . . , v 6 ) that is a zero of C K 4 (v) and satisfies |λ/s + v i | < λ/s for i = 1, . . . , 6. It then follows that the vector v ′ = (v ′ 1 , . . . , v ′ 6 ) defined by v ′ i = s i v i satisfies C H (v ′ ) = 0 and |λ + v ′ i | < λ for i = 1, . . . , 6. Therefore H does not have the multivariate property BC λ . By Lemma 5.4, G cannot have this property either.
(c) =⇒ (a): This is proven in [16, Remark 3 in Section 4.1], but for the convenience of the reader we repeat the proof here. Suppose that G is a loopless connected series-parallel graph; this means that G can be obtained from a tree by a finite sequence of series and parallel extensions of edges (i.e. replacing an edge by two edges in series or two edges in parallel). We will prove that G has the multivariate property BC 1 , by induction on the length of this sequence of series and parallel extensions. The base case is when G is a tree: then C G (v) = e∈E(G) v e and G manifestly has the multivariate property BC 1 . Now suppose that G is obtained from a smaller graph G ′ by replacing an edge e * of G ′ by two parallel edges e 1 , e 2 . Use the parallel reduction formula (3.1): since |1+v 1 | < 1 and |1+v 2 | < 1 imply |1+v * | < 1, we deduce that G has the multivariate property BC 1 if G ′ does. Suppose, finally, that G is obtained from a smaller graph G ′ by replacing an edge e * of G ′ by two edges e 1 , e 2 in series. Use the series reduction formula (3.2)/(3.3) and the fact that |1 + v| < 1 is equivalent to Re(1/v) < −1/2: then Re(1/v i ) < −1/2 for i = 1, 2 implies that Re(1/v * ) < −1 < −1/2, and moreover the prefactor v 1 + v 2 is nonzero; so we deduce that G has the multivariate property BC 1 if G ′ does.
For each graph G, let us define λ ⋆ (G) to be the maximum λ for which G has the multivariate property BC λ . Then Theorem 5.6 states a surprising (at first sight) dichotomy: either λ ⋆ (G) = 0 [when G is not series-parallel] or else λ ⋆ (G) ≥ 1 [when G is series-parallel].
Some series-parallel graphs have λ ⋆ (G) = 1 exactly: for example, the graphs K (n) 2 (a pair of vertices connected by n parallel edges) have C K (n) 2 (v) = (1 + v) n − 1 and hence even have univariate roots on the circle |1 + v| = 1. On the other hand, some series-parallel graphs have λ ⋆ (G) > 1: for example, the cycles C n have λ ⋆ (G) = n/2. [Proof: We have
C Cn (v) = n i=1 v i 1 + n i=1 1 v i ,(5.3)
Figure 1 :
1Curves for case (b). First plot shows the a-plane; second plot shows the b-plane. Dashed magenta curve is the circle |1 + v| = 1; solid blue curve is the locus of root a; solid red curve is the locus of root b.
Figure 2 :
2Blow-up of curves to show more clearly the "sliver" regions A + and B − . Top row shows the a-and b-planes for case (b); bottom row shows the a-and b-planes for case (d).
precisely the law for putting electrical conductances in parallel. And the series reduction formula (3.3) is precisely the law for putting electrical conductances in series! 4 The univariate Brown-Colbourn conjecture is false as well Let K (a,p 1 ,p 2 ) 4 be the graph obtained from K 4 by replacing one edge by p 1 parallel edges and replacing each of the other five edges by p 2 parallel edges. Let K (b,p 1 ,p 2 ) 4be the graph obtained from K 4 by replacing each of two nonintersecting edges by p 1 parallel edges and replacing each of the remaining four edges by p 2 parallel edges. Define in a similar manner K
edges: v ≈ −0.017476 ± 0.185846i, |1 + v| ≈ 0.999946
))
⊲⊳2 , which has 1512 vertices and 3016 edges, is a counterexample to the univariate Brown-Colbourn conjecture. Similarly, if we take (p 1 , p 2 ) = (1, 12) and s = 2, counterexamples can be obtained for k ≥ 36:• v 1 ≈ −0.112 358 418 620 + 0.453 757 934 703i, |1 + v 1 | ≈ 0.996 897 106 175• v 36 ≈ −0.000 172 469 038 + 0.013 125 252 246i, |1 + 2v 36 | ≈ 0.⊲⊳2 , which has 1804 vertices and 3600 edges, is a counterexample to the univariate Brown-Colbourn conjecture.Smaller counterexamples of the forms (K
Note
Added (April 2004). Building on the examples constructed in this section, Chang and Shrock [6, Sections 5.17 and 5.18] have recently devised families of strip graphs in which the limiting curve of zeros of C G (v)
Lemma 5. 2
2Let λ > 0 and let s be a positive integer. Then the following are equivalent for a graph G: (a) G has the univariate property BC λ . (b) G ⊲⊳s has the univariate property BC sλ . Lemma 5.3 Let λ > 0 and let s be a positive integer. Then the following are equivalent for a graph G: (a) G has the multivariate property BC λ . (b) G ⊲⊳s has the multivariate property BC sλ . (c) G ⊲⊳s has the multivariate property BC sλ for all vectors s satisfying s e ≥ s for all edges e.
: for all p > (1 − β)/α they should have a root v ∈ A + . 4 Likewise, the graphs K(b,1,p)
4
and K
(d,1,p)
4
(b,p,1)
4
and K
(d,p,1)
4
are expected to
Value of p
Graph
6
7
8
9
10
11
12
13
14
15
K
(b,1,p)
4
1
0.999765 0.997818 0.996996 0.996734 0.996749 0.996897 0.997102 0.997326 0.997547
K
(b,p,1)
4
0.998274 0.997234 0.997001 0.997083 0.997284 0.997519 0.997753 0.997971 0.998169 0.998345
K
(d,1,p)
4
1
1
1
0.999956 0.999813 0.999746 0.999718 0.999713 0.999718 0.999730
K
(d,p,1)
4
1
1
1
1
1
1
1
1
1
1
Table 1 :
1Minimum value of |1 + v| for a zero of C G (v) for selected graphs G = K ≤ p ≤ 5 the value equals 1. A value strictly less than 1 indicates a counterexample to the univariate Brown-Colbourn conjecture. For K(b/d,p 1 ,p 2 )
4
.
This proof is reproduced here as Theorem 5.6(c) =⇒ (a).3
We do not claim that this is a proof, though we suspect that a suitable topological argument might be able to turn it into a proof.
Brown and Colbourn [5, Proposition 4.4 and Theorem 4.5] have proven a result also for non-uniform subdivisions G ⊲⊳s : namely, for each graph G there exists an integer s such that G ⊲⊳s has the univariate property BC 1 whenever s e ≥ s for all e. This is significantly stronger than the just-mentioned trivial result, and it would be worth trying to understand it better. Brown and Colbourn's method looks very different from ours, at least at first glance; it would be interesting to try to translate it into our language. In particular, there may be a "partially multivariate" result hiding underneath their apparently univariate proof.
This can be done, for instance, by running breadth-first search with the vertices of H initially on the queue.
The relevant fact is the following [8, Exercise 8.16 and Proposition 1.7.2]: G is series-parallel ⇐⇒ G has no K 4 minor ⇐⇒ G has no K 4 topological minor. And the latter statement says precisely that G contains no subgraph H that is a subdivision of K 4 . See also[9,13].
AcknowledgmentsWe wish to thank Jason Brown, Robert Shrock and Dave Wagner for valuable conversations and correspondence. We also wish to thank Dario Bini for supplying us the MPSolve package[2,3]for finding roots of polynomials. Finally, we wish to thank an anonymous referee for helpful suggestions.This research was supported in part by U.S. National Science Foundation grant PHY-0099393 and by a University of Western Australia research grant.
It is an interesting open problem to characterize the graphs that have λ ⋆ (G) = 1 or, more ambitiously, to find a simple graph-theoretic formula for λ ⋆ (G). It is an interesting open problem to characterize the graphs that have λ ⋆ (G) = 1 or, more ambitiously, to find a simple graph-theoretic formula for λ ⋆ (G).
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Design, analysis and implementation of a multiprecision polynomial rootfinder. D A Bini, G Fiorentino, Numer. Algorithms. 23D.A. Bini and G. Fiorentino, Design, analysis and implementation of a multiprecision polynomial rootfinder, Numer. Algorithms 23 (2000), 127-173.
A Brandstädt, Le Van Bang, J P Spinrad, Graph Classes: A Survey. SIAM, PhiladelphiaA. Brandstädt, Le Van Bang and J.P. Spinrad, Graph Classes: A Survey (SIAM, Philadelphia, 1999).
Roots of the reliability polynomial. J I Brown, C J Colbourn, SIAM J. Discrete Math. 5J.I. Brown and C.J. Colbourn, Roots of the reliability polynomial, SIAM J. Discrete Math. 5 (1992), 571-585.
Reliability polynomials and their asymptotic limits for families of graphs. S.-C Chang, R Shrock, J. Statist. Phys. 112cond-mat/0208538 at xxx.lanl.govS.-C. Chang and R. Shrock, Reliability polynomials and their asymptotic limits for families of graphs, J. Statist. Phys. 112 (2003), 1019-1077, cond-mat/0208538 at xxx.lanl.gov.
C J Colbourn, The Combinatorics of Network Reliability. New York-OxfordOxford University PressC.J. Colbourn, The Combinatorics of Network Reliability (Oxford University Press, New York-Oxford, 1987).
R Diestel, Graph Theory. New YorkSpringer-VerlagR. Diestel, Graph Theory (Springer-Verlag, New York, 1997).
Topology of series-parallel graphs. R J Duffin, J. Math. Anal. Appl. 10R.J. Duffin, Topology of series-parallel graphs, J. Math. Anal. Appl. 10 (1965), 303-318.
Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefürht wird. G Kirchhoff, Ann. Phys. Chem. 72G. Kirchhoff,Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefürht wird, Ann. Phys. Chem. 72 (1847), 497-508.
Function Theory of Several Complex Variables. S G Krantz, Wadsworth & Brooks/Cole, Pacific Grove, California2nd ed.S.G. Krantz, Function Theory of Several Complex Variables, 2nd ed. (Wadsworth & Brooks/Cole, Pacific Grove, California, 1992).
An algebraic proof of Kirchhoff's network theorem. A Nerode, H Shank, Amer. Math. Monthly. 68A. Nerode and H. Shank, An algebraic proof of Kirchhoff's network theorem, Amer. Math. Monthly 68 (1961), 244-247.
Graphs and series-parallel networks. J Oxley, Theory of Matroids. N. WhiteCambridgeCambridge University Press6J. Oxley, Graphs and series-parallel networks, in N. White (editor), Theory of Matroids, Chapter 6, pp. 97-126 (Cambridge University Press, Cambridge, 1986).
J G Oxley, Matroid Theory. New YorkOxford University PressJ.G. Oxley, Matroid Theory (Oxford University Press, New York, 1992).
B Simon, The P (ϕ) 2 Euclidean (Quantum) Field Theory. PrincetonPrinceton University PressB. Simon, The P (ϕ) 2 Euclidean (Quantum) Field Theory (Princeton University Press, Princeton, 1993).
Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. A D Sokal, Combin. Probab. Comput. 10cond-mat/9904146 at xxx.lanl.govA.D. Sokal, Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions, Combin. Probab. Comput. 10 (2001), 41-77, cond-mat/9904146 at xxx.lanl.gov.
Chromatic roots are dense in the whole complex plane. A D Sokal, Combin. Probab. Comput. 13cond-mat/0012369 at xxx.lanl.govA.D. Sokal, Chromatic roots are dense in the whole complex plane, Combin. Probab. Comput. 13 (2004), 221-261, cond-mat/0012369 at xxx.lanl.gov.
Zeros of reliability polynomials and f -vectors of matroids. D G Wagner, Combin. Probab. Comput. 9D.G. Wagner, Zeros of reliability polynomials and f -vectors of matroids, Combin. Probab. Comput. 9 (2000), 167-190.
|
[] |
[
"SELF-SUPERVISED GUIDED HYPERGRAPH FEATURE PROPAGATION FOR SEMI-SUPERVISED CLASSIFICATION WITH MISSING NODE FEATURES",
"SELF-SUPERVISED GUIDED HYPERGRAPH FEATURE PROPAGATION FOR SEMI-SUPERVISED CLASSIFICATION WITH MISSING NODE FEATURES"
] |
[
"Chengxiang Lei \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Sichao Fu \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Yuetian Wang \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Wenhao Qiu \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Yachen Hu \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Qinmu Peng \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n",
"Xinge You \nSchool of Electronic Information and Communications\nHuazhong University of Science and Technology\n\n"
] |
[
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n",
"School of Electronic Information and Communications\nHuazhong University of Science and Technology\n"
] |
[] |
Graph neural networks (GNNs) with missing node features have recently received increasing interest. Such missing node features seriously hurt the performance of the existing GNNs. Some recent methods have been proposed to reconstruct the missing node features by the information propagation among nodes with known and unknown attributes. Although these methods have achieved superior performance, how to exactly exploit the complex data correlations among nodes to reconstruct missing node features is still a great challenge. To solve the above problem, we propose a self-supervised guided hypergraph feature propagation (SGHFP). Specifically, the feature hypergraph is first generated according to the node features with missing information. And then, the reconstructed node features produced by the previous iteration are fed to a two-layer GNNs to construct a pseudo-label hypergraph. Before each iteration, the constructed feature hypergraph and pseudo-label hypergraph are fused effectively, which can better preserve the higher-order data correlations among nodes. After then, we apply the fused hypergraph to the feature propagation for reconstructing missing features. Finally, the reconstructed node features by multi-iteration optimization are applied to the downstream semi-supervised classification task. Extensive experiments demonstrate that the proposed SGHFP outperforms the existing semi-supervised classification with missing node feature methods.
|
10.1109/icassp49357.2023.10094867
|
[
"https://export.arxiv.org/pdf/2302.08250v1.pdf"
] | 256,901,171 |
2302.08250
|
fe1d063e2c78b7c777e70e24a29a1632a951a17d
|
SELF-SUPERVISED GUIDED HYPERGRAPH FEATURE PROPAGATION FOR SEMI-SUPERVISED CLASSIFICATION WITH MISSING NODE FEATURES
Chengxiang Lei
School of Electronic Information and Communications
Huazhong University of Science and Technology
Sichao Fu
School of Electronic Information and Communications
Huazhong University of Science and Technology
Yuetian Wang
School of Electronic Information and Communications
Huazhong University of Science and Technology
Wenhao Qiu
School of Electronic Information and Communications
Huazhong University of Science and Technology
Yachen Hu
School of Electronic Information and Communications
Huazhong University of Science and Technology
Qinmu Peng
School of Electronic Information and Communications
Huazhong University of Science and Technology
Xinge You
School of Electronic Information and Communications
Huazhong University of Science and Technology
SELF-SUPERVISED GUIDED HYPERGRAPH FEATURE PROPAGATION FOR SEMI-SUPERVISED CLASSIFICATION WITH MISSING NODE FEATURES
Index Terms-Missing node featuresGraph neural net- worksSemi-supervised classification
Graph neural networks (GNNs) with missing node features have recently received increasing interest. Such missing node features seriously hurt the performance of the existing GNNs. Some recent methods have been proposed to reconstruct the missing node features by the information propagation among nodes with known and unknown attributes. Although these methods have achieved superior performance, how to exactly exploit the complex data correlations among nodes to reconstruct missing node features is still a great challenge. To solve the above problem, we propose a self-supervised guided hypergraph feature propagation (SGHFP). Specifically, the feature hypergraph is first generated according to the node features with missing information. And then, the reconstructed node features produced by the previous iteration are fed to a two-layer GNNs to construct a pseudo-label hypergraph. Before each iteration, the constructed feature hypergraph and pseudo-label hypergraph are fused effectively, which can better preserve the higher-order data correlations among nodes. After then, we apply the fused hypergraph to the feature propagation for reconstructing missing features. Finally, the reconstructed node features by multi-iteration optimization are applied to the downstream semi-supervised classification task. Extensive experiments demonstrate that the proposed SGHFP outperforms the existing semi-supervised classification with missing node feature methods.
INTRODUCTION
Graphs are widely applied for many real-world scenarios, such as social networks [1], citation networks [2], traffic networks [3], molecular networks [4]. In recent years, with the rapid development of deep learning, graph neural networks (GNNs) [5,6] have achieved remarkable success in graph-structured data with complex data correlations. GNNs typically operate by a message propagation scheme, where each node propagates its feature representations along the constructed edges. Meanwhile, the feature representation of each node is updated by aggregating the representations from its neighbors.
However, the existing GNNs variants [7] typically assume that all node feature attributes are fully observed during the training process. In fact, in many real-world applications, some feature attributes are unobserved due to resource limits or privacy concerns [8]. For example, in social networks, some users are unwilling to provide their ages and gender. Thus, when directly utilizing the existing GNNs variants to deal with the above tasks with missing node features, their performance will have a big deterioration.
To solve the above issue, many matrix factorization-based methods [9,10] have been proposed to reconstruct missing node features. However, these methods cannot make full use of the topological information between data. Recently, graph signal processing [11] provides several methods for interpolating signals on graphs, they generalize Fourier analysis to graphs to reconstruct missing signals on graphs. Such methods are too computationally intensive so they are infeasible for graphs with thousands of nodes. Very recently, several methods extend GNNs to tackle missing node features directly. For example, SAT [12] assumes that the structure and feature information on the graph share the same latent space and develops a distribution-matching strategy to reconstruct missing features. GCNMF [13] adapts GCN [14] to graphs with missing features by representing the missing features with a Gaussian mixture model. PaGNN [15] develops partial aggregation-based GNNs that only propagate the observed features. FP [8] propagates the known features to the nodes with unknown features iteratively to reconstruct missing features. However, FP only considers pairwise connection relationships between data and FP also assumes that each node has the same influence on all neighbors, which cannot accurately describe the local geometric distribution between data.
In this paper, we propose a self-supervised guided hypergraph feature propagation (SGHFP) for semi-supervised classification with missing node features. Specifically, the feature and pseudo-label hypergraph are first computed according to the node features with missing information and reconstructed node features generated by the previous iteration in turn. Be-
Hypergraph
Feature Propagation
GNNs Prediction Hypergraph Fusion fore each iteration, the obtained feature and pseudo-label hypergraph are further fused into an effective hypergraph. Compared to the single feature or pseudo-label hypergraph, the fused hypergraph can better describe the complex high-order structure information between data. Following, the fused hypergraph and FP are combined to reconstruct missing features in each iteration. After multi-iterations optimization, the reconstructed node features can be applied to downstream semisupervised classification tasks. To validate the effectiveness of SGHFP, we conduct extensive experiments on four benchmarks. Experiment results demonstrate that our approach outperforms many state-of-the-art methods. The main contributions are summarized as follows:
• Compared with the traditional graph, the proposed hypergraph can simultaneously utilize the higher-order correlations from feature and pseudo-label hypergraphs to update the inaccurate connection relationships.
• Our proposed SGHFP is an independent module, which can combine any GNNs variants for any graph representation learning tasks.
• Extensive experiments show that the proposed SGHFP outperforms many existing semi-supervised classifications with missing node feature methods.
SELF-SUPERVISED GUIDED HYPERGRAPH FEATURE PROPAGATION
Problem Definition and Notation Description
Give a simple graph G = (V, E), where V = {v i |i = 1, . . . n} is the set of nodes, and E = {e ij |i = 1, . . . n, j = 1, . . . n} is the set of edges. Let A ∈ R n×n denotes the adjacency relationship matrix, where A ij = 1 if e ij ∈ E, and A ij = 0 if e ij ∈ E. X ∈ R n×d denote the nodes features matrix, where d is the dimension of node features. Different from a simple graph, a hyperedge can connect two or more nodes. Let H = (Ṽ ,Ẽ, W ) denotes a hyper-graph, where W denotes the weight matrix of all hyperedges, and H ∈ R |Ṽ |×|Ẽ| is a incidence matrix, i.e.
h(ṽ,ẽ) = 1, ifṽ ∈ẽ 0, otherwise(1)
For a nodeṽ ∈Ṽ , its degree is defined as d(ṽ) = Σẽ ∈Ẽ w(ẽ)h(ṽ,ẽ). For a hyperedgeẽ ∈Ẽ, its degree is defined as δ(ẽ) = Σṽ ∈Ṽ h(ṽ,ẽ). Let D e denote all hyperedges' degree matrix and D v denote all nodes' degree matrix. In addition, we let
Θ = D −1/2 v HWD −1 e H D −1/2 v and ∆ = I − Θ,
where ∆ is called as hypergraph Laplacian. V k ⊆Ṽ denotes the set of nodes where the features are known, andṼ u = V c k =Ṽ \Ṽ k denote the unknown ones. In this paper, the problem we focus on is how to better reconstruct the unknown featuresx u , given the known featuresx k and the graph structure G.
Hypergraph Fusion
To accurately describe the local geometric distribution among nodes, We first generate a feature hypergraph G f according to the node features with missing information. Each time one node is selected as the centroid, and all its neighbors are linked as a hyperedge. Second, the reconstructed node features produced by the previous iteration are fed to a two-layer GNNs to construct a pseudo-label hypergraph G pl .
Denote v a and v b are two nodes, from [16], we can know that the label influence of v a on v b equals to the to the cumulative normalized feature influence of v a on v b after k iterations of propagation:
E[I l (v a , v b ; k)] = Σ k j=1Ĩf (v a , v b , j).(2)
Equation 2 shows that pseudo-label hypergraph is beneficial to increase the intra-class feature influence. To highlight the accuracy of connection relationships, the constructed feature hypergraph G f and pseudo-label hypergraph G pl are further combined to generate a fused hypergraph before each iteration. To reduce complexity, we fuse the two graphs into a sparse matrix. G f is used as the indices of specified elements, and G pl is used as the corresponding values. As shown in Fig.1, hyperedges connecting two nodes of the same class are bold so that features can be more easily propagated among nodes along hyperedges with stronger connections.
Hypergraph Feature Propagation
Similar to FP [8], we reconstruct the unknown node features x u through interpolation that minimizes Dirichlet energy:
(x, G) = 1 2 x ∆x = 1 2 Σ ij θ ij (x i − x j ) 2 ,
where θ ij are the individual entries of the normalized incidence matrix Θ. Dirichlet energy, which represents how much a function changes in a certain area, is widely used as a smoothness criterion.
For the convenience of derivation, the node feature matrix x is split into two sub-matrices. H and ∆ can be divided into four sub-matrices.
x = x k x u H = H kk H ku H uk H uu ∆ = ∆ kk ∆ ku ∆ uk ∆ uu .(3)
Letẋ(t) = −∇ (x(t)) denotes the associated gradient flow, and the known featuresx k =x k (t) is the boundary condition. Therefore, the solution at the missing nodes:x u = lim n→∞xu (t) is the interpolation. From [8], we can get a diffusion equation:
ẋ k (t) xu(t) =− 0 0 ∆ uk ∆uu x k xu(t) =− 0 ∆ ukxk + ∆uuxu(t) .(4)
and its solution in an iterative scheme:
x (k+1) = I 0 Θ uk Θ uu x (k) .(5)
Algorithm 1 SGHFP 1: Input:feature vector x, graph structure G, train epochs T 2: for x = 1 → T do 3: construct G f and G pl by x and G 4:
Θ ← f use(G f , G pl )
5:
x ← Θx Propagate features 6: x k ← y k Reset known features 7: end for
The update procedure in equation 4 is equivalent to the following two steps. First, the feature vector x is multiplied by the propagation matrix Θ. Second, the known features are reset to their original true values. This update procedure provides an iterative algorithm to reconstruct the unknown node features, as shown in Algorithm 1. Specifically, the feature hypergraph G f and pseudo-label hypergraph G pl are fused to generate a propagation matrix Θ (in the first iteration, only feature hypergraph G f is used). At each iteration, features are propagated among nodes by the propagation matrix Θ. After that, we clamp the known features by resetting them to their original true values.
EXPERIMENT
Datasets
We fed the reconstructed node features into many downstream semi-supervised node classification tasks, their classification performance can reflect the accuracy of feature reconstruction intuitively. In this paper, we evaluate our proposed SGHFP on four benchmark datasets including Cora, CiteSeer, PubMed [17], and Photo (Amazon) [18].
Experimental Setup
In all experiments, We randomly select 20 nodes per class as the training set, 1500 nodes for validation, and the rest for testing. In this paper, we use a two-layers GCN with a dropout rate of 0.5 as the downstream classifier. The Adam optimizer [19] with a learning rate of 0.005 is introduced to optimize the model parameters. Empirically, hypergraph feature propagation diffuses the features over 50 iterations. For all the baselines, the hyperparameters are the same as mentioned in the respective papers [8,13,15,20,21].
Results and Discussion
Comparison with State-of-the-art Methods
We report the mean classification accuracy with 10 runs for all methods. Each run has a different mask of missing features. Table 2 illustrates the classification performance of our proposed SGHFP and FP (combined with a downstream GCN model) under full features, 50%, 90%, and 99% missing node features. In addition, we also report the relative drop rate when features are partially missing compared to all features that are present. Table 2 shows that the proposed SGHFP outperforms FP under any missing rate and also has a lower In this part, we compare the proposed SGHFP with many state-of-the-art GNNs methods when 99% of the node features are missing. We additionally compare to featureagnostic baselines: Label Propagation (LP) [20], which only exploits the structure information of graphs by propagating labels iteratively, and Graph Positional Encodings (GPE) [21], which treats the first k eigenvector matrices of the Laplacian as node features. Table 4 shows that our proposed SGHFP outperforms the existing methods on all experimental datasets. For example, GCNMF and PaGNN have a large drop in relative accuracy when the feature missing rate is high. In comparison, SGHFP has only a 2.67% drop.
Ablation experiments
In this part, we investigate the impact of the self-supervised guided pseudo-label hypergraph (SGPH) and feature hypergraph (FH) module for feature reconstruction on Cora, Citseer, PubMed, and Photo datasets. Table 3 shows that our proposed SGPH and FH modules all improve the classification performance of downstream tasks with missing node fea-tures. For example, on the Cora dataset, FP with the FH module obtains gains of 0.55%. FP with SGPH and FH module (SGHFP) achieve 1.19% improvements in comparison to FP.
t-SNE visualization
To better demonstrate that the proposed SGHFP can better reconstruct the missing features, we use t-SNE and Silhouette scores to visualize the embedding of graphs with reconstructed features in 2-D space on Cora and Citeseer datasets. Nodes in the same class are expected to be clustered together and have higher Silhouette scores. Fig.2 shows that our proposed SGHFP can separate different categories and the nodes in the same class are clustered more compactly. All experiments are performed with a 90% missing rate of node features.
CONCLUSION
In this paper, we present a novel approach for semi-supervised classification with missing node features. The feature hypergraph and pseudo-label hypergraph are constructed to describe the local geometric distribution between data in turns. Then, a fused hypergraph generated by an effective strategy is further applied to the feature propagation model for reconstructing the missing features. Experimental results on several datasets demonstrate that the proposed SGHFP is useful to reconstruct the missing features and also outperforms many existing state-of-the-art methods. While SGHFP is designed for homogeneous graphs, it does not perform well on heterogeneous graphs. Learning heterogeneous feature-missing graphs with a more general learnable diffusion matrix could also be an interesting problem.
This work was supported in part by the National Natural Science Foundation of China under Grant 62172177, in part by the Fundamental Research Funds for the Central Universities under Grant 2022JYCXJJ034 and YCJJ202204016. * Work was done when Sichao Fu was interning at JD Retail POMC. ‡ Sichao Fu and Qinmu Peng are the corresponding authors.
Fig. 1 .
1A diagram illustrating our Self-supervised Guided Hypergraph Feature Propagation framework.
Fig. 2 .
2The t-SNE visualization and Silhouette score of the node embeddings with reconstructing features. Each color represents one class. relative drop rate. On average, SGHFP with 99% missing features loses only 2.67% of relative accuracy compared to the same GNNs model with no missing features.
Table 1 .
1Dataset statistics.Dataset Nodes Edges Features Classes
Cora
2485
5069
1433
7
Citeseer
2120
3679
3703
6
PubMed 19717 44324
500
3
Photo
7487 119043
745
8
Table 2 .
2Classificationperformance of our proposed SGHFP
and FP under different rates of missing features. The best
results are highlighted.
Dataset
Method
Missing Node Features Rate
0
50%
90%
99%
Cora
FP
80.39%
79.7% (-0.86%)
79.77% (-0.77%) 78.22% (-2.70%)
SGHFP 81.23% 80.9% (-0.41%) 80.41% (-1.01%) 79.41% (-2.25%)
Citeseer
FP
67.48% 65.74% (-2.57%) 65.57% (-2.82%)
65.4% (-3.08%)
SGHFP 67.45% 66.86% (-0.87%) 66.76% (-1.02%)
66.5% (-1.41%)
PubMed
FP
77.36% 76.68% (-0.89%) 75.85% (-1.96%) 74.29% (-3.97%)
SGHFP 77.45% 77.15% (-0.37%) 76.63% (-1.06%) 75.17% (-2.94%)
Photo
FP
91.73% 91.29% (-0.48%) 89.48% (-2.46%) 87.73% (-4.36%)
SGHFP 92.08% 91.46% (-0.67%) 90.04% (-2.22%) 88.40% (-3.99%)
Table 3 .
3Ablation experiments on Cora, Citseer, PubMed, and Photo dataset with 99% missing node features. 22±0.32 78.77±0.24 79.41±0.21 65.40±0.54 65.92±0.28 66.50±0.11 74.29±0.55 74.56±0.36 75.17±0.10 87.73±0.27 88.02±0.18 88.40±0.10Module
Cora
Citeseer
PubMed
Photo
FP
FH
SGPH
Accuracy 78.Silhouette_score:0.290
(a) FP on Cora
Silhouette_score:0.368
(b) SGHFP on Cora
Silhouette_score:0.354
(c) FP on Citeseer
Silhouette_score:0.438
(d) SGHFP on Citeseer
Table 4 .
4Classification performance of GCNMF, PaGNN, LP, GPE and SGHFP under 99% of features missing. The best results are highlighted.Dataset
GCNMF
PaGNN
LP
GPE
SGHFP
Cora
34.54±2.07 58.03±0.57 74.68±0.36 76.33±0.26 79.41±0.21
Citeseer 30.65±1.12 46.02±0.58 64.60±0.40 65.87±0.37 66.50±0.11
PubMed 39.80±0.25 54.25±0.70 73.81±0.56 73.70±0.29 75.17±0.10
Photo
29.64±2.78 85.41±0.28 83.45±0.94 83.45±0.26 88.40±0.10
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|
[] |
[
"A Quantum Bayes' Rule and Related Inference",
"A Quantum Bayes' Rule and Related Inference"
] |
[
"Huayu Liu \nSchool of Statistics and Mathematics\nCentral University of Finance and Economics\nSouth Road100081BeijingChina\n"
] |
[
"School of Statistics and Mathematics\nCentral University of Finance and Economics\nSouth Road100081BeijingChina"
] |
[] |
In this work a quantum analogue of Bayesian inference is considered. Based on the notion of instrument, we propose a quantum analogue of Bayes' rule, which elaborates how a prior normal state updates under observations. Besides, we investigate the limit of posterior normal state as the number of observations goes to infinity. After that, we generalize the fundamental notions and results of Bayesian inference according to quantum Bayes' rule. It is noted that our theory not only retains the classical one as a special case but possesses many new features as well.
| null |
[
"https://export.arxiv.org/pdf/2204.08845v3.pdf"
] | 255,546,227 |
2204.08845
|
f18fe574636bb69a19f0638134e4a2bd25165f95
|
A Quantum Bayes' Rule and Related Inference
9 Jan 2023
Huayu Liu
School of Statistics and Mathematics
Central University of Finance and Economics
South Road100081BeijingChina
A Quantum Bayes' Rule and Related Inference
9 Jan 2023(Dated: 10 January 2023)instrumentposterior normal statesequential measurement scheme
In this work a quantum analogue of Bayesian inference is considered. Based on the notion of instrument, we propose a quantum analogue of Bayes' rule, which elaborates how a prior normal state updates under observations. Besides, we investigate the limit of posterior normal state as the number of observations goes to infinity. After that, we generalize the fundamental notions and results of Bayesian inference according to quantum Bayes' rule. It is noted that our theory not only retains the classical one as a special case but possesses many new features as well.
I. INTRODUCTION
Bayesian inference was born in the 1920s and reached its heyday in the 1950s, and is now widely used in science and engineering. In the past century, quantum physics has flourished and merged with many fields. About fifty years ago, an interdisciplinary field called quantum statistics was born 1,2 . In this field there are many studies in which Bayesian analysis are applied to quantum physics, as shown in [3][4][5][6][7][8] . What is at the heart of Bayesian analysis is Bayes' rule, which elaborates how a prior distribution updates under observations. Intuitively, the update of a prior distribution according to Bayes' rule is quite similar to the update of a quantum state according to a specific measurement. We will see that the former is a special case of the latter. To understand this, first we have to embed a probability space into a complex Hilbert space. A probability space is a triad (X, A , P) where (X, A ) is a measurable space and P a probability measures on (X, A ). For simplicity, here we consider a probability space with finite samples.
Let (X, 2 X , P) be a probability space where X = {x 1 , · · · , x n }, 2 X the power of X (i.e. the set of all subsets of X) and P a probability measure on (X, 2 X ). Let H be a
= tr(ρP A ) (2) for all A ∈ 2 X , where P A = j∈{i:x i ∈A} |ψ j ψ j | is a projector. Therefore, the conditional probability P(B|A) = P(B ∩ A)
P(A)(3)
= tr(ρP B P A ) tr(ρP A )
= tr(P B P A ρP A P B ) tr(P A ρP A )
= tr(P B ρ A P B ) (6) for all A, B ∈ 2 X such that P(A) = 0, where ρ A = P A ρP A /tr(P A ρP A ) is a density operator. Readers familiar with quantum measurement may have recognized that operations ρ → P A ρP A , ∀A ∈ 2 X indeed defines a Lüders instrument, which describes the statistical properties of a specific family of measurements. The definition of instrument will be introduced in Sect.II.
With the ingredients above, we now turn to the connection between Bayes' rule and instrument. Assume that the unknown parameter θ takes values in the set Θ that consists of m elements and has a prior distribution Π. Assume that the observation x takes values in the set X that consists of n elements and has the conditional distribution P θ . Let H 0 be a m-dimensional complex Hilbert space with {φ l } m l=1 an orthonormal basis and H 1 a n-dimensional complex Hilbert space with {ψ j } n j=1 an orthonormal basis. Denote by c 0 (resp. c 1 ) the one-to-one correspondence between {θ k } and the projector |φ k φ k | for all k ∈ [m] (resp. {x i } and |ψ i ψ i | for all i ∈ [n]). In order to understand the connection between Bayes' rule and instrument, first we have to embed the product probability space (Θ × X, 2 Θ × 2 X , Π × P θ ) into a finite-dimensional complex Hilbert space, where Θ × X is the Cartesian product, 2 Θ × 2 X the product σ-algebra, Π × P θ the product probability measure on the measurable space (Θ × X, 2 Θ × 2 X ). But for brevity, we denote Π × P θ as P × in the following. A convenient choice for the finite-dimensional complex Hilbert space for embedding is the tensor product H 0 ⊗ H 1 since {φ l ⊗ ψ j : l ∈ [m], j ∈ [n]} is an orthonormal basis of H 0 ⊗ H 1 so that there is a natural one-to-one correspondence c 01 between {(θ k , x i )} and the projector |φ k ⊗ ψ i φ k ⊗ ψ i | for all k ∈ [m], i ∈ [n]. As before, there is a density operator ρ = k,i P × ({(θ k , x i )})|φ k ⊗ ψ i φ k ⊗ ψ i | such that P × (E) = (l,j)∈{(k,i):(θ k ,x i )∈E} tr(ρ|φ l ⊗ ψ j φ l ⊗ ψ j |)
= tr(ρP E ) (8) for all E ∈ 2 Θ × 2 X , where P E = (l,j)∈{(k,i):(θ k ,x i )∈E} |φ l ⊗ ψ j φ l ⊗ ψ j | is a projector. Then The analysis above suggests that the Bayesian update π(θ) → π(θ|x) can be realized by first carrying out the instrumental update ρ → ρ {(·,x)} and then applying Born's rule. In other words, through embedding, the update of a prior distribution according to Bayes' rule can be identified with the update of a prior normal state (describing a quantum system) according to a Lüders instrument (describing a specific family of measurements). We assert that this is also true for the general version of Bayesian update and will give a proof in Sect.III.
Quantum theory has been used outside of physics for quite some time. In cognition, decision-making and even economics, quantum theory excels at modeling non-commutative phenomena. Recently, 9 successfully modeled both Question Order Effect and Response
Replicability Effect in human cognition with a quantum system, suggesting that human cognitive processes are probably similar to the evolution of quantum states. In classical Bayesian inference, a Bayesian's knowledge of an object is modeled by a classical system and Bayes' rule tells a Bayesian how to update her knowledge of an object based on observations.
Since there is evidence that quantum system models human cognition better, we wonder if there is a quantum analogue of Bayes' rule together with a quantum analogue of Bayesian inference, in which a Bayesian's knowledge of an object is modeled by a quantum system.
To the best of our knowledge, although a few attempts have been made to quantize Bayes' rule [10][11][12][13][14][15] , there is still no research addressing a quantum analogue of Bayesian inference. In the following we will briefly review these articles.
In 10 , a quantum analogue of Bayes' rule is proposed and the exact condition with respect to the validity of it is explored. Bayes' rule is generalized in 11 with the prior being a density matrix and the likelihood being a covariance matrix. In 12 , an inherently diagrammatic formulation of quantum Bayes' rule is proposed and a necessary and sufficient condition for the existence of Bayesian inverse in the setting of finite-dimensional C * -algebras is provided.
A quantum analogue of Bayes' rule is put forward in 13 based on the notion of operator valued measure and quantum random variable. The graphical framework for Bayesian inference raised in 14 is sufficiently general to cover both the standard case and the proposals for quantum Bayesian inference in which the degrees of belief are considered to be represented by density operators instead of probability distributions. The approach of maximizing quantum relative entropy is followed in 15 to study a quantum analogue of Bayes' rule, resulting in some generalizations.
In this work we focus on a quantum analogue of Bayesian inference. We find that through Our manuscript is organized as follows. Sect.II is devoted to presenting some facts about operator valued measure, instrument and a family of posterior normal states. Then in Sect.III and Sect.IV, we focus on a quantum Bayes' rule and the limit of posterior normal state, respectively. Next in Sect.V, we move to a quantum analogue of Bayesian inference.
Finally Sect.VI is a discussion.
II. PRELIMINARIES
In general the sample space may have infinitely many elements. In order to understand the connection between Bayes' rule and instrument in this case, first we have to embed a probability space with an arbitrary measurable space into a complex Hilbert space. It is not difficult to find that the key to achieving this is to construct a map from the σ-algebra A to the set of projectors {P A : A ∈ A } so that there is a one-to-one correspondence between the probability measure P and the density operator ρ. In fact this map is an observable.
Observable is one of the important notions in quantum measurement that we will introduce (ii) For any countable collection of sets This guarantees that we can embed any probability space into a complex Hilbert space.
{A k } k∈N + ⊆ A with A i ∩ A j = ∅ for i = j we have ν ∞ k=1 A k = ∞ k=1 ν(A k ), weakly.(11)
As we discussed in Sect.I, there is a connection between Bayes' rule and instrument. The (ii) For any countable collection of sets
{A k } k∈N + ⊆ A with A i ∩ A j = ∅ for i = j we have I ∞ k=1 A k = ∞ k=1 I(A k ),(13)
where the convergence on the right side of the equation ( The observable ν induced by I is defined by
ν(A) = I * (A)1, ∀A ∈ A .(14)
Then for any ϕ ∈ S(M ), the probability measure P ϕ induced by ν is defined by
P ϕ (A) = ϕ, ν(A) , ∀A ∈ A .(15)
Note that different instruments may induce the same observable. It is shown in 17 that each instrument induces a unique observable and each observable can be induced by a unique class of instruments.
To measure a quantum system one needs an apparatus. In standard experimental scenarios, an apparatus will reveal the following two facts to the surveyor, called the statistical properties of an apparatus: (i) Probability that each measurement outcome occurs given a quantum state; (ii) Update of quantum states given each measurement outcome. However, different apparatuses may have the same statistical properties so that it would be a little redundant to distinguish them if we only care about their statistical properties. Hence the notion of instrument was put forward. It is shown in 18 that each apparatus corresponds to a unique instrument and each instrument corresponds to a unique statistical equivalence class of apparatuses.
Let I : A → B + (M * ) be an instrument. For any A ∈ A , P ϕ (A) is the probability that A occurs (iff the measurement outcome x ∈ A) when a quantum system S described by a von Neumann algebra M is in the normal state ϕ ∈ S(M ) and a measurement with an apparatus corresponding to I is performed. Furthermore, ϕ A = I(A)ϕ/P ϕ (A) is the normal state that the quantum system S is in immediately after A occurs (provided that
P ϕ (A) > 0).
Unfortunately, not all apparatuses are physically realizable. It is shown in 16 that the corresponding instrument of a physically realizable apparatus should be CP. Furthermore, there is a one-to-one correspondence between the statistical equivalence classes of physically realizable apparatuses and CP instruments. In other words, each CP instrument can be realized physically by at least one apparatus.
It is known that the measurement outcome {x} is an elementary event. However, it may happen that P({x}) = 0. So how will quantum states update given a measurement outcome
(I, ϕ) iff (i) ϕ x is a normal state for all x ∈ X; (ii) {ϕ x : x ∈ X} is weakly* P ϕ measurable (i.e. the function x → ϕ x , a is P ϕ measurable for any a ∈ M ); (iii) For any a ∈ M and A ∈ A , A ϕ x , a P ϕ (dx) = I(A)ϕ, a .(16)
If then for any a ∈ M , ϕ x , a = ϕ ′ x , a P ϕ -a.s.. The existence of a family of posterior normal states w.r.t. (I, ϕ) is discussed thoroughly in 19 .
{ϕ x : x ∈ X}
III. QUANTUM BAYES' RULE
To illustrate our theory, and to answer the question remained in Sect.I, we begin this section by exploring the connection between the general version of Bayes' rule and instrument. Recall that in Sect.I the product probability space
(Θ × X, 2 Θ × 2 X , P × ) is em- bedded into a (m × n)-dimensional complex Hilbert space H through a sharp observable B → P B
, as a result of which, the update of a prior distribution according to Bayes' rule can be identified with the update of a prior density operator according to a Lüders instrument. However, it suffices to reach the same conclusion by embedding the probability spaces (Θ, 2 Θ , Π) and (X, 2 X , 1 (·) P θ dΠ) into an arbitrary complex Hilbert space whose dimension is no less than m × n through a sharp observable ν : 2 Θ → B + (H) and a Lüders instrument
I : 2 X → B + (B 1 (H)) inducing a sharp observable λ compatible with ν (i.e. the commutator [ν(E), λ(A)] = 0 for all E ∈ 2 Θ and A ∈ 2 X )
, respectively. Following the idea above we will prove that through proper embedding the update of a prior distribution according to the general version of Bayes' rule is nothing but the update of a prior normal state according to a specific instrument.
Let (Θ, E , ν) and (X, A , µ) be σ-finite measure spaces and p(x|θ) a nonnegative E × A measurable real valued function satisfying
X p(x|θ)µ(dx) = 1, ∀θ ∈ Θ,(17)
so that the function
K : (θ, A) → A p(x|θ)µ(dx), ∀(θ, A) ∈ Θ × A(18)
is a Markov kernel. Denote by L ∞ (ν) ⊆ B(L 2 (ν)) the abelian von Neumann algebra of ν-a.e.
equivalence classes of essentially bounded E measurable functions from Θ to the complex field C, where L 2 (ν) is the complex Hilbert space of ν-a.e. equivalence classes of square integrable E measurable functions from Θ to C. Identify L ∞ (ν) * with L 1 (ν) and denote by L 1 +,1 (ν) the set of normal states. Let τ : E → L ∞ (ν) be a map defined by
E → 1 E , ∀E ∈ E ,(19)
where 1 E is an indicator. In fact τ is an observable since (i) 1 E ≥ 0 for all E ∈ E ; (ii) 1 Θ is the identity and (iii) 1 ∪ k≥1 E k = k≥1 1 E k for any countable collection of mutually disjoint measurable sets {E k } k≥1 . Therefore, we can embed any probability space (Θ, E , Π) satisfying Π ≪ ν (i.e. Π is absolutely continuous with respect to ν) into the complex Hilbert space L 2 (ν) through the observable τ . Note that in the case mentioned in Sect.I we can not determine the normal state (density operator) uniquely if we only embed (Θ, 2 Θ , Π) into H through a sharp observable (even after embedding both if the dimension of H is larger than m × n). However, the normal state can be uniquely determined if we embed (Θ, E , Π) into L 2 (ν) through τ since by Radon-Nikodým theorem there is a one-to-one correspondence between probability measures that are absolutely continuous with respect to ν and normal states on L ∞ (ν).
Let P • be a probability measure on (X, A ) defined by P • (A) = Θ 1(θ)K(θ, A)π(θ)ν(dθ) for all A ∈ A , where π(θ) ∈ L 1 +,1 (ν) (i.e. π(θ) ≥ 0 and |π|dν = 1) is a normal state. We have to embed (X, A , P • ) into L 2 (ν) through an instrument. Let I : A → B + (L 1 (ν)) be a map defined by A → K(θ, A)·, ∀A ∈ A ,(20)K[f ] := [Kf ], ∀[f ] ∈ L 1 (ν).(21)
Needless to say, I is an instrument, as verified below.
(i) Θ 1(θ)K(θ, A)f (θ)ν(dθ) ≤ Θ 1(θ)f (θ)ν(dθ) for all A ∈ A and L 1 (ν) ∋ f ≥ 0 since 0 ≤ K(θ, A) ≤ 1 for all θ ∈ Θ and A ∈ A . (ii) Θ 1(θ)K(θ, X)f (θ)ν(dθ) = Θ 1(θ)f (θ)ν(dθ) for all A ∈ A and L 1 (ν) ∈ f ≥ 0 since K(θ, X) = 1 for all θ ∈ Θ. (iii)
The σ-additivity of I follows directly from the σ-additivity of K(θ, ·) as a probability measure for all θ ∈ Θ.
The following result shows that {π(θ|x) : x ∈ X} is indeed a family of posterior normal states w.r.t. (I, π(θ)), where π(θ|x) = p(x|θ)π(θ)/ Θ p(x|θ)π(θ)ν(dθ) and π(θ) ∈ L 1 +,1 (ν).
Theorem 1. {π(θ|x) :
x ∈ X} is a family of posterior normal states w.r.t. (I, π(θ)).
Moreover, it is unique in the sense that if {π(θ|x) ′ : x ∈ X} is another family of posterior normal states w.r.t. (I, π(θ)), then π(θ|x) and π(θ|x) ′ only differ on a (ν × µ)-null set.
Proof. Denote by π(θ) an element of π(θ). (i) Obviously p(x|θ)π(θ) is a nonnegative (ν ×µ)a.e., E × A measurable real valued function such that Θ p(x|θ)π(θ)ν(dθ) ≥ 0 for all x ∈ X and Θ |π(θ|x)|ν(dθ) = 1. Thus π(θ|x) is a normal state for all x ∈ X. (ii) Denote by N 0 the ν-null set such that π(θ) ≥ 0 on N c 0 . We first show that pπd(ν × µ) < ∞.
pπd(ν × µ) = 1 N c pπd(ν × µ) (22) = Θ π(θ)ν(dθ) X 1 N c (θ, x)p(x|θ)µ(dx) (by Tonelli theorem) (23) = Θ K(θ, N c θ )π(θ)ν(dθ) (24) = Θ K(θ, N c θ )Π(dθ) < ∞,(25)
where N c is the complement of the (ν × µ)-null set N = N 0 × X and N c θ the section of N c at θ. By Tonelli theorem the function x → 1 N c pπdν = pπdν is nonnegative and A measurable. Let g ∈ L ∞ (ν) and denote by g an element of g. Then we have to show
that the function x → gpπdν is P • measurable. Since g = ℜ(g) + iℑ(g) = [ℜ(g)] + − [ℜ(g)] − + i{[ℑ(g)] + − [ℑ(g)] − }, the integral gpπdν = [ℜ(g)] + pπdν − [ℜ(g)] − pπdν + i{ [ℑ(g)] + pπdν − [ℑ(g)] − pπdν}, where ℜ(·), ℑ(·), (·) + , (·) − denote the real part, imaginary
part, positive part and negative part of a number, respectively. Thus it suffices to show that
the function ℓ : x → [ℜ(g)] + pπdν is P • measurable. To do this, we first show that [ℜ(g)] + pπd(ν × µ) < ∞. [ℜ(g)] + pπd(ν × µ) = 1 N c [ℜ(g)] + pπd(ν × µ)(26)= Θ ℜ + [g(θ)]π(θ)ν(dθ) X 1 N c (θ, x)p(x|θ)µ(dx)(27)= Θ ℜ + [g(θ)]K(θ, N c θ )π(θ)ν(dθ) (28) ≤ Θ |ℜ(g)|K(θ, N c θ )Π(dθ) (29) ≤ Θ g ∞ K(θ, N c θ )Π(dθ) < ∞,(30)
where · ∞ is the essential supremum norm on L ∞ (ν). Then by Fubini theorem the nonnegative function ℓ is A measurable but only finite µ-a.e.. Denote by N 1 the µ-null set such that ℓ is finite on N c 1 . Note that 1 N c 1 ℓ is a A measurable real valued function and equal to ℓ µ-a.e.. This implies that 1 N c 1 ℓ is equal to ℓ P • -a.s. since P • is absolutely continuous with respect to µ. Due to the fact that a function f from Θ to the extended real line is µ measurable if and only if there is A measurable extended real valued function h such that f = h µ-a.e., we have actually proved that ℓ is P • measurable. By the same way, the functions
x → [ℜ(g)] − pπdν, x → [ℑ(g)] + pπdν and x → [ℑ(g)] − pπdν are all P • measurable so that
the function x → gpπdν is P • measurable. Hence the function x → Θ g(θ)π(θ|x)ν(dθ) is P • measurable (and also µ measurable) for all g ∈ L ∞ (ν). (iii) Let g ∈ L ∞ (ν) and denote by g an element of g. Without loss of generality, assume that g is nonnegative. By (ii), the function x → Θ g(θ)π(θ|x)ν(dθ) is nonnegative and µ measurable. Consequently, there is a nonnegative A measurable function r that is equal to x → Θ g(θ)π(θ|x)ν(dθ) µ-a.e.. Thus
the integral A Θ g(θ)π(θ|x)ν(dθ) Θ 1(θ)K(θ, dx)π(θ)ν(dθ) = A r(x) Θ 1(θ)K(θ, dx)π(θ)ν(dθ) (31) = Θ Π(dθ) A r(x)K(θ, dx) (32) = Θ Π(dθ) A r(x)p(x|θ)µ(dx) (33) = A r(x)µ(dx) Θ p(x|θ)Π(dθ). (34) And the integral Θ g(θ)K(θ, A)π(θ)ν(dθ) = Θ g(θ)Π(dθ) A p(x|θ)µ(dx) (35) = A µ(dx) Θ g(θ)p(x|θ)Π(dθ) (by Tonelli theorem).(36)
Apparently r(x) Θ p(x|θ)Π(dθ) = Θ g(θ)p(x|θ)Π(dθ) µ-a.e.. Hence for all A ∈ A and
g ∈ L ∞ (ν) we have A Θ g(θ)π(θ|x)ν(dθ) Θ 1(θ)K(θ, dx)π(θ)ν(dθ) = Θ g(θ)K(θ, A)π(θ)ν(dθ).(37)
Conversely, assume that a family of posterior normal states {π(θ|x) : x ∈ X} w.r.t.
(I, π(θ)) exists, where π(θ) ∈ L 1 +,1 (ν) is a normal state. Then π(θ|x) shall satisfy the following equation
A Θ g(θ)π(θ|x)ν(dθ) Θ 1(θ)K(θ, dx)π(θ)ν(dθ) = Θ g(θ)K(θ, A)π(θ)ν(dθ)(38)
for all A ∈ A and L ∞ (ν) ∋ g ≥ 0. By Tonelli theorem we have
A Θ g(θ)π(θ|x)ν(dθ) µ(dx) Θ p(x|θ)π(θ)ν(dθ) = A µ(dx) Θ g(θ)p(x|θ)π(θ)ν(dθ) (39)
for all A ∈ A and L ∞ (ν) ∋ g ≥ 0, which implies that
π(θ|x) Θ p(x|θ)π(θ)ν(dθ) = p(x|θ)π(θ)(40)
holds on a set M satisfying for µ-a.e. x the section (M c ) x of M c is a ν-null set. Since ν and µ are σ-finite, this is equivalent to M c is a (ν × µ)-null set.
It is now clear that the Bayesian update π(θ) → π(θ|x) of a prior density π(θ) is nothing but the update π(θ) → π(θ|x) of a prior normal state π(θ) according to the instrument K.
This inspired us to propose a quantum analogue of Bayes' rule. Indeed Bayes' rule says if a Bayesian uses an apparatus corresponding to the instrument K to measure a quantum system S described by the von Neumann algebra L ∞ (ν), she will obtain a measurement Apparently quantum Bayes' rule retains the classical one as a special case. In practice, one may measure a quantum system S described by a von Neumann algebra M in a sequential measurement scheme, i.e. perform n measurements with n apparatuses (corresponding to n instruments I i :
A i → B + (M * ), i ∈ [n]
) on S one immediately after another. If the quantum system S is in the prior normal state ϕ before these sequentially performed measurements, then by induction the probability that A 1 , · · · , A n occurs one by one is I n (A n ) • · · · • I 1 (A 1 )ϕ, 1 and according to quantum Bayes' rule S will be in the posterior normal state ϕ x 1 ···xn if the measurement outcomes are {x 1 }, · · · , {x n }.
In classical Bayesian inference the probability space (X, A , P θ ) is usually a product of
n probability spaces (X i , A i , P (i) θ ), i ∈ [n]
. This implies that the components of x ∈ X is independent. Let p(x i |θ) be a nonnegative E × A i measurable real valued function satisfying
X i p(x i |θ)µ i (dx i ) = 1 for all i ∈ [n], where µ i is a σ-finite measure on (X i , A i ). Then the independence of the components of x is equivalent to p(x|θ) = n i=1 p(x i |θ) ( n i=1 µ i )-a.e.
for all θ ∈ Θ. But for convenience, the independence of the components of x is usually guaranteed by p(x|θ) = n i=1 p(x i |θ) for all θ ∈ Θ. And since the point-wise multiplication of two real valued functions is commutative, the update of a prior normal state π(θ) according to the instrument K = 1 (·) p(x|θ) ( n i=1 µ i ) (dx) is equivalent to the sequential update of π(θ) according to n instruments K i = 1 (·) p(x i |θ)µ i (dx i ), i ∈ [n] in any order.
In the case above it holds that
K(θ, n i=1 A i ) = n i=1 K i (θ, A i ) for all A i ∈ A i , i.e.
K is the unique kernel-like instrument that is equal to the composition of n instruments K i , i ∈ [n] on the measurable rectangles n i=1 A i . In fact under some topological assumptions, the composition of a finite number of instruments can be uniquely extended to an instrument in the following sense. Let (X i , B(X i )), i ∈ [n] be measurable spaces where X i is a second countable locally compact Hausdorff space and B(X i ) the Borel σ-algebra on X i . Let
instrument I ∈ Ins(Π n i=1 B(X i ), M * ) such that I(A 1 × · · · × A n ) = I n (A n ) • · · · • I 1 (A 1 ), ∀A i ∈ B(X i ).(41)
Thus if for all i ∈ [n] X i is a second countable locally compact Hausdorff space and A i the Borel σ-algebra on X i , then K is the unique instrument that agrees with the composition
of K i , i ∈ [n] on the measurable rectangles n i=1 A i . Note that ϕ n i=1 A i = ϕ A 1 ···An provided that P ϕ ( k i=1 A i ) > 0, k ∈ [n]
so it is natural to ask whether a family of posterior normal states w.r.t. (I, ϕ) is {ϕ x 1 ···xn :
x i ∈ X i , i ∈ [n]}.
To see this, it suffices to consider the case n = 2. Let I i : B(X i ) → B + (M * ), i ∈ [2] be instruments. Proof. (i) Apparently, ϕ x 1 x 2 is a normal state for all (x 1 , x 2 ) ∈ X 1 × X 2 . (ii) Since {ϕ x 1 :
Theorem 2. Assume that the map x 1 → I 2 (A 2 )ϕ x 1 , 1 is B(X 1 ) measurable for all A 2 ∈ B(X 2 ). Then {ϕ x 1 x 2 : x i ∈ X i , i ∈ [2]} is
x 1 ∈ X 1 } is a family of posterior normal states w.r.t. (I 1 , ϕ), we have A 1 ϕ x 1 , a P ϕ (dx 1 ) = I 1 (A 1 )ϕ, a for all a ∈ M and A 1 ∈ B(X 1 ). Setting a = I * 2 (A 2 )b, 0 ≤ b ∈ M and noticing that {ϕ x 1 x 2 : x 2 ∈ X 2 } is a family of posterior normal states w.r.t. (I 2 , ϕ x 1 ), we have
A 1 P ϕ (dx 1 ) A 2 ϕ x 1 x 2 , b P ϕx 1 (dx 2 ) = I 2 (A 2 ) • I 1 (A 1 )ϕ, b(42)= A 1 ×A 2 ϕ x , b I(dx)ϕ, 1 ,(43)
for all A 1 ∈ B(X 1 ) and A 2 ∈ B(X 2 ). Since the map x 1 → I 2 (A 2 )ϕ x 1 , 1 is B(X 1 ) measurable (of course P ϕ measurable) for all A 2 ∈ B(X 2 ), I 2 (A 2 )ϕ x 1 , 1 is a Markov kernel. As a result, P ϕ × P ϕx 1 is the unique probability measure on the measurable space
(X 1 × X 2 , B(X 1 × X 2 )) such that P ϕ × P ϕx 1 (A 1 × A 2 ) = A 1 I 2 (A 2 )ϕ x 1 , 1 P ϕ (dx 1 ) (44) = I 2 (A 2 ) • I 1 (A 1 )ϕ, 1(45)= I(A 1 × A 2 )ϕ, 1 .(46)
Hence by Carathéodory theorem P ϕ × P ϕx 1 is indeed Iϕ, 1 . Therefore
A 1 P ϕ (dx 1 ) A 2 ϕ x 1 x 2 , b P ϕx 1 (dx 2 ) = A 1 ×A 2 ϕ x , b I(dx)ϕ, 1(47)= A 1 P ϕ (dx 1 ) A 2 ϕ (x 1 ,x 2 ) , b P ϕx 1 (dx 2 )(48)
for all A 1 ∈ B(X 1 ) and A 2 ∈ B(X 2 ), which implies that there is a nonnegative B(X 1 × X 2 ) measurable function that is equal to ϕ (x 1 ,x 2 ) , b Iϕ, 1 -a.s. and agrees with ϕ x 1 x 2 , b on a set M satisfying for P ϕ -a.s. x 1 the section M c x 1 of M c is a P ϕx 1 -null set. Since P ϕ and P ϕx 1 are finite, this is equivalent to M c is a (P ϕ × P ϕx 1 )-null set. Thus ϕ x 1 x 2 , b is Iϕ, 1 measurable for all 0 ≤ b ∈ M (hence for all b ∈ M ). (iii) Apparently it holds that
A ϕ x 1 x 2 , b I(dx)ϕ, 1 = I(A)ϕ, b(49)
for all b ∈ M and A ∈ B(X 1 × X 2 ).
This gives a way to find out
{ϕ (x 1 ,x 2 ) : (x 1 , x 2 ) ∈ X 1 × X 2 } by iterating {ϕ x 1 : x 1 ∈ X 1 } and {ϕ x 2 : x 2 ∈ X 2 } sequentially.
However, in general the composition of two instruments satisfies [I 1 • I 2 , I 2 • I 1 ] = 0 so that the sequential update of a prior normal state according to some instruments in different orders varies.
In classical Bayesian inference, A → K(θ, A)1(θ) is a joint observable of the observables
A i → K i (θ, A i )1(θ), i ∈ [n] in the sense that K(θ, A)1(θ) = [ n i=1 K i (θ, A i )] 1(θ) for all measurable rectangles n i=1 A i .
This means that we can measure the observables A i → K i (θ, A i )1(θ), i ∈ [n] jointly by measuring them sequentially. However, this often fails in the quantum case since I * 1 is a joint observable of the observables
I * 1 (X 1 ) • · · · • I * i−1 (X i−1 ) • I * i (·) • I * i+1 (X i+1 ) • · · · • I * n (X n )1, i ∈ [n] and generally I * 1 (X 1 ) • · · · • I * i−1 (X i−1 ) • I * i (·) • I * i+1 (X i+1 ) • · · · • I * n (X n )1 = I * i 1 for i ∈ {2
, · · · , n}. A sufficient condition that one can measure the observables I * i 1, i ∈ [n] jointly by measuring them sequentially is given below and its proof is straight forward.
Proposition 3. If I * i commutes with I * 1 •· · ·•I * i−1 for any i ∈ {2, · · · , n} then the observables ν i , i ∈ [n] induced by I i , i ∈ [n] have I * 1 as a joint observable, where I is the composition of
I i , i ∈ [n].
IV. LIMIT OF POSTERIOR NORMAL STATE
In classical Bayesian inference, under some regularity conditions the posterior distribution is asymptotically normal, i.e. it can be approximated by an appropriate normal distribution when the number of independently identically distributed observations is sufficiently large.
However, if a quantum system S described by a von Neumann algebra M is in the prior normal state ϕ before sufficiently many sequentially performed measurements with apparatuses corresponding to instruments I i , i ∈ [n], then after these measurements S will probably not be in a converging posterior normal state. For example, identify B(H) * with B 1 (H) and define an instrument I ∈ Ins({∅, X}, B 1 (H)) by
I(X)· = u(·)u * ,(50)
where {∅, X} is the trivial σ-algebra on X and u ∈ B(H) a unitary operator. We assert
that {ρ x = uρu * : x ∈ X}ρ x 1 ···xn − ρ * 1 ≤ Ce −αn , ∀ρ ∈ G(C k ),(51)
where ρ x 1 ···xn = I n ρ is the posterior normal state after performing the sequential measurement scheme I n and ρ * ∈ G(C k ) satisfies I(X)ρ * = ρ * .
Proof. We first show that {ϕ x = I(X)ϕ : x ∈ X} is the unique family of posterior normal Let (Ω, F , P) be a probability space and T : Ω → Ω an invertible, measure preserving, ergodic map (i.e for all F ∈ F satisfying T −1 (F ) = F we have P(F ) = 0 or 1). We say that
a map f ∈ B + (M k (C)) is strictly positive iff f (a) > 0 for all a ∈ M + k (C) \ {0}. Let f 0 be a CPIns({∅, X}, M k (C))-valued F measurable map. Define f n (ω) = f 0 [T n (ω)], ∀ω ∈ Ω, n ∈ Z.
Theorem 5. Assume that there is a positive integer N such that
(1) P[f N (X) • · · · • f 0 (X) is strictly positive] > 0; (53) (2) P[ker(f * 0 (X)) ∩ M + k (C) = {0}] = 1. (54)
Then there is a positive number α > 0, a G(C k ) valued F measurable map ρ 0 and a sequence of F measurable maps {ρ n = ρ 0 [T n (ω)], ∀ω ∈ Ω, n ∈ Z} such that for any (m, z, n) ∈ Z 3 satisfying m < n, m ≤ z, z ≤ n we have
ρ xm···xn − ρ n 1 ≤ C α,z e −α(n−m) , ∀ρ ∈ G(C k ),(55)
where ρ xm···xn = f n (X) • · · · • f m (X)ρ is the posterior normal state after performing the sequential measurement scheme f n • · · · • f m and C α,z is finite P-a.s.
Proof. Since a CP instrument I : {∅, X} → B + (M * ) is uniquely determined by its channel I(X), there is a natural one-to-one correspondence between CP instruments I : {∅, X} → B + (M k (C)) and channels Ψ ∈ C(M k (C)). The rest of the proof follows from Lemma 3.8, Lemma 3.11 and Lemma 3.14 in 22 .
In classical Bayesian inference, a Bayesian is able to learn about an unknown parameter θ by updating her prior, which to some extent, can be viewed as her prior knowledge about θ, to her posterior. In other words, if an oracle were to know the true value of the parameter, a Bayesian shall ensure that with enough observations she would get close to this true value. This is guaranteed by the weak consistency of posterior distribution. Let X n be a random n-tuple (i.i.d. observations) whose conditional distribution is P n θ (an n-fold product of P θ ), Π a prior distribution and Π Xn the posterior distribution. Assume that the true value of θ is θ 0 (iff the ground truth distribution of θ is the Dirac measure Π 0 = 1 (·) (θ 0 )). Then under some conditions, Π X (n) converges weakly to Π 0 in probability P n θ 0 , as n → ∞. In the quantum case, the weak consistency of posterior normal state might be defined as follows. Let ϕ be a prior normal state that a quantum Bayesian stands for and ϕ 0
the ground truth. Let {I n ∈ Ins ( n i=1 A i , M * )} ∞ n=1 be a sequence of instruments such that I n+1 (A (n) × X n+1 )ϕ, 1 = I n (A (n) )ϕ, 1 for all A (n) ∈ n i=1 A i and n ∈ N + . Denote by {ϕ x (n) : x (n) ∈ Π n i=1 X n } (resp. {ϕ 0,x (n) : x (n) ∈ Π n i=1 X n })
a family of posterior normal states w.r.t. (I n , ϕ) (resp. (I n , ϕ 0 )). Then under some conditions ϕ X (n) could converge to ϕ 0,X (n) with respect to an appropriate metrizable topology in probability I n ϕ 0 , 1 .
Assume that X consists of a single element and equipped with the trivial topology. By the definition of the weak consistency of posterior normal state, Theorem 2 and Theorem 4, we have lim n→∞ tr(I n ρ 0 ) ( ρ X 1 ···Xn − ρ 0,X 1 ···Xn 1 > ǫ) = 0, ∀ǫ > 0, ∀ρ, ρ 0 ∈ G(C k ).
(56)
In other words, given sequential measurement scheme {I n } ∞ n=1 , for any prior normal state ρ ∈ G(C k ), the posterior normal state ρ X 1 ···Xn is weakly consistent at any ρ 0 ∈ G(C k ).
Similarly, by the definition of the weak consistency of posterior normal state, Theorem 2 and Theorem 5, we have lim (n−m)→∞ tr(f n−m ρ 0 ) ( ρ Xm···Xn − ρ 0,Xm···Xn 1 > ǫ) = 0, P-a.s., ∀ǫ > 0, ∀ρ, ρ 0 ∈ G(C k ), (57)
where f n−m = f n • · · · • f m . In other words, given sequential measurement scheme f n−m , (m, n) ∈ Z 2 , m < n, for any prior normal state ρ ∈ G(C k ), the posterior normal state ρ Xm···Xn is weakly consistent at any ρ 0 ∈ G(C k ) almost surely.
Let H be a separable complex Hilbert space. Define a CP instrument I ∈ CPIns(A , B 1 (H)) by
I(A)· = i∈A a i (·)a * i , ∀A ∈ A ,(58)
where a i ∈ B(H) for all i ∈ N + and ∞ i=1 a * i a i = 1 in the strong operator topology of B(H). Define a Markov kernel K :
G(H) × B(G(H)) → [0, 1] by (ρ, G) → ∞ i=1 tr(a i ρa * i )1 G (ρ i ),(59)
where ρ i = a i ρa * i /tr(a i ρa * i ). Let (Ω, F , P) be a probability space and {X n :
Ω → G(H)} ∞ n=0
a Markov chain such that P(X n+1 ∈ G|X n = ρ) = K(ρ, G), ∀n ∈ N. Denote by P ρ 0 the probability measure of {X n } ∞ n=0 initializing at ρ 0 , i.e. P ρ 0 (F ) = P(F |X 0 = ρ 0 ) for all F ∈ F . Theorem 6. 23 Let {X n } ∞ n=0 be the Markov chain starting at ρ 0 . Then there is a {ρ ∈ G(H) :
∞ i=1 a i ρa * i = ρ} valued random variable X ρ 0 such that 1 n n i=1
X n → X ρ 0 , weakly * and P ρ 0 -a.s.
as n → ∞.
V. QUANTUM BAYESIAN INFERENCE
With quantum Bayes' rule, a quantum analogue of Bayesian inference is ready to debut.
R(θ, δ) = E x E y L(θ, y) = X ϕ(θ), ν(dx) Y L(θ, y)δ(x, dy).(61)
The quantum Bayes risk R Π : ∆ → R + of a quantum Bayesian decision problem is defined by
R Π (δ) = E θ R(θ, δ) = Θ Π(dθ) X ϕ(θ), ν(dx) Y L(θ, y)δ(x, dy).(62)
Moreover, if there is a randomized decision ruleδ ∈ ∆ such that
R Π (δ) = inf δ∈∆ R Π (δ),(63)
thenδ is called a quantum Bayes solution w.r.t. (Π, ∆).
In the following we would like to explore the admissibility of quantum Bayesian solution.
We say that a randomized decision rule δ is inadmissible iff there is another randomized (ii) R Π (δ) < ∞;
decision rule δ ′ ∈ ∆ such that (i) R(θ, δ ′ ) ≤ R(θ, δ) for all θ ∈ Θ; (ii) There is a θ ′ ∈ Θ such that R(θ ′ , δ ′ ) < R(θ ′ , δ).
(iii) R(θ, δ) is a continuous function of θ for all δ ∈ ∆, thenδ is admissible.
Proof. Ifδ is inadmissible, then there is a randomized decision rule δ ′ ∈ ∆ such that
R(θ, δ ′ ) ≤ R(θ,δ), ∀θ ∈ Θ; (64) ∃ θ ′ ∈ Θ s.t. R(θ ′ , δ ′ ) < R(θ ′ ,δ).(65)
By condition (iii), there is a positive number ǫ > 0 together with an open ball S ǫ (θ ′ ) such that R(θ, δ ′ ) < R(θ,δ) − ǫ, ∀θ ∈ S ǫ (θ ′ ).
Then we have
R Π (δ ′ ) = Sǫ(θ ′ ) R(θ, δ ′ )Π(dθ) + Θ\Sǫ(θ ′ ) R(θ, δ ′ )Π(dθ) (67) < Sǫ(θ ′ ) [R(θ,δ) − ǫ]Π(dθ) + Θ\Sǫ(θ ′ ) R(θ,δ)Π(dθ) (68) = R Π (δ) − ǫΠ[S ǫ (θ ′ )] (69) < R Π (δ),(70)
which contradicts thatδ is a quantum Bayes solution w.r.t. (Π, ∆). Proof. Ifδ is inadmissible, then there is a randomized decision rule δ ′ ∈ ∆ such that
R(θ, δ ′ ) ≤ R(θ,δ), ∀θ ∈ Θ; (71) ∃ θ ′ ∈ Θ s.t. R(θ ′ , δ ′ ) < R(θ ′ ,δ).(72)
Then we have Proof. On the one hand,
R Π (δ ′ ) = Θ R(θ, δ ′ )Π(dθ) ≤ Θ R(θ,δ)Π(dθ) = R Π (δ),(73)c = R Π (δ) = inf δ∈D R Π (δ) (75) ≤ sup Π ′ ∈P inf δ∈D R Π ′ (δ) (76) ≤ inf δ∈D sup Π ′ ∈P R Π ′ (δ) (77) ≤ inf δ∈D sup θ∈Θ R(θ, δ),(78)
where P is a class of prior distributions such that Π ∈ P. On the other hand,
c = sup θ∈Θ R(θ,δ) ≥ inf δ∈D sup θ∈Θ R(θ, δ).(79)
Theorem 10. Let {Π n } ∞ n=1 be a sequence of prior distributions andδ n a quantum Bayes solution w.r.t. (Π n , D). Let δ ∈ D be a decision rule. If
sup θ∈Θ R(θ, δ) ≤ lim sup n→∞ R Πn (δ n ),(80)
then δ is a minimax decision rule.
Proof. If δ is not a minimax decision rule, then there is a decision rule δ ′ ∈ D such that
sup θ∈Θ R(θ, δ ′ ) < sup θ∈Θ R(θ, δ).(81)
For any n ≥ 1 we have
R Πn (δ n ) ≤ R Πn (δ ′ ) (82) ≤ sup θ∈Θ R(θ, δ ′ ) (83) < sup θ∈Θ R(θ, δ),(84)
which contradicts the condition (80).
R ϕ (δ) = E θ|x E y L(θ, y) = Θ ϕ x , λ(dθ) Y L(θ, y)δ(x, dy).(85)
Moreover, if there is a randomized decision rule δ * ∈ ∆ such that πdν, ∆) shall minimize the posterior risk Θ π(θ|x)ν(dθ) Y L(θ, y)δ(x, dy) for P almost all x ∈ X, where π ∈ L 1 +,1 (ν) and ν is a σ-finite measure on (Θ, E ). Since probability measures ϕ x , λ(·) and (·) π(θ|x)ν(dθ) can be quite different, it is likely to happen that there is a quantum Bayes solution w.r.t. (Π, ∆) not minimizing the quantum posterior risk Θ ϕ x , λ(dθ) Y L(θ, y)δ(x, dy) for P almost all x ∈ X.
R ϕ (δ * ) = inf δ∈∆ R ϕ (δ),(86)
In the following we will concentrate on a quantum analogue of Bayesian inference. This topic mainly consists of four parts: quantum posterior point estimation, quantum posterior credible interval, quantum posterior hypothesis testing and quantum posterior prediction.
One could see that in terms of framework, quantum Bayesian inference is almost the same as the classical one.
Assume that (X, A ), (Y, B) and (Θ, E ) are all (R, B(R)). Denote by P ϕx the probability measure ϕ x , λ(·) and by F(θ|x) the cumulative distribution function corresponding to P ϕx .
Theorem 11. Suppose the loss function L(θ, y) = c(θ)(θ −y) 2 , where c(θ) ≥ 0 for all θ ∈ Θ.
Then the quantum posterior solution δ * w.r.t. (ϕ, D) is
δ * (x, B) = 1 B E θ|x θc(θ) E θ|x c(θ) .(87)
Specifically, if c(θ) = 1 for all θ ∈ Θ, then the quantum posterior solution δ * w.r.t. (ϕ, D) is
δ * (x, B) = 1 B (E θ|x θ).(88)
Proof. Straight calculation shows that the quantum posterior risk
R ϕ (δ) = E θ|x E y c(θ)(θ − y) 2 (89) = E θ|x c(θ)[θ − δ(x)] 2 ,(90)
where δ(x) ∈ Y is the point such that δ(x, B) = 1 B [δ(x)], ∀B ∈ B. Apparently R ϕ (δ) takes its minimum when
δ(x) = E θ|x θc(θ)/E θ|x c(θ).(91)
Theorem 12. Suppose the loss function L(θ, y) =
k 0 (θ − y) if θ > y k 1 (y − θ) if θ ≤ y , where k 0 , k 1 ≥ 0.
Then the quantum posterior solution δ * w.r.t. (ϕ, D) is
δ * (x, B) = 1 B ξ θ|x k 0 k 0 + k 1 ,(92)
where ξ θ|x (p) := inf{θ ∈ Θ : F(θ|x) ≥ p} is the p quantile of F(θ|x). Specifically, if k 0 = k 1 = 1, then the quantum posterior solution δ * w.r.t. (ϕ, D) is
δ * (x, B) = 1 B [ξ θ|x (0.5)].(93)
Proof. Let p = k 0 /(k 0 + k 1 ). Assume that y > ξ θ|x (p). A little calculation shows that
L[θ, ξ θ|x (p)] − L(θ, y) = k 1 [ξ θ|x (p) − y] if θ ≤ ξ θ|x (p) (k 0 + k 1 )θ − k 0 ξ θ|x (p) − k 1 y if ξ θ|x (p) < θ ≤ y k 0 [y − ξ θ|x (p)] if θ > y ,(94)
and
L[θ, ξ θ|x (p)] − L(θ, y) ≤ k 1 [ξ θ|x (p) − y]1 (−∞,ξ θ|x (p)](95)+ k 0 [y − ξ θ|x (p)]1 (ξ θ|x (p),+∞)(96)
since (k 0 + k 1 )θ − k 0 ξ θ|x (p) − k 1 y < (k 0 + k 1 )y − k 0 ξ θ|x (p) − k 1 y = k 0 [y − ξ θ|x (p)]. Let δ 0 ∈ D such that δ 0 (x) = ξ θ|x (p). Then for any δ ∈ D such that δ(x) > ξ θ|x (p) we have
R ϕ (δ 0 ) − R ϕ (δ) = E θ|x {L[θ, ξ θ|x (p)] − L[θ, δ(x)]} (97) ≤ k 1 [ξ θ|x (p) − δ(x)]P ϕx [θ ≤ ξ θ|x (p)](98)+ k 0 [δ(x) − ξ θ|x (p)]P ϕx [θ > ξ θ|x (p)](99)≤ k 0 k 1 k 0 + k 1 [ξ θ|x (p) − δ(x)](100)+ k 0 k 1 k 0 + k 1 [δ(x) − ξ θ|x (p)] = 0,(101)
which is equivalent to
R ϕ (δ 0 ) ≤ R ϕ (δ).(102)
The inequality above also holds when δ(x) < ξ θ|x (p) and thus δ 0 is the quantum posterior
δ * (x, B) = 1 B [max θ∈Θ p x (θ)].(103)
Proof. Straight calculation shows that
R ϕ (δ) = E θ|x E y L(θ, y)(104)= 1 − δ(x)+ǫ δ(x)−ǫ p x (θ)µ x (dθ).(105)
Apparently minimizing R ϕ (δ) is equivalent to maximizing δ(x)+ǫ δ(x)−ǫ p x (θ)µ x (dθ) and thus δ(x) = max θ∈Θ p x (θ) as ǫ → 0.
We say thatθ E := E θ|x (θ) is the quantum posterior mean estimator of θ;θ Q := ξ θ|x (0.5) is the quantum posterior median estimator of θ andθ M := max θ∈Θ p x (θ) is the quantum posterior mode estimator of θ. Remark 1. Under the premise of Theorem 4, further assume that Θ is a metric space, E is the Borel σ-algebra on Θ, X consists of a single element and equipped with the trivial topology, and tr(ρ * λ) is a Dirac measure. Then by Theorem 4 we have 0 ≤ |tr[(ρ X 1 ···Xn − ρ * )λ]| ≤ λ ρ X 1 ···Xn − ρ * 1 → 0,
as n → ∞. This implies that tr(ρ X 1 ···Xn λ) converges weakly to tr(ρ * λ) in probability tr(I n ρ 0 ).
Thusθ E ,θ Q andθ M are all weakly consistent.
Assume that Y is the set of closed intervals on the real line and the loss function L(θ, y) =
= k 0 l[δ(x)] + k 1 P ϕx [θ / ∈ δ(x)].(108)
Here a quantum posterior solution δ * w.r.t. (ϕ, D) is generally not easy to obtain. A common strategy is to keep l[δ(x)] as small as possible while controlling P ϕx [θ / ∈ δ(x)] not to exceed a given small positive number. We say that a decision rule δ is a 1 − α quantum posterior credible interval iff
P ϕx [θ ∈ δ(x)] ≥ 1 − α, ∀x ∈ X,(110)
where 0 < α < 1.
Assume that (Y, F ) = ([n], 2 [n] ) and the loss function L(θ, y) = n i=1 δ iy 1 Θ c i (θ), where δ (··) is the Kronecker delta and Θ i , i ∈ [n] are mutually disjoint P ϕx measurable subsets of Θ. A little calculation shows that the quantum posterior risk R ϕ (δ) = E θ|x E y L(θ, y)
= n i=1 δ iδ(x) P ϕx (θ / ∈ Θ i )(111)
and apparently R ϕ (δ) takes its minimum when δ(x) = arg max i∈[n] P ϕx (θ ∈ Θ i ) so that the quantum posterior solution δ * w.r.t. (ϕ, D) is
δ * (x, B) = 1 B [arg max i∈[n] P ϕx (θ ∈ Θ i )].(113)
We say that the δ * above is the quantum posterior testing rule of the multiple hypothesis testing
H i : θ ∈ Θ i , i ∈ [n].(114)
In classical Bayesian inference, the posterior predictive distribution of a Z-valued random variable is given by the Markov kernel Θ K(θ, x, C)π(θ|x)ν(dθ), where (Z, C ) is a measurable space, K : (θ, x, C) → K(θ, x, C), ∀(θ, x, C) ∈ Θ × X × C a Markov kernel and π(θ|x) the posterior density. Note that the section K x of K is not only an instrument but an observable as well. Let η : C → M be an observable. We say that ϕ x , η is the quantum posterior predictive distribution of η.
VI. DISCUSSION
In a sense, quantum Bayes' rule tells a quantum Bayesian how to update her knowledge of an object based on observations. According to quantum Bayes' rule, a quantum analogue of Bayesian inference, which not only retains the classical one as a special case but possesses many new features as well, is put forward. However, there is still a lot to explore. (ii) In classical Bayesian inference, if the true value of the unknown parameter θ is θ 0 , then Π 0,x = 1 (·) p(x|θ)Π 0 (dθ) Θ p(x|θ)Π 0 (dθ)
= Π 0 ,(115)
where p(x|θ) is a nonnegative E ×A measurable real valued function satisfying X p(x|θ)µ(dx) = 1, ∀θ ∈ Θ, µ is a σ-finite measure on (X, A ). This indicates that Π 0 is never disturbed by a measurement with an apparatus corresponding to an instrument K = 1 (·) p(x|θ)µ(dx).
However, in the definition of the weak consistency of posterior normal state, the ground truth ϕ 0 is likely to be disturbed by a measurement with an apparatus corresponding to an instrument I n . So is it necessary for ϕ 0 to satisfy for a sequence of instruments
{I n ∈ Ins ( n i=1 A i , M * )} ∞ n=1
such that I n+1 (A (n) × X n+1 )ϕ, 1 = I n (A (n) )ϕ, 1 for any A (n) ∈ n i=1 A i and n ∈ N + we have ϕ 0,X (n) = ϕ 0 (a.s.), n ∈ N + and ϕ 0 , λ = Π 0 ?
(iii) What is the sufficient and necessary conditions for the weak consistency of posterior normal state?
(iv) Topics concerning covariant quantum posterior point estimator.
(v) Choice and robustness of prior normal state, e.g. global or local sensitivity measures of a class of prior normal states.
ACKNOWLEDGMENTS
The author would like to thank Professor Naihui Chen and Professor Weihua Liu for their discussion and advice on this paper.
n-dimensional complex Hilbert space and {ψ j } n j=1 an orthonormal basis of H. Denote by c the one-to-one correspondence between {x i } and the projector |ψ i ψ i | for all i ∈ [n], where [n] := {1, 2, · · · , n}. Then there is a density operator ρ = n i=1 P({x i })|ψ i ψ i | such that P(A) = j∈{i:x i ∈A} tr(ρ|ψ j ψ j |)
the posterior density of θ under the observation x (given P × ({(·, x)}) > 0) isπ(θ|x) = P × ({(θ, x)}) P × ({(·, x)}) (9) = tr(P {(θ,·)} ρ {(·,x)} P {(θ,·)} )(10)for all θ ∈ Θ, where {(·, x)} := {(θ k , x) : k ∈ [m]} and ρ {(·,x)} = P {(·,x)} ρP {(·,x)} /tr(P {(·,x)} ρP {(·,x)} )is a density operator. Again operations ρ → P {(·,x)} ρP {(·,x)} defines a Lüders instrument.
below. Let (X, A ) be a measurable space and H a complex Hilbert space. Denote by B + (H) the set of positive bounded linear operators on H and by G(H) the set of density operators on H. Definition 1. A map ν : A → B + (H) is called an observable iff (i) ν(X) = 1;
Moreover, ν is called a sharp observable iff ν * (A) = ν(A) = ν 2 (A) for all A ∈ A . The convex set of observables ν : A → B + (H) is denoted by O(A , H). A convex combination of ν 1 and ν 2 in O(A , H) (i.e. tν 1 + (1 − t)ν 2 , 0 < t < 1) can be viewed as a randomization of measuring processes described by ν 1 and ν 2 . Let ν : A → B + (H) be an observable. For any density operator ρ ∈ G(H), the probability measure ν ρ induced by ν is defined by ν ρ (A) = tr[ρν(A)], ∀A ∈ A . (12) One may question the capacity of a complex Hilbert space H. In other words, one may wonder whether there is a probability measure P on the measurable space (X, A ) such that for all observables ν ∈ O(A , H) and all density operators ρ ∈ G(H) we have P(A) = tr[ρν(A)], for some A ∈ A . Fortunately this will never happen. It is shown in 16 that for each probability measure P on the measurable space (X, A ) and each density operator ρ ∈ G(H) there is a unique observable ν : A → B + (H) such that P(A) = tr[ρν(A)], ∀A ∈ A .
notion of instrument is a bit more complicated. To introduce this, first we have to agree on some notations. Let M ⊆ B(H) be a von Neumann algebra. Denote by M * the predual of M (i.e. the set of σ-weakly continuous bounded linear functionals on M ), by S(M ) the set of normal states on M (i.e. the set of positive and unit elements of M * ), by ·, · the duality pairing between M * and M and by B + (M * ) the set of positive bounded linear maps on M * .A map Ψ ∈ B + (M * ) is called a subtransition iff it satisfies Ψϕ, 1 ≤ ϕ, 1 for all 0 ≤ ϕ ∈ M * . Moreover, if Ψ satisfies Ψϕ, 1 = ϕ, 1 for all ϕ ∈ M * then Ψ is called a transition. The dual of a subtransition (resp. transition) Ψ, denoted Φ, is defined by ϕ, Φa = Ψϕ, a for all ϕ ∈ M * and a ∈ M . It is a normal positive linear map on M such that Φ1 ≤ 1 (resp. Φ1 = 1). If Ψ is completely positive (CP), then Ψ is called an operation (resp. channel).Definition 2. A map I : A → B + (M * ) is called an instrument (normalized subtransition valued measure) iff (i) I(X) is a transition;
13) is with respect to the strong operator topology of B + (M * ). The dual of an instrument I, denoted I * , is defined by ϕ, I * (A)a = I(A)ϕ, a for all A ∈ A , ϕ ∈ M * and a ∈ M . It is a map from A to L + (M ) (i.e. the set of positive linear maps on M ). The convex set of instruments I : A → B + (M * ) is denoted by Ins(A , M * ). If I(A) is an operation for all A ∈ A , then I is called a CP instrument. The convex set of CP instruments I : A → B + (M * ) is denoted by CPIns(A , M * ).An instrument naturally induces an observable. Let I : A → B + (M * ) be an instrument.
{x} such that P({x}) = 0? This question is settled by the notion of a family of posterior normal states. Let H be a complex Hilbert space, M ⊆ B(H) a von Neumann algebra, I : A → B + (M * ) an instrument and ϕ ∈ S(M ) a normal state.Definition 3. The set {ϕ x : x ∈ X} is called a family of posterior normal states w.r.t.
is a family of posterior normal states w.r.t. (I, ϕ), then it exists uniquely in the sense that if {ϕ ′ x : x ∈ X} is another family of posterior normal states w.r.t. (I, ϕ)
outcome {x} and disturb the state of S according to {x}. This disturbance is described by an update of the normal state ϕ → ϕ x , where {ϕ x : x ∈ X} is a family of posterior normal states w.r.t. (K, π(θ)). But what if a Bayesian uses an apparatus corresponding to an arbitrary instrument to measure a quantum system described by an arbitrary von Neumann algebra? To answer this, we put forward the following quantum Bayes' rule by analogy to the classical one. Quantum Bayes' Rule. If a quantum system S described by a von Neumann algebra M is in the prior normal state ϕ before a measurement with an apparatus corresponding to an instrument I : A → B + (M * ), then after the measurement S will be in the posterior normal state ϕ x if the measurement outcome is {x}, where {ϕ x : x ∈ X} is a family of posterior normal states w.r.t. (I, ϕ).
I i : B(X i ) → B + (M * ), i ∈ [n]be instruments. Then according to20 , there is a unique
a family of posterior normal states w.r.t. (I, ϕ).
is the unique family of posterior normal states w.r.t. (I, ρ). By repeatedly applying I to the prior normal state ρ one sees that the posterior normal state ρ x does not converge. However, in some cases the posterior normal state converges as the sample size goes to infinity. Let I : {∅, X} → B + (M k (C)) be a CP instrument where M k (C) is the von Neumann algebra of k × k complex matrices.
Theorem 4 .
4Assume that the spectrum of I(X) has the unique eigenvalue {1} on the unit circle and this eigenvalue is simple. Then there are positive numbers C, α > 0 such that for any n ∈ N + we have
states w.r.t. (I, ϕ), where I : {∅, X} → B + (M * ) is an instrument and ϕ is a prior normal state. Assume that {ϕ x : x ∈ X} is a family of posterior normal states w.r.t. (I, ϕ). Thenfor all a ∈ M , the function x → ϕ x , a is P ϕ measurable and the following equation holdsX ϕ x , a P ϕ (dx) = I(X)ϕ, a .(52)Since the probability space (X, {∅, X}, P ϕ ) is complete, the function x → ϕ x , a is {∅, X} measurable for all a ∈ M . Therefore x → ϕ x , a shall be constant for all a ∈ M . This implies that for all x ∈ X, ϕ x = ϕ 0 for some ϕ 0 ∈ S(M ). Again due to the arbitrariness of a, ϕ 0 = I(X)ϕ. The rest of the proof follows from 21 .
A
quantum statistical model is a pair (ν, Λ) where ν : A → M is an observable and Λ ⊆ S(M ) a set of normal states. Moreover, if Λ = {ϕ(θ) ∈ S(M ) : θ ∈ Θ}, then (ν, ϕ(θ)) is called a parametric quantum statistical model. A quantum Bayesian decision problem consists of five elements: a parameter space (Θ, E , Π), which is a probability space; a random experiment (ν, ϕ(θ)), which is a parametric quantum statistical model; an action space (Y, B), which is a measurable space; a randomized decision rule δ : X × B → [0, 1], which is a Markov kernel; a loss function L : Θ × Y → R + such that L(θ, y) is B measurable for all θ ∈ Θ. Denote by ∆ a class of randomized decision rule δ : X × B → [0, 1]. If a randomized decision rule δ satisfying δ(x, ·) is a Dirac measure for all x ∈ X, then δ is called a non-randomized decision rule or decision rule in brief. Denote by D a class of decision rule δ : X × B → [0, 1]. The risk function R : Θ × ∆ → R + of a quantum Bayesian decision problem is defined by
Theorem 7 .
7Assume that Θ is a metric space and E the Borel σ-algebra on Θ. Denote by δ a quantum Bayes solution w.r.t. (Π, ∆). If(i) Π(E) > 0 for all open subsets E of Θ;
Theorem 8 .
8Ifδ is the unique quantum Bayes solution w.r.t. (Π, ∆), thenδ is admissible.
Theorem 9 .
9which shows that δ ′ is a quantum Bayes solution w.r.t. (Π, ∆) as well. But this contradicts thatδ is the unique quantum Bayes solution w.r.t. (π, ∆).We say thatδ ∈ D is a minimax decision rule iff it Letδ be a quantum Bayes solution w.r.t. (Π, D). If R(θ,δ) = c for all θ ∈ Θ, thenδ is a minimax decision rule.
Let ϕ ∈ S(M ) be a prior normal state, λ : E → M an observable, I : A → B + (M * ) an instrument and {ϕ x : x ∈ X} a family of posterior normal states w.r.t. (I, ϕ).
Definition 4 .
4The quantum posterior risk R ϕ : ∆ → R + of a quantum Bayesian decision problem is defined by
then δ * is called a quantum posterior solution w.r.t. (ϕ, ∆).Apparently our framework retains the classical one as a special case. In classical statistics, a statistical model is a triad (X, A , P) where (X, A ) is a measurable space and P a class of probability measures on (X, A ). Moreover, if P = {P θ : θ ∈ Θ}, then (X, A , P θ ) is called a parametric statistical model. If we replace the parametric quantum statistical model (ν, ϕ(θ)) in a quantum Bayesian decision problem with a parametric statistical model (X, A , P θ ) where P θ is a Markov kernel, we will get a classical Bayesian decision problem.It is noted that in classical Bayesian decision, a posterior solution w.r.t. (Π, ∆) is always a Bayes solution w.r.t. (Π, ∆) and each Bayes solution w.r.t. (Π, ∆) minimizes the posterior risk for P = P θ dΠ almost all x ∈ X. However, this is not true in the quantum case. To see this, assume that the map (θ, x) → Y L(θ, y)δ(x, dy) is E × A measurable for all δ ∈ ∆ and ϕ(θ), ν(A) is a Markov kernel defined by A p(x|θ)µ(dx), ∀A ∈ A , where p(x|θ) is a nonnegative E × A measurable real valued function satisfying X p(x|θ)µ(dx) = 1, ∀θ ∈ Θ, and µ is a σ-finite measure on (X, A ). Then each quantum Bayes solution w.r.t. (Π =
solution w.r.t. (ϕ, D). Theorem 13. Suppose the loss function L(θ, y) |θ − y| > ǫ 0 if |θ − y| ≤ ǫ and P ϕx (·) = (·) p x (θ)µ x (dθ) for a Borel measurable real valued function p x (θ) and a σ-finite measure µ x on (R, B(R)) and p x (θ) has maximum value. Then as ǫ → 0 a quantum posterior solution δ * w.r.t. (ϕ, D) is
k 0 l(y)+k 1 [1−1 y (θ)], where l(y) is the length of y and k 0 , k 1 ≥ 0. Then the quantum posteriorrisk R ϕ (δ) = E θ|x E y {k 0 l(y) + k 1 [1 − 1 y (θ)]} (107) = k 0 l[δ(x)] + k 1 E θ|x [1 − 1 δ(x) (θ)]
( i )
iWhat is the sufficient and necessary conditions that a quantum posterior solution w.r.t. (ϕ, ∆) is a quantum Bayes solution w.r.t. (Π, ∆) and vice versa?
proper embedding, the Bayesian update of a prior distribution can be identified with the instrumental update of a prior normal state. Based on the notions of von Neumann algebra, normal state, observable, and instrument, we put forward a quantum analogue of Bayes' rule, which is concise and has clear physical meaning. Besides, as with Bayes' rule, it is compatible with sequential measurement schemes. By the way, we obtain a sufficient condition for sequential measurements to be joint measurements. Parallel to the asymptoticnormality of posterior distribution, we obtain two sufficient conditions for the convergence of posterior normal state, and give the definition of the weak consistency of posterior normal state by analogy to the definition of the weak consistency of posterior distribution, and thus obtain two sufficient conditions for the weak consistency of posterior normal state. Then we generalize the fundamental notions and results of Bayesian inference according to quantum Bayes' rule. Fortunately, our theory retains the classical one as a special case, although we note that for a given quantum Bayesian decision problem, a quantum Bayes solution and a quantum posterior solution are generally no longer equivalent.
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|
[] |
[
"Proper motions of young stars in Chamaeleon I. A Virtual Observatory study of spectroscopically confirmed members",
"Proper motions of young stars in Chamaeleon I. A Virtual Observatory study of spectroscopically confirmed members"
] |
[
"Belén López Martí \nDepartamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain\n",
"Francisco Jiménez-Esteban \nDepartamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain\n\nSpanish Virtual Observatory\nSpain\n\nDivision of Science and Engineering\nSaint Louis University\nAvenida del Valle 34, E28003Madrid Campus, MadridSpain\n",
"Amelia Bayo \nEuropean Southern Observatory\nAlonso de Córdova 3107, VitacuraSantiagoChile\n",
"David Barrado \nDepartamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain\n\nCalar Alto Observatory\nCentro Astronómico Hispano-Alemán, C/ Jesús Durbán Remón 2-2, E04004AlmeríaSpain\n",
"Enrique Solano \nDepartamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain\n\nSpanish Virtual Observatory\nSpain\n",
"Carlos Rodrigo \nDepartamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain\n\nSpanish Virtual Observatory\nSpain\n"
] |
[
"Departamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain",
"Departamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain",
"Spanish Virtual Observatory\nSpain",
"Division of Science and Engineering\nSaint Louis University\nAvenida del Valle 34, E28003Madrid Campus, MadridSpain",
"European Southern Observatory\nAlonso de Córdova 3107, VitacuraSantiagoChile",
"Departamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain",
"Calar Alto Observatory\nCentro Astronómico Hispano-Alemán, C/ Jesús Durbán Remón 2-2, E04004AlmeríaSpain",
"Departamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain",
"Spanish Virtual Observatory\nSpain",
"Departamento de Astrofísica\nCentro de Astrobiología (INTA-CSIC)\nP.O. Box 78E-28261Villanueva de la Cañada, MadridSpain",
"Spanish Virtual Observatory\nSpain"
] |
[] |
Context. The study of the motion of the members of a given open cluster or stellar association provides key information about their formation and early evolution. The Chamaeleon cloud complex constitutes one of the closest and best studied low-mass star-forming regions in the Galaxy. Aims. We want to provide further evidence of the origin of the proposed stellar members of Chamaeleon and to identify interlopers from the foreground Cha and η Cha associations. Methods. We compile lists of spectroscopically confirmed members of Chamaeleon I and II, Cha and η Cha, and of background objects in the same line of sight. Using Virtual Observatory tools, we cross-match these lists with the UCAC3 catalogue to get the proper motions of the objects. In the vector point diagram, we identify the different moving groups, and use this information to study the membership of proposed candidate members of the associations from the literature. For those objects with available radial velocities, we compute their Galactic space velocities. We look for correlations between the known properties of the objects and their proper motions. Results. The members of the dark clouds exhibit clearly different proper motions from those of the foreground associations and of the background stars. The data suggest that Chamaeleon II could have different dynamical properties from Chamaeleon I. Although the two foreground clusters and η Chamaeleontis constitute two different proper motion groups, they have similar spatial motions, which are different from the spatial motion of Chamaeleon I. On the other hand, the space motions of the Chamaeleon II stars look more similar to those of the foreground clusters than to the Chamaeleon I stars, but the numbers are low. We find no correlations between the proper motions and the properties of the objects in either of the clouds. Conclusions. On the basis of proper motion, Chamaeleon I and II constitute two physical entities unrelated to the foreground and η Chamaeleontis clusters, but with the available data it is unclear to what extent the stellar populations in both clouds are physically connected to each other.
|
10.1051/0004-6361/201220128
|
[
"https://arxiv.org/pdf/1302.7167v1.pdf"
] | 56,298,438 |
1302.7167
|
ae44310b6101594cf40239c0a1019178440253ec
|
Proper motions of young stars in Chamaeleon I. A Virtual Observatory study of spectroscopically confirmed members
March 1, 2013
Belén López Martí
Departamento de Astrofísica
Centro de Astrobiología (INTA-CSIC)
P.O. Box 78E-28261Villanueva de la Cañada, MadridSpain
Francisco Jiménez-Esteban
Departamento de Astrofísica
Centro de Astrobiología (INTA-CSIC)
P.O. Box 78E-28261Villanueva de la Cañada, MadridSpain
Spanish Virtual Observatory
Spain
Division of Science and Engineering
Saint Louis University
Avenida del Valle 34, E28003Madrid Campus, MadridSpain
Amelia Bayo
European Southern Observatory
Alonso de Córdova 3107, VitacuraSantiagoChile
David Barrado
Departamento de Astrofísica
Centro de Astrobiología (INTA-CSIC)
P.O. Box 78E-28261Villanueva de la Cañada, MadridSpain
Calar Alto Observatory
Centro Astronómico Hispano-Alemán, C/ Jesús Durbán Remón 2-2, E04004AlmeríaSpain
Enrique Solano
Departamento de Astrofísica
Centro de Astrobiología (INTA-CSIC)
P.O. Box 78E-28261Villanueva de la Cañada, MadridSpain
Spanish Virtual Observatory
Spain
Carlos Rodrigo
Departamento de Astrofísica
Centro de Astrobiología (INTA-CSIC)
P.O. Box 78E-28261Villanueva de la Cañada, MadridSpain
Spanish Virtual Observatory
Spain
Proper motions of young stars in Chamaeleon I. A Virtual Observatory study of spectroscopically confirmed members
March 1, 2013Received; acceptedAstronomy & Astrophysics manuscript no. lopezMarti˙cham c ESO 2013stars:low-massbrown dwarfs − stars: kinematics and dynamics − stars: formation − stars: luminosity functionmass function − astronomical databases: miscellaneous − astronomical databases: virtual observatory tools
Context. The study of the motion of the members of a given open cluster or stellar association provides key information about their formation and early evolution. The Chamaeleon cloud complex constitutes one of the closest and best studied low-mass star-forming regions in the Galaxy. Aims. We want to provide further evidence of the origin of the proposed stellar members of Chamaeleon and to identify interlopers from the foreground Cha and η Cha associations. Methods. We compile lists of spectroscopically confirmed members of Chamaeleon I and II, Cha and η Cha, and of background objects in the same line of sight. Using Virtual Observatory tools, we cross-match these lists with the UCAC3 catalogue to get the proper motions of the objects. In the vector point diagram, we identify the different moving groups, and use this information to study the membership of proposed candidate members of the associations from the literature. For those objects with available radial velocities, we compute their Galactic space velocities. We look for correlations between the known properties of the objects and their proper motions. Results. The members of the dark clouds exhibit clearly different proper motions from those of the foreground associations and of the background stars. The data suggest that Chamaeleon II could have different dynamical properties from Chamaeleon I. Although the two foreground clusters and η Chamaeleontis constitute two different proper motion groups, they have similar spatial motions, which are different from the spatial motion of Chamaeleon I. On the other hand, the space motions of the Chamaeleon II stars look more similar to those of the foreground clusters than to the Chamaeleon I stars, but the numbers are low. We find no correlations between the proper motions and the properties of the objects in either of the clouds. Conclusions. On the basis of proper motion, Chamaeleon I and II constitute two physical entities unrelated to the foreground and η Chamaeleontis clusters, but with the available data it is unclear to what extent the stellar populations in both clouds are physically connected to each other.
Introduction
The kinematic properties of the members of a given stellar association hold important clues to its history. Some formation models predict that the early dynamical evolution of the parent protostellar cluster should lead to mass-dependent kinematic distributions, and, eventually, to an efficient mass segregation (Kroupa & Bouvier 2003), while other numerical simulations predict similar kinematic properties over the whole mass spectrum (? Bate 2012). Various authors have used radial velocity measurements to study the kinematic properties of young low-mass objects (e.g. Jeffries et al. 2006;Maxted et al. 2008), and a number of surveys have used proper motions to identify and confirm new lowmass members in young associations and clusters (e.g. Moraux et al. 2001;Kraus & Hillenbrand 2007;Bouy & Martín 2009;Caballero 2010). In particular, our recent proper motion study of the Lupus star-forming region shows that it is possible to distinguish between probable low-mass members of the complex and likely contaminants using kinematic information from available astrometric catalogues (López Martí et al. 2011, hereafter LJS11).
The Chamaeleon complex is one of the closest and best studied low-mass star-forming regions. At an estimated distance in the range 115-215 pc, it is composed of three dark clouds, named Chamaeleon I, II and III (Schwartz 1977;Schwartz et al. 1991). The oldest one is Chamaeleon I, with a mean age of about 2 Myr; it contains more than 300 known young stars (see Luhman 2008, for the latest census), most of them clustered in two cloud cores containing two intermediate-mass stars, HD 97048 and HD 97300. Chamaeleon II seems to be at an earlier stage of evolution than Chamaeleon I, because it contains more embedded than visual objects (Gauvin & Strom 1992). Surveys by several authors have identified more than 60 stars in this cloud Alcalá et al. 2008, and references therein). No active star formation seems to be taking place in Chamaeleon III.
The Chamaeleon clouds have also been the target of kinematic studies, generally based on radial velocities and focused on the brightest stars in the complex (e.g. Dubath et al. 1996). The exceptions are a few very low-mass stars and brown dwarf candidates in Chamaeleon I (Joergens 2006, and references therein). Proper motion and parallax information from Hipparcos has been used to assess the origin of the star formation in the clouds and their relation to other neighbouring associations (Sartori et al. 2003). X-ray observations during the past two decades, combined with Hipparcos astrometry, have shown that the young stars observed towards the Chamaeleon sky area can be grouped into several distinct associations: the Chamaeleon dark cloud complex itself, and the foreground (at around 100 pc) Chamaeleontis ( Cha) and η Chamaeleontis (η Cha) young associations (Feigelson et al. 2003;Mamajek et al. 1999). The observations have shown that these two associations are older than the Chamaeleon star-forming complex (∼15-20 Myr), but it is unclear whether there is a relation between the two, or whether they are somehow related to the dark clouds.
In this paper, we make use of proper motions from the Third US Naval Observatory CCD Astrograph Catalog (UCAC3; Zacharias et al. 2010) and of Virtual Observatory 1 (VO) tools to investigate the kinematic properties of members and candidate members of the Chamaeleon dark clouds. Our goals are to provide further evidence of a common origin for all these objects and to test their association with the dark clouds and between the clouds themselves.
Available data
Object compilation
We compiled a list of Chamaeleon confirmed members (through spectroscopy) from the literature. For Chamaeleon I, we retrieved the list from the most recent census by Luhman (2007), and completed it with the eight new members reported by Luhman & Muench (2008a). The final list contained 304 objects down to R = 24 mag, with extinctions in the range A V 0-20 mag). For Chamaeleon II, we used the list of confirmed members from Spezzi et al. (2008), which includes all objects studied in previous works. In total, we had 41 Chamaeleon II spectroscopically confirmed members, with R-band magnitudes down to about 23 mag and extinctions in the range A V 0-11 mag.
To assess how clearly we can separate the different populations based on the available proper motions, and to check the possible presence of interlopers from the foreground clusters in our Chamaeleon I and II lists, we compiled two lists of members for the Cha and η Cha associations, merging the catalogues provided by Luhman (2004) and Fernández et al. (2008) for both groups, and adding the objects from the lists by Luhman & Steeghs (2004) in η Cha and by da Silva et al. (2009) in Cha. Membership of these objects in the corresponding cluster has been confirmed by several different means, including spectroscopy and proper motions. The final compiled lists contained 36 stars in Cha and 20 in η Cha down to R ∼ 18 mag. No estimations of extinction are available for these stars, but their location outside areas of dense dust suggests that it is very low in most cases.
For control purposes, we also considered the list of 88 background sources from Table 4 of Luhman (2007). These objects were initially proposed in the literature as Chamaeleon I members, but they were later discarded through spectroscopy. We note that the magnitude limit of most of these compilations is much fainter than the UCAC3 magnitude limit (see Sect. 2.2 below); therefore, a significant fraction of objects are not expected to have counterparts in that catalogue. Since the faintest objects are also the lowest mass ones, and given that the substellar mass limit is estimated to be around R ∼ 21 mag at the age and distance of the Chamaeleon clouds, this implies that only stellar members of the associations will have measured proper motions. In particular, the objects at the substellar boundary studied by López Martí et al. (2004) in Chamaeleon I and Barrado y Navascués & Jayawardhana (2004) in Chamaeleon II are not included in UCAC3.
In a second step in our study, we also investigated those candidate members of the Chamaeleon associations from the literature whose true nature is still unclear. This list included 63 Xray emitting stars unveiled by the ROSAT mission (Alcalá et al. 1995) that remained unclassified, as well as some objects from other previous works still lacking spectroscopic confirmation of youth: the list of uncertain objects provided by Luhman et al. (2004, his Table 8; 21 objects) for Chamaeleon I, and the candidate members of Chamaeleon II without spectroscopic confirmation of membership from Spezzi et al. (2008, 11 objects) and the DENIS survey (Vuong et al. 2001, 41 objects).
In total, our list of unconfirmed candidates amounted to 136 objects (63 ROSAT stars and 73 objects from other works), but the vast majority of them are too faint to be included in the proper motion study, and only a handful of them have proper motion measurements available in UCAC3, as explained in Sect. 4.
Proper motion data
To get proper motion measurements, we cross-matched our compiled catalogues with the UCAC3 catalogue, available within the Virtual Observatory. This is an all-sky survey containing about 100 million objects, 95% of them with proper motions, covering a dynamical range of about 8-16 mag in a single bandpass between V and R. Its positional accuracy is about 15 to 100 mas per coordinate, depending on magnitude. The proper motion errors range from 1 to 10 mas/yr, depending on magnitude and observing history.
As in our previous work on Lupus (LJS11), we followed a VO-based methodology to cross-match and analyse the data. We made use of the Multiple Cone Search utility of TOPCAT 2 . A matching radius of 2 was used. We retained all sources whose proper motion errors were not set to zero, which had been computed using more than two epoch positions, and which had an object classification flag (ot) between 0 and 3. After purging the data in this way, the number of counterparts with proper motion data in Chamaeleon I and II were 81 and 25 (corresponding to the 27% and 61% of the total number of stars included in our compiled lists), respectively. In addition, UCAC3 provided proper motions for 19 objects in Cha and 12 in η Cha, respectively (53% and 60% of the compiled lists, respectively). As for the background sources, the number of objects with available proper motions was 52 (59%). These samples were further cleaned by removing bad proper motion measurements and interlopers identified during the analysis (see Sect. 3 for details). The proper motion data for the members of the dark clouds, the foreground clusters, and the contaminants are presented in Tables 1, 2, and 3, respectively. In LJS11, a comparison of the cross-matching results of the Lupus catalogues with different astrometric catalogues (USNO-B1, SuperCOSMOS, PPMX) led to the conclusion that UCAC3 provided the best compromise between the number of counterparts (dependent among others on the magnitude limit of the catalogue) and the typical proper motion errors. For the present work, we further compared the analysis presented in Sect. 3.1 with the results obtained using the recently released PPMXL catalogue (Röser et al. 2010). Although this catalogue is deeper in magnitude than UCAC3, and thus included much more sources from our initial member lists, it was not possible to distinguish the different populations due to the larger errors in the proper motions for most of the objects. Thus, we concluded that UCAC3 is still the best option available for our study.
Complementary data
Radial velocities
Aiming at better insight into the kinematics of the members of the Chamaeleon associations, we combined the UCAC3 proper motions with radial velocity measurements from the literature Joergens & Guenther 2001;Biazzo et al. 2012) and from the SIMBAD database. We only considered measurements with errors better than about 30%. If several measurements were available for the same star, the average was computed. Known and suspected spectroscopic binaries were discarded.
Our final sample contains radial velocity information for 18 stars in Chamaeleon I and nine in Chamaeleon II. These data are summarized in Table 1. We also retrieved radial velocity data for 15 stars in the Cha association and four stars in the η Cha association, which are listed in Table 2. We also considered radial velocities from Covino et al. (1997) for the candidate members of the associations discussed in Sect. 4.1.
Photometry
We compiled multiwavelength photometry for the confirmed members and candidate members of the Chamaeleon associations discussed in this work. Most of the data were retrieved using the latest version of the VO SED Analyzer tool 3 (VOSA; Bayo et al. 2008, submitted). They include ultraviolet photometry from GALEX (Bianchi & GALEX Team 2000), optical photometry (B, V and I) from the Tycho-2 and DENIS catalogues DENIS Consortium 2005), near-infrared photometry from 2MASS (Skrutskie et al. 2006), and mid-infrared photometry from AKARI and WISE Cutri & et al. 2012).
More photometry data from the literature were retrieved through the VizieR catalogue service (Ochsenbein et al. 2000). For Chamaeleon II members and candidate members, we retrieved the compilation of optical and infrared photometry by Alcalá et al. (2008), which includes Johnson U-band data from Hughes & Hartigan (1992), Cousins R C I C and Sloan z photometry from Spezzi et al. (2007), and mid-and far-infrared photometry from IRAS and from the "cores to disks" (c2d) Spitzer legacy programme (Evans et al. 2003). 4 For Chamaeleon I members and candidate members, we retrieved the Spitzer/IRAC and MIPS 24 µm photometry provided by Luhman & Muench (2008a). Optical photometry from the survey by López Martí et al. (2004) was also included in the data compilation for this cloud, as well as IRAS fluxes for three objects. The photometric information is summarized in Table 4. 5
Kinematic groups towards Chamaeleon
3.1. Identification of proper motion groups Figure 1 shows the vector point diagram for the young sources seen towards the Chamaeleon sky area that are included in the UCAC3 astrometric catalogue. In Fig. 2 we can see the histograms of the total proper motion modulus for the Chamaeleon I and II members, for the members of the foreground associations, and for the known background contaminants. As expected, the young objects are clearly separated from the older background objects in these plots, and several distinct kinematic groups can be seen, which correspond to different spatial locations in the sky (see Fig. 3).
The spectroscopically confirmed members of Chamaeleon I (hereafter "the Cha I moving group") are clustered in the same area of the vector-point diagram, roughly around the position (−19, +2) mas/yr. The Chamaeleon II objects, on the other hand, are seen mostly clustered around the position (−25, −7) mas/yr, slightly shifted from the Chamaeleon I sources (Fig. 1). This seems to suggest that these objects (hereafter "the Cha II moving group") are moving, on average, faster than the Chamaeleon I stars in the tangential direction. This is also seen in the shift of the peak of the total proper motion distribution in Chamaeleon II with respect to Chamaeleon I (Fig. 2). If real, this result is interesting, because all estimations in the literature place Chamaeleon II at the same distance or further away than Chamaeleon I; hence, the difference in proper motion between both populations cannot be attributed to distance. However, the difference in mean proper motions is of little significance owing to the low number of Chamaeleon II sources with measured proper motions, and to the large dispersion.
The two foreground associations form in turn two distinct kinematic groups in the vector point diagram, which are clearly different from those in the dark clouds. The mean UCAC3 proper motions for these two groups are in good agreement with previously published values (Kharchenko et al. 2005;Wu et al. 2009). Interestingly, the total proper motion is very similar in both associations, as shown by the histograms in Fig. 2. As shown later in Sect. 3.3, the radial velocities of the stars in both clusters are also very similar.
Finally, with a few exceptions, the objects classified as background contaminants have lower proper motion values than any of the young moving groups and are clustered in a different location of the vector point diagram. This confirms our ability to distinguish members and non-members of the Chamaeleon associations with the UCAC3 data. Even so, the separation between the groups is not complete, and there is some overlapping between young stars and contaminants and between young stars from different groups.
A few spectroscopically confirmed members from both Chamaeleon clouds present UCAC3 proper motions that are clearly discrepant from those of their attributed associations. To check the reliability of the reported measurements, we followed the same procedure as in LJS11: Using Aladin, we visually inspected these objects, blinking and comparing two sets of images separated several decades in time, from the optical POSS and the near-infrared 2MASS surveys (Skrutskie et al. 2006). The 2MASS sources and the UCAC3 counterparts were superimposed on the images to assess the reliability of the cross-match. Besides, we used other available astro-photometric databases, namely the Astrographic Catalogue AC2000.2 (Urban et al. 1998) and the SuperCOSMOS Sky Survey (Hambly et al. 2001) to verify the peculiar proper motions, and the USNO-B Catalog (Monet et al. 2003) and the PPMX and PPMXL catalogues (Röser et al. 2008(Röser et al. , 2010 for comparison. This procedure showed that the clearly discrepant proper motions were mostly caused by errors in the measurements, mainly from centroiding errors or source confusion. A rough estimation of the proper motions of the objects, based on the comparison of the positions measured by the different catalogues, and with the motion of other nearby cloud members, suggests that most of them have proper motions in relatively good agreement with the corresponding moving groups.
The reliability of the proper motions is difficult to assess in some cases (e.g. HD 104237D and E) because these sources are part of visual binaries or multiples not always resolved in the images. And in a few cases (e.g. CW Cha), we could not test the proper motions in this way because SuperCOSMOS provides no data for the stars, or else the data were wrong. The sources with clearly wrong proper motions (7 from Chamaeleon I and 2 from Chamaeleon II) were removed from the vector point diagram of Fig. 1. They were also excluded from Tables 1 and 2. Objects with dubious proper motions (because they could not be tested or the checks were unconclusive) were not removed from the tables, but a flag was added.
We also identified four spectroscopically confirmed members (2MASS J11183572-7935548, CM Cha, Sz 60W, and CP-68 1388) whose UCAC3 proper motions seem to be right but do not agree with the means of their attributed associations; these interlopers are discussed in Sect. 3.4 below.
Stars with Hipparcos and Tycho measurements
Five Chamaeleon I and three Cha bona fide members from our UCAC3 samples have proper motion and parallax measurements provided by the latest Hipparcos reduction (van Leeuwen 2007). In addition, six Cha and two η Cha members have proper motions from the Tycho catalogue . These data are summarized in Table 5, together with the parallax, if available, and the distance derived from it. We used this information as a further check of the accuracy of the UCAC3 measurements.
We note that the proper motions of bright stars in UCAC3 are based on about 140 catalogues, including Hipparcos and Tycho. Colours as in Fig. 1. Also shown are the expected displacement of these objects within 10 5 Myr. The background image is a dust map by Schlegel et al. (Schlegel et al. 1998). Only objects with reliable proper motions are considered. Therefore, we generally expect good agreement between the UCAC3 measurements and the proper motions listed in Table 5, as is indeed found. However, problems may have affected the measurements in some of the images used by UCAC3, either due to saturation, nebulosity, or the presence of nearby fainter stars. This seems to be the case for two Chamaeleon I stars (HD 97300 and CW Cha). For these two objects, the Hipparcos proper motions are in much better agreement with membership in the Cha I moving group than the UCAC3 values.
In particular, the membership in the dark cloud of HD 97300 (the intermediate-mass star seen towards the northern core of Chamaeleon I) and its evolutionary status have sometimes been questioned. However, most authors agree that this is most likely a Herbig Ae/Be star similar to HD 97048 in the southern Chamaeleon I core. The proper motion seems to confirm that this star is a member of the Chamaeleon I cloud.
It is also interesting to note that HD 97048 has been included in the catalogue of potential young runaway stars by Tetzlaff et al. (2011), albeit with not very high probability (61%). However, we notice that the selection by these authors is only based on Hipparcos stars, which are relatively bright. Therefore, the sample used by Tetzlaff et al. (2011) included only a handful of Chamaeleon I members. Indeed, Table 5 shows that the proper motion of this object is lower, in modulus, than the rest of Chamaeleon I stars with Hipparcos or Tycho measurements, what could be interpreted as a hint for a different motion pattern. However, when the larger sample of cloud members provided by the UCAC3 catalogue are considered, HD 97048 does not stand up as having remarkably different kinematical properties from the rest (see also the discussion in Sect. 6.1). In view of our analysis, we think it unlikely that HD 97048 is actually running away from its birth place. There is also overall good agreement between the individual parallaxes of the stars in Table 5 and the usually quoted distances to the associations.The only possible exception is T Cha, whose Hipparcos parallax suggests a greater distance. However, given the large error, this estimation is still consistent with membership to the Cha cluster (but see also discussion in Sect. 5). Figure 4 shows two plots of the radial velocities versus the proper motion components of the stars. With a couple of exceptions, all the objects from the same proper motion group display similar radial velocities, and are therefore clustered in these diagrams. The radial velocity data therefore confirm that there is a physical relationship between these stars. However, given that all the associations have similar mean radial velocities, in the range 10-20 km/s, it is not possible to distinguish between members of the dark clouds and of the foreground associations on the basis of radial velocity alone.
Radial velocities of the members of the proper motion groups
The largest individual deviations from the mean group values are observed for EM Cha in the η Cha cluster (V r = 4.3±2 km/s) and GSC 9420-0948 in the Cha cluster (V r = 5 ± 2 km/s). The radial velocities of these stars differ by more than 3σ from the mean cluster values, which challenges their membership to the quoted associations. We therefore flag these objects in Table 2. Also, T Cha has a reported radial velocity discrepant in more than 2σ from the mean value for the Cha association (V r = 20.0 ± 2.0 km/s). The discrepancy may be related to the presence of an unresolved companion, as there is indeed one reported in the literature (Huélamo et al. 2011). We discuss this object further in Sect. 5. Table 6. Mean and weighted mean proper motion components and radial velocities for the moving groups identified towards the Chamaeleon sky area a
Group
Arithmetic means Weighted means
< µ α cos δ > < µ δ > < µ > < V r > < µ α cos δ > < µ δ > < µ > < V r > (mas/yr) (mas/yr) (mas/yr) (km/s) (mas/yr) (mas/yr) (mas/yr) (km/s) Cha I −19.5 ± 4.8 +1.7 ± 3.9 20.0 ± 4.8 14.6 ± 1.2 −20.7 ± 0.3 +0.9 ± 0.3 19.80 ± 0.12 14.02 ± 0.06 Cha II −23.5 ± 5.3
−7.7 ± 2.9 24.9 ± 5.3 12.9 ± 1.6 −23.9 ± 0.7 −7.8 ± 0.7 25.9 ± 0.9 14.5 ± 0.2 Cha −36.8 ± 11.7 −1.7 ± 13.2 39.8 ± 9.3 13.5 ± 2.6 −37.8 ± 0.5 −1.5 ± 0.7 38.5 ± 0.5 11.7 ± 0.2 η Cha −29.1 ± 4.9 +27.4 ± 4.9 40.4 ± 3.4 17.0 ± 1.4 −29.9 ± 0.5 +25.9 ± 0.5 38.69 ± 0.03 14 ± 1
Notes. a Interlopers and objects with uncertain proper motion measurements have been excluded from the calculations.
Interlopers
Once the proper motion lists are purged from inaccurate measurements, three true interlopers remain, namely one Chamaeleon I object (2MASS J11183572-7935548) and two Chamaeleon II objects (2MASS J11183572-7935548 and CM Cha), whose proper motions seem to be in better agreement with membership to the Cha group. The source 2MASS J11183572-7935548 was included as a Chamaeleon I member in the list of Luhman (2007). However, in a later work, Luhman et al. (2008) reclassified it as an Cha member on the basis of its UCAC2 proper motion. Our analysis of the UCAC3 data confirms their result. This object is therefore included in Table 2 as a probable member of the Chamaeleontis cluster.
CM Cha had been listed as a Chamaeleon II member by Spezzi et al. (2008). However, it has UCAC3 proper motion components µ α cos δ = −43.3 ± 10.6 and µ δ = 7.5 ± 9.1 mas/yr. These values are only marginally compatible with the Cha II moving group, and seem to be in better agreement with membership in the foreground Cha association. Unfortunately, AC2000.2 provides no data for this star, and the data provided by SuperCOSMOS are wrong, probably because this object appears blended with other star in the POSS image, so we cannot test its proper motion directly in the way explained above. In the literature we find a reported proper motion of (−60, 21) mas/yr for this object, with a quoted accuracy of 5-10 mas/yr (Teixeira et al. 2000). The PPMXL catalogue reports a similar value of (−62, 9) mas/yr. These measurements are quite discrepant from the UCAC3 measurement, even when the large errors are considered, but indicate in any case that CM Cha is probably located closer to the Sun than the Chamaeleon II members. On the other hand, there is spectroscopic evidence that the object is young: Spezzi et al. (2008) report an equivalent width for the Li I absorption line of 0.38 Å, which is similar to many other cloud members, but close to the lowest values measured by these authors. (Their range encompasses 0.35 to 0.61 Å, and typical values in star forming regions are in the range 0.3-0.8 Å.) They also report an Hα equivalent width of −29 Å, which is consistent with low accretion. This suggests that, though young, CM Cha is among the most evolved objects in the Spezzi et al. sample. On the other hand, the reported radial velocity for this star (15.9 ± 1.1 km/s; Torres et al. (2006) quite well agrees with the values measured for other stars in the Chamaeleon clouds, but also with the radial velocities reported for Cha members (see Sect. 3.3). The space motion of this star (as computed using the UCAC3 proper motions) also seems to be in better agreement with this cluster (see Sect. 5 below). Taking all this into account, we tentatively reclassify CM Cha as a member of the Cha young association, and list it accordingly in Table 2. We note, however, that if the proper motion reported by Teixeira et al. (2000) were correct, this star would probably be located even closer to the Sun than the Cha members, and thus could be an unrelated young star from the solar neighbourhood.
The situation is also ambiguous for Sz 60W, because the large error in µ δ makes its UCAC3 proper motion compatible with both the Cha and the Cha II moving groups. This star is part of a binary object with a separation of 3.6 , and the secondary has no UCAC3 counterpart. This may affect the proper motion measurement of the primary, if the system is not completely resolved in the images used by UCAC3 to compute the proper motion, or if source confusion has occurred at some stage. Indeed, secondaries at separations lower than 10 are often not resolved in UCAC3. More precise measurements are required to clarify the nature of Sz 60W, but we note that the spatial location of this source, within the dark cloud (in contrast with CM Cha that is located in the outskirts), suggests a physical connection with Chamaeleon II. For the time being, Sz 60W is listed in Table 1 as a member of this cloud.
In addition, we identify one object, namely CP-68 1388, whose reported UCAC3 proper motion differs from the rest of Chamaeleontis members, and is seen detached from moving group in the vector-point diagram of Fig. 1. This is due to the value of the proper motion in the δ direction (µ δ = 7.6 ± 2.3 mas/yr), because its µ α cos δ value (−35.8 ± 2.2 mas/yr) and its radial velocity (15.9 km/s) are in good agreement with those of other cluster members. This star was observed by Tycho, and the proper motions reported in Table 5 agree closely with the UCAC3 values (although the Tycho µ δ value is somewhat lower, µ δ = 4.70 mas/yr). The different proper motion and the spatial location of CP-68 1388, to the North of Chamaeleon I and more than 6 • away from the closest Chamaeleontis member, makes its membership to the cluster dubious. It is interesting, however, that this object is placed halfway between the and η Chamaeleontis cluster in the vector point diagram. As shown later in Sect. 5, the space velocities of CP-68 1388 agree well with other stars in both clusters. We therefore keep this object in the list of Cha members for the time being.
VW Cha also deserves some comments. This star has been listed in the literature both as a Chamaeleon I member (Luhman 2007) and as a member of the Cha group (Fernández et al. 2008). We find that both its spatial location within the Chamaeleon I southern cloud core and its UCAC3 proper motion (µ α cos δ = −19.7 ± 1.3 mas/yr, µ δ = −0.8 ± 3.4 mas/yr) are in better agreement with VW Cha belonging to the dark cloud. The proper motion values reported by SIMBAD (µ α cos δ = −23 mas/yr and µ δ = −7 mas/yr; Teixeira et al. (2000) agree well with our classification. This star also has a reported radial velocity (V r = 17.6 ± 3.3 km/s; Torres et al. 2006) that is slightly higher than the mean value for Chamaeleon I, but similar to the values reported for other member stars (see Sect. 3.3).
Final member lists
Once the proper motion lists are purged of incorrect measurements, and once the interlopers assigned to the associations they are likely to belong to, the definitive member lists for the Cha I, Cha II, Cha and η Cha contain 73, 22, 21, and 12 stars, respectively. These are the objects presented in Tables 1 and 2. These lists still contain objects with dubious measurements that could not be proved wrong, as discussed in Sect. 3.1.
After this analysis, we computed the means of the proper motion and values for the different kinematic groups, which are listed in Table 6. Interlopers and objects with wrong or dubious proper motion measurements were not considered in the calculations. Because of the large proper motion errors for some of the objects, we also computed the weighted means, which are also given in Table 6. The values of both the arithmetic and weighted means for each parameter are fully consistent with each other, within the estimated errors. Table 6 also lists the arithmetic and weighted mean radial velocities that we derive for three of the four groups. For the and η Cha clusters, the mean radial velocities are in good agreement with previously published values (Kharchenko et al. 2005;Wu et al. 2009).
Proper motions of candidate Chamaeleon members
In the next step, we made an attempt to confirm candidate members of Chamaeleon I and II on the basis of their proper motion. As in our Lupus study, we compared the proper motions of candidate members proposed in the literature with those of the confirmed members.
ROSAT stars
In the 1990s, the ROSAT mission unveiled a dispersed population of X-ray emitting stars towards the Chamaeleon area, but not confined to the dark clouds (Alcalá et al. 1995). Alcalá et al. (1997) provided the first spectroscopic evidence that these stars were young. Covino et al. (1997) performed radial velocity measurements for most of these stars, showing that the sample included objects belonging to different kinematical populations. This result was confirmed for a subsample of these objects by Frink et al. (1998) combining information from several proper motion catalogues. In addition, these authors used the Hipparcos parallaxes available for some of the stars to show that they were located at different distances(from 60 to about 200 pc). The different stellar populations identified by these authors were thus assigned to the Chamaeleon I cloud, the foreground Cha cluster, or to an undetermined field population. However, many stars remained unclassified owing to the lack of proper motions. Only 18 out of the 81 stars studied by Covino et al. (1997) are included in our compilations of members of the associations seen towards the Chamaeleon area. These objects are flagged in Tables 1 and 2.
To clarify the nature of the remaining ROSAT sources, we performed the same proper motion analysis on them described in Sect. 3.1. UCAC3 provides proper motion measurements for 38 of these objects, listed in Table 7. The radial velocities and evolutionary status of these stars according to Covino et al. (1997) are also indicated. We show the vector point diagram for these objects in Fig. 5, compared to the loci of the known young populations in the Chamaeleon area. We also show the spatial location of the ROSAT sources in Fig. 3.
The UCAC3 vector point diagram confirms that the ROSAT stars are a mixture of different kinematical populations. From their position on this diagram, some sources present completely discrepant proper motions from any of the moving groups identified in Sect. 3.1. The radial velocity measurements from Covino et al. (1997) confirm that these stars follow a completely different motion pattern with respect to the Chamaeleon associations. Other sources can be nicely assigned to one of the moving groups in the Chamaeleon sky area; however, a few of them are also discarded because their radial velocities, as provided by Covino et al. (1997), are in complete disagreement with membership to any of the associations, or because their location in the sky does not coincide with that of the group with similar proper motion (especially in the case of the dark clouds). All the ROSAT objects with discrepant proper motions and/or radial velocities are classified as zero-age main sequence (ZAMS) stars or stars of unclear nature by these authors. This left us with 13 stars (11 PMS stars and two stars of unclear nature) whose spatial location, proper motions and radial velocities (except for RXJ1150.4-7704, which has no available radial velocity measurement) all overlap with one of the Chamaeleon moving groups. We assigned two of these objects to the Cha I moving group. The eleven other sources are classified as candidate members of the Cha association (three objects) or the η Cha association (eight objects). We do not find any good Chamaeleon II candidate members. The membership of all these sources is indicated in Table 7.
In two cases, namely RXJ1123.2-7924 and RX J1158.5-7913, membership in a single association cannot be clearly attributed. The spatial location of these two objects (two PMS stars according to the study by Covino et al. (1997) seems more consistent with membership in the Cha moving group than in either of the dark clouds. However, this is not so straightforward from their UCAC3 proper motions. RXJ1123.2-7924 is located in the vector point diagram closer to the Cha II moving group than to the Cha members, but its proper motion values are still marginally consistent with membership in this group. Given its location in the sky, in the area between the Chamaeleon I and the Chamaeleon III clouds, and about 5 • from the known Chamaeleon II members, we tentatively assign this object to the Cha moving group, and list it in Table 7. RX J1158.5-7913 is seen towards the small cloud [DB2002b] G300.23-16.89, located halfway between the Chamaeleon I and II clouds and about 2.8 • south-east of the former. The star T Cha and other members and candidate members of the Cha association are also seen in this area. However, the proper motion components of RX J1158.5-7913 (µ α cos δ = −18.3 ± 3.8 mas/yr, µ δ = −3.1 ± 1.9 mas/yr) place this star in the overlapping area between Chamaeleon I and II members in the vector point diagram, but closer to the Chamaeleon I mean value given in Table 6. This looks completely incompatible with membership in the Cha cluster, even though the radial velocity reported by Covino et al. (V = 13.1 ± 2 km/s) is very close to the mean value we obtain for the Cha members (see Table 6). Because the proper motion and radial velocity look consistent with the mean values derived for the Cha I moving group, we tentatively assign RX J1158.5-7913 to this cloud for the time being. More accurate kinematic data will help clarify the nature of this object. to be included in the UCAC3 catalogue and, therefore, we cannot test their membership on the basis of their proper motion. There are, however, a few exceptions, which we discuss in this section. The UCAC3 proper motions for these candidates are listed in Table 8, and they are plotted in the vector point diagram shown in Fig. 5. provides a list of objects whose membership in Chamaeleon I is uncertain (cf. his Table 8), 21 of them lacking confirmation in later works. Only one of these objects has a valid UCAC3 counterpart, namely C1-14, an infrared source first quoted by Prusti et al. (1991), and also identified with the X-ray source CHX 15a. 6 This object is located within the northern cloud core of Chamaeleon I and, as shown in Fig. 5, its proper motion is compatible with the Cha I moving group. The spectral type of this source is badly constrained, but quoted in the range F0-A7, which would make it one of the most massive objects in the dark cloud, together with HD 97300 and HD 97048.
Other candidate members from the literature
In addition to the confirmed Chamaeleon II members, Spezzi et al. (2008) list a number of objects still lacking spectroscopic confirmation of membership to the cloud, but considered likely members on the basis of the multiwavelength photometric analysis reported in a previous work . Only two of these sources have UCAC3 counterparts, namely IRAS 12589-7646 (ISO-Cha II 89) and IRAS 12448-7650. The proper motion for the latter object clearly disagrees with membership in the Cha II moving group, and strongly suggests that IRAS 12448-7650 belongs to the background. As for IRAS 12589-7646, its proper motion (µ α cos δ = −21.2 ± 3.9 mas/yr, µ δ = 7.9 ± 3.9 mas/yr) places this source in the area of the Cha I moving group rather than Cha II. From our own check, however, the proper motion seems to be around (5, 1.1) mas/yr, which would place it in the same area as the background objects in the vector-point diagram. In view of these results, the classification of IRAS 12589-7646 as a Chamaeleon II member is dubious, and we conservatively flag it as "rejected" in Table 8. 6 In addition to C1-14, SIMBAD lists three other sources within 2 (the value of our cross-matching radius) from the UCAC3 position: CD-75 522, Glass R, and WKK F 32. The former has a reported proper motion suggesting ia foreground star. From the intercomparison of their properties as listed in SIMBAD, we are led to think that Glass R, WKK F 32, and C1-14 must actually be the same object. Vuong et al. (2001) performed a search for candidate members of Chamaeleon II based on DENIS I JK s photometry. Only five objects from their list are included in UCAC3, namely their sources [VCE2001] C18, C29, C64, X3, and X4. As seen in Fig. 5, their proper motions do not agree with membership in the Cha II moving group. Therefore, it seems unlikely that these objects belong to the Chamaeleon II dark cloud. The only possible exception is the X-ray source [VCE2001] X4, whose location in the vector point diagram is marginally compatible with those of the Chamaeleon II members. While deceiving, this result is not totally surprising, since spectroscopic studies of about 30 sources from the Vuong et al. (2001) selection also failed to confirm most of them as Chamaeleon II members (Barrado y Navascués & Jayawardhana 2004; Spezzi et al. 2008), thus showing the high contamination of the original sample (which amounted to 76 sources). None of the five DENIS sources discussed above with available proper motions was observed in these works. To date, there have only been five confirmed young stars from the Vuong et al. (2001)) list (see compiled list in Spezzi et al. 2008), but only one, [VCE2001] C61, is included in UCAC3 and listed in Table 1. The proper motion of this object is compatible with Chamaeleon II membership.
Space velocities
We computed the galactic space velocities for the stars (members and candidate members) with available radial velocity measurements. For this purpose we made use of an IDL routine retrieved from the IDL Astronomy Library, 7 originally written by W. Landsman and later modified by S. Koposov. The procedure follows the formulation in Johnson & Soderblom (1987), except for two details: the U component is positive toward the Galactic anticentre, and the Hipparcos transformation matrix is used. We modified the routine to include the computation of the velocity errors in the way prescribed by Johnson & Soderblom (1987).
Since most of the stars lack parallax measurements, we assumed the canonical distances to the associations for this calculation: 160 ± 15 pc and 178 ± 18 pc to Chamaeleon I and II, respectively (Whittet et al. 1997), and 111 ± 5 pc and 97 ± 5 pc to the Cha and η Cha clusters, respectively (Feigelson et al. Fig. 1. 2003). For T Cha and CV Cha, we used the distances inferred from their Hipparcos parallaxes as given in Table 5.
The resulting space velocities are listed in Table 9, as well as the velocities corrected for the velocity of the Sun with respect to the local standard of rest (LSR), assumed to be (U , V , W )=(−8.5, 13.38, 6.49) km/s. Figure 6 shows the histograms for the space velocities of the stars in the different associations.
Although this exercise is certainly affected by lots of uncertainties, especially in the stellar distances, the resulting space velocities confirm the membership of the stars in the different Chamaeleon associations to the young disk population, according to the prescriptions listed in Leggett (1992). The histograms displayed in Fig. 6 further confirm the presence of different kinematical populations towards Chamaeleon. In particular, the stars in the Chamaeleon I cloud seem to display, as a sample, a different motion pattern from the rest. This is most evident in the Galactic centre-anticentre direction (U component), especially when the velocities are corrected with respect to the LSR (upper right panel of Fig. 6): While the mean U LS R -component of the Cha I moving group is negative (i.e. pointing to the Galactic centre), for the rest of associations this component is positive (i.e. pointing to the Galactic anticentre).
The stars in the two foreground clusters seem to have very similar space velocities, so the differences in proper motion observed in the vector point diagram of Fig. 1 seem to be mostly related to the different spatial location of both groups in the projected sky (separated by about 8.5 • ). This fact, together with the similar ages and distances of both associations, hints at a con-nection between them, as already suggested in the literature (e.g. Feigelson et al. 2003). Indeed, the space velocities we obtain are in good agreement with published values (e.g. Zuckerman & Song 2004), and with the membership of both clusters to the so-called "Local Association" (or "Pleiades moving group"), a coherent kinematic stream of young stars (ages ∼20-150 Myr) with embedded clusters and associations (such as the Pleiades, α Per or Scorpius-Centaurus) first postulated by Eggen (1975). This is shown in Fig. 7, where we show the location of the stars from Table 9 in the (U, V) and (V, W) planes, together with the location of the mean space velocities for several nearby young moving groups (ages 20-600 Myr; Montes et al. 2001).
Interestingly, the stars in Chamaeleon II seem to have space velocities more similar to those of the foreground clusters than to the Chamaeleon I stars. However, the numbers are low, and the errors for most stars are large, as is the dispersion in the data. Better precision and statistics with larger samples are required to investigate the eventual relation of Chamaeleon II with the and η Cha associations, and to other young moving groups.
We close this section with some remarks on particular objects. First of all, we note that the usually quoted parallax for T Cha is 15 ± 3 mas, corresponding to a distance of only 66 ± 15 pc. This is the value included in the Hipparcos Main Catalogue (ESA 1997). However, the new reduction performed by (van Leeuwen 2007) provides a parallax of 6 ± 3 mas, corresponding to a distance of 160 ± 80 pc. The first value would place this star closer to the Sun than most of the members of the Cha association, and with quite discrepant space velocities, even when the errors are considered. In contrast, the most recent distance value is compatible, within the error, with the canonical distance to the cluster, and provides space velocities in relatively good agreement with T Cha being an Cha member. The same trend is observed if the space velocities are computed using the Hipparcos proper motions from Table 5 instead of the UCAC3 ones. Nevertheless, T Cha is seen detached from the group of Cha members in both planes represented in Fig. 7, and, given the large uncertainties in the distance (i.e. in the space velocities), the membership of this star to the Cha moving group cannot be assigned with certainty.
Second, we note that the space velocities of CM Cha show relatively good agreement with those of other Cha members, when the mean distance to this cluster is used in the calculation. If the distance to Chamaeleon II is used instead, the resulting space velocities are quite discrepant from those of the only confirmed Cha II member with available space velocities. This further supports the hypothesis that CM Cha is a member of the Cha association, as suggested in Sect. 3.4. Even so, it must be noted that the velocity component in the direction of the Galactic poles (W) of CM Cha is quite different from the rest of Cha members, as it is pointing towards the northern instead of the southern Galactic pole. We note, though, that a handful of η Cha cluster members are also located in the same area of the (V, W) plane.
We made the same check with the two ROSAT sources discussed at the end of Sect. 4.1. RX J1158.5-7913, a potential interloper from the Chamaeleon I cloud in the sky area of the Cha members, has space velocities that are fully consistent with membership in the dark cloud, if the mean distance to it is used in the calculation. In contrast, the resulting space velocities when the distance to Cha is used are not consistent with membership in this cluster. On the other hand, RXJ1123.2-7924, whose proper motion looks compatible with both the Cha cluster and the Chamaeleon II cloud, has space velocities fully compatible with the foreground association. Fig. 8. Enlargements of the map from Fig. 3 showing the Chamaeleon I and II areas in more detail (upper and lower panels, respectively). Figure 8 shows two enlargements of Fig. 3 in the location of the Chamaeleon I and II clouds. As noted in previous works, the sources belonging to the Cha I moving group tend to cluster around the two main cores containing the intermediate-mass stars HD 97300 (the northern core) and HD 97048 (the southern core). Luhman (2007) reports on a difference in isochronal ages of a few Myr between both cores, which he interprets as the result of two different bursts of star formation. We therefore checked whether any proper motion difference could be seen between the two cores. To this purpose, all the stars within a radius of 50 from one of the intermediate-mass star were defined as members of the corresponding core. This distance was chosen after inspecting the distance distributions from both intermediate-mass stars. At the UCAC3 precision, we found no obvious difference between the proper motions of both groups. No evident difference is seen either between the in-and the off-cloud populations, neither in proper motion moduli nor in proper motion directions. This seems to confirm that both samples belong to the same structure. It also suggests that the location of the outer sources is not the consequence of these objects being ejected from their parental birth sites.
Properties of the kinematic populations of the Chamaeleon dark clouds
Spatial location
We performed a similar analysis in Chamaeleon II, dividing the population in a northern group, located north-west of a dense core in the northern part of the cloud, and a southern group, located in and around a smallest core to the south. Again, we found no obvious difference between the proper motions of both groups. A comparison between the in-and off-cloud population is not possible in this case due to the low number of objects and the dubious proper motions of most of the sources located offcloud (leftmost sources in the lower panel of Fig. 8).
Physical parameters
As a next step, we investigated the possible correlations between the proper motions and the physical parameters of the objects. Luhman et al. (2008) performed an analysis of the spectral energy distributions (SEDs) of confirmed members of Chamaeleon I. A similar study of Chamaeleon II was presented by Alcalá et al. (2008) and Spezzi et al. (2008). These works provide a classification of the confirmed members of the dark clouds into class III (diskless) sources, class II (disk) sources and so-called "flat-spectrum sources", which are thought to be halfway between class I (envelope) sources and class II sources. They also provide estimations of the effective temperature and bolometric luminosity of the objects. In total, we compiled values of physical parameters for 47 stars in Cha I and 14 stars in Cha II, whose properties are summarized in Table 10. Only objects with reliable UCAC3 proper motions are included.
We used this information to look for correlations between the proper motions and the physical parameters of the objects. Figures 9 and 10 show the plots of the proper motion components versus T e f f and log L/L for the objects in both clouds. No trends are seen in the proper motion with either of the considered parameters. Since these properties are directly correlated with mass in young low-mass stars, this result suggests that the motion of the objects does not depend on their mass to the precision that we can prove with the UCAC3 data.
In Chamaeleon I, where this can be investigated, we do not find any correlation either between proper motion and the presence of a (primordial) circumstellar disk, or for objects with and without disks (class II and class III sources) separately. We note that similar studies in other star-forming regions also failed to find any correlation between the proper motion and the properties of the objects (e.g. Bouy & Martín 2009, LSJ11). More accurate proper motion measurements and better statistics are required to reveal any eventual trend with the physical properties of young low-mass stars in these regions.
Conclusions
We have performed a kinematic study of the Chamaeleon clouds based on UCAC3 proper motions and radial velocities from the literature. Our analysis has led to the following conclusions:
-With the UCAC3 data, we were able to separate the several distinct kinematic groups that coexist in the Chamaeleon sky area: the Chamaeleon I and II dark clouds, the young Cha and η Cha associations, and the field contaminants. -Compiled radial velocity data from the literature showed that all the young stars have similar radial velocities, regardless of the association the objects belong to. It is not possible to discern membership to the different groups seen towards Chamaeleon on the basis of radial velocity information alone. -The stars in Chamaeleon I and II may have different kinematical properties, but this result must be confirmed with more precise proper motions. If true, the Chamaeleon II members would be moving faster in the tangential direction than the Chamaeleon I members. Because they are located at the same or a further distance, this would question the physical connection between both clouds. -The kinematical analysis confirmed the membership of almost all the studied objects to the corresponding associa-tions. We only identified two possible interlopers from the Cha association in our list of Chamaeleon II members: CM Cha and Sz 60W. With the available data, we reclassified CM Cha as an Cha member. The nature of Sz 60W is more uncertain due to the big proper motion error, but given its spatial location and its isochronal age (2 Myr) we think it more likely that this object belongs to Chamaeleon II.
-We reanalysed a sample of X-ray detected stars towards the Chamaeleon sky area and confirmed that they constitute a mixture of different populations. After purging the sample of foreground and background contaminants and excluding the objects previously classified in the literature, we identified two stars as likely Chamaeleon I members, three as Cha members, and eight as η Cha members. We did not identify any Chamaeleon II members in this sample. -We checked the proper motions of candidate members of the Chamaeleon dark clouds from the literature, and confirmed one object, C1-14, as a probable member of Chamaeleon I. This would be one of the earliest type (i.e. most massive) stars in this dark cloud. In Chamaeleon II, we checked the proper motions of two candidate members listed by Spezzi et al. (2008) and of five candidate members from the list by Vuong et al. (2001), and found it unlikely that any of them belongs to the dark cloud. -For those objects with available radial velocities, we computed the corresponding Galactic space motions. Our results confirm the different spatial motion of Chamaeleon I from the foreground associations. The data also hint at a difference in spatial motion between Chamaeleon I and II, further suggesting different origins for the populations in both clouds, but this has to be confirmed with better datasets. Fig. 10. Same as Fig. 9 for Chamaeleon II.
-The space velocities for the and η Cha members and candidate members are in very good agreement with previous estimations in the literature, which identify these two clusters as part of the Local Association. Interestingly, the space motions of most of the Cha II members with available radial velocity measurements also look quite similar to those of the Local Association stars. It would be highly desirable to have a larger sample of stars with estimated space velocities in this dark cloud, in order to investigate the eventual relation between Chamaeleon II and the Local Association and to confirm or discard the different motion pattern from Chamaeleon I. -We investigated the relation between the proper motions and the published properties of the members and candidate members of Chamaeleon I and II, such as spatial location, the presence of circumstellar material, effective temperature, or luminosity, but we found no evident correlations. More accurate proper motions, combined with radial velocity information, may unveil possible hidden trends with the physical properties of the stars, undetectable with the precision of the proper motions used here.
Our work has shown how the use of kinematical information can complement photometric and spectroscopic data to constrain the selection of members of young star-forming clusters, minimizing the contamination of the samples. Currently available astrometric catalogues are deep enough to provide proper motions for a significant number of candidate members of nearby starforming regions precisely enough to disentangle true members, not only from background sources, but also from young stars in clusters overlapping in the line of sight. In a forthcoming paper (López Martí et al. in preparation), we will report on a search for new candidate members of the Chamaeleon clouds based on proper motion. Furthermore, the results obtained both in the present work and in our previous Lupus study illustrate the potential of using a VO methodology for analysing heterogeneous datasets in an efficient manner.
Most of the issues left unsolved in the present study are expected to get a definitive answer with the advent of the Gaia mission of the European Space Agency, whose launch is foreseen in 2013. Gaia will provide astrometric information with unprecedented precision for most of the stars in the samples studied here, remarkably improving our understanding of the kinematics, physical properties and formation process of the objects in the Chamaeleon associations.
Acknowledgements. This work was funded by the Spanish MICINN through grant Consolider-CSD2006-00070. It also benefitted from funding from the Spanish government through grants ESP2007-65475-C02-02, AYA2008-02156, AYA2010-21161-C02-02, and AyA2011-24052, and from the Madrid regional government through grant PRICIT-S2009ESP-1496. A. B. was co-funded under the Marie Curie Actions of the European Commission (FP7-COFUND).
This publication made use of VOSA, developed under the Spanish Virtual Observatory project supported by the Spanish MEC through grants AyA2008-02156 and AyA2011-24052. It greatly benefitted from the use of the SIMBAD database and the VIZIER Catalogue Service, both operated at the CDS (Strasbourg, France). We used the VO-compliant tools Aladin, developed at the CDS, and TOPCAT, currently developed within the AstroGrid project.
This publication makes use of data products from the Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the US National Aeronautics and Space Administration and National Science Foundation, and from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the US National Aeronautics and Space Administration.
Notes.
(1) Dubious proper motion.
(2) Discrepant proper motion form Hipparcos.
(3) ROSAT source studied by Covino et al. (1997).
(4) Spectroscopic binary.
(5) Sometimes classified in the literature as a member of the Cha cluster (see text for discussion).
(6) Possible interloper from the Cha cluster (see text for discussion). (2) Notes.
(1) Possible interloper (see text for discussion).
(2) ROSAT source studied by Covino et al. (1997).
(3) Classified in the literature as a Chameleon I or II member.
(4) Dubious proper motion.
(5) Discrepant radial velocity respect the mean group value. Notes: C=confirmed as probable kinematical member; R=rejected candidate (proper motion not consistent with membership).
Fig. 1 .
1UCAC3 vector point diagram for the objects seen towards the Chamaeleon sky area. Only spectroscopically confirmed members are considered. Sources belonging to different kinematic associations and to the background have been plotted with different symbols and colours. Objects with reliable and dubious proper motions have been indicated with solid and open symbols, respectively (see Sect. 3.1 for details). To help the eye, the moving groups have been labelled and marked with ellipses. Some interlopers (objects with discrepant proper motions that are more likely members of other associations) are identified (see discussion in Sect. 3.4).
Fig. 3 .
3Current spatial location of the members of the Chamaeleon associations (circles) and of the ROSAT sources discussed in Sect. 4.1 (squares).
Fig. 4 .
4Plots of radial velocity vs. proper motion components for stars in the Chamaeleon associations. Only objects with reliable proper motions are considered. Symbols and colours as inFig. 1.
Fig. 5 .
5Vector point diagram for Chamaeleon candidate members from the literature. The ellipses indicate the approximate location of the kinematical groups identified in Fig. 1.
Several works have provided lists of candidate members in the Chamaeleon clouds that still lack spectroscopic confirmation of youth. Unfortunately, most of the postulated objects are too faint 9 Belén López Martí et al.: Proper motions of young stars in Chamaeleon
Fig. 6 .
6Histograms of space velocities, uncorrected and corrected for the solar reflex motion with respect to the LSR (left and right panels, respectively) for stars in the different Chamaeleon associations. Symbols as inFig. 2.
Fig. 7 .
7(U, V) and (V, W) planes for stars in the different Chamaeleon association. The big crosses indicate the location of known nearby moving groups. Rest of symbols and colours as in
Fig. 9 .
9Proper motion components versus bolometric luminosity (left) and effective temperature (right) for Chamaeleon I members with and without disks (red circles and blue asterisks, respectively).
References: 1 )
1;Spezzi et al. (2008);Vuong et al. (2001) b
Table 5 .
5Hipparcos and Tycho proper motions and Hipparcos parallaxes for Chamaeleon stars The 1997 Main Hipparcos Catalogue provides a parallax π = 15.06 mas for this object (d∼ 66 pc).UCAC3
Hipparcos/Tycho
Name
µ α cos δ
µ δ
µ α cos δ
µ δ
π
d
Reference a Remarks b
(mas/yr)
(mas/yr)
(mas/yr)
(mas/yr)
(mas)
(pc)
Chamaeleon I cloud
HD 93828
-25.2 ± 0.9 -7.1 ± 1.0 −26.72 ± 0.99 −8.91 ± 0.84
5.70 ± 0.96
175 ± 30
H
HD 97048
-17.8 ± 0.8
1.5 ± 0.8 −18.64 ± 0.72 +2.18 ± 0.62
6.31 ± 0.62
158 ± 16
H
HD 97300
-17.7 ± 1.0 -2.7 ± 1.1 −21.63 ± 0.94 −0.72 ± 0.78
5.60 ± 0.88
178 ± 28
H
d
CV Cha
-21.3 ± 1.1
1.9 ± 1.0 −20.82 ± 2.95 +1.32 ± 2.78
6.97 ± 2.86
143 ± 59
H
CW Cha
-13.6 ± 3.2
4.2 ± 3.4 −26.05 ± 3.82 −6.46 ± 3.55
3.92 ± 3.56
255 ± 232
H
d
Chamaeleontis cluster
CP-68 1388
-35.8 ± 2.2
7.6 ± 2.3
−36.3 ± 1.9
+4.7 ± 1.6
T
T Cha
-41.2 ± 3.2
-6. ± 1.4 −33.45 ± 3.87 −9.36 ± 2.97 6.11 ± 2.96 c
164 ± 79
H
d
DW Cha
-40.4 ± 2.0
-2. ± 1.0
−41.7 ± 1.62
+0.13 ± 1.4 11.06 ± 1.52
90 ± 12
H
RX J1159.7-7601 -40.9 ± 2.1
-6. ± 1.8 −40.53 ± 1.66 −5.83 ± 1.49
9.89 ± 1.72
101 ± 18
H
HD 104467
-41.3 ± 1.0
-5. ± 1.0
−39.8 ± 1.7
−6.3 ± 1.6
T
HD 105923
-37.9 ± 2.5
-8. ± 1.6
−37.6 ± 1.7
−7.7 ± 1.6
T
RX J1239.4-7502 -40.7 ± 2.8
-13 ± 2.4
−45.9 ± 2.8
−13.6 ± 2.6
T
CD-69 1055
-40.9 ± 2.6
-17 ± 1.1
−40.6 ± 2.1
−19.1 ± 2.0
T
MP Mus
-40.4 ± 1.0
-20 ± 1.6
−40.8 ± 1.6
−23.3 ± 1.6
T
η Chamaeleontis cluster
EG Cha
−27.1 ± 1.0 25.3 ± 1.0
−29.5 ± 2.1
27.6 ± 1.6
T
HD 75505
−30.3 ± 1.4 25.0 ± 1.8
−29.9 ± 2.0
27.9 ± 1.7
T
Notes.
a H=Hipparcos (van Leeuwen 2007); T=Tycho (Høg et al. 1998)
b d=discrepant proper motion respect to UCAC3
c
Table 1 .
1Confirmed members of Chamaeleon I and II with UCAC3 proper motions (excluding wrong measurements and interlopers) Belén López Martí et al.: Proper motions of young stars in ChamaeleonTable 1.2MASS J
Other names
UCAC3
µ
α cos δ
µ
δ
V
r
R
I
J
H
K
Notes
(mas/yr)
(mas/yr)
(km/s)
(mag)
(mag)
(mag)
(mag)
(mag)
Chamaeleon I Cloud
10463795-7736035 HD 93828
025-022840
-25.2 ±
0.9
-7.1 ±
1.0
13.298 12.637
7.844 ±
0.02
7.655 ±
0.06
7.477 ±
0.02
10555973-7724399 T3A
026-024097
-5.7 ±
10.9
1.4 ±
7.0
13.4 ±
0.9 14.049 12.982 10.778 ±
1.27
9.832 ±
0.04
8.685 ±
0.02
(1)
10563044-7711393 SY Cha, T4
026-024109
-19.0 ±
4.7
7.4 ±
2.6
12.7 ±
0.1 13.339 12.421
9.969 ±
0.02
9.102 ±
0.02
8.631 ±
0.02
10574219-7659356 Sz 4, T5
027-026396
-9.2 ±
7.6
0.0 ±
7.7
16.5 ±
1.3 14.754 13.329 10.433 ±
0.02
9.563 ±
0.02
9.246 ±
0.02
(1)
10581677-7717170 T6
026-024168
-22.5 ±
6.9
1.7 ±
2.7
12.182 10.893
9.253 ±
0.03
8.406 ±
0.04
7.758 ±
0.03
10590108-7722407 SW Cha, T7
026-024197
-18.1 ±
1.8
4.6 ±
1.8
15.7 ±
1.2 13.496 12.891 10.135 ±
0.02
9.232 ±
0.02
8.616 ±
0.02
10590699-7701404 CR Cha, T8
026-024202
-20.0 ±
1.3
4.7 ±
1.9
14.9 ±
0.8 14.231 13.295
8.462 ±
0.03
7.815 ±
0.04
7.31 ±
0.02
11004022-7619280 T10
028-028320
-17.1 ±
8.2
-1.1 ±
5.5
16.027 14.561 11.859 ±
0.03 11.238 ±
0.02 10.871 ±
0.02
11011875-7627025 CHXR 9C
028-028370
-14.1 ±
4.8
-7.5 ±
13.4
13.305 11.663
10.08 ±
0.03
9.307 ±
0.03
8.994 ±
0.02
11022491-7733357 CS Cha, T11
025-023391
-20.2 ±
1.2
5.6 ±
2.7
14.7 ±
0.3 11.714 11.047
9.105 ±
0.03
8.452 ±
0.06
8.199 ±
0.03
(4)
11022610-7502407
030-032402
-17.9 ±
4.7
-4.5 ±
4.5
16.64
14.243 11.761 ±
0.03 11.115 ±
0.02 10.806 ±
0.02
11023265-7729129 CHXR 71
026-024313
-19.3 ±
2.7
7.0 ±
5.9
16.357 14.764 11.268 ±
0.03 10.459 ±
0.02 10.129 ±
0.03
11025504-7721508 T12
026-024325
-20.7 ±
3.2
8.5 ±
3.3
16.084 15.282 11.565 ±
0.03 10.858 ±
0.03 10.451 ±
0.02
11035682-7721329 CHXR 12
026-024352
-19.3 ±
2.2
-2.5 ±
2.2
15.505 13.686 10.796 ±
0.02
9.994 ±
0.03
9.712 ±
0.02
11040909-7627193 CT Cha, T14
028-028608
-18.9 ±
1.6
1.5 ±
1.6
15.1 ±
0.5 12.162 11.417
9.715 ±
0.02
8.944 ±
0.05
8.661 ±
0.02
11045100-7625240 CHXR 14N
028-028660
-18.1 ±
1.9
4.3 ±
1.9
13.463
12.7
10.543 ±
0.03
9.831 ±
0.03
9.598 ±
0.02
11045285-7625514 CHXR 14S
028-028661
-18.6 ±
2.5
1.1 ±
2.4
14.007
13.22
10.723 ±
0.02
9.981 ±
0.02
9.747 ±
0.02
11050752-7812063
024-022792 -19.2 ±
13.2
-4.7 ±
5.3
16.535 14.615 11.776 ±
0.02 11.096 ±
0.02 10.816 ±
0.02
11051467-7711290 Hn 4
026-024389
-23.7 ±
3.6
3.0 ±
3.5
16.425 15.139 10.944 ±
0.02 10.035 ±
0.03
9.608 ±
0.02
11052472-7626209
028-028699
-21.1 ±
3.0
0.4 ±
3.1
15.539 14.009 11.474 ±
0.02 10.741 ±
0.02 10.522 ±
0.02
11055261-7618255 T20
028-028735
-21.9 ±
2.0
0.7 ±
1.9
13.367 12.807 10.312 ±
0.02
9.593 ±
0.02
9.339 ±
0.02
11061540-7721567 T21
026-024403
-19.2 ±
1.0
5.0 ±
1.6
17.0
7.597 ±
0.02
6.828 ±
0.05
6.419 ±
0.02
11064346-7726343 T22
026-024409
-21.4 ±
3.5
-0.5 ±
3.7
16.252 14.776 10.813 ±
0.03
9.788 ±
0.03
9.392 ±
0.02
11064510-7727023 CHXR 20
026-024411
-17.9 ±
2.4
5.6 ±
7.1
15.535 14.781 10.184 ±
0.02
9.197 ±
0.02
8.88 ±
0.02
11065906-7718535 T23
026-024418
-18.3 ±
3.4
1.7 ±
4.6
16.199 15.368 11.204 ±
0.03 10.423 ±
0.02 10.003 ±
0.02
11071148-7746394 CHXR 21
025-023408
-22.3 ±
3.9
3.0 ±
3.7
16.162 14.964 11.084 ±
0.03 10.083 ±
0.02
9.657 ±
0.02
11071915-7603048 T25
028-028831
-22.1 ±
2.8
-2.0 ±
2.9
14.781 13.468 10.961 ±
0.02 10.092 ±
0.02
9.769 ±
0.02
11072040-7729403 [LES2004]ChaI 448, ISO 99
026-024421
-20.8 ±
2.8
2.8 ±
2.5
16.241 15.254 11.134 ±
0.02 10.547 ±
0.02 10.259 ±
0.02
11072074-7738073 DI Cha, T26
025-023410
-15.8 ±
1.6
-5.1 ±
1.7
13.5 ±
0.6
7.819 ±
0.03
6.942 ±
0.04
6.217 ±
0.02
11072825-7652118 VV Cha, T27
027-026870
-20.0 ±
1.9
-1.8 ±
1.9
15.4 ±
1.3 14.455 13.013 10.663 ±
0.03
9.905 ±
0.03
9.519 ±
0.02
11074366-7739411 T28
025-023415
-19.8 ±
2.2
-0.3 ±
3.9
15.719
14.11
10.165 ±
0.02
8.975 ±
0.02
8.258 ±
0.03
11075588-7727257 CHXR 28
026-024437
-28.8 ±
3.3
6.1 ±
3.4
14.907 13.734
9.215 ±
0.03
8.052 ±
0.03
7.691 ±
0.03
11075792-7738449 T29
025-023419
-26.7 ±
5.7
13.7 ±
7.7
15.038 13.725
9.502 ±
0.03
7.919 ±
0.04
6.83 ±
0.02
11080148-7742288 VW Cha, T31
025-023420
-19.7 ±
1.3
-0.8 ±
3.4
15.1 ±
0.1 14.915
13.48
8.703 ±
0.03
7.637 ±
0.04
6.962 ±
0.03
(5)
11080329-7739174 HD 97048, T32
025-023421
-17.8 ±
0.8
1.5 ±
0.8
7.267 ±
0.02
6.665 ±
0.05
5.941 ±
0.03
11081509-7733531 T33A
025-023422
-2.1 ±
5.6
-4.4 ±
5.5
14.726 13.679
8.638 ±
0.03
7.729 ±
1.27
6.876 ±
1.27
(1)
11081648-7744371 T34
025-023423
-17.0 ±
3.1
-1.2 ±
4.1
15.975 14.716 11.198 ±
0.03 10.339 ±
0.03 10.024 ±
0.02
11083905-7716042 T35
026-024448
-19.7 ±
3.0
1.2 ±
3.1
15.822 17.073 11.173 ±
0.03
9.904 ±
0.02
9.109 ±
0.02
11084069-7636078 CHXR 33
027-026898
-17.3 ±
2.6
-0.8 ±
6.0
15.044
13.57
10.558 ±
0.02
9.658 ±
0.02
9.276 ±
0.02
11085367-7521359 PU Car
030-033240
-5.5 ±
5.2
2.9 ±
5.1
10.98 ±
0.03
10.072 ±
0.03
9.556 ±
0.03
(1)
11085464-7702129 T38
026-024450
-18.4 ±
3.1
6.1 ±
2.7
16.279 14.393 11.208 ±
0.03 10.175 ±
0.02
9.457 ±
0.02
11091769-7627578 CHXR 37, RXJ1109.4-7627
028-028916
-22.3 ±
1.8
-0.8 ±
9.9
13.1 ±
2.0 15.032 14.147
9.998 ±
0.02
9.044 ±
0.02
8.701 ±
0.02
(3)
11092379-7623207 VZ Cha, T40
028-028926
-20.9 ±
3.3
1.7 ±
2.6
14.7
13.513 11.686
10.44 ±
0.03
9.214 ±
0.02
8.242 ±
0.04
11094006-7628391 CHXR 40
028-028940
-22.4 ±
6.8
4.0 ±
2.6
15.058 13.993 10.072 ±
0.02
9.23 ±
0.02
8.961 ±
0.02
11095003-7636476 HD 97300, T41
027-026916
-17.7 ±
1.0
-2.7 ±
1.1
7.641 ±
0.03
7.347 ±
0.06
7.149 ±
0.02
(2)
11095407-7629253 T43
028-028953
-25.5 ±
6.4
2.5 ±
5.8
16.736 15.258 11.295 ±
0.03 10.003 ±
0.02
9.254 ±
0.02
11095873-7737088 T45
025-023440
-20.9 ±
1.7
-0.6 ±
2.3
13.104 12.147
9.835 ±
0.03
8.768 ±
0.05
7.97 ±
0.02
Table 2 .
2Kinematic members of the foreground and η Chamaeleontis clusters (excluding wrong measurements and interlopers)2MASS J
Other names
UCAC3
µ
α cos δ
µ
δ
Rv
R
I
J
H
K
Notes
(mas/yr)
(mas/yr)
(km/s)
(mag)
(mag)
(mag)
(mag)
(mag)
Chamaeleontis cluster
10574936-6913599 CP-68 1388
042-083475
-35.8 ±
2.2
7.6 ±
2.3
15.9
13.941 12.741
8.484 ±
0.03
8.009 ±
0.05
7.792 ±
0.02 (1)
11080148-7742288 VW Cha, RXJ1108.1-7742
025-023420
-19.7 ±
1.3
-0. ±
3.4
14.915
13.48
8.703 ±
0.03
7.637 ±
0.04
6.962 ±
0.03 (2)
11183572-7935548
Cha 13
021-021275
-46.6 ±
4.5
4.4 ±
2.6
14.201 12.374
10.50 ±
0.03
9.891 ±
0.02
9.62 ±
0.02
(3)
11493184-7851011 DZ Cha
023-025652
-43.4 ±
2.9
-8. ±
1.8
12.2 ±
2.0 11.997 10.513
9.449 ±
0.02
8.721 ±
0.05
8.486 ±
0.02 (2)
11571348-7921313 T Cha, RXJ1157.2-7921
022-024424
-41.2 ±
3.2
-6. ±
1.4
20.0 ±
2.0 14.981 13.477
8.955 ±
0.03
7.86 ±
0.05
6.954 ±
0.02 (2)
11582681-7754450 GSC 9415-2676
025-027033
-38.5 ±
2.9
-7. ±
2.2
13.0 ±
2.0
10.34 ±
0.03
9.715 ±
0.03
9.436 ±
0.02
11582816-7754294 DW Cha, RXJ1158.5-7754
025-027034
-40.4 ±
2.0
-2. ±
1.0
13.1 ±
2.0 14.084 12.967
8.219 ±
0.03
7.556 ±
0.04
7.404 ±
0.02 (2)
11594226-7601260 RXJ1159.7-7601
028-033151
-40.9 ±
2.1
-6. ±
1.8
13.1 ±
2.0
10.32
9.978
9.14 ±
0.03
8.469 ±
0.04
8.304 ±
0.03 (2)
12000829-7811395 HD 104237D
024-027122
-42.9 ±
7.1
27. ±
7.1
9.615 ±
0.06
8.744 ±
0.11
8.124 ±
0.21 (4)
12000931-7811424 HD 104237E
024-027125
7.3 ±
5.2
5.9 ±
5.2
9.104 ±
1.27
8.254 ±
1.27
7.488 ±
0.02 (4)
12013915-7859168 HD 104467, RXJ1201.7-7859
023-026618
-41.3 ±
1.0
-5. ±
1.0
10.0 ±
2.0
7.263 ±
0.03
6.967 ±
0.04
6.848 ±
0.02 (2)
12020369-7853012 GSC 9420-0948, RXJ1202.1-7853 023-026647
-41.0 ±
3.5
-4. ±
1.7
5.0 ±
2.0 14.944 13.629
9.215 ±
0.02
8.456 ±
0.04
8.307 ±
0.02 (2, 5)
12043615-7731345 GSC 9416-1029, RXJ1204.6-7731 025-027530
-38.8 ±
1.5
-2. ±
2.4
10.4 ±
2.0 12.879 11.306
9.765 ±
0.02
9.125 ±
0.02
8.881 ±
0.02 (2)
12074597-7816064 2MASS J12074597-7816064
024-027765 -53.4 ±
14.5
-15 ±
7.0
15.827 14.409
11.55 ±
0.02
10.98 ±
0.03
10.67 ±
0.02
12113815-7110360 HD 105923, HIP 105923
038-075763
-37.9 ±
2.5
-8. ±
1.6
14.2
8.506
5.818
7.674 ±
0.02
7.308 ±
0.04
7.175 ±
0.03
12194369-7403572 RXJ1219.7-7403, GSC 9239-1495 032-049103
-42.4 ±
4.2
-6. ±
1.9
13.1 ±
2.0
12.48
11.386
9.746 ±
0.03
9.048 ±
0.03
8.858 ±
0.02 (2)
12202177-7407393 GSC 9239-1572, RXJ1220.4-7407 032-049216
-40.6 ±
2.6
-5. ±
1.8
18.0 ±
2.0 11.836 10.565
9.26 ±
0.02
8.607 ±
0.03
8.367 ±
0.02 (2)
12392124-7502391 RXJ1239.4-7502, CD-74 712
030-043659
-40.7 ±
2.8
-13 ±
2.4
13.1 ±
2.0
9.537
8.346
8.434 ±
0.02
7.953 ±
0.03
7.777 ±
0.02 (2)
12582559-7028490 CD-69 1055
040-110662
-40.9 ±
2.6
-17 ±
1.1
12.8
9.558
7.938
8.184 ±
0.03
7.702 ±
0.06
7.545 ±
0.03
13021351-7637577 CM Cha, IRAS 12584-7621
027-035419 -43.3 ±
10.6
7.5 ±
9.1
12.04
10.936
10.02 ±
0.02
9.165 ±
0.03
8.515 ±
0.02 (3)
13220753-6938121 MP Mus
041-134775
-40.4 ±
1.0
-20 ±
1.6
11.6 ±
0.2
8.277 ±
0.03
7.641 ±
0.02
7.293 ±
0.02
η
Chamaeleontis cluster
08365623-7856454 EG Cha, RECX 1
023-014604
−27.1 ±
1.0
25.3 ±
1.0
18.0 ±
2.0
9.359
8.525
8.155 ±
0.02
7.498 ±
0.05
7.338 ±
0.02 (2)
08413703-7903304 EH Cha, RECX 3
022-014105
−29.5 ±
2.2
33.6 ±
2.2
15.595 13.998 10.349 ±
0.02
9.647 ±
0.02
9.415 ±
0.02
08414471-7902531 HD 75505
022-014112
−30.3 ±
1.4
25.0 ±
1.8
7.148
6.776
7.059 ±
0.03
6.987 ±
0.04
6.928 ±
0.02
08422372-7904030 EI Cha, RECX 4
022-014146 −27.9 ±
10.7
17.9 ±
2.4
14.993 13.508
9.535 ±
0.02
8.779 ±
0.06
8.615 ±
0.02
08422710-7857479 EK Cha, RECX 5
023-014912
−30.8 ±
2.7
35.1 ±
4.7
14.229 12.114 10.777 ±
0.02 10.099 ±
0.02 9.855 ±
0.02
08423879-7854427 EL Cha, RECX 6
023-014921
−35.7 ±
2.3
25.0 ±
2.4
13.023
11.39
10.232 ±
0.03
9.584 ±
0.02
9.29 ±
0.02
08430723-7904524 EM Cha, RECX 7
022-014185
−29.7 ±
1.9
25.3 ±
3.1
4.3 ±
2.0
8.42 ±
0.02
7.758 ±
0.03
7.635 ±
0.03 (2, 5)
08431857-7905181 ET Cha, RECX 15
022-014197 −14.2 ±
20.9 36.0 ±
28.9
15.144 13.742 10.505 ±
0.03
9.834 ±
0.02
9.431 ±
0.02
08441637-7859080 EN Cha, RECX 9
023-015007
−29.5 ±
2.4
27.5 ±
2.7
13.795 11.695
10.26 ±
0.03
9.668 ±
0.03
9.335 ±
0.02
08443188-7846311 EO Cha, RECX 10
023-015019
−31.6 ±
2.0
25.2 ±
1.9
15.0 ±
2.0 11.659 10.287
9.653 ±
0.02
8.919 ±
0.06
8.732 ±
0.02 (2)
08470165-7859345 EP Cha, RECX 11
023-015143
−30.7 ±
1.2
26.0 ±
1.2
10.311
9.438
8.729 ±
0.02
8.025 ±
0.06
7.655 ±
0.04
08475676-7854532 EQ Cha, RECX 12
023-015181
−32.1 ±
1.8
26.8 ±
1.8
18.0 ±
2.0 12.076 10.534
9.323 ±
0.02
8.683 ±
0.08
8.41 ±
0.03
Table 3 .
3Background objects.2MASS J
Other names
UCAC3
µ α cos δ
µ δ
(mas/yr)
(mas/yr)
10452780-7715335
026-023750
1.2 ± 2.2
7.2 ± 4.6
11001473-7757103
025-023372 -10.7 ± 4.5
15.1 ± 2.7
11014949-7412268
032-038265
-4.8 ± 3.9
1.7 ± 3.2
11035902-7743349 ISO 35
025-023396
1.5 ± 3.8
-4.4 ± 2.9
11045890-7656550
027-026781
-6.5 ± 4.0
16.1 ± 16.8
11050937-7706578
026-024388
-8.6 ± 1.6
6.0 ± 1.7
11051798-7706565
026-024390
5.3 ± 3.0
8.3 ± 2.1
11052489-7620373
028-028700
-1.4 ± 3.4
0.3 ± 1.7
11052850-7639489
027-026798 -11.3 ± 2.3
2.3 ± 3.1
11053587-7638034
027-026805
-8.9 ± 3.8
-2.0 ± 1.8
11054972-7640462
027-026814
-0.1 ± 1.8
4.9 ± 1.8
11055683-7656438
027-026817 -15.1 ± 1.9
11.0 ± 1.9
11063479-7645404
027-026840 -43.9 ± 2.1
15.5 ± 3.5
11063841-7612032
028-028775
-1.9 ± 2.6
4.1 ± 2.6
11064235-7632450
027-026847
-4.0 ± 3.1
2.2 ± 3.2
11064274-7636124
027-026848
-5.9 ± 1.8
9.8 ± 2.4
11064586-7625317
028-028787
-0.8 ± 1.8
2.3 ± 3.4
11065986-7651181
027-026858
-4.4 ± 1.6
1.5 ± 1.6
11070350-7631443
027-026861
-1.0 ± 2.9
7.4 ± 2.9
11070380-7635440
027-026862 -72.5 ± 1.4
30.2 ± 1.3
11073280-7748594 ISO 104
025-023413
1.5 ± 2.4
1.9 ± 2.3
11074763-7711156
026-024434
-7.1 ± 1.6
2.1 ± 1.6
11075699-7741558 ISO 118
025-023418 -21.2 ± 6.7
-1.3 ± 5.3
11081069-7637424
027-026887
-0.3 ± 2.2
-3.1 ± 2.2
11081916-7656525 ISO 139
027-026889 -13.1 ± 2.5
3.0 ± 3.4
11083897-7614457
028-028890
-3.9 ± 1.9
2.4 ± 1.9
11085527-7704502 ISO 166
026-024451 -13.4 ± 1.6
6.3 ± 1.6
11085813-7646392 ISO 170
027-026908 -34.0 ± 2.5
12.3 ± 2.5
11090332-7700495 ISO 172
026-024453 12.7 ± 7.4
9.6 ± 7.4
11090367-7707456 ISO 175
026-024454
5.3 ± 1.9
0.6 ± 1.9
11090766-7618144
028-028907
-4.3 ± 1.4
6.2 ± 1.4
11091379-7637531 ISO 183
027-026913
4.3 ± 2.0
10.9 ± 2.1
11092068-7713591 ISO 188
026-024457
-5.3 ± 2.7
-5.6 ± 2.8
11092482-7642073 ISO 191
027-026915
-5.5 ± 3.5
1.2 ± 3.5
11095119-7658568 ISO 213
027-026917 -82.0 ± 1.0
17.6 ± 1.1
11095465-7625333
028-028954
1.8 ± 3.0
1.1 ± 3.1
11095822-7659147 [LM04]ChaI 616, ISO 229 027-026919
-3.2 ± 2.6
4.8 ± 2.6
11102819-7626164 ISO 242
028-028979 -22.1 ± 1.3
5.0 ± 1.3
11102990-7736064 ISO 243
025-023448
-3.0 ± 4.1
7.9 ± 4.4
11103067-7637186 ISO 245
027-026923
-6.3 ± 3.3
0.5 ± 2.8
11105038-7631440 ISO 255
027-026928
-6.9 ± 5.9
8.4 ± 2.0
11105215-7709286 ISO 257
026-024459
-1.4 ± 1.8
-7.6 ± 1.8
11110383-7706261 ISO 263
026-024460
6.2 ± 3.2
1.7 ± 4.4
11110670-7717594 ISO 264
026-024461
6.3 ± 2.2
10.7 ± 2.2
11110714-7722056 ISO 265
026-024462
-9.3 ± 4.8
-0.3 ± 2.9
11111062-7657205 ISO 267
027-026938
-4.9 ± 2.1
8.5 ± 2.1
11111680-7628573
028-029023 7.3 ± 15.6
4.8 ± 21.1
11131188-7647392
027-027030
-6.7 ± 2.5
-3.3 ± 2.4
11173079-7627078
028-029481
-2.6 ± 1.2
5.8 ± 1.0
11195060-7628492
028-029694
-7.5 ± 6.9
0.1 ± 4.3
11213026-7805240
024-023578
-2.8 ± 2.5
2.6 ± 4.5
11225703-7649560
027-027766
-8.7 ± 1.1
6.4 ± 2.6
Table 7 .
7Kinematic properties of ROSAT stars not included in our initial member lists Notes. aCovino et al. (1997) b This work (a question mark if dubious, see text)RXJ
2MASS J
Other names
UCAC3
µ α cos δ
µ δ
V r
Evolutionary Group b
(mas/yr)
(mas/yr)
(km/s)
status a
1129.2-7546
11291261-7546263
029-033458
−24.5±1.5
−2.1±1.5
11.4±2.0
PMS
Cha I
1158.5-7913
11583429-7913175
022-024487
−18.3±3.8
−3.1±1.9
13.1±2.0
PMS
Cha I?
1123.2-7924
11225562-7924438
Cha 14
022-022912
−29.8±2.1 −19.0±1.7
10.0±2.0
PMS
Cha?
1150.4-7704
11502829-7704380
Cha 19
026-027460
−40.3±1.2
−8.3±1.5
PMS
Cha
1216.8-7753
12164593-7753333
025-028496
−37.5±1.6
−8.0±1.6
14.0±2.0
PMS
Cha
0850.1-7554
08500540-7554380
029-020107
−15.9±1.1
33.0±0.9
15.5±2.0
PMS
η Cha
0902.9-7759
025-017479
−34.6±1.7
22.4±1.7
11.0±2.0
PMS
η Cha
0915.5-7608
09152912-7608471
028-021037
−29.3±2.0
18.9±1.4
21.0±2.0
PMS
η Cha
0928.5-7815
09282116-7815352 HD 82879
024-017885
−25.4±1.0
18.0±1.5
16.8±2.0
?
η Cha
0935.0-7804
09345604-7804193
024-018258
−28.6±1.3
19.2±1.3
14.0±2.0
PMS
η Cha
0951.9-7901
09515069-7901377 HD 86356
022-017379
−26.4±0.8
38.5±0.8
12.2±2.0
PMS
η Cha
1001.1-7913
10010873-7913074
022-017956
−25.0±1.5
38.7±2.3
12.2±2.0
PMS
η Cha
1140.3-8321
11401658-8321003
014-014885
−41.7±1.1
27.7±1.1
10.5±2.0
?
η Cha
1017.9-7431
10175369-7431172
031-030277
−6.5±2.1
6.6±1.3 104.0±2.0
ZAMS
Field
1035.8-7859
10354856-7858583
023-020966
−65.9±5.5
6.2±3.0
70.8±2.0
ZAMS
Field
1044.6-7849
10443523-7849244
023-021597
−87.6±2.3
25.9±4.3
−8.4±2.0
ZAMS
Field
0842.4-8345
08422284-8345248
013-008994
−49.5±1.3
91.2±1.2
−8.0±2.0
?
Field
0849.2-7735
08491110-7735585
025-016705
−11.5±1.2
16.3±1.1
−3.4±2.0
?
Field
0853.1-8244
08530528-8243597
015-010451
0.5±0.9 −14.4±1.1
7.7±2.0
?
Field
0936.3-7820
09361783-7820417 HD 84075
024-018347
−73.4±0.8
50.1±0.8
3.2±2.0
?
Field
0946.9-8011
09465051-8011352
020-015930
−16.8±1.9
−3.9±3.9
5.4±2.0
?
Field
0952.7-7933a 09531375-7933285 HD 86588
021-016442
−13.5±0.9
2.0±0.9
2.0±2.0
?
Field
1007.7-8504
10072939-8504329 DR Oct, HD 89499 010-008728
−566.0±
395.0± −67.1±2.0
?
Field
1009.6-8105
10093502-8105509
018-015349
−15.4±4.5
12.0±3.8
8.8±2.0
?
Field
1014.2-7636
10140807-7636327
027-023785
−47.2±1.7
30.6±3.6
8.0±2.0
?
Field
1014.4-8138
10142048-8138423
017-014665
−36.5±1.3
17.4±2.0
6.6±2.0
?
Field
1048.9-7655
10485467-7655450
027-025899
−95.6±1.4
45.7±2.1 −19.9±2.0
?
Field
1207.9-7555
12075118-7555159
029-037031 −156.7±2.6
−1.5±2.0
−3.4±2.0
?
Field
1209.8-7344
12094282-7344414
033-053854
−9.3±2.3
−3.2±1.3
1.0±2.0
?
Field
1217.4-8035
12172694-8035069 HD 106772
019-022395
−0.4±1.5 −10.4±1.4 −13.0±2.0
?
Field
1220.6-7539
12203437-7539286
029-038424 −111.2±1.5
4.0±1.0
6.0±2.0
?
Field
1223.5-7740
12232906-7740514 HD 107722
025-029020
−66.8±1.2
11.9±1.4
10.4±2.0
?
Field
1225.3-7857
12251340-7857347
023-028311
−27.0±1.0 −16.4±1.6
−6.7±2.0
?
Field
1233.5-7523
12332981-7523112 HD 109138
030-042831
−94.4±1.1
16.3±1.0
14.0±2.0
?
Field
1307.3-7602
13072289-7602360
028-039286
−19.1±1.5
1.7±1.2 −62.8±2.0
?
Field
1349.2-7549E 13491293-7549475
029-049787
−61.7±1.0 −32.3±1.7
1.0±2.0
?
Field
Table 8 .
8UCAC3 proper motions for candidate members of the Chamaeleon dark clouds from the literature2MASS J
Other names
UCAC3
µ
α cos δ
µ
δ
R
phot
I
phot
J
H
K
Ref. (a)
Notes (b)
(mas/yr)
(mas/yr)
mag
mag
mag
mag
mag
Chamaeleon I Cloud
11094646-7628575 C1-14
028-028944 -27.4 ±
2.0
10.2 ±
1.0 13.977 13.111 8.432 ±0.04
8.035 ±0.06
7.854 ±0.02
1
C
Chamaeleon II Cloud
12482571-7706366 IRAS 12448-7650
026-031454
-5.4 ±
1.8
4.2 ±
1.7
13.507
9.872
6.789 ±0.02
5.674 ±0.02
5.177 ±0.02
2
R
12540028-7624251 [VCE2001] C18
028-038152
-1.4 ±
4.4
0.6 ±
5.9
14.894 11.308 8.194 ±0.03
7.139 ±0.05
6.606 ±0.02
3
R
12551447-7622005 [VCE2001] C29
028-038240 -6.6 ±
10.2
7.1 ±
4.6
15.36
12.238 10.053±0.02 9.063 ±0.02
8.673 ±0.02
3
R
13024772-7702399 IRAS 12589-7646, Iso-Cha II 89 026-031795 -21.2 ±
3.9
7.9 ±
9.3
13.179
9.554
7.196 ±0.02
6.199 ±0.03
5.716 ±0.02
2
R
13024919-7705448 [VCE2001] X3
026-031797 -24.6 ±
5.3
5.9 ±
6.0
15.889 15.483 14.315±0.03 13.839±0.04 13.597±0.04 3
R
13030903-7700328 [VCE2001] X4
026-031820 -11.3 ±
5.8 -13.7 ±
4.0 13.597 12.954 12.465±0.02
12.1 ±0.03
11.991±0.03 3
R
13074107-7805434 [VCE2001] C64
024-032467 -12.9 ±
5.4
0.8 ±
14.7 15.605 13.072 10.458±0.02 9.118 ±0.02
8.641 ±0.02
3
R
Notes.
arXiv:1302.7167v1 [astro-ph.SR] 28 Feb 2013 Belén López Martí et al.: Proper motions of young stars in Chamaeleon
The VO is a project designed to provide the astronomical community with the data access and the research tools needed to enable the exploration of the digital, multiwavelength universe resident in the astronomical data archives. http://www.ivoa.net
TOPCAT is an interactive graphical viewer and editor for tabular data that allows the user to examine, analyse, combine, and edit astronomical tables. http://www.star.bris.ac.uk/∼mbt/topcat/
http://svo2.cab.inta-csic.es/svo/theory/vosa/ 4 The c2d data are now retrievable through VOSA(Bayo et al. submitted).
Available online.
Belén López Martí et al.: Proper motions of young stars in Chamaeleon
Belén López Martí et al.: Proper motions of young stars in Chamaeleon
http://idlastro.gsfc.nasa.gov/ftp/pro/astro/gal uvw.pro 10 Belén López Martí et al.: Proper motions of young stars in Chamaeleon
Notes.(1) The distance inferred from the Hipparcos parallax(Table 5) was used in the calculation.(2) Assuming the distance to Cha for this star, the following space velocities are obtained (U, V, W) = (2.0 ± 2.0, −14.5 ± 2.0, −7.2 ± 1.2) km/s (3) If the distance to Cha II is assumed for this object, the resulting space velocities are (U, V, W) = (20.6 ± 8.0, −34.5 ± 5.5, 3.9 ± 7.7) km/s (4) With a parallax π = 15.06 ± 3.31, we obtain for this star (U, V, W) = (1.5 ± 7.0, −21.5 ± 8.1, −10.2 ± 3.7) km/s (5) Assuming the distance to Cha II for this star, we obtain(Spezzi et al. (2008)
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|
[] |
[
"Higher metal abundances do not solve the solar problem",
"Higher metal abundances do not solve the solar problem"
] |
[
"G Buldgen \nDépartement d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland\n\nSTAR Institute\nUniversité de Liège\nLiègeBelgium\n",
"P Eggenberger \nDépartement d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland\n",
"A Noels \nSTAR Institute\nUniversité de Liège\nLiègeBelgium\n",
"R Scuflaire \nSTAR Institute\nUniversité de Liège\nLiègeBelgium\n",
"A M Amarsi \nTheoretical Astrophysics\nDepartment of Physics and Astronomy\nUppsala University\nBox 516751 20UppsalaSweden\n",
"N Grevesse \nSTAR Institute\nUniversité de Liège\nLiègeBelgium\n\nCentre Spatial de Liège\nUniversité de Liège\nAngleur-LiègeBelgium\n",
"S Salmon \nDépartement d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland\n"
] |
[
"Département d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland",
"STAR Institute\nUniversité de Liège\nLiègeBelgium",
"Département d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland",
"STAR Institute\nUniversité de Liège\nLiègeBelgium",
"STAR Institute\nUniversité de Liège\nLiègeBelgium",
"Theoretical Astrophysics\nDepartment of Physics and Astronomy\nUppsala University\nBox 516751 20UppsalaSweden",
"STAR Institute\nUniversité de Liège\nLiègeBelgium",
"Centre Spatial de Liège\nUniversité de Liège\nAngleur-LiègeBelgium",
"Département d'Astronomie\nUniversité de Genève\nChemin Pegasi 51CH-1290VersoixSwitzerland"
] |
[] |
Context. The Sun acts as a cornerstone of stellar physics. Thanks to spectroscopic, helioseismic and neutrino flux observations, we can use the Sun as a laboratory of fundamental physics in extreme conditions. The conclusions we draw are then used to inform and calibrate evolutionary models of all other stars in the Universe. However, solar models are in tension with helioseismic constraints. The debate on the "solar problem" has hitherto led to numerous publications discussing potential issues with solar models and abundances. Aims. Using the recently suggested high-metallicity abundances for the Sun, we investigate whether standard solar models, as well as models with macroscopic transport reproducing the solar surface lithium abundances and analyze their properties in terms of helioseismic and neutrino flux observations. Methods. We compute solar evolutionary models and combine spectroscopic and helioseismic constraints as well as neutrino fluxes to investigate the impact of macroscopic transport on these measurements. Results. When high-metallicity solar models are calibrated to reproduce the measured solar lithium depletion, tensions arise with respect to helioseismology and neutrino fluxes. This is yet another demonstration that the solar problem is also linked to the physical prescriptions of solar evolutionary models and not to chemical composition alone. Conclusions. A revision of the physical ingredients of solar models is needed in order to improve our understanding of stellar structure and evolution. The solar problem is not limited to the photospheric abundances if the depletion of light elements is considered. In addition, tighter constraints on the solar beryllium abundance will play a key role in the improvement of solar models.
|
10.1051/0004-6361/202245448
|
[
"https://export.arxiv.org/pdf/2212.06473v1.pdf"
] | 254,591,496 |
2212.06473
|
002343385435fa37e1486ca1d6cbb04e97b7ed5e
|
Higher metal abundances do not solve the solar problem
13 Dec 2022 December 14, 2022 October, 2022
G Buldgen
Département d'Astronomie
Université de Genève
Chemin Pegasi 51CH-1290VersoixSwitzerland
STAR Institute
Université de Liège
LiègeBelgium
P Eggenberger
Département d'Astronomie
Université de Genève
Chemin Pegasi 51CH-1290VersoixSwitzerland
A Noels
STAR Institute
Université de Liège
LiègeBelgium
R Scuflaire
STAR Institute
Université de Liège
LiègeBelgium
A M Amarsi
Theoretical Astrophysics
Department of Physics and Astronomy
Uppsala University
Box 516751 20UppsalaSweden
N Grevesse
STAR Institute
Université de Liège
LiègeBelgium
Centre Spatial de Liège
Université de Liège
Angleur-LiègeBelgium
S Salmon
Département d'Astronomie
Université de Genève
Chemin Pegasi 51CH-1290VersoixSwitzerland
Higher metal abundances do not solve the solar problem
13 Dec 2022 December 14, 2022 October, 2022Astronomy & Astrophysics manuscript no. ArticleMaggSun: helioseismology -Sun: oscillations -Sun: fundamental parameters -Sun: abundances
Context. The Sun acts as a cornerstone of stellar physics. Thanks to spectroscopic, helioseismic and neutrino flux observations, we can use the Sun as a laboratory of fundamental physics in extreme conditions. The conclusions we draw are then used to inform and calibrate evolutionary models of all other stars in the Universe. However, solar models are in tension with helioseismic constraints. The debate on the "solar problem" has hitherto led to numerous publications discussing potential issues with solar models and abundances. Aims. Using the recently suggested high-metallicity abundances for the Sun, we investigate whether standard solar models, as well as models with macroscopic transport reproducing the solar surface lithium abundances and analyze their properties in terms of helioseismic and neutrino flux observations. Methods. We compute solar evolutionary models and combine spectroscopic and helioseismic constraints as well as neutrino fluxes to investigate the impact of macroscopic transport on these measurements. Results. When high-metallicity solar models are calibrated to reproduce the measured solar lithium depletion, tensions arise with respect to helioseismology and neutrino fluxes. This is yet another demonstration that the solar problem is also linked to the physical prescriptions of solar evolutionary models and not to chemical composition alone. Conclusions. A revision of the physical ingredients of solar models is needed in order to improve our understanding of stellar structure and evolution. The solar problem is not limited to the photospheric abundances if the depletion of light elements is considered. In addition, tighter constraints on the solar beryllium abundance will play a key role in the improvement of solar models.
Introduction
The Sun plays a key role in stellar physics. Thanks to the numerous high-quality observations available, its acts as both a laboratory of fundamental physics and a calibrator for stellar evolution models (Christensen-Dalsgaard 2021). However, the modelling of solar structure is still a subject of debate, fueled in part by uncertainties in the solar chemical composition. Striking disagreements exist with helioseismic constraints, when adopting the "low-metallicity" compositions presented in Asplund et al. (2005) and Asplund et al. (2009), and more recently in Asplund et al. (2021) and Amarsi et al. (2021) -hereafter, the "solar problem" (see e.g. Basu & Antia 2008;Buldgen et al. 2019a, and references therein). These lowmetallicty compositions are based on spectroscopic analyses of the solar disc-centre intensity, utilising three-dimensional radiative-hydrodynamic simulations of the solar atmosphere (3D RHD) and where available non-local thermodynamic equilibrium radiative transfer (non-LTE).
Recently, Magg et al. (2022) presented a new spectroscopic analysis of the solar chemical composition. In contrast to the papers above, their analysis was based on the solar disc-integrated flux (in which, spectral lines form higher up in the atmosphere, and thus are potentially more sensitive to non-LTE effects as well as to blends that can be exacerbated by the extra broadening due to rotation), and 1D model atmospheres (derived from spatial and temporal averages of 3D RHD models). They inferred a "high-metallicity" chemical composition similar to the canonical 1D LTE compilations of Grevesse & Noels (1993) and Grevesse & Sauval (1998), hereafter GN93 and GS98. Using this high-metallicity composition and standard solar models, they found better agreement with helioseismic constraints. They concluded that the solar problem is solved, without need for any revision of fundamental physical ingredients.
We consider whether the solar problem is in fact solved. We show that a revision of abundances simply affects the magnitude of the corrections required in solar models, but does not validate a physical prescription for modelling the Sun. Recently, Eggenberger et al. (2022) demonstrated that stepping away from the standard solar models is required to simultaneously reproduce both helioseismic inversions of the solar internal rotation and the spectroscopic measurement of the lithium photospheric abundance; a result that was foreseen by Christensen-Dalsgaard et al. (1996). In the last decades, numerous studies have investigated the implications of revising the physical ingredients of solar models, such as accretion, massloss, transport of chemicals, convection, opacities, dark matter, dark energy and nuclear reactions (see e.g. Guzik et al. 2001;Brun et al. 2002a;Guzik et al. 2005;Guzik & Mussack 2010;Vinyoles et al. 2015;Spada et al. 2018;Zhang et al. 2019;Yang 2019;Bellinger & Christensen-Dalsgaard 2022;Saltas & Christensen-Dalsgaard 2022;Yang 2022, and refs therein) and multiple generations of standard and non-standard models were computed (e.g. Serenelli 2010;Vinyoles et al. A&A proofs: manuscript no. ArticleMagg 2017;Christensen-Dalsgaard et al. 2018a;Jørgensen & Weiss 2018, amongst other).
We focus on the changes induced by reproducing the lithium abundance with various parametric diffusion coefficients using solar models built with the Magg et al. (2022) abundances, similarly to Richard et al. (1996). We discuss the impact on neutrino fluxes and helioseismic constraints of reproducing all available spectroscopic constraints and how improved beryllium abundances will help us pin down the properties of macroscopic transport at the base of the solar convective zone.
Standard and non-standard solar models
We present our set of solar evolutionary models and discuss the changes of properties induced by the inclusion of macroscopic transport at the base of the solar convective zone (BCZ). We used both Standard Solar Models (SSM) and non-standard models, computed with the Liège Stellar Evolution Code (Scuflaire et al. 2008). We used the following physical ingredients: the solar abundances were the Magg et al. (2022) solar abundances, the opacity were the OP opacities (Badnell et al. 2005), supplemented at low temperature by opacities of Ferguson et al. (2005), the FreeEOS equation of state was adopted (Irwin 2012), and the nuclear reaction rates were from Adelberger et al. (2011).
The first model, "Model Std", is a SSM, including microscopic diffusion following Thoul et al. (1994), with the screening coefficients of Paquette et al. (1986) and the effects of partial ionization. As seen from Table 1, the results for this setup are almost identical to those illustrated in Table 6 of Magg et al. (2022). The slight differences in the positioning of the BCZ are due to differences in the prescription for microscopic diffusion that can alter the metallicity profile close to the BCZ (see also Table 1 in Buldgen et al. (2019b) for an illustration).
All other models in Table 1 were computed with an additional parametric diffusion coefficient, aiming at reproducing the solar lithium abundance (Wang et al. 2021). To do so, we assumed that the initial lithium and beryllium abundances were provided by the meteoritic values (Lodders et al. 2009). We start with Model DT R that is fitted to reproduce the transport induced by the combined effect of shear-induced turbulence, meridional circulation and the magnetic Tayler instability (Eggenberger et al. 2022). We adopted a simple parametrization in power law of the density following Proffitt & Michaud (1991)
D T (r) = D ρ BCZ ρ(r) n ,(1)
with D a constant in cm 2 s −1 , ρ the local value of the density, ρ BCZ the value of the density at the BCZ of the model and n a fixed constant number. In their recent paper, Eggenberger et al. (2022) found that the behaviour of rotating models could be well reproduced with n = 1.3, which is used in Model DT R . In addition, we also tested different values of n, from 2 to 5 to investigate the impact of reproducing the solar lithium depletion in solar models. We note that the depletion is highly significant, as lithium is reduced by 2.29 dex with an uncertainty on the current photospheric abundance of 0.06 dex. Therefore it constitutes an important constraint to consider when studying solar evolution.
The results of the calibrated models with parametric transport are provided in Table 1. The BCZ position in the model is altered when macroscopic mixing is included, thus we investigated in model DT R +Ov the effect of including adiabatic overshooting to replace the transition in temperature gradient at the helioseismic value of 0.713 ± 0.001R ⊙ (Basu & Antia 1997). The following sections will discuss the results obtained and their consequences for the current issues in solar modelling.
Lithium, helium, neutrinos and convective envelope position
The evolution of the photospheric lithium and beryllium abundances as a function of the solar age are shown in Fig. 1, with A(X) = log(X/H) + 12. A first result confirmed here is that SSMs are unable to reproduce the lithium depletion in the Sun. As mentioned in Proffitt & Michaud (1991), additional mixing at the BCZ is required to reproduce the observed depletion.
The calibration of this mixing was done for various values of n, changing the value of D simutaneously to reproduce the lithium abundance. Each leads to a different beryllium depletion at the age of the Sun. In the right Panel of Fig. 1, we show that a higher value of n leads to a lower depletion of beryllium at the age of the Sun. This is a direct consequence of the higher burning temperature of beryllium at ≈ 3.5×10 6 K. A higher n value leads to a steeper diffusion coefficient and thus a less efficient transport of beryllium down to ≈ 3.5 × 10 6 K, despite the recalibration of the factor D to reproduce the lithium depletion. Thus beryllium acts as a strong constraint on the functional form of the macroscopic transport coefficient at the BCZ and is thus of highest importance to constrain the physical origin of the lithium depletion.
The final beryllium abundance will also be affected by the presence or absence of strong adiabatic overshooting at the BCZ. The inclusion of this additional mixing has strong consequences for solar models. First, the position of the BCZ is significantly shifted by about 0.002 R ⊙ (hence 2σ) with respect to the position obtained in the SSM framework. This shift in the position of the BCZ is also linked to a small change in the mass coordinate of the convective zone. It is actually due to a change in the metallicity profile close to the BCZ. When only microscopic diffusion is included in the models, the competition between pressure diffusion and thermal diffusion leads to a drop in diffusion velocities close to the BCZ that induces an accumulation of elements (see Baturin et al. 2006, for a discussion).
This is particularly visible for the Z profile of SSMs. The accumulation of metals close to the BCZ locally increases the opacity, as the major contributors at the BCZ are Oxygen, Iron and Neon. Therefore, the temperature gradient is locally steepened, leading to a deeper convective zone. Once macroscopic transport is included in the models, this local maximum is erased, which leads to a shallower convective zone. This result is obtained for all models including transport with a n ≤ 5. From Fig 2, can see that n should be much higher than 5 to avoid this issue, meaning that D should be increased significantly to compensate and the transport should probably behave almost instantaneously in a shallow region, coming closer to the behaviour of the model with overshooting and showing tension with the results of young solar-like stars in open clusters (taken from Dumont et al. 2021). Meanwhile, beryllium would provide a definitive answer on the behaviour of the mixing and thus its physical origin.
A second consequence of the inclusion of macroscopic mixing is the reduction of the efficiency of gravitational settling. As the microscopic diffusion velocities drop very fast in the radiative interior, if mixing is included, settling is inhibited, as seen in Fig. 2. To reproduce the solar luminosity and radius at the solar age, a higher initial hydrogen abundance is required and thus the core metallicity at the age of the Sun is reduced.
Due to the lower core metallicity, neutrino fluxes are significantly affected, as shown in Table 2. The pp flux, φ pp , is unchanged as it is mostly related to reproducing the solar luminosity. The beryllium and boron neutrino fluxes, φ Be and φ B , are significantly affected by the inclusion of macroscopic transport, as the core metallicity, temperature and temperature gradient are not high enough and steep enough to reproduce the observations. We refer the reader to Salmon et al. (2021) for an in-depth discussion in the case of SSM using various physical ingredients as well as to the seminal works by Bahcall et al. Appel et al. (2022) ment with the lower end of the helioseismically inferred interval in the SSM, it is now in marginal agreement or disagreement with the upper end of the interval in the models with transport. This impacts the agreement with the first adiabatic exponent profile, Γ 1 = ∂ ln P ∂ ln ρ | S , in the solar convective envelope determined from helioseismology. As seen from Vorontsov et al. (2013) (e.g. Fig 6 and 7), a Helium mass fraction above 0.25 is never in agreement with a metal mass fraction above 0.012, whatever the equation of state used. Further investigations of the properties of the Γ 1 profile in the solar envelope with the most recent equations of state are required to restrict the Y − Z interval allowed in solar models, as it would provide strong constraints on the transport of chemical elements during solar evolution.
Helioseismic inversions
The second point to investigate is the impact of transport on helioseismic inversions. As shown in Magg et al. (2022), the increased metallicity brought back the models to the level of agreement of the SSMs of the 1990s. As shown in the left panel of Table 1, the red cross indicating the seismic value and the 1σ interval from Basu & Antia (1995). Right panel: metallicity profile as a function of normalized radius for the evolutionary models of Table 1. GN93 models in Buldgen et al. (2019b). This is no surprise as the Magg et al. (2022) abundances are almost the same as the GS98 abundances. They are however in strong disagreement with the surface lithium abundance.
The models based on high-metallicity abundances that are able to reproduce the surface lithium abundance draw a more complex picture. As illustrated in previous works (e.g. Brun et al. 2002b;Christensen-Dalsgaard et al. 2011, 2018b, macroscopic transport reduces the glitch at the BCZ, but also leads to increased discrepancies in the deeper radiative zone and in the core. While still small, these remain significant. From the entropy proxy inversions, we see that macroscopic transport does not improve the agreement with helioseismic inversions. The entropy plateau in the convective zone is now too high, and some deviations are seen in the radiative zone, particularly at the BCZ due to the less steep temperature gradient induced by the erasement of the metallicity peak resulting from microscopic diffusion. This further emphasizes the tension between the models including macroscopic transport and constraints such as neutrino fluxes and helioseismic inversions.
Overall, the results simply illustrates the importance of putting helioseismic inversions in their context. The final sound speed and entropy proxy profiles will be the result of the whole calibration procedure and therefore of all the ingredients entering the model computation.
The only way to directly constrain the solar metallicity using helioseismology is by using Γ 1 , which is highly sensitive to the equation of state of the solar plasma and has a tendency to favour a low metallicity (Vorontsov et al. 2013;Buldgen et al. 2017a) value in the most recent studies. Therefore, determining solely the solar metallicity from helioseismic data is currently restricted by the data available (further analyses using the sets of modes by Reiter et al. (2020) are required) and by the reliability of the solar equation of state in the deep convective envelope.
Conclusion
In this study, we discussed in details the implications of the physical ingredients of solar models regarding the conclusion of the solar problem in light of the high-metallicity compo-sition presented by Magg et al. (2022). We have shown that, as it stands, further constraints on the solar beryllium abundance are key to better understand the properties of macroscopic transport at the base of the solar convective zone. We have also shown that standard solar models are in complete disagreement with the observed solar lithium abundance. However, our work only tackles one aspect of the issue, following previous works (e.g. Richard et al. 1996;Brun et al. 2002a;Christensen-Dalsgaard et al. 2018a;Zhang et al. 2019), and numerous other processes can be demonstrated to impact the results of comparisons between solar models and helioseismic constraints, as shown in the extensive litterature on the subject (see Christensen-Dalsgaard 2021, and refs therein).
One solution to erase this disagreement is to introduce macroscopic transport at the base of the envelope, which could originate in the combined actions of rotation and magnetic instabilities, as shown in Eggenberger et al. (2022). We tested various parametrizations and their impact on various key constraints of solar models. We find the following points:
-The inclusion of macroscopic transport leads to tension with the helioseismic Y CZ value, as well as with the Γ 1 profile inversions of Vorontsov et al. (2013). Looking at the results of Table 1 in Buldgen et al. (2019b), using other key ingredients such as the OPAL opacities or the SAHA-S equation of state might lead to a higher Y value that would further increase these tensions. -The r CZ value is significantly impacted by the inclusion of macroscopic transport to reproduce the photospheric lithium abundance, leading to a shift by about 2σ of its position. We also note that it is significantly affected by the physical prescription used for microscopic diffusion. -Some tension is also observed with the lithium abundance of young solar twins in stellar clusters. Repositioning r CZ at the helioseismic value using adiabatic overshooting further increases this tension. -By changing the initial conditions of the calibration, macroscopic transport also leads to a significantly lower Z abundance in the core, leading to tension with the CNO Borexino neutrino fluxes and disagreement at 1σ of the beryllium and boron fluxes. Table 1. Right panel: relative differences in entropy proxy between the Sun and the evolutionary models of Table 1.
-Sound speed and entropy proxy profiles in the solar interior are overall worse when macroscopic transport is included.
In their work, Magg et al. (2022) conclude that "While SSMs offer an incomplete description of the physics in the solar interior, current results alleviate the need for more complex physics, such as accretion of metal-poor material (Serenelli et al. 2011), energy transport by dark matter particles (Vincent et al. 2015), revision of opacities (Bailey et al. 2015), enhanced gravitational settling and other effects (Guzik & Mussack 2010).". We find this assessment to be incorrect, as reproducing the lithium abundance with various parametrizations essentially reduces and even destroys the agreement they find for some constraints.
The situation for solar models using the Asplund et al. (2021) abundances is going to be slightly different. Including transport helps reproducing the helium abundance in the CZ (Yang 2019(Yang , 2022Eggenberger et al. 2022) and agrees with Γ 1 inferences by Vorontsov et al. (2013). However, it does not solve the issues with the positioning of the BCZ, the sound speed profile and the neutrino fluxes. Those will likely require revision of physical ingredients. Ultimately, the only difference between the Asplund et al. (2021) models and the Magg et al. (2022) models is the required magnitude of these revisions.
Therefore, we find that the need for additional physics such as the effects of rotation, of which we have a clear map (e.g. Schou et al. 1998), planetary formation, of which the impact can be measured (Kunitomo & Guillot 2021;Kunitomo et al. 2022) and opacity revision, which can be quantified and tested (e.g. Ayukov & Baturin 2017;Buldgen et al. 2019b) is not alleviated. On the contrary, the discussion related to the metal content of the Sun cannot be separated from other fundamental physical ingredients entering solar models such as the equation of state, the radiative opacities, the transport of chemicals or the evolutionary history. In this context, further refining fundamental physics computations, experimental setups and helioseismic techniques are paramount to provide the most accurate description of the internal structure of the Sun and use it as a stepping stone for the modelling of solar-like stars. Indeed, the need for additional transport at the base of convective envelopes is found in solar twins (Deal et al. 2015) and F-type stars (Verma & Silva Aguirre 2019) and often considered a key ingredient to determine reliable stellar ages. Further exploring such processes for our very own Sun thus appears a natural and meaningful endeavour.
Fig. 2 :
2Fig 3, we reach similar conclusions. In the right panel of Fig 3, we illustrate the inversion result for the entropy proxy, S 5/3 = P/ρ 5/3 , introduced in Buldgen et al. (2017b), which provides a complementary view on solar models. The level of agreement of the standard model is excellent and similar to that of the GS98 Left panel: evolution of the Helium abundance in the convectine zone Y CZ for the evolutionary models of
Fig. 3 :
3Left panel: relative differences in squared adiabatic sound speed between the Sun and the evolutionary models of
Table 1 :
1Parameters of the solar evolutionary modelsName (r/R) BCZ (m/M) CZ
Y CZ
Model Std
0.7148
0.9760
0.2443
Model DT R
0.7182
0.9769
0.2522
Model DT 2
0.7178
0.9768
0.2513
Model DT 3
0.7177
0.9768
0.2510
Model DT 4
0.7176
0.9768
0.2507
Model DT 5
0.7174
0.9767
0.2503
Model DT R + Ov
0.7133
0.9759
0.2516
Fig. 1: Left panel: lithium photosperic abundance as a function of age for the models of Table 1, the red cross is the solar value and the dark blue and teal crosses are the values for young solar-like stars in open clusters from Dumont et al.we
-1
0
1
2
3
1
1.5
2
2.5
3
1
2
3
4
0.9
1
1.1
1.2
1.3
1.4
(2021). Right panel:
beryllium photospheric abundance as a function of age for the models of Table 1, the red cross indicates the solar value from
Asplund et al. (2009).
Another consequence of the inhibition of settling seen from the left panel ofFig 2 isthat the mass fraction value of Helium in the CZ is significantly changed. While it was in marginal disagree-; Bahcall & Serenelli (2005) and to Villante & Serenelli
(2021) for a review. Similarly, the CNO neutrino flux is signifi-
cantly reduced and now in disagreement with the observed value
from Borexino (Appel et al. 2022). Therefore, additional pro-
cesses such as planetary formation (Kunitomo & Guillot 2021;
Kunitomo et al. 2022) or modification to key physical ingre-
dients such as opacity or the electronic screening formalism
(Mussack & Däppen 2011; Mussack 2011) might be required to
reproduce the neutrino observations when the lithium depletion
is reproduced, particularly the fluxes from the Borexino experi-
ment.
Table 2 :
2Neutrino fluxes of the evolutionary modelsName
φ(pp)
φ(Be)
φ(B)
φ(CNO)
Model Std
5.96
4.89
5.42
6.11
Model DT R
5.98
4.73
5.04
5.53
Model DT 2
5.98
4.74
5.07
5.57
Model DT 3
5.98
4.75
5.08
5.59
Model DT 4
5.98
4.75
5.09
5.61
Model DT 5
5.98
4.76
5.10
5.63
Model DT R + Ov
5.98
4.74
5.06
5.56
O-G21 1 5.97 +0.0037
−0.0033
4.80 +0.24
−0.22
5.16 +0.13
−0.09
−
Borexino 2
6.1 +0.6
−0.7
4.99 +0.13
−0.14
5.68 +0.39
−0.41
6.6 +2.0
−0.9
Note: 1 Orebi Gann et al. (2021) , 2 Borexino Collaboration et al.
(2018), Borexino Collaboration et al. (2020),
We acknowledge support by the ISSI team "Probing the core of the Sun and the stars" (ID 423) led by Thierry Appourchaux.
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[] |
[
"Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces Universal Quantum Computation in Waveguide QED using DFS 2",
"Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces Universal Quantum Computation in Waveguide QED using DFS 2"
] |
[
"V Paulisch \nMax-Planck-Institute of Quantum Optics\nHans-Kopfermann-Strasse 185748GarchingGermany\n",
"H J Kimble \nNorman Bridge Laboratory of Physics\n12-33\n\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n",
"A González-Tudela [email protected] \nMax-Planck-Institute of Quantum Optics\nHans-Kopfermann-Strasse 185748GarchingGermany\n"
] |
[
"Max-Planck-Institute of Quantum Optics\nHans-Kopfermann-Strasse 185748GarchingGermany",
"Norman Bridge Laboratory of Physics\n12-33",
"California Institute of Technology\n91125PasadenaCAUSA",
"Institute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"Max-Planck-Institute of Quantum Optics\nHans-Kopfermann-Strasse 185748GarchingGermany"
] |
[] |
The interaction of quantum emitters with one-dimensional photon-like reservoirs induces strong and long-range dissipative couplings that give rise to the emergence of so-called Decoherence Free Subspaces (DFS) which are decoupled from dissipation. When introducing weak perturbations on the emitters, e.g., driving, the strong collective dissipation enforces an effective coherent evolution within the DFS. In this work, we show explicitly how by introducing single-site resolved drivings, we can use the effective dynamics within the DFS to design a universal set of one and two-qubit gates within the DFS of two-level atom-like systems. Using Liouvillian perturbation theory we calculate the scaling with the relevant figures of merit of the systems, such as the Purcell Factor and imperfect control of the drivings. Finally, we compare our results with previous proposals using atomic Λ systems in leaky cavities. arXiv:1512.04803v1 [quant-ph]
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10.1088/1367-2630/18/4/043041
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[
"https://arxiv.org/pdf/1512.04803v1.pdf"
] | 29,885,260 |
1512.04803
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2987c37e3894d9956fe21f6e1f77ad1656e219c2
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Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces Universal Quantum Computation in Waveguide QED using DFS 2
15 Dec 2015
V Paulisch
Max-Planck-Institute of Quantum Optics
Hans-Kopfermann-Strasse 185748GarchingGermany
H J Kimble
Norman Bridge Laboratory of Physics
12-33
California Institute of Technology
91125PasadenaCAUSA
Institute for Quantum Information and Matter
California Institute of Technology
91125PasadenaCAUSA
A González-Tudela [email protected]
Max-Planck-Institute of Quantum Optics
Hans-Kopfermann-Strasse 185748GarchingGermany
Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces Universal Quantum Computation in Waveguide QED using DFS 2
15 Dec 2015
The interaction of quantum emitters with one-dimensional photon-like reservoirs induces strong and long-range dissipative couplings that give rise to the emergence of so-called Decoherence Free Subspaces (DFS) which are decoupled from dissipation. When introducing weak perturbations on the emitters, e.g., driving, the strong collective dissipation enforces an effective coherent evolution within the DFS. In this work, we show explicitly how by introducing single-site resolved drivings, we can use the effective dynamics within the DFS to design a universal set of one and two-qubit gates within the DFS of two-level atom-like systems. Using Liouvillian perturbation theory we calculate the scaling with the relevant figures of merit of the systems, such as the Purcell Factor and imperfect control of the drivings. Finally, we compare our results with previous proposals using atomic Λ systems in leaky cavities. arXiv:1512.04803v1 [quant-ph]
Introduction
Recent theoretical and experimental work has shown that an attractive configuration to engineer strong collective dissipation is given by one-dimensional (1d) photonic-like systems such as photonic crystal waveguides [1,2,3,4,5,6,7,8,9], optical fibers [10,11,12,13,14], metal [15,16,17,18] and graphene plasmonic [19,20,21,22] waveguides or superconducting circuits [23]. Their interaction with quantum emitters, usually referred to as waveguide QED, offers interesting characteristics: i) the density of modes of the waveguide is inversely proportional to the group velocity 1/v g (ω a ), and therefore is strongly enhanced when the atomic frequency is in a region of slow light, e.g., in photonic crystal waveguides close to a band edge. This enhancement implies achieving regions of a large decay rate into the waveguide, Γ 1d , compared to other decay channels, denoted by Γ * , characterized through the Purcell Factor, P 1d = Γ 1d /Γ * ; ii) the 1d guided modes retain a small modal area λ 2 a , for propagation lengths L prop λ a (the wavelength of the 1d mode associated to the atomic frequency considered); iii) the interaction is strongly long-range, favoring individual adressing, and it can even be homogeneous if the positions of the atom-like systems are chosen properly [24,25], in contrast to 2d or 3d system. This collective dissipation leads to the emergence of subradiant states that form the so-called Decoherence-Free Subspace (DFS) [26,27].
Previous works have already considered how to use the DFS of two atoms trapped in leaky cavities to design one and two-qubit gates using three-level Λ-type schemes [28,29,30], where two atomic hyperfine levels are used to encode the qubit. In the light of the variety of systems available nowadays that allows to engineer robust onedimensional DFS [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], which may couple to different types of quantum emitters, e.g., atoms, quantum dots, NV centers or superconducting qubits, it is interesting to revisit the problem and fill some of the gaps that have not been considered so far, namely,: i) how to encode decoherencefree qubits using only two-level systems (TLS) (as Λ-schemes might not be available for all platforms); ii) extend the proposal to systems with more than two atoms; iii) analyze the scaling of the fidelities with the relevant figures of merit of the system, e.g., P 1d , that in previous works [28,29,30] was mainly done through numerical analysis; iv) discuss the consequences of imperfect addressing on the fidelity of the gates.
In this work, we show an implementation of universal quantum gates by using N TLS strongly coupled to 1d photon-like reservoirs. First, we show that by pairing the TLS, we can define decoherence free-qubits in the singlet (i.e., antisymmetric) states of each pair. The combination of these singlets form the so-called computational subspace where we define our operations. Then, we explicitly show how to build single qubit (e.g., phase gates and Pauli-X gates) and two-qubit (e.g., controlled Z-gates) operations within the computational subspace without coupling to the other states in the DFS. By using Liouvillian perturbation theory, we obtain analytical expressions for the scaling of the fidelities of the operation (1 − F ∝ 1/ √ P 1d ) and estimate the error when increasing the number of atoms. Finally, we revisit the problem of the implementation with Λ Figure 1. (a) TLS (in blue) trapped along a one-dimensional waveguide, addressed by single-site resolved control fields. In green, we depict the pairing that we will use to engineer the computational qubits.(b) Level structure of a TLS with a coherent driving with amplitude (detuning) Ω n (∆ n ) with an additional level to which transitions can be driven off-resonantly to engineer the ∆ n σ n ee term of the hamiltonian. (c) A three-level system in which the excited state is driven far off-resonantly can be made approximately equivalent to a TLS with modified parameters as shown in the legend.
systems in leaky cavities [28,29,30] and show how both lead to the similar scaling.
The paper is divided as follows: in section 2, we introduce the set-up where we implement our proposal and establish the general formalism that we use to characterize the operations. In section 3, we describe the logical qubits and computational subspace and show how to build a set of universal quantum gates in the ideal case, that is, without considering decay into other non-guided modes or deviations from Quantum Zeno dynamics [26,27,31]. Then, in sections 4 and 5, we analyze possible error sources, both analytically and numerically, including spontaneous emission and imperfect addressing for the different gates of our proposal. Finally, in section 6, we compare the scaling with the proposal of three-level atoms in leaky cavities already explored in the literature [28,29,30].
General set-up and formalism
Set-up: waveguide QED
The general set-up that we consider is depicted in figure 1a; namely N TLSs, {|g n , |e n } n=1...N , placed at positions z n and coupled to a 1d field with bosonic annihilation operators a q . Due to the variety of implementations available nowadays, we will keep the discussion as general as possible without making further assumptions on the nature of the TLS and/or 1d waveguides.
The composite system is described by the Hamiltonian H = H 0 + H I , where H 0 is the free term given by H 0 = H qb + H field , (using = 1)
H qb = ω a N n=1 σ n ee , H field = q ω q a † q a q ,(1)
where ω a is the TLS energy, σ n ij = |i n j| n are atomic operators, and ω q is the energy dispersion relation of the waveguide modes. We consider a dipolar coupling of the form
H I = n σ n ge E(z n ) + H.c. ,(2)
with E(z) = q g q (a q e iqz + a † q e −iqz ), and g q the single photon coupling constant. When the system-reservoir coupling is weak (Born approximation) and the evolution of the 1d-reservoir is much faster that the one of the system (Markov approximation), the evolution of ρ, the reduced density matrix for the atom-like system, can be described by a Markovian master equation of the form dρ/dt = L [ρ] [24,25,32,33], with the superoperator
L [ρ] = n,m Γ n,m σ n ge ρσ m eg − ρσ m eg σ n ge + H.c. ,(3)
where
Γ n,m = Γ 1d 2 e iq(ωa)|zn−zm| ,(4)
where Γ 1d , the decay into waveguide modes, that we will assume to be larger than the rate of spontaneous emission into all other modes, Γ * Γ 1d as this is the regime we are interested in. Moreover, as the propagation lengths of the waveguide modes for many implementations are long (L p λ a ), the atoms can be separated several wavelengths apart and can therefore be individually addressed as depicted in figure 1(a). In particular, we assume to control the TLS state through the Hamiltonian (in the interaction picture with respect to H qb )
H las = n 1 2 Ω n σ n ge + H.c. + ∆ n σ n ee ,(5)
where Ω n is the amplitude of the coherent driving (that we consider to be resonant, i.e., ω L = ω a ) which controls the number of excitations of the system, and ∆ n is a phase shift interaction term. The latter can be obtained, e.g., in atomic systems, by adding an off-resonant driving to another excited state |e , as depicted in figure 1(b), which results in an Stark shift ∆ n = |Ω | 2 /(ω a − ω L ). In general, the way of implementing Ω n and ∆ n will depend on the particular system. For completeness, it is worth mentioning that Λ systems can also be mapped to effective TLS by using an off-resonant Raman transition as depicted in figure 1(c). By adiabatically eliminating the excited state |e , one can formally project the dynamics to the two metastable states, {|g , |e }, and find a similar light-matter hamiltonian as the one of equation 2, with the advantage that the effective TLS defined by {|g , |e } will be long-lived as they are encoded in metastable states. For example, by switching both Ω g and Ω e at the same time with detuning δ( |Ω g |, |Ω e |) as depicted in figure 1(c), we can implement a coherent driving term with effective Ω = ΩgΩ * e 4δ . By switching δ in this case big enough one can neglect spontaneous emission processes as they will be proportional to Γ * |Ωe| 2 +|Ωg| 2 4δ 2 . Moreover, if we switch only Ω e and adiabatically eliminate the photonic modes we also obtain an irreversible transition from |e → |g , but with a renormalization of the decay rates Γ 1d → Γ 1d | Ωe 2δ | 2 and Γ * → Γ * | Ωe 2δ | 2 . Hence, the Purcell factor P 1d = Γ 1d /Γ * is unchanged. In that situation our analysis is an alternative implementation to the one developed in Refs. [28,29,30].
Decoherence-Free Subspaces
In the case of equidistant spacing at positions commensurate with the wavelength of the guided mode, i.e. z n = n2π/q(ω a ), the effective interaction induced by the waveguide modes yields a pure Dicke model [34] decay described by
L D [ρ] = Γ 1d 2 (S ge ρS eg − S eg S ge ρ) + H.c.,(6)
where we have introduced the collective spin operator S ge = N n=1 σ n ge . The states satisfying S ge |Ψ = 0 are decoherence-free with respect to the collective dissipation L D . These states can be easily described in the collective spin basis {|J, m J , α J }, that is the eigenstates of the collective operators S 2 = i=x,y.z S 2 i and S z with
S 2 |J, m J , α J = J(J + 1)|J, m J , α J , (7a) S z |J, m J , α J = m J |J, m J , α J ,(7b)
where J = N/2, N/2 − 1, . . ., m J = −J, −J + 1, . . . , J. The index α J is introduced because the states in the collective spin basis are degenerate, with degeneracy given by:
α J = 1, . . . , N J − N J−1 .
It is easy to observe in this basis that the states |J, −J, α J are decoherence free, and therefore span the decoherence-free subspace (DFS).
The DFS has a dimension of N N/2 (assuming even atom number N ), and is composed of all the possible states which are antisymmetric with the permutation of two atoms. Thus, an alternative way of characterizing the DFS is to consider all possible (tensor products) of singlet states
|A m,n = (|e m ⊗ |g n − |g m ⊗ |e n ) / √ 2 ,(8)
where m, n denote the atomic positions of the pair of atoms that form the singlet. This characterization makes it more difficult to describe an orthonormal basis of the DFS. However, we show in the next section that it is convenient to define our computational subspace.
Quantum Zeno dynamics using Liouvillian perturbation theory
We are interested in the regime where the collective dissipation induced by L D , with characteristic timescale Γ −1 1d , dominates over any possible perturbation of the system, L pert , with characteristic timescale τ Γ −1 1d . Under these assumptions, any state outside of the DFS will only be virtually populated due to the strong dissipation and therefore the dynamics will be restricted to the slow subspace, i.e., the DFS. Mathematically, we formalize this intuitive picture by defining a projection superoperator P (with P 2 = P) satisfying: PL D = L D P = 0 that projects out the fast dynamics yielding only the effective evolution in the slow subspace. It is then possible to integrate out the fast dynamics (see Appendix A) arriving to an effective master equation given by
∂Pρ ∂t = L eff [Pρ] = PL pert P + PL pert Q 1 −L D QL pert P + O τ −3 /Γ 2 1d ρ .(9)
This result to first order (left hand term in the brackets) accounts for the ideal Quantum Zeno dynamics [26,27,31]. The second order in perturbation theory then yields correction terms mainly coming from slightly populating the (super)radiant states.
In our case, there will be two types of perturbations, namely,
• The Hamiltonian L pert [·] = −i [H las , ·] to control the atomic state. This results to first order in an effective Hamiltonian H eff = PH las P that couples only atomic states within the DFS. Here, we introduced the projection onto the DFS for pure states P = i |d i d i |, where the states |d i form an orthonormal basis of the DFS. We use this effective laser coupling to control the atomic state of the ensemble. Besides, there is a second order correction resulting from H las that will be relevant for the analysis of the error probability of our proposal as we show in section 4.
• The contribution of the emission of photons to other radiative modes different from the guided mode of the waveguide that we embed into a single decay rate, Γ * and describe through the Liouvillian
L pert [ρ] = L * [ρ] = n Γ * 2 (σ n ge ρσ n eg − ρσ n ee + H.c.).(10)
This contribution is relevant for the error analysis of the gates in section 4.
Universal Single-and Two-Qubit Gates
In this Section, we show how to engineer a set of universal gates, i.e., defined by any arbitrary single-qubit rotation and a controlled gate [35], using the effective evolution H eff within the DFS that appear in our waveguide QED setup. Firstly, due to the large degeneracy of the DFS, we need to define a set of logical qubits that will expand our computational subspace. Then, we show how to choose {Ω n , ∆ n } such that they define a set of universal one and two-qubit gates, namely, the phase and Pauli-X (and Y ) gate and the controlled-(−Z) gate. A summary of the parameters for these gates can be found in table 1. For completeness, we also give the parameters for other gates such as the Hadamard or SWAP gates. The former can be easily constructed because all single qubit rotations can be performed and the latter is constructed through the same idea as the controlled-(−Z) gate. Due to the degeneracy of the DFS, the challenge lies in defining operations within the computational subspace, without populating the rest of the states within the DFS. In section 4, we revisit the problem and consider the effect of spontaneous emission and second order corrections to the Zeno dynamics that ultimately limit the fidelity of the operations.
Definition of Logical Qubits and Computational Subspace
Before finding the appropriate gates, we first need to define the logical qubits within the DFS. In principle, assuming an even number of atoms N , the dimension of the DFS is N N/2 and therefore, it is possible to encode log 2 N N/2 logical qubits. However, it is more useful to restrict the computational subspace to a smaller set of states in order to achieve universal quantum computation. As the DFS is spanned by all (tensor) products of singlet states over two atoms, it is natural to define the logical qubits as
|0 L j ≡ |G j,j+1 = |g j ⊗ |g j+1 (11a) |1 L j ≡ |A j,j+1 = (|e j ⊗ |g j+1 − |g j ⊗ |e j+1 ) / √ 2. (11b)
It is instructive to consider particular examples to see how the DFS and computational space look, i.e., for the case of N = 2 and N = 4 atoms.
Two atoms: In this case it is easy to plot the complete Hilbert space (including states outside DFS) as it consists only of 4 states as depicted in figure 2(a). The separation into DFS states and non-DFS states is easily done in the familiar singlet-triplet basis. The DFS consists of two states: the one with two atoms in the ground state and the singlet state, i.e., the antisymmetric combination of one single excited state. Thus, we can encode one logical decoherence free qubit. The other two states are superradiant, i.e., they decay with an enhanced decay rate of 2Γ 1d .
Four atoms:
The complete Hilbert space consists of 2 4 = 16 states, and the dimension of the DFS, shown in figure 2(b), is 4 2 = 6. As aforementioned, we want to use as computational subspace the tensor product of the antisymmetric pairs described in equation 11a and 11b , which consist only of 2 2 = 4 states. This is why Table 1. Summary of the optimal parameter settings for the π/8-gate T , the Pauli gates X, Y, Z, the Hadamard gate H, the SWAP-gate and the controlled-(−Z) gate. The subindex denotes on which logical qubit the gate acts. The settings for the Rabi couplings and detunings are denoted by
x ± ij = 1 2 (x i ± x j ).
T is the duration for which the operation is applied to obtain the corresponding gate. In general ∆ D is a large detuning (see equation 17), that prevents transitions to other states and Ω n = 0 for n ≥ 3.
Gate
Ω − 12 Ω + 12 ∆ + 12 ∆ − 12 ∆ + 34 ∆ − 34 T ∆ n≥5 T 1 0 0 -∆ T 0 0 0 π 4∆T 0 Z 1 0 0 ∆ Z 0 0 0 π ∆ Z 0 X 1 Ω X ∈ R 0 0 0 ∆ D 0 π √ 2ΩX ∆ D Y 1 Ω Y ∈ iR 0 0 0 ∆ D 0 π √ 2|ΩY| ∆ D ∝ H 1 -Ω H ∈ R 0 √ 2Ω H 0 ∆ D 0 π 2ΩH ∆ D SWAP 12 0 0 0 ∆ S 0 ∆ S π ∆S ∆ D C(−Z) 12 0 0 0 ∆ C 0 0 π2 √ 2 ∆C ∆ D
in this situation we have to distinguish within the DFS between the computational space and the additional states that must be either decoupled or used as auxiliary states. In particular, for N = 4 the auxiliary states are |A 1,2 G 3,4 + A 3,4 G 1,2 / √ 3 and
|A 1,3 A 2,4 + A 1,4 A 2,3 / √ 2.
For N > 4 atoms: In general (for N > 2) the dimension of the DFS, N N/2 (for even N ), is larger than the one of the computational subspace, 2 N/2 . Thus, one can split the projection onto the DFS, P, into two orthogonal projections, i.e. one into the computational subspace P CS and its orthogonal counterpart Q CS :
P = P CS + Q CS ,(12a)P CS = j odd |0 L j 0| + |1 L j 1| ,(12b)
such that the effective Hamiltonian can be written as follows
H eff = P CS H las P CS + (P CS H las Q CS + H.c.) + Q CS H las Q CS ,(13)
which separates the transitions within the computational [auxiliary] subspace P CS HP CS [Q CS HQ CS ] and the coupling between these two subspaces: P CS HQ CS . This separation will be useful to argue that we can make operations within the DFS even in the situations with N > 4 as we will show afterwards. For a general situation, it is easy to show that by projecting H las into the computational subspace we obtain an effective evolution inside the computational subspace given by:
P CS H las P CS = j odd Ω − j,j+1 √ 2 |1 L j 0| + h.c. + ∆ + j,j+1 |1 L j 1|,(14)
where we used the following notation x ± i,j = 1 2 (x i ± x j ) to abbreviate the combination of parameters.
Single-Qubit Gates
The goal is to find the {Ω n , ∆ n } such that they define both the phase and Pauli-X (and Y ) gates over the computational subspace.
Two atoms:
This is the simplest situation because the size of the computational space is the same as the one of the DFS. In this case (see also [29]), a phase shift gate on the logical qubit (α|0 L + β|1 L → α|0 L + βe −iφ |1 L ) is obtained by applying Ω − 12 = 0, ∆ + 12 = 0 for a time T = φ ∆ +
12
. Pauli-X rotations (plus a phase) are obtained for 0 = Ω − 12 ∈ R, ∆ + 12 = 0 and time T = π √ 2Ω −
12
. Note that to avoid errors in both cases, one should also set Ω + 12 = ∆ − 12 = 0 as will be discussed in section 4. The Pauli Y can be obtained as the X just by using iΩ − 12 ∈ R, so that we will not discuss it further. Four atoms: In this case, the way to do phase gates and rotations is the same as for the two atom case. However, in the case of the rotations, states within the computational subspace couple to auxiliary states for more than two atoms. In particular, the state |10 L is coupled to the auxiliary state |A 1,3 A 2,4 + A 1,4 A 2,3 / √ 2 for Ω − 12 = 0 as shown in figure 2. However, this transition can be made far off-resonance by setting |∆ + 34 | |Ω − 12 |. This results in an additional error rate |Ω − 12 | 2 2∆ + 34 that will be considered when calculating the fidelity of the operation. ‡ For N > 4 atoms: Again in the case of rotations, transitions to states outside the computational subspace in the ideal case (Ω + 12 = ∆ − 12 = 0) are possible when Ω − 12 = 0, that is when
Q CS HP CS = Q CS H eff P CS = 0 ,(15)
where we use that P CS P = P CS . However, the transitions to these states can be made far off-resonant by setting
Ω n = 0, and |Ω − 12 | ∆ n = ∆ D Γ 1d , n ≥ 3,(16)
because the auxiliary states inside the DFS that the computational subspace couples to extend over more than two atoms § and can therefore be detuned as Ω − 12 . § The auxiliary states necessarily extend over more than two atoms, because it is orthogonal to the logical qubits and therefore contains excited (superradiant) triplet states in the "pairing" of the atoms. An antisymmetric combination of such states can be in the DFS, but not in the computational subspace, and necessarily extends over multiple atom "pairs".
Q CS HQ CS ∼ ∆ D Q CS ,(17)
while keeping the desired transition driven by Ω − 12 as resonant. One has to make sure that the Stark-shift introduced by this off-resonant transition is small and possibly correct the detuning that it will induce by choosing appropriately the applied laser frequency ω L .
Controlled Pauli-Z
For universal quantum computation, a controlled two-qubit gate is required. In this case, the minimal system to encode the operation is the N = 4 atom case, where two decoherence-free logical qubits can be obtained.
Four atoms:
In order to build the controlled-Z gate, we use one of the auxiliary states, |A 1,2 G 3,4 + A 3,4 G 1,2 / √ 2. Now, it is possible to drive only the transition between this state and |10 L without affecting the other states within the DFS by the choice Ω n = 0, ∆ 3 = ∆ 4 = 0 and ∆ 1 = −∆ 2 = 0. A π-pulse on the state |10 L leads to a relative phase of −1 on this state, i.e.
|10 L 1,3 −→ −|10 L 1,3 ,(18)
for 1 √ 2 ∆ − 12 T = 2π without affecting the other states of the computational subspace. Hence, we have defined a a controlled controlled-(−Z) gate which is equivalent up to single qubit unitaries to a CNOT-gate [35].
For N > 4 atoms: One can restrict the dynamics to the subspace of four atoms in a similar way as for the single-qubit rotations. With the choice of Ω n = 0, and |∆ − 12 | ∆ n = ∆ D Γ 1d , n ≥ 5,
transitions to states over more than four atoms are far off-resonant. As before, this adds an error rate proportional to |∆ − 12 | 2 ∆ D with a proportionality factor depending on the coupling strength after the projection onto the DFS.
Error analysis: spontaneous emission and imperfect addressing.
So far we have considered only the interaction within the ideal Quantum Zeno Dynamics, where the only possible sources of error were due to the larger dimension of the DFS with respect to the computational space. In this section, we take into account other sources of errors that will be present in most of the implementations, namely, i) errors coming from spontaneous emission to other modes, with rate Γ * , included through L * [ρ] as in equation 10; ii) errors from deviations from the Zeno Hamiltonian, attributed to photons emitted to the waveguide from the small population present in the states outside the DFS; iii) errors that may arise from an imperfect control of the laser parameters {Ω n , ∆ n }. In what follows, we assume to work in a regime with P 1d 1, such that the following parameter hierarchy can be satisfied: Γ * ||H eff || Γ 1d . (f) (e) This section discusses, for each gate separately, the numerical results and their analytical approximation of the fidelity between the theoretical final (goal) state, |ψ f , and the real atomic state, ρ, obtained after the gate operation, i.e., F = ψ f |ρ|ψ f 1/2 . The numerical results are obtained by solving the master equation in second order perturbation theory (see equation 9). We have checked numerically that this is a good approximation in the parameter ranges considered throughout this manuscript. To obtain the analytical approximations, we used the effective non-hermitian Hamiltonian that can be obtained from the second order master equation (see appendix A).
Phase Shift Gate
For the phase shift gate we must set Ω n = 0 for all n and ∆ n = ∆ D ∆ − 12 for all n ≥ 3 to avoid errors from transitions to auxiliary states. By choosing ∆ 1 = ∆ 2 , i.e., ∆ − 12 = 0, no errors (from second order perturbation theory) occur because the computational states do not couple to the radiant ones. However, it is instructive to consider the errors that appear for situations where ∆ − 12 = 0, e.g, because of imperfect addressing, as this yields a useful understanding about how to deal with situations where the second order correction cannot be avoided.
Two atoms: For the simplest situation the additional errors due to imperfect addressing, i.e., ∆ − 12 = 0, enter at a rate proportional to |∆ − 12 | 2 /Γ 1d through the same error channel as the spontaneous emission into all other modes with rate Γ * , that is, via the quantum jump operator |0 L 1|. Then, the infidelity, i.e., 1 − F , for a π/2 phase shift of the first logical qubit on the normalized state α|0 L + β|1 L can be approximated by
1 − F ≈ |β| 2 4 π ∆ + 12 Γ * + 4 |∆ − 12 | 2 Γ 1D .(20)
One observes, that in the ideal case, ∆ − 12 = 0, the infidelity can be arbitrarily close to 0 for large ∆ + 12 . If ∆ − 12 is not negligible, the transition strength ∆ + 12 cannot be chosen arbitrarily large to decrease the infidelity. For example, in the worst case scenario where ∆ − 12 = ∆ + 12 this results in an optimal infidelity scaling |β| 2 π 2 P −1/2 1d
for ∆ + 12 = ∆ − 12 = 1 2 √ Γ * Γ 1d .
Four atoms:
A similar behaviour can be obtained by choosing ∆ − 12 = 0 such that the infidelity is arbitrarily close to 0 (see figure 3a). Slight deviations from this ideal value do not change this behaviour drastically (see figure 3b). However, when ∆ − 12 is not negligible, it leads to two types of errors that decrease the fidelity (see figure 3c): i) virtual population of non-DFS states, which leads to an error rate proportional to |∆ − 12 | 2 /Γ 1d as for two atoms; and ii) transitions to auxiliary states, in particular |A 1,2 G 3,4 + A 3,4 G 1,2 / √ 2. The latter can be made far off-resonance by applying a detuning on the second qubit such that |∆ − 12 | |∆ + 34 | Γ 1d . With a large off-resonance ratio r ∆ = |∆ + 34 /∆ − 12 | 1, one still achieves a small infidelity (see figure 3d). As shown in figures 3a-d, the detuning of the third and fourth atom is important when ∆ − 12 cannot be neglected. As expected, the infidelity decreases by increasing the off-resonance ratio r ∆ (see figure 4a). For large enough r ∆ , the infidelity can be analytically approximated by
1 − F r ∆ →∞ −→ π 8 Γ * |∆ + 12 | + 2 |∆ + 12 | Γ 1d .(21)
This leads to a minimal infidelity ∝ P −1/2 1d (see figure 4b) for ∆ + 12 = ∆ − 12 = Γ * Γ 1d /2.
Pauli-X Gate
For rotations around the x-axis, we set ∆ 1 = ∆ 2 = 0, and ∆ n = ∆ D Ω − 12 for all n ≥ 3 to avoid transitions to auxiliary states. In contrast to the phase shift gate, even in the ideal case, Ω + 12 = 0, errors will occur because Ω − 12 couples to state outside the DFS, as shown schematically in figure 2(a) for the two atom case. Moreover, we also include a short discussion on deviations due to imperfect control on ∆ 1(2) = 0 and Ω + 12 .
Two atoms:
Using Ω + 12 = 0, the error rate from deviations from the Zeno Hamiltonian enters in the same way as from the spontaneous emission into all other modes, that is via the quantum jump operator |0 L 1|. The corresponding decay rate is (|∆ − 12 | 2 + |Ω − 12 | 2 /2)/Γ 1d . The error from Ω + 12 = 0 enters differently, but can still be included in the estimation of the infidelity. Neglecting the errors from ∆ + 12 = 0, the infidelity for a Pauli-X gate ( 1 √ 2 |Ω − 12 |T = π/2) on state |1 L can be approximated by
1 − F ≈ 1 2 π √ 2|Ω − 12 | Γ * + |Ω − 12 | 2 2Γ 1D + |∆ − 12 | 2 Γ 1D + |Ω + 12 | 2 2Γ 1D ≡ ε 0 .(22)
In the ideal case of perfect control of addressing parameters, i.e., ∆ − 12 = Ω + 12 = 0, the minimal value of the infidelity, proportional to P −1/2 1d , is obtained at |Ω − 12 | = √ 2Γ * Γ 1D as shown in red circles of figure 6. Note, that even for Ω − 12 = Ω + 12 and ∆ − 12 = 0, the infidelity is still proportional to P −1/2 1d . Four atoms: In this case apart from the transitions out of the DFS, Ω − 12 also couples states inside the DFS, but out of the computational space (see figure 2) such that we need to detune these processes to achieve the rotations. As already explained in the previous section, this can be done by setting |Ω − 12 | |∆ + 34 | Γ 1d . As expected, the infidelity decreases when increasing the off-resonance ratio r Ω = |∆ + 34 /Ω − 12 | (see figure 5). For large enough ratios r Ω ( 4), the infidelity can be approximated by
1 − F ≈ ε 0 + α r 2 Ω ,(23)
where the constant α = O(1) can be obtained through a numerical fit. The infidelity of a π/2-pulse on the state |00 L is plotted in figure 3e, whereas the minimal infidelity is shown to scale with P −1/2 1d in figure 6. for different values of the off-resonance ratio, that is r Ω = 2 (black), r Ω = 5 (red) and r Ω = 10 (blue). (b) Scaling of the minimal infidelity for the same values as in (a) for different Purcell Factors, that is P 1d = 10 3 (black), P 1d = 10 4 (red) and P 1d = 10 5 (blue).
Controlled Pauli-Z
For the controlled-(−Z) gate, we set Ω n = 0 for all n, ∆ + 12 = 0 and ∆ n = ∆ D |∆ − 12 | for all n ≥ 5. As ∆ − 12 couples |10 L and |11 L also to states outside the DFS, the fidelity shows a similar behaviour as the Pauli-X gate (see figure 3(f) for example with N = 4), i.e., there is an optimal ∆ − 12 that sets the maximum fidelity. The infidelity can be approximated similarly to equation 20, i.e., after a controlled Pauli-Z gate (|∆ − 12 |T / √ 2 = π) acting on the state |10 + 11 L / √ 2 can be approximated by
1 − F ≈ 3π 2 √ 2|∆ − 12 | Γ * + 3 4 |∆ − 12 | Γ 1D ,(24)
which attains its minimal value, 3π/ √ 2P 1d , for |∆ − 12 | = 4Γ * Γ 1d /3. As for the single qubit gates, the infidelity scales with P −1/2 1d , shown in blue circles figure 6.
Summary of analysis
Summing up, from the explicit analysis with two and four TLS, we have shown both numerically and analytically that both the single-qubit rotations and the control (-Z) gate show a scaling of the infidelity as P −1/2 1d (see figure 6). Only for small values of the Purcell Factor P 1d does the minimal infidelity deviate slightly from the theoretical analysis because the hierarchy Γ * Ω − 12 , ∆ − 12 Γ 1d is no longer well satisfied. Moreover, in the N = 4 case, we also showed how to deal with the errors that come from the larger size of the DFS with respect to the computational one. For single qubit rotations in a system of four emitters, the choice |Ω − 12 |, |∆ + 12 | |∆ + 34 | Γ 1d ensures that the dynamics can be restricted to two atoms. In the extreme case where |∆ + 34 | Γ 1d the perturbation analysis is no longer valid. However, in this case the levels are so strongly shifted, that they are decoupled from the collective dissipation, so that the system can be described as a system of only two emitters. The same is true, if the emitters can be completely decoupled from the waveguide by other means available in a particular implementation. For more atoms the same arguments hold as the second order correction introduced from deviations from Zeno dynamics satisfies
PL pert Q 1 L D QL pert Pρ L pert 2 Γ 1d ,(25)
where L pert is the perturbation to the purely collective decay and · denotes the maximum norm. This is independent of the atom number N , because L pert does not increase with the number of atoms for one and two-qubit gates. So there is an upper limit on the second order correction, which leads to the P −1/2 1d -scaling. Finally, the error rate stemming from spontaneous emission of the logical states |1 L = |A is proportional to the number of excited states in the system. Therefore, the gate fidelity does depend on the full state, and can be upper bounded by considering the worst-case state, that is the state with |1 L in all other computational qubits, which indeed will depend on the atom number.
Further error analysis: finite propagation length of 1d modes.
For completeness, it is interesting to consider another source of error that may be very relevant for some implementations with short propagation lengths, e.g., plasmonic waveguides [15,16,17,18]. The finite propagation length enters into the decay matrix [17] as
Γ n,m = Γ 1d 2 e iq(ωa)|zn−zm| e −|zn−zm|/Lprop = Γ 1d 2 e −x|n−m| ,(26)
if the atoms are equidistantly placed a multiple of a wavelength apart, d and where we introduced x = d/L prop as the perturbation parameter. For simplicity, we restrict our discussion to the case with N = 2, where analytical expressions can be obtained. In that situation, the finite propagation length only leads to the replacements
Γ * → Γ * + Γ 1d (1 − e −x ) ≈ Γ * + Γ 1d x,(27)2Γ 1d → Γ 1d 2 − (1 − e −x ) ≈ Γ 1d (2 − x) ,(28)
when x 1. Therefore, the scaling of the infidelity is then given by
1 − F ∝ Γ * + Γ 1d x Γ 1d (2 − x) ≈ P −1/2 1d + 1 2 P 1/2 1d x,(29)
which scales as 1/ √ P 1d as long as xP 1d 1,that is, that the distance between neighbouring emitters satisfies d = |z n − z n+1 | L prop P −1 1d . For more atoms, it is difficult to obtain the analytical scaling as the superradiant state is not an eigenstate of the modified decay matrix and thus the DFS states change as well. However, because the finite propagation length enters as e −|zm−zn|/Lprop ≈ 1 − |z m − z n |/L prop it can be treated as a perturbation to the Liouvillian of equation 6 that will be kept small as long as N d L prop [36]. Depending on the particular implementation other errors have to be considered, e.g., for atoms trapped close to a dielectric waveguide the separation condition |z n − z m | = n2π/q(ω a ) might not be satisfied exactly or because its position is changing over time due to atomic motion. However, its main effect can be approximated as an effective increase of Γ * that is small with current state of the art parameters for photonic crystal waveguides, as discussed in reference [36].
Comparison to Three-Level Atoms
The use of the DFS of atomic Λ-systems in cavity QED setups has already been considered in detail in the literature [28,29,30]. In that case, a three-level system with a Λ-type level structure is used to define a logical qubit in the two metastable states |0 and |1 . The excited state |e decays to one of the metastable states, say |1 . When two atoms are inside the cavity an additional decoherence-free state emerges, i.e., (|1e − |e1 ) / √ 2, that can be used to define a CNOT gate in the so-called bad-cavity limit, where the atom-cavity coupling (g) is smaller than the cavity losses κ, but the decay into the cavity (g 2 /κ) is still bigger than into the rest of the decay channels (Γ * ). The ratio between the good/bad processes is the so-called cooperativity C = g 2 κΓ * , which therefore plays a similar role as P 1d in our proposal. The errors in the CNOT gate come both from Γ * , and from deviations from the Zeno Hamiltonian, giving rise to an optimal infidelity proportional to 1/ √ C, which is similar to the one that we found using only TLSs.
We note that using TLS the computational qubits have a finite lifetime compared to the implementations using atomic metastable states. However, i) there are situations in which one would like to use gates to build a given atomic state within the DFS in order to map it immediately into a photonic state in the waveguide [36] such that long lifetimes are not required; ii) some of the implementations have extremely long-lived qubits, e.g. superconducting systems [23]. iii) Moreover, if Λ schemes are available, as in atoms, we can also implement our single and two-qubit gates with metastable states with the equivalence shown in figure 1c. In that case, our proposal just constitutes a complementary way of doing universal quantum computation within DFS.
Conclusion & Outlook.
Summing up, we have shown how to implement a universal set of quantum gates using the decoherence-free subspaces appearing within TLS interacting with one-dimensional photon-like reservoirs. We have given an explicit construction of single and two-qubit gates for logical qubits defined in the DFS and analyzed possible sources of errors such as spontaneous emission to other modes, coupling to states outside of the DFS, imperfect addressing and finite propagation lengths. Through both analytical and numerical analysis, we have shown the fidelities of the gates scale generally with (1−F ) min ∝ P −1/2 1d , analogous to the one using Λ schemes [28,29,30]. Thus, this work widens up the zoology of quantum emitters that can be used to implement quantum gates within waveguide QED setups. An interesting outlook for the application of these gates is to use them for generating entangled states of many emitters within the DFS, which afterwards can be mapped into waveguide multiphoton states in a very efficient way [36].
can be formally projected into the DFS of the Liovillian L D , by using a projector operator P satisfying: PL D = L D P = 0. This projector can be found via the right (left) eigenoperators ρ ij (χ ij ) corresponding to the eigenvalue 0 of L D , which are combined to
Pρ = i,j ρ ij χ ij , ρ , (A.1)
where A, B = Tr A † B is the inner product on the space of density matrices. The orthogonal eigenoperators are indexed such that χ ij , ρ kl = δ i,k δ j,l = ρ ij , ρ kl . We also define the orthogonal part of P, Q = 1 − P. The left eigenoperators can be derived from the right ones by
χ ij = ρ ij + α(1)
ij S eg ρ ij S ge + α (2) ij S eg S eg ρ ij S ge S ge + . . . ,
(A.2)
where the coefficients α where the last approximation is obtained by i) applying a Markov approximation ρ(τ ) ≈ ρ(t) in the integral; ii) neglecting terms of higher order in τ −1 /Γ 1d ; and iii) extending the integral to infinity. Plugging this into the equation for the DFS-part of the state, that is d dt Pρ = PL pert Pρ + PL pert Qρ = L eff Pρ, (1.4) yields an effective Liouvillian of the atomic system within the DFS given (up to second order in τ −1 /Γ 1d ) by
L eff = PL pert P + PL pert Q 1 −L D QL pert P + O τ −3 /Γ 2 1d .
(1.5)
The first order of this Liouvillian, i.e., PL pert P, is the effective evolution induced within the DFS induced by the strong collective dissipation. This is commonly referred to as the ideal Quantum Zeno dynamics [26,27,31] as it can be understood as the effective dynamics enforced by the continuous monitoring of the atomic system due to the interaction of the waveguide modes. The second order term stems from slight population of (super)radiant modes that generates some corrections on the ideal Quantum Zeno dynamics.
It is instructive to write the effective master equation derived in Eq. 1.4 in a form that separates the non-hermitian evolution dynamics and the contribution coming from quantum jump processes. For our particular situation, considering the perturbation of L * and H las as defined in the main text, it can be shown after some algebra thaṫ
ρ = −i [H eff , ρ] + PL * ρ + P o 1 ρo † 2 + o 2 ρo † 1 − o † 2 o 1 ρ − ρo † 2 o 1 ,(1.6)
where H eff = PH las P, where we used the projection P [Q] for pure states inside [outside] of the DFS as defined in the main text. Furthermore, o 1 = Q Γ 1d 2 S + S − −1 QH las P, o 2 = QH las P can be obtained by noting that the second order term reduces to simple matrix multiplication in the corresponding subspace because PPAQ = 0 = PQAP for all operators A. Although this does not look like a Liouvillian in Lindblad form, it is trace-preserving, as
Tr PA = i,j Tr ρ ij χ ij , A = i χ ii , A = Tr A,(1.7)
because i χ ii = 1. From Eq. 1.6, it is straightforward to define a non-Hermitian Hamiltonian from the above master equation, that is
H nh = P H eff − i Γ * 2 n σ n eg σ n ge − io † 2 o 1 P,(1.8)
which describes the no-jump evolution, and that we use to get the analytical estimations of the infidelity.
Figure 2 .
2(a) Hilbert space of two TLS in the basis of non-DFS and DFS states with energies in the interaction picture with respect to H qb . The blue arrows denote the transitions necessary for the Pauli-X gate. The triplet states |E = |ee and |S = |eg + ge / √ 2 are not inside the DFS. (b) DFS of 4 TLS consists of 6 states that split into the 2 2 -dimensional computational subspace and two states in the auxiliary subspace. The blue [green] arrows denote the transitions necessary for the Pauli-X [and Controlled-(-Z)] gate.
Figure 3 .
3Infidelities of single and two qubit gates for N = 4 atoms. (a)-(d) Infidelity of a π/2 phase shift on the first logical qubit of four atoms on the state |10 + 00 L / 12 . (e) Infidelity of a single qubit Pauli-X gate on the first qubit of four atoms on the state |00 L for ∆ + 34 = 10Ω − 12 . (f) Infidelity of the controlled-(−Z) gate for four atoms (2 qubits) acting on the state |10 + 11 L / √ 2. The black lines in (d)-(f) represent the scaling of the coupling strength ∆ − 12 , Ω − 12 and ∆ − 12 , respectively, for the minimal infidelity with P
Figure 4 .+
4Dependence of infidelity on off-resonance ratio r ∆ = |∆ + 34 /∆ − 12 | for π/2 phase shift on first qubit with ∆ − 12 = ∆ + 12 . (a) Infidelity varied over the coupling strength ∆ + 12 for P 1d = 10 4 . The results correspond to r ∆ = 5 (black), r ∆ = 10 (red) and r ∆ = 15 (blue). (b) Minimal infidelity depending on the Purcell Factor P 1d . The numerical results (points) corresponding to the values as in (a) fit well with the approximation (line) of (1 − F ) min ∝ P C(r ∆ ), where the C(r ∆ ) is a number which does not depend on P 1d .
Figure 5 .
5(a) Numerical calculation of infidelity of Pauli-X gate on the first qubit on the state |00 L for N = 4 atoms with ∆ + 34 = 10Ω − 12 and P 1d = 10 4
Figure 6 .
6Scaling behaviour of the minimal infidelity for the Pauli-X (red squares) and controlled-(−Z) gates (blue circles) with the same parameters as in figure 3e-f. The scaling fits well with the scaling P −1/2 1d (black line) for large enough values of the Purcell Factor P 1d . For the phase shift gate the infidelity is arbitrarily close to 0 in the ideal case.
are determined by χ ij , L D ρ = 0. With this choice, the projector is independent of the choice of ρ ij and hence, one can choose ρ ij = |d i d j |, where |d i are orthonormal states from the DFS. Using these projectors, one can formally integrate out the fast dynamics outside the DFS, described by Qρ:d dt Qρ = Q (L D + L pert ) Qρ + QL pert Pρ , [Q(L D + L pert )Q(t − τ )] QL pert Pρ(τ ) ≈ Q(−L −1 D )QL pert Pρ + O(τ −2 /Γ 2 1d ), (1.3b)
‡ In fact, this argument can be reversed to excite the auxiliary state from the computational state |10 L with the choice ∆ + 34 = 0 and ∆ + 12
would vanish and the system would be described by pure states except for the spontaneous emission into non-guided modes.
AcknowledgmentsWe gratefully acknowledge discussions with I. Cirac. The work of AGT and VP was funded by the European Union integrated project Simulators and Interfaces with Quantum Systems (SIQS). AGT also acknowledges support from Alexander Von Humboldt Foundation and Intra-European Marie-Curie Fellowship NanoQuIS (625955).Appendix A: Second order Liouvillian perturbation theory.When the system evolves under a very strong collective decay, the driving term H las and the decay into other bath modes may be treated as a perturbation to the collective dissipation given by L D[37,38]. In order to describe these perturbations as generally as possible, we denote them by a Liouvillian L pert , and assume that it has a relevant timescale τ . If the timescale satisfies, τ 1/Γ 1d , the dynamics of the atomic system
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Radiation from an N -Atom System. I. General Formalism. R H Lehmberg, Phys. Rev. A. 23R. H. Lehmberg. Radiation from an N -Atom System. I. General Formalism. Phys. Rev. A, 2(3):883-888, Sep 1970.
Radiation from an N -Atom System. II. Spontaneous Emission from a Pair of Atoms. R H Lehmberg, Phys. Rev. A. 2R. H. Lehmberg. Radiation from an N -Atom System. II. Spontaneous Emission from a Pair of Atoms. Phys. Rev. A, 2:889-896, Sep 1970.
Noiseless quantum codes. Paolo Zanardi, Mario Rasetti, Physical Review Letters. 79173306Paolo Zanardi and Mario Rasetti. Noiseless quantum codes. Physical Review Letters, 79(17):3306, 1997.
Decoherence-Free Subspaces for Quantum Computation. D A Lidar, I L Chuang, K B Whaley, Phys. Rev. Lett. 812594D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-Free Subspaces for Quantum Computation. Phys. Rev. Lett., 81:2594, 1998.
Quantum computing using dissipation to remain in a decoherence-free subspace. Almut Beige, Daniel Braun, Ben Tregenna, Peter L Knight, Phys. Rev. Lett. 8581762Almut Beige, Daniel Braun, Ben Tregenna, and Peter L Knight. Quantum computing using dissipation to remain in a decoherence-free subspace. Phys. Rev. Lett., 85(8):1762, 2000.
Driving atoms into decoherence-free states. Almut Beige, Daniel Braun, Peter L Knight, New Journal of Physics. 2122Almut Beige, Daniel Braun, and Peter L Knight. Driving atoms into decoherence-free states. New Journal of Physics, 2(1):22, 2000.
Quantum computing in a macroscopic dark period. Ben Tregenna, Almut Beige, Peter L Knight, Physical Review A. 65332305Ben Tregenna, Almut Beige, and Peter L Knight. Quantum computing in a macroscopic dark period. Physical Review A, 65(3):032305, 2002.
Quantum zeno subspaces. P Facchi, S Pascazio, Phys. Rev. Lett. 8980401P. Facchi and S. Pascazio. Quantum zeno subspaces. Phys. Rev. Lett., 89:080401, Aug 2002.
Cavity qed with atomic mirrors. De Chang, Jiang, H J Gorshkov, Kimble, New Journal of Physics. 14663003DE Chang, L Jiang, AV Gorshkov, and HJ Kimble. Cavity qed with atomic mirrors. New Journal of Physics, 14(6):063003, 2012.
Quantum Noise. G W Gardiner, P Zoller, Springer-VerlagBerlin2nd editionG. W. Gardiner and P. Zoller. Quantum Noise. Springer-Verlag, Berlin, 2nd edition, 2000.
Coherence in Spontaneous Radiation Processes. R H Dicke, Phys. Rev. 9399R. H. Dicke. Coherence in Spontaneous Radiation Processes. Phys. Rev., 93:99, 1954.
Quantum computation and quantum information. M A Nielsen, I L Chuang, Cambridge University PressM. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2000.
Deterministic generation of arbitrary photonic states assisted by dissipation. A González-Tudela, V Paulisch, D E Chang, H J Kimble, J I Cirac, Phys. Rev. Lett. 115163603A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac. Deterministic generation of arbitrary photonic states assisted by dissipation. Phys. Rev. Lett., 115:163603, Oct 2015.
Effective operator formalism for open quantum systems. Florentin Reiter, Anders S Sørensen, Physical Review A. 85332111Florentin Reiter and Anders S Sørensen. Effective operator formalism for open quantum systems. Physical Review A, 85(3):032111, 2012.
Generalized Schrieffer-Wolff formalism for dissipative systems. E M Kessler, Phys. Rev. A. 8612126E. M. Kessler. Generalized Schrieffer-Wolff formalism for dissipative systems. Phys. Rev. A, 86:012126, Jul 2012.
|
[] |
[
"Nearly Static Magnetized Kerr Black-hole in Non-linear Electrodynamics",
"Nearly Static Magnetized Kerr Black-hole in Non-linear Electrodynamics"
] |
[
"K G Managave \nDepartment of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia\n",
"H A Redekar \nDepartment of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia\n",
"R B Kumbhar \nDepartment of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia\n",
"S P Das \nDepartment of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia\n",
"K Y Rajpure \nDepartment of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia\n"
] |
[
"Department of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia",
"Department of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia",
"Department of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia",
"Department of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia",
"Department of Physics\nShivaji University\nKolhapur-416004MaharashtraIndia"
] |
[] |
We have analyzed the thermodynamics of slowly rotating magnetized Kerr blackhole, with typical spin parameter a ≤ 0.1 (nearly static) in the background of non-linear electrodynamics. In particular we have studied the Bekenstein-Hawking entropy, Hawking temperature, angular momentum, specific heats and identified regions of parameters for possible phase-transitions. It turns out that once the stability lost the black-hole never come back to its original stable position.1
| null |
[
"https://export.arxiv.org/pdf/2303.07736v1.pdf"
] | 257,504,755 |
2303.07736
|
b05ae5b64b070d51eff4c7a4759219fdc7ac17aa
|
Nearly Static Magnetized Kerr Black-hole in Non-linear Electrodynamics
March 15, 2023
K G Managave
Department of Physics
Shivaji University
Kolhapur-416004MaharashtraIndia
H A Redekar
Department of Physics
Shivaji University
Kolhapur-416004MaharashtraIndia
R B Kumbhar
Department of Physics
Shivaji University
Kolhapur-416004MaharashtraIndia
S P Das
Department of Physics
Shivaji University
Kolhapur-416004MaharashtraIndia
K Y Rajpure
Department of Physics
Shivaji University
Kolhapur-416004MaharashtraIndia
Nearly Static Magnetized Kerr Black-hole in Non-linear Electrodynamics
March 15, 2023
We have analyzed the thermodynamics of slowly rotating magnetized Kerr blackhole, with typical spin parameter a ≤ 0.1 (nearly static) in the background of non-linear electrodynamics. In particular we have studied the Bekenstein-Hawking entropy, Hawking temperature, angular momentum, specific heats and identified regions of parameters for possible phase-transitions. It turns out that once the stability lost the black-hole never come back to its original stable position.1
Introduction
The solutions of Black-hole (BH) are natural outcome of the General theory of Relativity(GTR). Generally the BH characterized by their radius, masses, charges, angular momentum etc. Depending upon the parameter choices BH could have very different types, e.g., the Schwarzschild, Kerr, Kerr-Newman, Reissner Nordström [1], [2] . In our analysis we considered the Kerr solutions, i.e., the BH with rotation parameters.
The evolution of the different composite stars estimates that the rotation parameters could have varied ranges of 0.01 to 0.60 with very different characteristics features [3]. The nearly static in our consideration is with spin parameter a is 0.1 or less.
Nonlinear electrodynamics (NLED) has been attracted attention to astrophysicists since sometime back. NLED could have combined with General theory of relativity(GTR) -we called as NLED-GTR. This theory has some remarkable features, e.g., consistent theory of inflationary model of the Universe [4,5,6,7,8,9,10].
This NLED-GTR model has few characteristic features, e.g., absence of initial singularities, putting an upper limit on the Electric field at the origin of point-like particles, the finite selfenergy of the charged particles [8], [11] [12].
It is to be noted that in Quantum Electrodynamics(QED) the non-linear terms arises due to the loop corrections [14,15,16]. The NLED-GTR model follows the correspondence principle, i.e., in the weak-field limit we get the linear electrodynamics.
The physics of BH (both electrically and magnetically charged) in presence of NLED has been studied since some time [12], [17], [18]. We have studied the thermodynamics properties of the slowly rotating magnetized Kerr BH in NLED-GTR scenario.
The paper is organized as follows: we briefly outline the NLED-GTR model in section 2. In section 3 we discussed the Kerr BH relevant for the NLED-GTR and estimated the magnetized mass density that affects the relevant thermodynamic parameters for slowly rotating (with spin parameter a) for γ = 1 2 , 4 5 and 1. We conclude in Section 6. We adopted natural units, i.e., c = = 1, ε 0 = µ 0 = 1 and with metric signature η = diag(−1, 1, 1, 1)
Non-linear Electrodynamics (NLED)-model
The Lagrangian density is as (following [12]):
L = − F 1 + (βF) γ ,(1)
where the parameter β has the dimensions of 4D-volume (manifests the strength of the coupling), and γ (manifests the order of interaction) is the dimensionless parameter [12].
The F = (1/4)F µν F µν = (B 2 − E 2 )/2, where F µν = ∂ µ A ν − ∂ ν A µ
is the field strength tensor, with A µ = (φ, A) the φ and A are the scalar and vector potential respectively. One can obtain the field equations
∂ µ (L F F µν ) = 0,(2)
where L F = ∂L/∂F by using the Euler-Lagrange (EL) equations. The EL equation on Lagrangian density leads to
L F = (γ − 1)(βF) γ − 1 (1 + (βF) γ ) 2 .(3)
The group velocity over the background is less than photon speed requires L F ≤ 0 [19] so that the causality principle holds and confirm the absence of tachyon. In our analysis we imply that 0 ≤ γ ≤ 1 and we consider only magnetized black holes (E=0, F = B 2 /2). For simplicity in our analysis we set β = 1.
The symmetrical energy-momentum tensor (is obtained from Eqn. 1 ) is:
T µν = (γ − 1)(βF) γ − 1 [1 + (βF) γ ] 2 F α µ F να − g µν L.(4)
The trace of the energy-momentum tensor Eqn.4 leads to
T ≡ T µ µ = 4γF(βF) γ [1 + (βF) γ ] 2 .(5)
The trace of Eqn.?? is non-zero due to the presence of dimensional parameter β = 1(as we considered throughout our analysis) and it leads the breaking of scale invariance.
The models exhibits Maxwell's Electrodynamics at the weak field limit, βF 1, L → −F, i.e., holds the correspondence principle.
Magnetized Kerr black holes
We consider the electromagnetic energy density to be small enough that the Kerr vacuum solution is valid. We have also considered the rotation parameter is small enough (consistent with astrophysical evidence of evolution of stars) that the Kerr-solution is almost nearly static (with rotation parameter a ≤ 0.1) over the magnetized NLED background.
The geometry of Kerr space-time metric is
ds 2 = −dt 2 + 2M r Σ dt − a sin 2 θdφ 2 + Σ ∆ dr 2 + Σdθ 2 +(r 2 + a 2 ) sin θdφ 2 ,(6)
where Σ = r 2 + a 2 cos 2 θ, ∆ = r 2 + a 2 − 2M r.
Here M is the mass-energy parameter of the BH and a is its rotation parameter. The thermodynamic quantities of a Kerr BH can be expressed in terms of its horizon radius r + = M + √ M 2 − a 2 , which is defined by taking ∆ = 0. The magnetic mass of the Kerr BH is
M (r) = r 0 ρ M (r)r 2 dr,(8)
where ρ M is the magnetic energy density [12]. The magnetic mass of the BH is m M = ∞ 0 ρ M (r)r 2 dr. The magnetic energy density (E=0), found from Eq.??, is
ρ M = T 0 0 = F 1 + (βF) γ .(9)
For the magnetized black hole [13] one can find the field-strength with magnetic charge(q) and at distance r is F = q 2 /(2r 4 ). We set for simplicity the value of q = 1.
Similar to [12] we have also considered γ = 1 2 . We have cross-checked our results with rotation parameter a = 0 and found consistent with results in [12].
Thermodynamic Parameters
In this section we are summarizing few important thermodynamic parameters [22] relevant for our analysis.
• The Bekenstein-Hawking entropy is
S BH = π(r 2 + + a 2 ),(10)
• The Hawking temperature of the black hole horizon is
T H = 1 2π r + r 2 + + a 2 − 1 2r + ,(11)
• The angular momentum(J) of the black hole is
J = a 2r + (r 2 + + a 2 ),(12)
• The specific heat capacity at constant angular momentum is
C J = T H ∂S BH ∂T H J = 2π(r 2 + − a 2 )(r 2 + + a 2 ) 2 3a 4 + 6r 2 + a 2 − r 4 + .(13)
Numerical analysis
In our numerical analysis we have considered three different values of γ = 1 2 , 4 5 and 1. We assume the value of spin parameter (a) is slowly rotating or nearly static. Let us comment briefly on that. Some massive stars especially stars with initial masses M > 20M Sun , produce BHs upon core-collapse. The resulting stellar remnant is a rotating Kerr BH, whose dimensionless spin is defined as a = Jc GM 2 , where J is the angular momentum of the BH, c is the speed of light, G is Newton's Gravitational constant and M is the mass of the BH [21]. The values of spin parameter (a) can be in wide ranges: from 0.006 to 0.549 [21]. The numerical values of a = 0.1(0.01) could be possible for a BH formed from a single massive star irrespective of the angular momentum. The BH speed rates are measured in X-ray binaries (XRBs), and current estimates suggest a broad range of spin-rates 0.1 ≤ a ≤ 1 [3].
γ = 1 2
We have plotted the entropy (in the left-panel of Fig.1 ) as a function of radial co-ordinates. The entropy is almost identical for both of the two-spin parameters under consideration.
We have plotted the Hawking temperature (right-panel of Fig.1) as a function of radial co-ordinates. It seems that the Hawking temperature is positive approximate at r ≥ 0.02 (0.2) for spin parameter a = 0.01(0.1) with solid-blue(dashed-red).
In the left(right)-panel of Fig.2 we have plotted the angular momentum (specific heat at constant angular momentum). It reflects the fact that with low(high) spin parameter the angular momentum is small(large). The specific heat is discontinuous at r = 0.02(0.2) for the spin parameter a = 0.01(0.1) and the BH undergoes second order phase-transition. In this region of r the Heat capacity is negative and the BH is unstable. Moreover for larger r the specific heat (C J ) is always negative it suggests that once the BH undergoes instability it never comes back to its original stable position. We have plotted the Entropy (in the left-panel of Fig.3 ) as a function of radial co-ordinates for γ = 4 5 . The entropy is almost identical for both of the two-spin parameters. The values of spin parameter for nearly static Kerr solutions are small enough to show-up any appreciable changes.
We have plotted the Hawking temperature (right-panel of Fig.3) as a function of radial coordinates. It seems that the Hawking temperature is positive approximate at r ≥ 0.225 (0.7) for spin parameter a = 0.01(0.1) with solid-blue (dashed-red). It turns out that the large values of γ with respect to Fig.1 the BH instability occurs for larger values of radial parameters.
In the left(right)-panel of Fig.4 we have plotted the angular momentum (specific heat at constant angular momentum). It reflects the fact that with low(high) spin parameter the angular momentum is small(large). The specific heat is discontinuous at r = 0.0225(0.7) for the spin parameter a = 0.01(0.1) and the BH undergoes second order phase-transition. In this region of radial distance the specific heat capacity (C J ) is negative and the BH is unstable. The allowed values of γ is 0 ≤ γ ≤ 1 and we considered the extreme value. We have plotted the Entropy (in the left-panel of Fig.5 ) as a function of radial co-ordinates. The entropy is almost identical for both of the two-spin parameters. As the values of spin parameter for near static Kerr solutions are small enough to show-up any appreciable changes.
We have plotted the Hawking temperature (right-panel of Fig.5) as a function of radial coordinates. It seems that the Hawking temperature is positive approximate at r ≥ 0.31 (0.71) for spin parameter a = 0.01(0.1) with solid-blue (dashed-red). It seems that the large values of γ with respect to Fig.1 the BH instability occurs for the large values of radial parameters.
In the left(right)-panel of Fig.6 we have plotted the angular momentum (specific heat at constant angular momentum). It reflects the fact that with low(high) spin parameter the angular momentum is small(large). The specific heat is discontinuous approximately at r = 0.31(0.71) for the spin parameter a = 0.01(0.1) and the BH undergoes second order phasetransition. For larger r the specific heat capacity (C J ) never comeback to positive values so we can say that BH is unstable.
Summary and Conclusion
We have analyzed the thermodynamic properties of magnetized Black-hole in presence of the non-linear electrodynamics with two independent parameters β and γ. In our numerical analysis we set β = 1 throughout. We assumed the magnetized Black-hole is rotating very slowly with two values of rotation parameter (a) 0.01 and 0.1 (both normalized to mass), so that nearly static Kerr metric could be used for the magnetized BH. The correspondence principle takes place so that for the non-rotational model the metric is converted into the Schwarzschild metric in the vanishing limits of a-parameter. We have analyzed several thermodynamic parameters, for example Black-hole entropy, Hawking temperature, angular momentum and specific heat capacity at constant angular momentum. We have also demarcated the regions where the Black-holes undergoes second-order phase transitions and regions of instability.
Figure 1 :
1The Entropy (left) and Hawking temperature (right) for γ
Figure 2 :
2The angular momentum(J) and the specific heat at constant angular momentum (C J ) for slowly rotating Kerr BH (with spin parameter, a=0.01 with solid-blue and 0.1 with dashed-red) in non-linear electrodynamics. The value of γ is
Figure 3 :Figure 4 :
34The Entropy (left) and Hawking temperature (right) for γ The angular momentum(J) and the specific heat at constant angular momentum (C J ) for slowly rotating Kerr BH (with spin parameter, a=0.01 with solid-blue and 0.1 with dashed-red) in non-linear electrodynamics. The value of γ is
Figure 5 :Figure 6 :
56The Entropy (left) and Hawking temperature (right) for γ The angular momentum(J) and the specific heat at constant angular momentum (C J ) for slowly rotating Kerr BH (with spin parameter, a=0.01 with solid-blue and 0.1 with dashed-red) in non-linear electrodynamics. The value of γ is 1.
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|
[] |
[
"Interpretable Preference-based Rein-forcement Learning with Tree-Structured Reward Functions",
"Interpretable Preference-based Rein-forcement Learning with Tree-Structured Reward Functions"
] |
[
"Tom Bewley [email protected] \nUniversity of Bristol Bristol\nUnited Kingdom\n",
"Freddy Lecue [email protected] \nThales Montréal\nCanada\n"
] |
[
"University of Bristol Bristol\nUnited Kingdom",
"Thales Montréal\nCanada"
] |
[
"IFAAMAS"
] |
The potential of reinforcement learning (RL) to deliver aligned and performant agents is partially bottlenecked by the reward engineering problem. One alternative to heuristic trial-and-error is preference-based RL (PbRL), where a reward function is inferred from sparse human feedback. However, prior PbRL methods lack interpretability of the learned reward structure, which hampers the ability to assess robustness and alignment. We propose an online, active preference learning algorithm that constructs reward functions with the intrinsically interpretable, compositional structure of a tree. Using both synthetic and human-provided feedback, we demonstrate sample-efficient learning of tree-structured reward functions in several environments, then harness the enhanced interpretability to explore and debug for alignment.We now introduce the key assumption that differentiates our approach from prior work and enables interpretability. That is, we constrain the feature function so that for all ( , ) ∈ S × A, ( , ) is a one-hot vector. This effectively induces a partition of the state-action space into disjoint subsets, which map to the 2 The importance of -bounding is revealed here: it ensures that applying Φ −1 cannot yield infinite values. 3 See Appendix A for a brief discussion of the psychological assumptions underlying the feature expectation decomposition.
|
10.5555/3535850.3535865
|
[
"https://arxiv.org/pdf/2112.11230v1.pdf"
] | 245,353,680 |
2112.11230
|
6e86872620166491d75eec2c2306495b764ddc5a
|
Interpretable Preference-based Rein-forcement Learning with Tree-Structured Reward Functions
May 9-13, 2022
Tom Bewley [email protected]
University of Bristol Bristol
United Kingdom
Freddy Lecue [email protected]
Thales Montréal
Canada
Interpretable Preference-based Rein-forcement Learning with Tree-Structured Reward Functions
IFAAMAS
. of the 21st International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2022)Online18May 9-13, 2022
The potential of reinforcement learning (RL) to deliver aligned and performant agents is partially bottlenecked by the reward engineering problem. One alternative to heuristic trial-and-error is preference-based RL (PbRL), where a reward function is inferred from sparse human feedback. However, prior PbRL methods lack interpretability of the learned reward structure, which hampers the ability to assess robustness and alignment. We propose an online, active preference learning algorithm that constructs reward functions with the intrinsically interpretable, compositional structure of a tree. Using both synthetic and human-provided feedback, we demonstrate sample-efficient learning of tree-structured reward functions in several environments, then harness the enhanced interpretability to explore and debug for alignment.We now introduce the key assumption that differentiates our approach from prior work and enables interpretability. That is, we constrain the feature function so that for all ( , ) ∈ S × A, ( , ) is a one-hot vector. This effectively induces a partition of the state-action space into disjoint subsets, which map to the 2 The importance of -bounding is revealed here: it ensures that applying Φ −1 cannot yield infinite values. 3 See Appendix A for a brief discussion of the psychological assumptions underlying the feature expectation decomposition.
INTRODUCTION
It has recently been argued that the paradigm of reinforcement learning (RL) , in which agents learn action-selection policies by exploration to maximise future reward, is sufficiently general to give rise to most, if not all, aspects of natural and artificial intelligence [24]. However, the origin of the reward signal itself has received limited research attention compared with the algorithms used to optimise it. Singh et al. [25] argue that the traditional presentation of reward, as a known and hard-wired property of an agent's environment, is misleading with respect to both biological plausibility and real-world technical applications. Motivated by evolutionary biology, they propose a differentiation between a latent fitness function , which produces a scalar evaluation of the true quality of a behavioural trajectory, and the reward function , which is a mere proxy to convert global pressures on fitness into local pressures on immediate decision-making. may have a nontrivial relationship to because it must be tailored to both the learning dynamics of the agent and the structure of the environment. During the learning process, two nested optimisation loops are at work: inner adaptation of the RL agent within a given reward structure , and outer adaptation of to better align it with fitness.
As RL becomes more powerful, the engineering of aligned reward functions will become "both more important and more difficult" [10]. The prevailing reliance on heuristic approaches, which already presents challenges to real-world deployment [19] and hampers the use of RL non-experts [28], is likely to prove inadequate in the face of vastly more capable agents which can exploit any specification errors. Several alternatives to heuristic reward engineering have been proposed, including inverse RL [1], inverse reward design [15], and advice-taking mechanisms [20], all of which involve interaction with a human with (possibly tacit) knowledge of . In this paper, we focus on yet another human-in-the-loop approach, preference-based RL (PbRL) [28], in which a reward function is inferred from preferences expressed by the human over sets of candidate behaviours, indicating which have higher fitness. Given a dataset of preference labels, is constructed to reward commonly-preferred states and actions, and this function is used to train the agent. While ostensibly restrictive, preference feedback places low cognitive demands on the human, requires little domain expertise or training, and tends to yield lower variance than direct fitness labelling since it does not require the human to fix an absolute scale [14].
The outcome of PbRL is a policy whose true fitness depends on its expected reward under , and the alignment of to . Without ground-truth knowledge of , it is hard to define quantitative metrics for the latter, which instead becomes a fuzzy, multi-faceted judgement, requiring the assessor to build an intuitive understanding of the structure of and its effect on agent learning. In large part, this is an interpretability problem. The importance of interpretability for human-in-the-loop RL has been highlighted in surveys [3,19], and some post hoc analysis has been applied to learnt reward functions to gain some insight into feature influence [21,23], but to our knowledge, there have been no efforts to make intrinsically interpretable (loosely speaking, human-readable) by constraining its functional form. Indeed, prior works implement reward functions as deep neural networks, or ensembles thereof [8], which are notoriously opaque to scrutiny.
We present a PbRL algorithm that learns intrinsically interpretable reward functions from human preferences. Specifically, it yields tree-structured reward functions, formed of independent components associated with disjoint subsets of the state-action space, and defined hierarchically as a binary tree. The tree is incrementally refined as new preference labels arrive, and the traceability of these changes provides a powerful mechanism for monitoring and debugging. Trees afford both diagrammatic and geometric visualisation, textual description as a rule set in disjunctive normal form, and the efficient computation of feature importance metrics. Maintaining an explicit uncertainty estimate for each reward component also facilitates active preference learning based on upper confidence bounds. We evaluate our algorithm on four benchmark RL problems using both synthetic and human feedback, and in both offline and online learning settings. We observe aligned and sample-efficient learning of tree-structured reward functions in each of these contexts, alongside some informative failure cases. We then harness the enhanced interpretability to explore and debug for alignment. This paper is structured as follows. Section 2 reviews the general PbRL problem definition, section 3 introduces the additional treestructuring constraint that we apply to improve interpretability, and section 4 presents our algorithm for approximately solving the constrained problem. Section 5 gives experimental results with a focus on quantitative performance metrics, while section 6 uses case studies to demonstrate the qualitative interpretability benefits of the tree structure for the purpose of alignment. Finally, section 7 briefly concludes and discusses directions for future work.
PBRL PROBLEM DEFINITION
The PbRL problem is formalised within a Markov Decision Process without reward (MDP\R) [1], in which at discrete time , an agent's action ∈ A influences the evolution of an environmental state ∈ S according to a Markovian dynamics function ( +1 | , ). We specifically consider fixed-length episodic MDP\Rs, in which is initially 0, 0 is sampled from an initial distribution 0 , and the process deterministically terminates at a fixed = . The events of an episode are described by a trajectory in state-action space, = (( 0 , 0 ), ..., ( −1 , −1 )) ∈ (S × A) . There also exists a human observer, who evaluates the quality of trajectories according to the latent fitness function : (S × A) → R. The ultimate goal of the agent is to learn an action selection policy ( | ) that maximises the expected fitness over induced trajectories:
argmax E ∼Pr( | 0 , , ) ( ).
(1)
In order to learn about , the agent must interact with the human. In PbRL, we assume the human cannot specify the analytical form of , or even evaluate it absolutely for a given trajectory, but can only assess the relative fitness of a trajectory pair , and provide a label ∈ [ , 1 − ] indicating their assessment of the probability that has higher fitness than (denoted by ≻ ). Here, ∈ (0, 0.5] is a noise parameter preventing extreme probabilities. Agent-human interaction therefore consists of sampling trajectory pairs from a distribution : ((S × A) ) 2 → [0, 1] and obtaining preference labels in response. The inference of reduces to minimising some loss ℓ over labelled pairs:
argmin E ( , )∼ ℓ ( , ( ≻ | )).(2)
Here, is a statistical model of human preference labels given estimated fitness values for a trajectory pair, whose definition is an assumption of the modelling process. Applying a basic rationality hypothesis, we can assume that the probability of the human preferring to is a monotonically non-decreasing function of the fitness difference ( ) − ( ). Specifically, we adopt Thurstone's law of comparative judgement [27], which models the fitness of a set of trajectories ( 1 ), ..., ( ) as a multivariate normal distribution with mean ∈ R and covariance ∈ R × . 1 This leads to 1 1+exp( − ) . Both models are well-established in the preference modelling literature and often behave similarly in practice, but Thurstone's better matches the statistical assumptions of our method. As we explore later, the estimation of also provides a natural mechanism for uncertainty-driven active learning. the following preference model:
( ≻ | , ) = Φ − √︁ + − 2 ,(3)
where Φ is the standard normal cumulative distribution (CDF). Following the classic approach of Mosteller [22], we note that under Thurstone's model, a label implies that the variance-scaled difference in fitness between and is proportional to Φ −1 ( ), where Φ −1 is the inverse normal CDF. 2 Therefore, a suitable choice for the labelling loss ℓ is the squared error in this variance-scaled fitness difference. Equation 2 can be rewritten as
argmin , E ( , )∼ Φ −1 ( ) − − √︁ + − 2 2 .(4)
For the agent to perform the optimisation in equation 1, and must be parameterised in a way that generalises to unlabelled trajectories. We adopt a linear model = r ⊤ n , = n ⊤ Σn , ∀ , ∈ {1.. }, where n ∈ R is a feature vector summarising the trajectory, r ∈ R is a vector of weights, and Σ ∈ R × is a covariance matrix associated with r. As in several prior works [1,2,8], we add a second level of decomposition by defining : S × A → R as a function that constructs feature vectors from individual state-action pairs, and n = −1 =0 ( , ) as the unweighted feature expectation over . 3 By the linearity of the normal distribution, it follows that − = r ⊤ (n −n ) and
+ −2 = (n −n ) ⊤ Σ(n −n ). The final form of equation 2 is thus argmin ,r,Σ E ( , )∼ Φ −1 ( ) − r ⊤ (n − n ) √︁ (n − n ) ⊤ Σ(n − n ) 2 . (5)
The decomposition also allows us to rewrite equation 1 as
argmax E ∼Pr( | 0 , , ) −1 ∑︁ =0 r ⊤ ( , ),(6)
which is structurally identical to the conventional RL objective of maximising (undiscounted) return in an MDP with reward. Thus, once the agent has inferred the function and vector r it can define a reward function ( , ) = r ⊤ ( , ), then employ any unmodified RL algorithm to learn a policy . As alluded to in the Introduction, thereby serves as a proxy for , with the true fitness of being a function of both its expected return under (equation 6) and the alignment of to , which is approximated by the labelling loss (equation 5). Given that equation 6 is the domain of standard RL, the contribution of this paper is an algorithm for approximately solving equation 5, subject to the particular interpretability constraint outlined in the following section. possible one-hot vectors. For trajectory , n can be interpreted as the number of timesteps spent in each subset, and the reward vector r as a set of components that reward state-action pairs according to the subsets they fall within. We model reward components as independent, so that Σ is a diagonal matrix. Furthermore, the partition induced by has a binary tree structure, with the subsets as leaves, connected by a hierarchy of internal nodes emanating from a root. Each internal node applies a test to the state-action pair ( , ). If the test is passed, the logical flow proceeds to the "right" child node. Otherwise, it proceeds to the "left" child. Testing continues until a leaf node ∈ {1.. } is reached, the state-action pair is mapped to the corresponding one-hot vector, and the reward is given as the corresponding component r with variance Σ . Although most of our algorithm does not rely on this assumption, we focus here on Euclidean state and action spaces S = R , A = R , in which a state-action pair is a vector ( , ) ∈ R , where = + . Internal node tests have the form ( , ) ≥ : ∈ {1.. }, which evaluates whether the th element of ( , ) meets or exceeds a threshold . Consequently, each state-action subset has the geometry of an axisaligned hyperrectangle. Figure 1 provides an illustrative example. For a trajectory , n counts the timesteps (white circles) spent in each subset, and its dot product with r gives the mean fitness estimate . Each component also has a variance Σ , which can be used to compute the trajectory fitness variance = Σ (not shown).
PROPOSED ALGORITHM
We now present an algorithm for optimising equation 5 subject to the tree-structuring constraint on , alongside an active learning scheme that adapts the sampling distribution to focus on trajectories with high-variance fitness estimates and correct overestimation errors. As the algorithm iterates over stages of preference elicitation, reward component fitting, tree structure refinement and distribution updates, there is no single first step. We have tried to order the following subsections to maximise comprehensibility.
Remark. The indirect, multi-stage optimisation approach described here was converged on after extensive experimentation with alternatives, which we outline in Appendix B. The final approach is computationally efficient, easy to implement, and yields reward functions that are significantly more robust to small data changes than the alternatives.
Preference Elicitation and Representation
We first outline the process of sampling trajectory pairs and storing preference labels. We assume a finite data setting, in which the trajectory space (S × A) is approximated by a sequence of trajectories T = ( 1 , ..., ), and the domain of is restricted to T 2 . This distribution can thus be written as a matrix Ψ ∈ [0, 1] × :
Ψ = 1, Ψ = 0, ∀ ∈ {1.. }.
For the moment, let us take both T and Ψ as given; we discuss their origins in section 4.5. An instance of preference feedback is obtained by sampling a trajectory pair , with probability Ψ , presenting the pair (e.g. by visualisation) to a human, and recording the resultant preference label ∈ [ , 1− ]. We represent a set of feedback instances by three data structures, a set P, a matrix ∈ {−1, 0, 1} × and a vector y ∈ R , which are incrementally assembled as preferences arrive. After sampling the th pair , and observing , we add { , } to P, append a row to in which the th element is 1, the th element is −1 and all other elements are 0, and append y = . P thus serves as a record of which pairs have been sampled. and y allow us to express equation 5 in matrix form as follows:
argmin ,r,Σ Φ −1 (y) − (diag( ⊤ Σ ⊤ ) − 1 2 ) ⊤ ⊤ r 2 .(7)
Here, is the × matrix of columnar feature vectors n . This notation is used in equations throughout this section.
Trajectory-Level Fitness Estimation
Direct optimisation of equation 7 subject to the tree-structuring constraint is computationally intractable, so we approximate the global problem by a sequence of local ones. First, we temporarily apply Thurstone's Case V reduction [27], which assumes unit standard deviations for all fitness differences, i.e. diag( ⊤ Σ ⊤ ) 1 2 = 1 , so simplifies the bracketed expression in equation 7 to Φ −1 (y) − ⊤ r. Following the method developed by Morrissey and Gulliksen [13], we then use least squares to compute a somewhat biased (by the Case V assumption) estimate of ⊤ r = , which is the × 1 vector of mean fitness estimates at the level of complete trajectories:
⊤ r = ≈˜= ( ⊤ ) −1 ⊤ Φ −1 (y).(8)
Independent Reward Component Fitting
We then use the trajectory-level estimates˜to fit the means and variances of the reward components. This is a kind of temporal credit assignment problem: how does each component contribute to the overall fitness of a trajectory ? To solve it, we recall that under our tree-structuring constraint, is the number of timesteps spends in the th state-action subset. A priori, we cannot know which timesteps are responsible for the fitness of , so the least strong assumption is a uniform temporal prior, that the reward component for this subset contributes a fraction / of the fitness. We can thereby compute an empirical estimate of the th reward component mean by taking a weighted sum over trajectories:
r = ∈ P˜ ∈ P .(9)
Note that this summation is only over trajectories for which at least one preference label has been provided (denoted, in a slight abuse of notation, by ∈ P). We then use˜and r to empirically estimate the variance of the th component, Σ :
Σ = RSS( ) ∈ P ; RSS( ) = ∑︁ ∈ P ˜− r 2 .(10)
The residual sum of squares RSS(·) is a useful intermediate function that we use again below. By this method, reward components are fitted independently, using only the corresponding rows of . The independent treatment of components leaves Σ as a diagonal matrix.
Tree Structure Refinement
So far, we have assumed a fixed matrix, and thus a fixed feature function , but this function can be modified by adding or removing internal nodes in the tree structure described above. Consider splitting the th leaf node into two by replacing it with an internal node which tests whether the th element of the ( , ) vector exceeds a threshold . The effect of this split on the feature matrix is to replace the th row of with a new 2 × matrix, denoted by [ ] , representing how the number of timesteps each trajectory spends in the th state-action subset is distributed between two new child subsets contained within it. Using the two rows of [ ] , reward components for the child subsets can be fitted using equations 9 and 10. All other reward components remain unchanged. Repeatedly making such splits grows the tree, each time increasing the number of leaves, subsets and associated reward components by 1. Whenever the trajectory-level fitness estimates˜are updated, our algorithm grows the existing tree by recursive splitting up to a maximum = max , at each step choosing a component to split , splitting dimension and threshold to greedily minimise the total residual sum of squares =1 RSS( ), and thus achieve a better fit to˜. Since reward components are fitted independently, a single step of this optimisation process can be expressed as follows:
max 1≤ ≤ max 1≤ ≤ max ∈ C RSS( ) − RSS( [ ] 1 ) − RSS( [ ] 2
). (11) C is a set of candidate split thresholds along dimension . In our experiments, where dataset sizes are moderate, we exhaustively search over all values that occur in labelled trajectories:
C = {( , ) , ∀( , ) ∈ , ∀ ∈ P}.
Crucially, the RSS-based splitting criterion in equation 11 is precisely the one used in classical regression tree learning [6]. Our algorithm thus utilises a virtuallyunmodified, highly-optimised, regression tree implementation.
Once max is reached, we then iterate backwards through the growth process, pruning the tree back until = 1. At each step in this backward pass, we use the corresponding , r and Σ to evaluate the labelling loss expression given in equation 7 (i.e. the global objective that we aim to optimise). To this labelling loss, we add a complexity regularisation term to modulate the tradeoff between predictive accuracy and interpretability (through compactness) and also to mitigate overfitting. We identify the tree size that minimises the regularised labelling loss, and use this tree as the updated .
Trajectory Pair Sampling Distribution
Our algorithm works in two data settings: offline and online. In the offline setting, the underlying trajectory dataset T remains fixed but the sampling matrix Ψ is modified over time. In the online setting, both Ψ and T change, with the latter being gradually augmented with new trajectories. This makes it possible to use trajectory data generated by the PbRL agent itself as it learns a policy in real-time.
Offline setting. A wide variety of active preference learning schemes have been proposed for PbRL [26]. Partly inspired by existing work in the bandit literature [7,30] we adopt an upper confidence bound (UCB) strategy, which weights trajectory pairs according to optimistic estimates of their summed fitness. This strategy prioritises trajectories with highly uncertain fitness under the current model, for which additional preference labels are likely to be most useful for reducing uncertainty (in this respect, it is similar to [8]). Additionally, the optimism induces a bias towards identifying and correcting cases where trajectory fitness is overestimated, ultimately yielding a conservative reward function which counteracts the well-known overestimation bias in value-based RL [12]. Finally, biasing the preference dataset towards promising trajectories leads the reward function to prioritise distinguishing between high and very high fitness behaviour (rather than low vs very low), which reduces the risk of an agent stagnating at mediocre fitness with no incentive to improve. To implement the UCB strategy we use , r and Σ to compute a vector of optimistic fitness values:
u = + diag( ) 1 2 = ⊤ r + diag( ⊤ Σ ) 1 2 ,(12)
where ≥ 0 determines the number of standard deviations added to the mean. We then construct an × weighting matrix as follows:
off = 0 if = or { , } ∈ P, or (P ≠ ∅ and ∉ P), u + u + otherwise, ∀ ∈ {1.. }, ∀ ∈ {1.. }.
(13) Here, the three "zeroing" conditions respectively prevent comparing a trajectory to itself, prevent repeated pairs, and ensure that one of any sampled pair has already received feedback. 4 The offset is calibrated so that the minimum element not matching a zeroing condition is set to 0. 5 If all elements match a zeroing condition, then all possible trajectory pairs have been sampled and the process must be halted. Otherwise, we define Ψ = off / off .
Online setting. If T monotonically expands with new trajectories over time, but preference labels are obtained at a constant rate, it is possible to show that a higher density of labels is given to trajectory pairs that appear earlier, creating a strong earliness bias in the preference dataset. Assuming a total labelling budget max and known final trajectory count max , we correct for this bias by collecting a batch of labels every time new trajectories are added, using monotonically-increasing batch sizes. On the th batch, we define on the same as off , with the extra condition that on = 0 if ≤ ( − 1) and ≤ ( − 1), which ensures that at least one of and are in the most recent trajectories. We compute Ψ by normalising on as above, then sample trajectory pairs, where
= round max 2 (2 − 1) − max ( max − 1) .(14)
We refer the reader to Appendix C for a more detailed justification of this approach and a derivation of equation 14. Note that the offline setting is recovered by setting = max = |T |. 4 The last condition ensures that the graph representing the set of pairwise comparisons P is connected, meaning there is a path between any two ∈ P, ∈ P. As shown by Csató [9], this is necessary for the least squares solution in equation 8 to be unique. 5 Unless this is also the maximum element, in which case it is offset to a positive value (nominally 1) to ensure that off > 0 and prevent a divide-by-zero error.
Complete Algorithm
Our algorithm initiates with one reward component ( = 1) and
r = [0], Σ = [0]
. During batch , labelling is paused every samples for an iteration of trajectory-level fitness estimation, reward component fitting and tree refinement. This leads to a modified u vector, which alters the sampling matrix for the rest of the batch.
In the online setting, where an RL agent uses the learnt reward to train a policy in real-time, the latest and r are used as soon as an update is made. Otherwise, the algorithm runs until the budget max is expended, and the final fixed reward is used to train a policy at a later time. A pseudocode algorithm is given in Appendix D.
PERFORMANCE RESULTS
We have evaluated our algorithm in four RL environments under various learning conditions. In all cases we used a feedback budget max between 600 and 620, equating to ≈ 1 hour of human time. Full experimental details are reported in Appendix E; below we discuss our findings by reference to the blue annotation letters in figures 2-5. Key findings are in bold and summarised at the end.
Offline with Oracle Feedback (Figure 2)
Here, T was fixed, and comprised of trajectories generated by an RL agent (the pilot agent) as it trained on each environment's default, hand-engineered reward function. Preference labels came from a synthetic oracle with query access to this reward function. We then trained a second agent (the PbRL agent) on the learnt reward, and finally measured the PbRL agent's alignment to the original reward function, which was taken to represent groundtruth fitness; see [16] for a similar method of quantitative evaluation. We show learning curves (time series of fitness per episode) for both learnt (a) and ground-truth (b) reward (5 repeats; mean and min-max range shown). The consistent monotonicity of the former indicates that tree-structured reward functions gave rise to stable agent learning in all cases, and the latter confirm that this learning was well-aligned with the ground-truth. For Pendulum and RoboCar, asymptotic fitness was indistinguishable from the pilot, while for LunarLander and FoodLava, it was slightly below. We show additional plots for FoodLava. (c) shows that the -regularised labelling loss was minimised by a tree with 9 leaves, hence 9 reward components. (d) gives another measure of alignment by plotting true vs learnt fitness (±1 std) for both the pilot trajectories T and those generated during PbRL agent training. In both cases there is a clear positive correlation, although the relationship for the latter is noisier, suggesting a degree of distributional shift. (e) visualises the 9 reward components over the two state dimensions as coloured rectangles. 6 The components are arranged isomorphically to the maze layout, with negative reward in the red "lava" region and positive reward around the green "food". However, some misalignment is visible as high-reward component 8 is too large in the vertical direction. Referring to (f), which plots the final 10 trajectories of all 5 PbRL repeats, we see that this misalignment led to policies that sometimes terminated just below the food. shows how the regularised labelling loss varied as a function during the tree pruning sweep performed after each batch. The overlaid white curve shows how the tree size was modified accordingly to track the optimal . As batches accumulated, the global pattern was that first increased to a maximum, then remained somewhat below that maximum thereafter, with large changes becoming less frequent. Inspecting the model at three checkpoints during training, we find that (c) the positive correlation between true and learnt fitness became less noisy over time, and (d) the reward components converged to an arrangement that positively rewarded both proximity to the goal and facing towards it (bearing ≈ 0), doing so in an almost-symmetric manner.
Online with Human Feedback (Figure 5)
Finally, we ran the algorithm online using feedback from a single human participant. Due to the labour-intensiveness of this experiment, we focused on two environments -FoodLava and LunarLanderboth of which revealed the risk of prematurely fixing the reward structure. For both, we initially used max = 100 as in the oraclebased experiments, and found the agent quickly converged to high fitness according to the learnt reward function (a; dotted lines) but after episode max , gradually lost fitness according to the groundtruth reward (b). This indicated that the reward structure had been fixed in a state that was only partly aligned, so that further optimisation hindered true performance. Using rectangle plots of the final reward components, we can see that for FoodLava (c), the maximum positive reward component was placed around the food region, but was "loosely" targeted as it exceeded the bounds of the food. The overlaid final trajectories show that the agent learnt to seek out this high reward, but sometimes stop short of the food itself. For LunarLander (d), a similarly "loose" reward function was learnt that gave high reward for a vertical position close to zero, regardless of the vertical velocity. The trajectory overlay indicates this reward function led the agent to maintain high negative velocity as it approached the ground, which was recognised as a crash landing by the ground-truth reward function. For both environments, we completed a second run using a higher max , thereby distributing the same feedback budget over a larger fraction of the training process, providing more time for the human to reactively fine-tune alignment. For these runs, we found that ground-truth fitness (b) almost matched the pilot agent from the offline oracle experiment, with no sign of a performance dropoff. For FoodLava, the rectangle plot of reward components (e) reveals a much smaller area of maximum reward that did not exceed the bounds of the food region and thus incentivised the agent to reliably enter it. For LunarLander (f), we had a very different reward structure to the first run, which gave positive reward for maintaining slow vertical velocity, and negative reward for exceeding a velocity threshold of −0.55, rather than merely rewarding reaching the ground. This incentivised the agent to gradually decelerate as its height decreased, resulting in softer landings that were not registered as crashes by the ground-truth reward function. For both environments, we show a learning timeline for the second run (g). We again see the trend of increasing to a maximum before stabilising at a lower value. The checkpoints also show the relationship between true and learnt fitness becoming less noisy over time (h) and several intermediate reward component layouts that emerged during training (i).
Summary of Key Findings. • With no exceptions, the tree-structured reward functions gave rise to stable, convergent reinforcement learning. • Using several hundred instances of oracle feedback based on ground-truth reward functions, our algorithm could reconstruct those reward functions sufficiently well to train PbRL agents whose performance nearly matched that of conventional RL agents. • The aggregated preferences of 62 human participants (offline), as well those of a single participant (online), yielded learnt reward functions that were similarly well-aligned to the groundtruth. This is despite participants having no direct knowledge of the ground-truth, instead relying on intuitive task understanding alone. • In the offline setting, dataset biases led to causal confusion, where the learnt reward incentivised state-action pairs that commonly appeared alongside high-fitness behaviours, as well as the behaviours themselves. Careful rebalancing of training data, or moving to an online learning setup, would both help to mitigate this problem. • In the online setting, the main failure mode was "loose" alignment due to fixing the reward prematurely. Increasing max gave more time to reactively correct for misaligned changes in behaviour. • In the online setting, the number of components tended to initially increase rapidly, then stabilise somewhat below the maximum later on, with large changes becoming less frequent.
• In all human experiments, we did not encounter a single tree with a split along an action dimension, so that all rewards were a function of state only. We are wary to make a general claim about this result, but it is consistent with a recent suggestion that human teaching focuses on states over actions [17], and indicates future PbRL work may succeed by learning state-dependent rewards only.
INTERPRETABILITY DEMONSTRATION
The analysis of rectangle plots above, which provides insight into the learnt reward functions and their effect on behaviour, exemplifies the interpretability benefits of the tree-structuring constraint. Figure 6 demonstrates these benefits further by focusing on two specific PbRL runs: one failure case and one success case.
Failure case: RoboCar using offline human feedback. As is visible in figure 3 (l), this reward function led to policies that sometimes reached the goal as desired, but other times made no move towards the goal and appeared to seek only to maintain a vertical position close to 0. We can diagnose this misalignment by examining the full reward function tree ( figure 6; a). Here, the splitting dimensions are = vertical position (initialised to 0), = distance to goal, and = bearing to goal in radians. The first two splits appear wellaligned, creating a component with maximum reward for achieving < 1.16 and a smaller positive reward for < 5.84. The remaining splits are problematic, creating components that penalise moving out of the region ∈ [−1.64, 1.68] and, otherwise, reward a bearing outside of ∈ [−2.15, 2.18] (i.e. facing away from the goal). To understand this, consider the design of the environment. In each episode, the goal position is randomised but the car is initialised facing to the right, making it easier to reach the goal when it is also to the right. Hence, a majority of goal-reaching trajectories in the pilot run dataset showed the car driving directly forward, rarely exiting a narrow corridor around = 0. The splits to penalise large absolute are thus an example of causal confusion, in which behaviour correlating with a high fitness outcome is mistaken for being high fitness in itself, and would not appear if the environment were differently initialised or the dataset better balanced. We give a similar, if subtler, causal confusion justification of the -based splits in Appendix F. The heatmaps (b), (c) and (d) provide fine-grained insight into the effect of the misaligned reward function on the learning dynamics of one of the 5 PbRL repeats. (b) represents the timesteps spent in each component -the matrix -over the 1000episode training history. Multiplying row-wise with the mean reward vector r we obtain (c), which gives per-episode reward from each component, and can be understood as a decomposed learning curve. Summing (c) column-wise gives (d), the total fitness for each episode, which is a conventional learning curve. From these, we find that the agent quickly (by episode 50) learned to avoid negative-reward components 3 and 8, inducing an early bias towards exiting the ∈ [−1.64, 1.68] corridor. With this bias in place, exploration was curtailed and the agent largely settled into the moderate positive rewards of components 4 and 7. Although there was a gradual increase in visitation to component 1 (the one corresponding to reaching the goal) in the first half of training, the agent never completely prioritised this component, with visitation peaking around episode 700 before dropping off again. In (e), we harness the component structure to construct textual report cards for two episodes near the end of training (950 and 975) which describe the state-action subsets that were visited. While both are in the top 10% of episodes by performance on the learnt reward, the former is aligned (obtaining positive reward from components 1 and 2) while the latter is not (staying entirely in component 6, thereby driving straight ahead despite the goal being behind it).
Success case: FoodLava using online oracle feedback. In this run (chosen randomly from the 5 repeats), we achieved aligned learning. For the first max = 100 episodes, a label batch was obtained at intervals of = 10 and the tree structure incrementally updated by growth and pruning. (f) depicts the net changes resulting from each update using both rectangle plots and a graph of the split/merge dependencies between components from batch-to-batch. Key events in the construction of the reward function included the merging of four components into one at = 4, and the corrective splitting, merging, and re-splitting (at a different threshold) of a component between = 8 and = 10, yielding the final maximum-reward component 9 whose subset boundaries (horizontal position ≥ 7.95, vertical position ≥ 8.06) lined up almost exactly with the food region ( ≥ 8, ≥ 8). From = 2 onwards, component 1 was persistent, being neither split nor merged. However, its mean and variance were continually refined as preference labels arrived, a process that we visualise in (g). For each batch, every trajectory that had been labelled so far (of which there are more for later batches) is shown as a black horizontal line, whose vertical position corresponds to its trajectory-level fitness estimate˜(divided by the episode length = 200) and whose width is proportional to the time spent in component 1, 1 . The method described in section 4.3 effectively fits a normal distribution to these lines, and the results for all batches are overlaid (mean as squares, ±1 std as shading). Between batches 2 and 10, component 1 saw a slight increase in its mean, and a gradual narrowing of its variance, as more trajectories were labelled. (h) depicts the fixed 9-component tree used for the final post fix = 300 training episodes, and (i) shows the timesteps spent in each component throughout this period. As of episode 100, roughly equal time was being spent in components 2 and 9, but the latter came to dominate around episode 250. Since component 9 corresponded to the food region, this indicated consistent, rapid solution of the navigation problem, with occasional failures (j) attributable to the agent becoming stuck in component 1.
CONCLUSION
We have presented an algorithm for interpretable PbRL using treestructured reward functions, demonstrating successful learning of compact and aligned reward functions across four environments, alongside informative and actionable failure cases due to causal confusion in the offline setting and premature reward fixing in the online setting. We have also shown the value of interpretability for exploring and debugging the learnt reward structure. In the offline setting, future work could construct trajectory datasets using unsupervised agents that optimise for behavioural diversity (e.g. [11]) instead of our somewhat contrived pilot agents. In the online setting, there is scope for larger human experiments, with a focus on ablation and hyperparameter tuning. An additional layer of interpretability could be realised by integrating our method with agent architectures that learn decomposed value functions [18].
APPENDIX A: DISCUSSION OF FEATURE EXPECTATION DECOMPOSITION
The feature expectation decomposition, in which a trajectory-level feature vector is constructed as a sum over constituent state-action pairs T −1 t =0 ϕ(s it , a it ), was popularised by Abbeel and Ng [1] and has become a standard component of work on inverse RL, PbRL and imitation learning in the years since. In the PbRL context, it encodes the assumption that a human evaluates the fitness of a behavioural trajectory by independently assessing each state-action pair, then performing an accurate unweighted sum over the trajectory length T . The validity of this assumption depends on many contextual factors, including whether a pair of trajectories is presented to the human concurrently or consecutively. In the latter case, there is a possiblity for ordering to affect a human's judgement, including via the anchoring effect [13]. Assuming a visual mode of presentation, another important distinction is whether visualisations are static (e.g. line plots) or dynamic (e.g. videos). If trajectory presentation is dynamic, evaluation is a temporally-extended experience, and the peak-end rule indicates that the human may depart from a uniform treatment of timesteps and exhibit a bias towards later (thus better-remembered) parts of the trajectory, as well as those with the highest intensity of positive or negative valence [10]. In human experiments for this paper, we present trajectories as short videos, arranged side-by-side in an on-screen window, so biases from temporal ordering should not be in effect. We also have the videos loop infinitely and enforce no time limit on the feedback process, so participants are able to review all parts of each trajectory many times. We aim for this to partially mitigate the peak-end effect.
A more fundamental issue is that humans commonly reason in terms of counterfactuals and hypotheticals. There is thus a risk of preference judgements being made on the basis of ancitipated future actions, or outcomes that an agent nearly realised but didn't, rather than the actual contents of a given trajectory as assumed in the defining equations of PbRL. In a review of best practice for human-agent teaching, Knox et al. [11] hypothesise that overall judgements draw on a mixture of observed past behaviour (about which the feature expectation vector is informative) and anticipated future behaviour (about which the vector contains no information), but also note that "positive results" have been attained by methods which ignore the latter.
The proceeding discussion serves to highlight that rather than any well-evidenced psychological validity, the advantage of the feature expectation decomposition lies in its simplicity and mathematical convenience. It is required to derive many of the equations in the main paper, not least equation 6, which expresses fitness in terms of a reward function and enables the use of a conventional RL algorithm for policy learning. Given that these conveniences come at the cost of a potentially significant mis-modelling of the human judgement process, we see the construction of a more psychologically plausible fitness decomposition for PbRL as an important direction for future work.
APPENDIX B: OTHER ATTEMPTED OPTIMISATION METHODS
Before converging on the optimisation approach given in the main paper (i.e. trajectory-level fitness estimation using Thurstone's Case V, uniform temporal credit assignment to reward components, RSSbased splitting criterion) we explored several less robust or efficient alternatives. We briefly describe these below.
Bradley-Terry model for trajectory-level fitness estimation. Prior to adopting Thurstone's Case V model for the calculation ofμ, we tried using the somewhat more popular Bradley-Terry model [2], computing maximum likelihood fitness estimates using both the iterative minorisation-maximisation (MM) algorithm [9] and the newer random walk algorithm proposed by Negahban et al. [12]. While we found the latter to be significantly faster and thus better suited to an online learning setting, both tended to yield highly skewed fitness distributions, with most trajectories assigned a fitness very close to 0, while a small number of outliers had fitness estimates several orders of magnitude larger. We found that this was problematic when it came to computing the means and variances of reward components, as the outliers dominated any smaller distinctions between the rest of the trajectory set. In contrast, the least squares method using Thurstone's Case V gave a far more even and unskewed spread of fitness estimates (following a roughly normal distribution), so that no single trajectory had an outsized impact on the reward components. We also found that Bradley-Terry fitness estimates were less robust to individual labelling "mistakes" than the least squares Case V solutions, with a single incorrect preference label often leading to a dramatic reordering of the fitness ranking over the set of trajectories. We concluded that such high sensitivity would be undesirable, especially in human experiments where the chance of a mistake was high.
Least squares regression for component fitting. Separate from the method used to estimateμ is the subsequent step of computing mean and variance estimates for individual reward components. Initially, we used the expression N ⊤ r ≈μ to frame the estimation of r as a second least squares regression problem: r ≈ (N N ⊤ ) −1 Nμ. This approach suffered from two major problems. Firstly, it tended to give brittle and poorly regularised results, with extremely high or low reward magnitudes assigned to state-action subsets that were only visited by especially high-or low-fitness trajectories (even if this was just for a single timestep) and near-zero reward for many others. Secondly, it fitted the reward components in a way that was not independent. This prevented our use of the highly-optimised classical regression tree algorithm for tree growth, and instead required a far more expensive algorithm that repeatedly computed a least squares solution for each candidate split threshold. Our final independent fitting method, based on the uniform temporal prior, solves both of these problems.
TrueSkill for end-to-end component fitting. We also tried bypassing the intermediate estimation of trajectory-level fitnessμ completely, instead translating directly from a set of preference labels {y i j } to reward component means and variances. For this, we used the TrueSkill algorithm [8], developed by Microsoft to assign skill scores to individual video game players on the basis of their history of team-level match results. We refer readers to the cited paper for algorithmic details. In our appropriation of TrueSkill, the "players" were the reward components, "skill scores" were reward values, "teams" were trajectories, "match results" were pairwise preference labels, and the algorithm's partial play feature was harnessed to weight each component x's contribution to a trajectory τ i by the fraction of time spent in the xth state-action subset, N xi /T (mirroring the uniform temporal credit assignment used in our final algorithm). TrueSkill is also grounded in the equations of Thurstone's model, and estimates a mean and variance for each player's score. The mapping of features of the algorithm into our problem context was thus remarkably well-motivated, and we found that it produced robust and highly plausible reward estimates for a fixed set of reward components (i.e. fixed tree structure ϕ). Crucially, however, the algorithm could not readily be adapted to handle the splitting or merging of reward components, as in our tree growth process, which effectively changes the number of players in a team. The only way of implementing this functionality would be to run the full algorithm from scratch whenever a change is made, which would induce a prohibitive runtime on the order of seconds for each split threshold considered during tree growth. This unfortunately meant that TrueSkill could not be used to solve the complete PbRL problem. We are nonetheless excited by the prospect of future work that uses TrueSkill as part of a preference learning framework, and are confident that the algorithm has productive applications far outside the domain of video gaming.
APPENDIX C: JUSTIFICATION AND DERIVATION OF FEEDBACK SCHEDULING
Let k max be the labelling budget and n max be the final trajectory count. Trajectories are added to T one at a time, with τ i : i ∈ {1..n max } denoting the ith trajectory added. In constant-rate labelling, we obtain k max /n max new preference labels each time a trajectory is added. Ignoring the effect of UCB weighting, and the conditions on Ψ that prevent duplicates and ensure connectivity, 1 the expected number of labels for a trajectory pair τ i , τ j is
ρ(τ i , τ j ) = n max n=max(i, j) k max /n max n(n − 1) = k max n max n max n=max(i, j) 1 n 2 − n = k max n max n max − (max(i, j) − 1) n max (max(i, j) − 1) ∝ n max max(i, j) − 1 − 1,(1)
which decreases as max(i, j) increases. Hence, earlier trajectory pairs obtain a higher density of labels. Our feedback scheduling method is designed to correct for this bias and produce a uniform label density prior to introducing UCB weighting and the conditions on Ψ. Concretely, we wish for the following to hold:
ρ(τ i , τ j ) = k max n max (n max − 1) , ∀i, j ∈ {1..n max } : i j.(2)
To achieve this, we organise the labelling process into batches, with a batch collected every time f l new trajectories are added, where n max %f l = 0. Within the bth batch (b ∈ {1..n max /f l }), we only permit the sampling of trajectory pairs for which at least one of τ i and τ j is in the most recent set of f l trajectories. This is practically achieved by adding the condition W on i j = 0 if i ≤ f l (b − 1) and j ≤ f l (b − 1). Crucially, each pair τ i , τ j only meets this condition for exactly one batch, namely the bth, where f l (b −1) < max(i, j) ≤ f l b.
The number of trajectory pairs that meet the condition for the bth batch, z b , is the difference between the current total number of possible pairs, f l b(f l b−1), and the number of pairs as of the previous batch, f l (b − 1)(f l (b − 1) − 1):
z b = f l b(f l b − 1) − f l (b − 1)(f l (b − 1) − 1) = f 2 l (2b − 1) − f l .
(3) Let k b denote the size of the bth batch. Again ignoring the UCB weighting and constraints, we assume uniform sampling from the set of condition-matching trajectory pairs. The expected label density for τ i , τ j can thus be written as
ρ(τ i , τ j ) = k b z b = k b f 2 l (2b − 1) − f l .(4)
We now wish to find an expression for k b such that equation 2 holds. This can be done by equating equations 2 and 4
ρ(τ i , τ j ) = k max n max (n max − 1) = k b f 2 l (2b − 1) − f l ,(5)
and rearranging for k b :
k b = k max f 2 l (2b − 1) − f l n max (n max − 1) .(6)
This equation holds for all b ∈ {1..n max /f l }, and thus can be used to schedule the rate of feedback so as to achieve uniform label density. Since in practice k b must be an integer, the equation given in the main paper includes a final round(·) operation.
APPENDIX D: COMPLETE ALGORITHM
Algorithm 1 provides a complete pseudocode outline of our PbRL algorithm. It includes a subfunction, updateRewardFunction, which performs an iteration of trajectory-level fitness estimation, reward component fitting and tree structure refinement given the latest preference dataset, and is written separately in algorithm 2.
Inputs. Algorithm 1 takes the following as input:
• T off : Offline trajectory dataset. In the online setting T off = [].
• n max : Final size of trajectory dataset after collection of online data. In the offline setting, n max = |T off |.
• f l : Batch frequency. In the offline setting, f l = n max = |T off |.
• λ: Number of standard deviations added to µ to create optimistic fitness estimates u for UCB sampling.
• k max : Total labelling budget.
• f u : Within-batch tree/reward function update frequency.
• m max : Maximum tree size allowed during growth stage.
• D s , D a : Dimensionality of state and action spaces.
• α: Tree complexity regularisation parameter.
• n post fix : Number of RL episodes after fixing reward function.
Subfunctions. In addition to updateRewardFunction, numerous other subfunctions are used. For brevity, we do not define these in detail, but describe them informally below, in order of appearance in algorithms 1 and 2:
• initPi: Initialise the agent's policy to some high-entropy state to encourage exploration.
• initTree: Initialise the tree with a single leaf (m = 1).
• rlOneEp: Run one episode of reinforcement learning on the current reward function, using any conventional online RL algorithm suitable for continuous action spaces.
• append: Append an element to a list or a row to a matrix.
• computeN: Use ϕ to compute the n vector for a trajectory τ .
• computeW: Compute the un-normalised weighting matrix using equation 13 of the main paper, as well as the recency condition for the online setting.
• sample: Sample pairs from T with probabilities Ψ.
• getPreferenceLabel: Query the human (or synthetic oracle) to obtain a preference label for trajectory pair τ i , τ j .
• ARow: Given trajectory pair indices i, j, construct a new row for the A matrix as described in section 4.1 of the main paper.
• numLeaves: Count the leaves of the tree.
• RSS: Perform the residual sum of squares calculation given in equation 10 of the main paper.
• computeNxdc: Compute the 2 ×n matrix N [xdc] by counting the number of timesteps each trajectory in T spends in two children of the xth leaf of ϕ, where the children are created by splitting at threshold c along dimension d.
• argmax: Return the index of the maximum of a list.
• applySplit: Update the tree by splitting the xth leaf at threshold c along dimension d.
• splitReplace: Replace the xth row of a matrix with a provided 2-row matrix, thereby increasing the number of rows by 1.
• getLastSplit: Return the index x of the most-recently split leaf in a tree.
• applyMerge: Update the tree by merging the xth and x + 1th leaves into a single one.
• mergeReplace: Replace the x and x + 1th elements of a list (respectively, rows of a matrix) with a single provided element (row), thereby decreasing the number of elements (rows) by 1.
• vecToDiag: Convert an m-dimensional vector to an m × m diagonal matrix.
• argmin: Return the index of the minimum of a list.
Offline setting as a special case. Algorithm 1 subsumes both online and offline PbRL settings, with the latter recovered by setting f l = n max = |T off | ≥ 2 and n post fix > 0. The effect of this is to ensure lines 8-11 are skipped (preventing the collection of online data) and lines 14-33 are run (creating one large label batch of size k b = k max ). The condition on line 35 is then triggered, breaking the outer loop on the first iteration, and RL training is run for n post fix episodes using a fixed reward function (lines 40-42).
APPENDIX E: EXPERIMENTAL DETAILS Environments
Our experiments were conducted in four episodic RL environments with Euclidean state-action spaces, all implemented in Python using OpenAI Gym [4]. Figure 1 contains an annotated visualisation of each environment.
Pendulum-v0. A built-in component of the Gym library and a classic feedback control problem. The task is to swing an inverted pendulum upright and hold it there for as long as possible up to a time limit of T = 200. The state dimensions are the cosine and sine of the pendulum's angle from upright θ and its angular velocity θ (D s = 3), and the pendulum is initialised in a random position. The action is a torque u on the rotational joint (D a = 1). The default reward function is R(s t , a t ) = θ 2 t + 0.1 θ 2 t + 0.001u 2 t . LunarLanderContinuous-v2. Another built-in Gym environment, in which the objective is to guide an aerial craft to a gentle landing on a landing pad surrounded by uneven terrain. We set a time limit of T = 300. The state dimensions are the craft's horiontal and vertical positions x, y and velocities v x , v y , its angle from vertical θ and angular velocity θ , and two binary contact detectors c l , c r indicating whether the left and right landing legs are in contact with the ground (D s = 8). The craft is initialised in a narrow zone above the landing pad, with slightly-randomised angle and velocities. The action is a pair of throttle values for two engines: main u m and side u s (D a = 2). For most timesteps, the default reward function gives R(s t , a t ) = shaping t − shaping t −1 , 2 where shaping t = −100
x 2 t + y 2 t + (v x t ) 2 + (v y t ) 2 + |θ | + 10 c l t + c r t .
In addition, a one-off reward of +100 is given if the craft successfully lands on the pad, and −100 is given if it crashes or drifts out-ofbounds (|x | ≥ 1). The distinction between a landing and a crash is based on a force analysis in an underlying rigid-body physics simulation, whose details are undocumented in the provided open source code. To convert LunarLander into a fixed-length episodic task, we disabled a default condition that terminates the episode immediately after a landing, crash or out-of-bounds event.
FoodLava. A simple holonomic navigation task created by ourselves to enable easy visualisation of trajectories for debugging during code development. The objective is to control a circular black sprite to escape or avoid a red "lava" region and move to the green "food" while navigating around a pair of walls (black lines). The time limit is T = 200. The state dimensions are the sprite's horizontal and vertical positions x, y (D s = 2) and are initialised randomly within the boundaries of the maze layout ([0, 10] 2 ). The action dimensions are horizontal and vertical velocities v x , v y (D a = 2), which are clipped if the resultant motion vector would intersect a wall or external boundary. The default reward function is
R(s t , a t ) = 1 if x t ≥ 8 ∧ y t ≥ 8 ("food" region) −1 if x t ≤ 3 ("lava" region) 0 otherwise.
RoboCar. Also created by ourselves using the PyBullet 3D physics simulator [6], the task in this environment is to drive a four-wheeled car to a green goal object while avoiding four black obstacles. The time limit is T = 200. The state dimensions are the coordinates of the car's centroid x, y, the cosine and sine of its orientation θ , its velocity components v x , v y , the distance and bearing in radians to the goal d, β (β = 0 when facing the goal) and a binary indicator of contact with an obstacle c o (D s = 9). The car is initialised at (x, y, θ, v x , v y ) = (0, 0, 0, 0, 0) and the obstacles are always the same, but the goal location is randomised on each episode. The action is a throttle u t and steering angle u s , which are applied subject to limits and a simple model of drag and mechanical resistance (D a = 2). The default reward function is R(s t , a t ) = −0.05d t − 0.1c o t + [d t < 2], where [·] is Iverson bracket notation; this final term adds a reward of +1 if the car is within a radius of 2 from the goal.
Common Setup and Parameters
Feedback budget. Since our survey (offline human experiment) received approximately 60 respondents, each of which provided 10 preference labels per environment, we used a feedback budget of k max = 600 for all other experiments. This figure enabled direct comparison of our algorithm's performance across environments and experiment types, and also reflects what we see as a reasonable demand on human labour (on the order of 1 hour, assuming ≈ 10 labels per minute).
Preference label noise. For the noise parameter used to prevent extreme probabilities, we used ε = 0.1, which effectively set a minimum 10% chance that the higher-fitness trajectory was erroneously preferred. A 10% random error rate matches that used in prior PbRL work by Christiano et al. [5].
Optimism parameter for UCB sampling. Throughout all experiments we used λ = 2, meaning the optimistic fitness estimates u were 2 standard deviations above the means µ.
Tree growth parameters. We used a maximum tree size of m max = 100 and a complexity regularisation parameter of α = 0.01. We found that the final reward structure was sometimes quite sensitive to the latter, which could easily have been tuned for each environment and experimental context. For the sake of simplicity and to avoid cherry-picking, we identified this single value as one that provided good performance across all four environments.
RL algorithm. For all RL agents -both pilot and PbRL -we used the soft-actor critic algorithm [7] with discount factor γ = 0.99, learning rates of 1e −4 and 1e −3 for the policy and value networks respectively, an entropy regularisation coefficient of 0.2, and an interpolation factor of 0.99 for Polyak averaging of the target networks. All networks had two hidden layers of 256 units each. Note that for the offline experiments, E = n max = n post fix , because these involved first training a pilot agent to create an offline trajectory dataset (E = |T off | = n max ), then training a PbRL agent on the resultant fixed reward function (E = n post fix ). For the online experiments, E = n max + n post fix , because the training of the PbRL agent consisted of a phase of online preference gathering and reward modification (up to n max ) followed by a phase of continued training on a fixed reward function (another n post fix episodes).
Repeated experimental runs.
In all experiments other than the most labour-intensive online study with human feedback, we trained 5 PbRL agents for each environment, with mean, minimum and maximum performance shown on the learning curve plots (main paper figures 2-4). In the offline setting, all repeats used the same learnt reward function, so the variation reflected the stochasticity of the RL process only. In the online setting, a reward function was constructed from scratch during each agent's training. Since this process naturally differed slightly between runs, it provided an additional source of variation compared with the offline experiments.
Offline with Oracle Feedback
Oracle implementation. To implement the oracles for automated preference labelling, we directly queried the environment's default reward function for each state-action pair in the two trajectories τ i , τ j , summed over the trajectory lengths, and returned y i j = 1−ε = 0.9 if the sum for τ i exceeded that for τ j , y i j = ε = 0.1 if vice versa, and y i j = 0.5 if the sums were equal. While adding more than these three preference levels would increase the information content of each label, it would have required further assumptions about the mapping from fitness differences to preferences, and we found that the algorithm performed well without this added complexity. Update frequency. As described in Appendix D, preference elicitation in the offline setting consists of a single batch of k max samples, but the batch can be paused every f u samples to refine the tree structure, reward components and sampling distribution. In our offline oracle experiment we used f u = 60. Given that k max = 600, this meant that a total of 10 updates were completed during the feedback process.
Offline with Human Feedback
Survey administration. Our survey was run via Google Forms, with trajectory pairs displayed in a separately-hosted web application; this was required to enable each participant to receive a randomised set of pairs. As outlined in Appendix A, pairs were displayed side-by-side as infinitely-looping videos (τ i on the left, τ j on the right). Videos were generated by enabling OpenAI Gym's monitor wrapper during the training of the pilot agents, and appeared visually as animated versions of the images in figure 1 above (for RoboCar, the bird's eye view was used). Survey participants gave ratings on a discretised 0-10 scale, with 0 corresponding to the strongest preference for τ i (interpreted as y i j = 1 − ε = 0.9), vice versa for 10, and a linear interpolation for intermediate values (so a rating of 5 corresponded to y i j = 0.5). Participants were given 10 trajectory pairs to rate for each environment. We publicised our survey among research groups in the fields of computer science and engineering, as well as non-expert personal and professional contacts. Over 1 week of data collection we had 62 respondents of a wide variety of experience levels (see main paper figure 3b), yielding a total preference dataset size of k max = 62 × 10 = 620 for each environment. Prior to any human experiments being run, our survey design received full university ethics approval.
Lack of periodic updates. Since the survey was run via the internet with asynchronous responses from many participants, it would have added significant technical complexity to perform tree and reward component updates on the back-end throughout the 1 week survey period. For this reason, we waited until all responses were gathered before doing a single update step, thereby effectively setting f u ≥ 620. Since our algorithm initiates with equal fitness estimates for all trajectories, the result was that pairs were sampled uniform-randomly throughout the survey, subject to the non-identity, duplication and connectivity conditions given in section 4.5 of the main paper. Unfortunately, this somewhat reduced the comparability of these results to those from the offline oracle experiment, where f u = 60. We would likely have attained better performance in this experiment if periodic updates were possible, as it would have enabled the UCB weighting method to work as designed. Developing a more sophisticated pipeline for large-scale user experiments is a clear priority for future work.
Online with Oracle Feedback
Oracle implementation. The oracles were identical to those used in the offline experiment.
Fixing of reward function. PbRL agents were trained for a total of E = n max +n post fix episodes, where E varied between environments as given in the table above. The training period was divided into two phases of pre-and post-fixing of the reward function as follows:
Pendulum LunarLander FoodLava RoboCar n max 100 100 100 200 n post fix 100 100 300 800
Batch and update frequencies. In all environments, we obtained a batch of preference labels every f l = 10 episodes until n max was reached, with batch sizes given by the scheduling equation (main paper equation 14). We found no measurable benefit to performing tree and reward component updates more than once per batch, so set f u to an arbitrarily large value to disable this feature.
Online with Human Feedback
User interface. Using the OpenCV Python library [3], we constructed a graphical user interface to obtain human preference labels over trajectories generated by a PbRL agent running locally on the same machine. Mirroring the survey used in the offline human experiment, the interface presented trajectory pairs side-byside as infinitely-looping videos, and user input was constrained to a 0-10 scale (mapped to numerical keys on the keyboard) with 0 and 10 representing maximal preference for τ i and τ j respectively.
Participant. This experiment was run with a single human participant: one of the authors of this paper.
Fixing of reward function. As discussed in the main paper, the first run for both FoodLava and LunarLander used the same values of n max and n post fix as the oracle-based experiment. After observing the premature convergence phenomenon, we updated to the following values for the second run: Batch and update frequencies. These were unchanged from the online experiment with oracle feedback.
APPENDIX F: CAUSAL CONFUSION OF BEARING INFORMATION IN ROBOCAR
In the failure case encountered when learning a reward function for RoboCar from offline human feedback (the survey responses), we find that part of the misalignment was due to a counterintuitive treatment of the bearing information β when the car was in the y ∈ [−1.64, 1.68] corridor with a distance-to-goal d ≥ 5.84. The offending subtree is shown in figure 2 below. The first two splits here created positive reward components for β ≥ 2.18 or β < −2.15, and negative reward otherwise. The agent was thus rewarded for facing away from its goal, which is heavily misaligned. The near-exact symmetry of these two splits implies that they were due to a reliable feature of the task rather than a statistical fluke, and our diagnosis is as follows.
In RoboCar, the agent must learn to navigate the car around four black obstacles in order to reach the goal. Thus, in the pilot trajectory dataset T , many successful goal-reaching trajectories featured the car initially moving away from the goal to bypass an obstacle, before turning to face the goal and moving rapidly towards it. As a result, a large proportion of timesteps belonging to trajectories with high inferred fitness fell outside of the β ∈ [−2.15, 2.18] region. Meanwhile, many unsuccessful trajectories involved the car driving directly at the goal, and becoming stuck against an obstacle for many timesteps in a row. Thus, if the car was both far from the goal (i.e. d ≥ 5.84) and facing towards it (β ∈ [−2.15, 2.18]) , it was statistically more likely that this timestep was a member of a low-fitness trajectory than a high-fitness one. The exact opposite was true for timesteps spent facing away from the goal. This led our algorithm to fall foul of causal confusion by creating a tree that rewarded facing away from the goal; a circumstance which merely correlated with high fitness rather actively than driving it.
An aligned solution to this problem would be to instead introduce a split on the obstacle contact indicator variable c o , creating a reward component that penalised collisions directly. It is not entirely clear why this option was not taken, but it may be that the human survey participants were not sufficiently consisent in the penalisation of obstacle contact in their responses. What we can be confident about is that an online learning setup would help to reduce the likelihood of such a confusion persisting, as it would provide an opportunity for the human(s) to reactively penalise early examples of behaviour that resulted from it.
We have not been able to meaningfully diagnose of the final, asymmetric split at β = −0.676, which suggests that this is likely due to a random statistical imbalance in the training dataset T .
Proc. of the 21st International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2022), P. Faliszewski, V. Mascardi, C. Pelachaud, M.E. Taylor (eds.), May 9-13, 2022, Online. © 2022 International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved.
Figure 1 :
1Tree and induced partition for a simple case with = = 1 and = 3.
Figure 2 :Figure 3 :
23Performance in offline setting using oracle feedback; additional plots shown for FoodLava. Performance in offline setting using human feedback; additional plots shown for Pendulum.
Figure 4 :Figure 5 :
45Performance in online setting using oracle feedback; learning timeline shown for RoboCar. Performance in online setting using human feedback; results with learning timelines for FoodLava and LunarLander.
Figure 6 :
6Interpretability demo for two exemplar cases.
1uWifFigure 1 :
1Input: Offline trajectory dataset T off , hyperparameters n max , f l , λ, k max , f u , m max , D s , D a , α, n post fix 2 Output: Final agent policy π 3 4 /* Initialise policy, tree and persistent data structures */5 π ← initPi(); ϕ ← initTree(); T = T off ; r ← [0]; Σ ← [0]; P ← {}; A ← []; y ← []; N ← []; b ← 0; done ← False; 6 while done = False do 7if |T | < n max then 8 /* Run one episode of RL using latest reward function and store trajectory */ 9 τ , π ← rlOneEp(π , ϕ, r);10 T ← append(T , τ );11 N ← append(N ⊤ , computeN(τ , ϕ)) ⊤ ; /* Double-transpose operation appends a column to N ← N ⊤ r + λ diag(N ⊤ ΣN ) ← computeW(P, u, f l , b); /* Un-normalised weighting matrix (paper eqn 13 + recency condition) */ 20 if i j W i j > 0 then 21 /* Sample pair and obtain label, unless all pairs have been sampled already (sum ofW = 0) */ 22 τ i , τ j ← sample(T ,W / i j W i j ); 23 y i j ← getPreferenceLabel(τ i , τ j ); 24 P ← P ∪ {τ i , τ j };25 A ← append(A, ARow(i, k % f u = 0 or k = k b then 29 /* Update reward function periodically, and at end of batch */ 30 ϕ, r, Σ, N ← updateRewardFunction(ϕ, T , P, N , A, y, m max , D s , D a , α) ; /* (algorithm 2) */ 31 u ← N ⊤ r + λ diag(N ⊤ ΣN ) |T | = n max then 36 done = T rue ; /* Fix reward function once n_max is reached */ 37 end 38 end 39 /* Run remaining RL episodes using fixed reward function */ 40 for i ∈ {1..n post fix } do 41 _, π ← rlOneEp(π , ϕ, r); x, d, c, N [xdc] ← X[argmax(Q)]; 23 ϕ ← applySplit(ϕ, x, d, c); 24 m ← m + 1; 25 N ← splitReplace(N , x, N [xdc] ); 26 end 27 r ← []; v ← []; 28 /* Compute reward component means and variances */ 29 for x ∈ {1..33 L ← []; Z ← []; 34 /* Prune tree back to root by iteratively undoing the most recent split */ 35 while m > 1 do 36 x ← getLastSplit(ϕ); 37 ϕ ← applyMerge(ϕ, x); 38 m ← m − 1; 39 /* Replace xth and x+1th rows/entries of N, r and v with values for merged leaf */ 40 N ← mergeReplace(N , x, N x + N x +1 ) ; /* Conservation: timesteps in merged leaf is sum of two children *quality is labelling loss (paper eqn 7) + complexity regularisation */ 45 L ← append(L, Φ −1 (y) − (diag(N A ⊤ ΣAN ⊤ ) − 1 2 ) ⊤ AN ⊤ r 2 + αm); 46 Z ← append(Z, (ϕ, r, Σ, N )) ; /* Store current state of tree and reward components for retrieval later */ 47 end 48 /* Find and return best tree */ 49 ϕ, r, Σ, N ← Z[argmin(L)]; The four RL environments used in experiments.
Figure 2 :
2r5 = -0.41 (std=0.482) r6 = -0.0497 (std=0.495) Subtree containing β-based splits.
The replay buffer capacity B, minibatch size M and total number of training episodes E were independently selected for each environment after an informal search, and held constant across all experiments:Pendulum LunarLander FoodLava RoboCarB
5000
20000
20000
40000
M
32
64
128
64
E
200
200
400
1000
Prior works[8,29] have adopted the Bradley-Terry model[5] which maintains no covariance estimate and uses a logistic function ( ≻ | ) =
This and subsequent visualisations use a projection method introduced in[4], which represents the state-action subset for each component by its rectangular projection onto two dimensions, coloured by its mean reward. Where the projections of multiple subsets overlap, their colours are averaged, weighted by the number of samples falling within them in the trajectory set T. In this first FoodLava visualisation, there are in fact no overlaps because the tree only contains splits along the two plotted dimensions.5.2 Offline with Human Feedback (Figure 3)Using the same pilot trajectories as T , we then gathered preference data from 62 human participants via a survey (a) then trained PbRL agents using the resultant reward functions. We asked participants to indicate their level of relevant expertise (b) and expectations of the likelihood of our method succeeding, both before and after completing the survey (c), and to rank the tasks by perceived difficulty of giving feedback (d). Notably, this ranking turned out to predict the success of our method in this setting, since for LunarLander and Pendulum we achieved asymptotic ground-truth fitness (e) only slightly below the oracle results. Information about the ground-truth was not used anywhere in this experiment, which instead relied on participants' intuitive understanding of the tasks. The fact that it could nonetheless be well-optimised by the resultant PbRL agents indicates that for LunarLander and Pendulum, human intuition was broadly aligned with the default reward functions. Again, the coloured rectangle plots (f) and (g) provide insight into the learnt reward structure, with high reward in LunarLander given in a column above the landing zone, and in Pendulum given when the pole is upright with small angular velocity. The final 10 trajectories from the PbRL runs are overlaid, showing that the agents sought out high-reward regions and consequently solved the respective landing and pole-balancing tasks. We were unable to achieve aligned learning in FoodLava and RoboCar, although in the former the outcome was not as catastrophic as the learning curve suggests. As can be seen in the rectangle/trajectory plot (h), the PbRL agent learnt to solve most of the maze but was not incentivised to proceed to the food, since a large positive reward component covered the entire upper third. This, we hypothesise, is evidence of a causal confusion problem: within the pilot run dataset, most trajectories that reached the upper third then went on to the food, so our uniform temporal credit assignment was unable to determine that the latter step was necessary for a favourable preference label. A similar issue arose in RoboCar, which we focus on in our interpretability analysis (section 6). For Pendulum, we include the labelling loss curve (i) and alignment plot (j), alongside box plots showing the agreement between provided preference labels and ground-truth fitness differences (k). Ratings from participants of all expertise levels generally aligned with fitness (above dotted line), with this trend becoming slightly more pronounced for more certain ratings (closer to10), and an indication that those with RL expertise exhibited somewhat lower variance than those without.5.3 Online with Oracle Feedback(Figure 4)Next, we deployed our algorithm in an online setting, using trajectories generated by the PbRL agent as it trained on the learnt reward in real-time. After max trajectories were gathered, the reward function was fixed and the agent continued to train until convergence. Initially, we used synthetic oracle feedback. The ground-truth learning curves (a) indicate that overall performance was similar to the oracle-based offline setting, with mean final fitness being slightly higher for Pendulum and FoodLava and slightly lower for LunarLander and RoboCar. We further examine one of the five RoboCar runs via a hybrid visualisation that we call a learning timeline (b). With max = 200 and = 10 we had a total of 20 labelling batches, over which the number of labels increased according to the scheduling equation 14. The heatmap
We ignore the UCB weighting in this analysis because our aim is to achieve uniform label density before adding this effect. In theory, the duplication and connectivity conditions should be included, but this would greatly complicate the analysis and would not yield a closed-form expression for k b .
Note that this expression abuses notation because it is dependent on the shaping quantity from the previous timestep. The fact that reward depends on more than the current state-action pair means that strictly speaking, LunarLander is a partiallyobservable MDP.
Algorithm 2: updateRewardFunction subfunction.1 Input: ϕ, T , P, N , A, y, m max , D s , D a , α 2 Output: Feature function ϕ, reward components r, Σ, feature matrix N
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|
[] |
[
"R-Parity Violation and Scalar Leptons Production at Lepton- Antilepton Colliders",
"R-Parity Violation and Scalar Leptons Production at Lepton- Antilepton Colliders"
] |
[
"R A Alanakyan [email protected] \nTheoretical Physics Department\nYerevan Physics Institute\nAlikhanian Brothers St.2375036YerevanArmenia\n"
] |
[
"Theoretical Physics Department\nYerevan Physics Institute\nAlikhanian Brothers St.2375036YerevanArmenia"
] |
[] |
In the framework of models with R-parity violation, scalar neutrinos and scalar leptons production in the processes l + i l − j →ν k,L Z 0 and l + i l − j → l ∓ k,L W ± is considered.We also consider within the Minimal Supersymmetric Standard Model Higgs bosons production in the processes µ + µ − → H 0Yerevan Physics InstituteYerevan 1998
| null |
[
"https://export.arxiv.org/pdf/hep-ph/9804247v1.pdf"
] | 18,537,342 |
hep-ph/9804247
|
97a20920fc1740b42f6a60312a1f8a35a43e5b60
|
R-Parity Violation and Scalar Leptons Production at Lepton- Antilepton Colliders
Apr 1998
R A Alanakyan [email protected]
Theoretical Physics Department
Yerevan Physics Institute
Alikhanian Brothers St.2375036YerevanArmenia
R-Parity Violation and Scalar Leptons Production at Lepton- Antilepton Colliders
Apr 1998arXiv:hep-ph/9804247v1 7 Preprint YeRPHI-1507(7)-98
In the framework of models with R-parity violation, scalar neutrinos and scalar leptons production in the processes l + i l − j →ν k,L Z 0 and l + i l − j → l ∓ k,L W ± is considered.We also consider within the Minimal Supersymmetric Standard Model Higgs bosons production in the processes µ + µ − → H 0Yerevan Physics InstituteYerevan 1998
As known, in theories with R-parity violation [1]- [7] supersymmetric particles may be produced singly as a result of their superpartners collision.In particularly, in [8]- [11] has been suggested scalar neutrino production virtually and in resonance at e + e − collisions, in [12] its production virtually and in resonance at µ + e − collisions 1 .
However, the scalar neutrino mass is not fixed in theory and, thus, we don't know which energies are necessary for scalar neutrinos production in resonance.
That is why it is necessary to study associated scalar neutrino production with gauge bosons.
In [14] scalar neutrino production with photon in lepton-antilepton colliders has been considered:
l + i l − j →ν k γ,(1)
Here we study scalar neutrinos and charged leptons production with W ± -boson and Z 0 -boson :
l + i l − j →ν k,L Z 0 ,(2)l + i l − j →l ∓ k,L W ± .(3)
It is interesting to notice that the result received below for the processes (2),(3) are applicable to the tree processes of charged and pseudoscalar Higgs 1 for references on µ + e − and µ + µ − -colliders see [12] and [13] respectively.
bosons production in assotiation with gauge bosons 2 :
µ + µ − → H 0 3 Z 0 ,(4)µ + µ − → H ± W ∓ ,(5)
in the framework of the Minimal Supersymmetric Standard Model(MSSM), see [15,16] and references therein) at large tan β limit.
We do not consider here the processes µ + µ − → H 0 1,2 Z 0 because the main contribution to this processes comes from s-channel Z 0 -bosons exchange diagram of the Fig.2, however at large tan β the s-channel Z 0 -boson exchange contribution is supressed as 1 tan β , whereas other diagrams of the Fig.2 are enhanced as tan β.The t, u-channel contribution to the process µ + µ − → H 0 Z 0 within the Standard Model has been considered in [18], for main s-channel Z 0 -boson exchange contribution see e.g. [20] references therein.
2.Results
Using R-parity violating interaction with electrons and neutrinos:
L = h ijk (l i P L l jνk +l i P L ν jlk ) + H.c.(6)
we obtain the following amplitudes of processes (2),(3):
M = gh ijk cos θ Wū (k 1 ) i a Lk 4Ẑ t − a RẐk 4 u − (k 4 Z) s − m 2 ν P L u(k 2 ) j .(7)
where a L = − 1 2 + sin 2 θ W , a R = sin 2 θ W ,
M = gh ijk √ 2ū (k 1 ) k 4Ŵ t + 2 (k 4 W ) s − m 2 ν + imνΓν P L u(k 2 ).(8)
Here we neglect the lepton masses and use the following notations: Z µ ,W µ is the polarization 4-vector of the Z 0 -and W ± L -bosons, s = (k 1 + k 2 ) 2 , t = (k 1 − k 4 ) 2 , u = (k 2 − k 4 ) 2 , ml, mν are the masses of the scalar lepton and scalar neutrino respectively, Γν is the width of the scalar neutrino,in case of the process (3) we denote h ijk = p h ipk V pj (we take into account that typically in SUSY models if we neglect leptons masses, the masses of scalar leptons of the all flavours are equal to each other).
For the differential cross sections of the processes (2),(3) we obtain the following results:
dσ(l + i l − j →ν k,L Z 0 ) dt = αh 2 ijk 8 sin 2 θ W cos 2 θ W s 2 (( a 2 L t 2 + a 2 R u 2 )(tu − m 2 ν m 2 Z ) + + 2a L a R (t − m 2 ν )(u − m 2 ν ) tu + sm 2 ν (s − m 2 ν ) ( a L t − a R u ) + +( 1 8 m 2 Z − 1 2 m 2 ν ) s (s − m 2 ν ) 2 ),(9)dσ(l + i l − j →l ∓ k,L W ± ) dt = αh 2 ijk 8 sin 2 θ W s 2 (a 1 t − m 2 W m 2 ν 1 t 2 + b),(10)a = m 2 W + m 2 l − s − 2sm 2 ν (s − mν) 2 + Γ 2 ν m 2 ν (s − mν) 2 ,(11)b = −1 − 2m 2 l s (s − m 2 ν ) 2 + m 2 ν Γ 2 ν + s 2m 2 W | −1 + (s − m 2 l − m 2 W ) s − m 2 ν + imν Γν | 2 ,(12)t − < t < t + ,(13)
where
t ± = m 2 ν,l + m 2 Z,W − s ± (m 2 ν,l + m 2 Z,W − s) 2 − 4m 2 ν,l m 2 Z,W 2(14)
for the processes (2),(3) respectively.
After performing integration within the limits (14), (15) we obtain for the total cross sections the following result:
σ(l + i l − j →ν k,L Z 0 ) = αh 2 ijk 8 sin 2 θ W cos 2 θ W s 2 (((a 2 L + a 2 R )(m 2 Z + m 2 ν − s) + + 4a L a R m 2 ν (s − m 2 Z ) (m 2 Z + m 2 ν − s) − − 1 2 sm 2 ν (s − m 2 ν ) ) log( t + t − ) + +(t + − t − )(2a L a R − 2(a 2 L + a 2 R ) + 1 8 s(m 2 Z − 4m 2 ν ) (s − m 2 ν ) 2 )),(15)σ(l + i l − j →l ∓ k,L W ± ) = αh 2 ijk 16 sin 2 θ W s 2 (a log( t + t − ) + (b − 1)(t + − t − ))
. (16) At √ s ≫ mν, ml, m W the previous formulas are reduced and we have:
σ(l + i l − j →ν k,L Z 0 ) = αh 2 ijk 4 sin 2 θ W cos 2 θ W s ((a 2 L +a 2 R ) log( s mν m Z )+a L a R −(a 2 L +a 2 R )). (17) σ(l + i l − j →l ∓ k,L W ± ) = α(h ijk ) 2 8 sin 2 θ W s (log( s mlm W ) − 1).(18)
On Fig. 3,4 we present the number of eventsν k,L Z 0 andl − k,L W − per year for the processes (2),(3) versus mν, mν at fixed √ s, at yearly luminosity
L = 1000f b −1 .
In our numerical results we suppose that scalar neutrino predominantly decay into W ±l∓ pairs (i.e. Γν = Γ(ν → Wl)).
3.Comparision with other mechanisms of scalar leptons production
The processes [21]- [26]:
e + e − →νν * ,l +l−(19)
which proceed through virtual neutral gauge bosons (and their superpartners exchanges) have a larger cross sections than the processes (2),(3) however, it becomes kinematically allowed at energies √ s > 2mν , (2ml), whereas process (3), (4) is kinematically allowed at lower energies √ s > mν +m Z , ml+ m W .
At LHC scalar leptons may be also produced in pairs (see [27] and ref-
4.Higgs bosons production with gauge bosons
Using Higgs bosons interactions with leptons (A4)-(A6) we obtain in the large tan β limit that the amplitudes of the processes (4),(5) are different only by coefficient from amplitudes of scalar lepton and neutrino production with W ± -boson and Z 0 -boson considered in formulas (7),(8) above :
M (µ + µ − → H ± W ∓ ) = g 2 2 m m W tan βū(k 1 )(k 4Ŵ t +2 (k 4 W ) s − m 2 H )P L u(k 2 ), (20) M (µ + µ − → H 0 3 Z 0 ) = M L − M R ,(21)M (µ + µ − → H 0 1,2 Z 0 ) = M L + M R + M 0 ,(22)
where
M L,R = g 2 2 cos θ W m m W tan βū(k 1 ) a L,Rk 4Ẑ t − a R,LẐk 4 u ∓ (k 4 Z) s − m 2 3 P L u(k 2 ),(23)
where M 0 is the contribution from s-chanel Z 0 -bosons exchange.
Amplitudes M L,R,0 do not interfere with each other and for differential cross section of the process (4) we obtain:
dσ(µ + µ − → H 0 3 Z 0 ) dt = πα 2 8 sin 4 θ W cos 2 θ W s 2 m 2 m 2 W tan 2 β((a 2 L + a 2 R ) ( 1 u 2 + 1 t 2 )(tu − m 2 3 m 2 Z ) + 4a L a R (t − m 2 3 )(u − m 2 3 ) tu + + sm 2 3 s − m 2 3 (a L + a R )( 1 t − 1 u ))(24)
Differential cross section of the process (5) and total cross section of processes (4),(5) may be obtained using formulas (10)-(18) by replacements:
h ijk → gm √ 2m W
tan β ,mν → m 3 , ml → m 4 and Γν = 0.
For instance, far from threshold we obtain:
σ(µ + µ − → H 0 3 Z 0 ) = πα 2 2 sin 4 θ W cos 2 θ W s m 2 m 2 W tan 2 β((a 2 L +a 2 R )(log( s m 3 m Z )−1)+a L a R ).(25)σ(µ + µ − → H ∓ W ± ) = πα 2 4 sin 4 θ W s m 2 m 2 W tan 2 β(log( s m 4 m W ) − 1).(26)
At m 4 > m t , m 3 ≈ m 4 with high accuracy and consequently for the processes Besides the contribution to the processes (2),(3) from the tree diagrams of the Fig.1 there is also a contribution from the loops with virtual W ±and t-quarks and with other heavy particles in various extensions of the Standard Model such as contributions from squarks, charged Higgs bosons, chargino.Some of such contributions (the ZW ∓ H ± vertex) have been calculated previously, see [29] and references therein. On the Fig.5 we depicted some of the box diagrams which also describe the loop contribution to the process (5).Naively the loop contribution is of order σ ∼ α 4 s , it is possible that various amplitudes from various contributions are partly compensate each other at some parameters (a similar situation takes place in the loop contribution [28] to the process µ + µ − → H 0 i γ) and at these parameters the cross section may be essentially lower and the tree contribution will dominate over the loop contribution especially at tan β ≫ 1.
It must be noted also, at s . Using result of the [14] for the process µ + µ − → H 0 i γ:
σ(µ + µ − → H 0 3 γ) = πα 2 2sin 2 θ W m 2 m 2 W tan 2 β 1 s − m 2 H ((1+ m 4 H s 2 ) log( s m 2 µ )−2 m 2 H s ),(27)
we see that near threshold the cross section of the tree process µ + µ − → H 0 i γ exceed the cross section of process (5),far from threshold, as seen from comparision of the (25) with (27)
m 2 4 = m 2 3 + m 2 W (A.1) m 2 1,2 = 1/2 m 2 3 + m 2 Z ± ((m 2 3 + m 2 Z ) 2 − 4m 2 Z m 2 3 cos 2 2β) 1/2 (A.2) tan 2α = m 2 3 + m 2 Z m 2 3 − m 2 Z tan 2β. (A.3)
It follows from (A1) that MSSM guarantes the existence of, at least, one light Higgs boson with m 2 < m Z .
Interactions of the Higgs bosons with muons are described by lagragnian:
L = i gm 2m W cos α cos βμ µH 0 1 + i gm 2m W sin α cos βμ µH 0 2 + + gm 2m W tan βμγ 5 µH 0 3 + (i gm √ 2m W tan βμP L νH + + h.c.) (A.4)
At tan β ≫ 1 the mass relation (A2),(A3) and formula (A4) are strongly reduced:
m 2 = m 3 , m 1 = m Z , sin α cos β = tan β ≫ cos α cos β at m 3 < m Z , (A.5) m 2 = m H Z , m 1 = m 3 , cos α cos β = tan β ≫ sin α cos β at m 3 > m Z . (A.6)
It must be noted, that radiative corrections [30]- [33] can strongly change relations (A1),(A2) however in the large tan β limit and at m 3 < m Z or at m 3 ≫ m Z formulas (A1),(A5),(A6) hold approximately true even after taking into account the radiative corrections.
Figures Cuption
erences therein), however with increasing masses of the scalar leptons the cross sections are decreases faster than in case of the reactions (2),(3).
( 4 )
4,(5) we can use numerical results depicted on the Fig.3,4 for h ijk = 10 −2 which are the same as for the processes (4),(5) at tan β = 17.5 (in accordance with the above-mentioned replacements).
√ s ≫ M (where M-is the mass of the heaviest particles in loop )that the loop contribution also decreases faster than the tree contribution, because loop integrals contain the additional degree of M 2
, the cross section of the process of the process µ + µ − → H 0 i γ exceed the cross section of the process (5) in several times.Appendix A In the MSSM, the Higgs sector contains two doublets of Higgs bosons with opposite hypercharge (Y = ±1 ). After spontaneous symmetry breaking the following physical states appear:charged Higgs bosons H ± , and three neutral ones, H 0 1 , H 0 2 , H 0 3 . At tree level the masses of charged Higgs bosons (m 4 ) and scalars H 0 1,2 and an angle α (which described the mixing of scalar states) are being expressed through the mass of pseudoscalar H 0 3 and tanβ = v 2 v 1 where v 2 ,v 1 are both doublets vacuum expectations by following relationships:
Fig. 1
1Diagrams corresponding to the processes (2),(3).
Fig. 2
2Tree diagrams corresponding to the processes (4),(5).
Fig. 3
3Number of eventsν k,L Z 0 per year (σL) (at yearly luminosity L = 1000f b −1 ) produced in reaction (2) as a function of √ s with h µµk = 10 −2 .Curves 1,2,3 correspond to the mν = 0.1, 0.5, 0.7T eV respectively.
Fig. 4
4Number of eventsl ∓ k,L W ± per year (σL) (at yearly luminos-ity L = 1000f b −1 ) produced in reaction(3)as a function of √ s with h µµk = 10 −2 .Curves 1-5 correspond to the mν = ml = 100GeV ;mν = ml = 300GeV ;mν = ml = 700GeV ;mν = 500, ml = 300GeV ;mν = 700, ml = 300GeV ; respectively.
Fig. 5
5Loop diagrams corresponding to the processes (4),(5).Shaded ring correspond to the diagrams with Z 0 W ∓ H ± -vertexes.
Figure 1 :Figure 2 :Figure 5 :
125Figure 1:
The tree processes of Higgs bosons production in association with photons in µ + µ −collisions has been considered in[17],[18],[19],[14].
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[] |
[
"Probing the Nature of the unidentified TeV Gamma-ray source HESS J0632+057 with Swift",
"Probing the Nature of the unidentified TeV Gamma-ray source HESS J0632+057 with Swift"
] |
[
"A D Falcone \nDepartment of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA\n",
"J Grube \nSchool of Physics\nUniversity College Dublin\nBeleld, Dublin 4Ireland\n",
"J Hinton \nSchool of Physics and Astronomy\nUniversity of Leeds\nLS2 9JTLeedsUK\n",
"J Holder \nDept. of Physics and Astronomy and the Bartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA\n",
"G Maier \nPhysics Department\nMcGill University\nH3A 2T8MontrealQCCanada\n",
"R Mukherjee \nDepartment of Physics and Astronomy\nBarnard College\nColumbia University\n10027NYUSA\n",
"J Skilton \nSchool of Physics and Astronomy\nUniversity of Leeds\nLS2 9JTLeedsUK\n",
"M Stroh \nDepartment of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA\n"
] |
[
"Department of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA",
"School of Physics\nUniversity College Dublin\nBeleld, Dublin 4Ireland",
"School of Physics and Astronomy\nUniversity of Leeds\nLS2 9JTLeedsUK",
"Dept. of Physics and Astronomy and the Bartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA",
"Physics Department\nMcGill University\nH3A 2T8MontrealQCCanada",
"Department of Physics and Astronomy\nBarnard College\nColumbia University\n10027NYUSA",
"School of Physics and Astronomy\nUniversity of Leeds\nLS2 9JTLeedsUK",
"Department of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA"
] |
[] |
New generation TeV gamma-ray telescopes have discovered many new sources, including several enigmatic unidentified TeV objects. HESS J0632+057 is a particularly interesting unidentified TeV source since: it is a point source, it has a possible hard-spectrum X-ray counterpart and a positionally consistent Be star, it has evidence of long-term VHE flux variability, and it is postulated to be a newly detected TeV/X-ray binary. We have obtained Swift X-ray telescope observations of this source from MJD 54857 to 54965, in an attempt to ascertain its nature and to investigate the hypothesis that it's a previously unknown Xray/TeV binary. Variability and spectral properties similar to those of the other 3 known X-ray/TeV binaries have been observed, with measured flux increases by factors of ∼3. X-ray variability is present on multiple timescales including days to months; however, no clear signature of periodicity is present on the timescales probed by these data. If binary modulation is present and dominating the measured variability, then the period of the orbit is likely to be ≥ 54 days (half of this campaign), or it has a shorter period with a variable degree of flux modulation on successive high states. If the two high states measured to date are due to binary modulation, then the favored period is approximately 35-40 days. More observations are required to determine if this object is truly a binary system and
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10.1088/2041-8205/708/1/l52
|
[
"https://arxiv.org/pdf/0912.0941v1.pdf"
] | 2,858,182 |
0912.0941
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49f10f7822bed568d7cc85eac64af36cbf6e950a
|
Probing the Nature of the unidentified TeV Gamma-ray source HESS J0632+057 with Swift
4 Dec 2009
A D Falcone
Department of Astronomy & Astrophysics
Pennsylvania State University
16802University ParkPAUSA
J Grube
School of Physics
University College Dublin
Beleld, Dublin 4Ireland
J Hinton
School of Physics and Astronomy
University of Leeds
LS2 9JTLeedsUK
J Holder
Dept. of Physics and Astronomy and the Bartol Research Institute
University of Delaware
19716NewarkDEUSA
G Maier
Physics Department
McGill University
H3A 2T8MontrealQCCanada
R Mukherjee
Department of Physics and Astronomy
Barnard College
Columbia University
10027NYUSA
J Skilton
School of Physics and Astronomy
University of Leeds
LS2 9JTLeedsUK
M Stroh
Department of Astronomy & Astrophysics
Pennsylvania State University
16802University ParkPAUSA
Probing the Nature of the unidentified TeV Gamma-ray source HESS J0632+057 with Swift
4 Dec 2009
New generation TeV gamma-ray telescopes have discovered many new sources, including several enigmatic unidentified TeV objects. HESS J0632+057 is a particularly interesting unidentified TeV source since: it is a point source, it has a possible hard-spectrum X-ray counterpart and a positionally consistent Be star, it has evidence of long-term VHE flux variability, and it is postulated to be a newly detected TeV/X-ray binary. We have obtained Swift X-ray telescope observations of this source from MJD 54857 to 54965, in an attempt to ascertain its nature and to investigate the hypothesis that it's a previously unknown Xray/TeV binary. Variability and spectral properties similar to those of the other 3 known X-ray/TeV binaries have been observed, with measured flux increases by factors of ∼3. X-ray variability is present on multiple timescales including days to months; however, no clear signature of periodicity is present on the timescales probed by these data. If binary modulation is present and dominating the measured variability, then the period of the orbit is likely to be ≥ 54 days (half of this campaign), or it has a shorter period with a variable degree of flux modulation on successive high states. If the two high states measured to date are due to binary modulation, then the favored period is approximately 35-40 days. More observations are required to determine if this object is truly a binary system and
Introduction
During the past 5 years, TeV gamma ray astronomy has experienced rapid growth due to the commissioning of a new generation of sensitive atmospheric Cherenkov telescopes. These telescopes have led to an expansion of the TeV catalog from ∼10 sources to nearly 100 sources in just a few years (Horan & Wakely 2009;Weekes 2009;Buckley et al. 2008;Aharonian et al. 2008). Many of these sources have come from the survey of the Galactic plane by the HESS telescopes in Namibia. One of the most exciting results of this survey is a large number of unidentified TeV sources, which offer a new region of discovery space at very high energies. HESS J0632+057, located within the Monoceros Loop region, is one of these unidentified sources.
HESS J0632+057 was first detected by HESS using 13.5 hours of data collected between March 2004and March 2006(Aharonian et al. 2007). It has a peak significance in the field of view of 7.1σ, and the post-trials significance is 5.3σ, after accounting for the trials associated with searching the entire field of view. The spatial distribution is consistent with a point source (RMS size < 2 arcmin at 95% confidence) near the edge of the Rosette Nebula, located at RA 06h 32' 58.3", Dec. +05 • 48' 20" (± 28" stat., 20" sys.). The source position is located on the edge of the 99% contour of the EGRET unidentified object 3EG J0634+0521, but no object at the position of HESS J0632+057 is listed in the the Fermi bright source list (Abdo et al. 2009). MWC 148, a Be star, is at the centroid of the HESS position. Aharonian et al. (2007) suggested that the TeV gamma ray emission could be associated with either: a) a shock front driven by the stellar wind of MWC 148, b) shock accelerated cosmic rays interacting with a molecular cloud at the edge of the Monoceros Loop, c) an association with an unidentified ROSAT source 1RXS J063258.3+054857, d) an association with EGRET unidentified gamma ray source 3EG J0634+0521, and/or e) a previously unknown X-ray binary source associated with MWC 148.
On 2007 September 17, XMM-Newton observed the region surrounding HESS J0632+057, obtaining 26 ksec of good data (Hinton et al. 2009). This resulted in the detection of several sources, the brightest being a point source, XMMU J063259.3+054801, positionally coincident with MWC 148 and HESS J0632+057 and was within the 99% error circle of 1RXS J063258.3+054857. During the relatively short observation period, the object exhibited significant variability in the form of a gradual decline from ∼130 c/ksec to ∼90 c/ksec. During this observation, it had a mean deabsorbed 1-10 keV flux of (5.3 ± 0.4) ×10 −13 erg cm −2 s −1 . The spectrum was well fit with an absorbed power law spectrum, and it had a hard photon index of 1.26 ± 0.04, similar to the spectra of known TeV binary systems.
Follow-up observations of HESS J0632+057 at gamma-ray energies above 1 TeV by VERITAS during December 2006 -Jan 2007 and December 2008 -January 2009 yielded upper limits well below the fluxes published by HESS (Acciari et al. 2009b). The nondetection of HESS J0632+057 by VERITAS together with the HESS results provides evidence for variability in the gamma-ray flux on time scales of months, although the available data do not allow any conclusion on a possible periodicity.
At optical wavelengths, no reports of variability have been found in the literature. Significant flux variability is reported in radio at 5 GHz on about month time-scales around a mean flux of 0.3 mJy ), but no periodic variability could be discerned from these data. This radio flux is orders of magnitude lower than the typical radio flux expected from a TeV blazar which makes that potential interpretation unlikely.
The point-like nature of the detected TeV source, the excellent positional coincidence with MWC 148 (chance coincidence of ∼ 10 −4 according to Aharonian et al. (2007)), the location on the Galactic plane with a low radio flux, and the variable X-ray and gammaray emission from a location coincident with MWC 148 are all facts that argue in favor of an X-ray binary in association with MWC 148. There are currently just three known binary systems that are confirmed as producers of detected TeV gamma ray emission, namely PSR B1259-63 (Aharonian et al. 2005b), LS 5039 (Aharonian et al. 2005a), and LS I +61 303 (Albert et al. 2006;Acciari et al. 2008Acciari et al. , 2009. There is also an accreting black hole binary system, Cygnus X-1, with evidence of detection of a GeV/TeV gamma-ray flare (Albert et al. 2007), but this result relies upon marginal significance which would benefit from future confirmation if a high state occurs in the future. The three confirmed TeV binaries are powered by pulsar winds driving shock acceleration or by accretion onto the compact object driving a microquasar jet. Confirmation of HESS J0632+057/MWC 148 as a TeV binary would add a 4th object to this short and special list and allow us to begin studying its interesting properties; whereas, refutation of the binary hypothesis would establish this as an even more mysterious and interesting class of TeV source. The most direct way to test this binary hypothesis is to search for periodic emission signatures. In this paper, we report on recent monitoring data taken with Swift-XRT with the aim of advancing our understanding of this source.
The Observations
Swift was launched on 2004 November 20 (Gehrels et al. 2004) with a primary goal to provide detailed measurements of numerous gamma ray bursts (GRBs) and their afterglows with unprecedented reaction times. An ability to slew quickly and frequently has also made Swift an excellent multiwavelength observatory for many non-GRB sources through pre-planned monitoring observations and rapid response target of opportunity (ToO) observations. The narrow field instruments used for these studies are the X-ray telescope (XRT; Burrows et al. (2005)) and the Ultraviolet-Optical Telescope (UVOT; Roming et al. (2005)). Bright, hard-spectrum sources can also be studied with the burst alert telescope (BAT) in the ∼ 15 − 150 keV energy band. For these HESS J0632+057 observations, UVOT was overwhelmed by the bright star in the field of view. The source was not bright enough to be detected by BAT at high energies, or to allow BAT to produce a constraining upper limit. All of the analysis in this paper is based upon XRT data.
The XRT observations were typically taken as 4-6 ksec exposures. The initial set of exposures, from MJD 54857 to MJD 54874, were taken with various sampling intervals chosen primarily to overlap with the simultaneous observations at the VERITAS very high energy gamma ray observatory (see Acciari et al. (2009b) for results). These observations included a series of 4 observations spaced on daily time intervals, as well as two more observations with 1 week spacing. One month later, starting on MJD 54904, a series of monitoring observations was initiated with ∼3-4 day time spacing. This resulted in 20 additional observations extending from MJD 54904 to MJD 54965, at which time the spacecraft had to discontinue observations due to a Sun constraint. The total Swift-XRT data set includes 103 ksec of observations extending from 2009 January 26 to 2009 May 20. All observations beyond January 29 are being reported here for the first time.
Analysis
The Swift-XRT data were processed using the most recent versions of the standard Swift tools and the most recent calibration files available at the time of data processing. In particular, we utilized Swift Software version 3.0, FTOOLS version 6.6.3, and XSPEC version 12.5.0. Light curves were generated using xrtgrblc version 1.3.
All of the observations were obtained in photon counting (PC) mode. Circular and annular regions are used to describe the source and background areas respectively, and the radii of both regions depend on the measured count rate. In order to handle cases where the sources land on bad CCD detector columns, point spread function correction is handled using xrtlccorr. Since these observations always resulted in XRT count rates ∼0.01-0.06 c/s, there was no significant pile-up, which occurs in photon counting mode at count rates above ∼0.5 c/s. The full light curves use a bin size of one observation per bin (observations were typically 4-5 ksec duration). We also evaluated single observation light curves, and in these cases, we used a bin size minimum of 20 cts/bin. All error bars are reported at the 1-sigma level, unless otherwise specified.
As described in the results section below, we calculated spectral fits for the entire combined data set. We also calculated spectral fits for a series of observations that were binned temporally with large enough bins to provide enough photon statistics to allow reasonable spectral fits. Following the temporal binning, each observation was fit spectrally using a minimum energy binning ratio of 12 photons/bin, and in most cases the binning ratio was 20 photons/bin. Once the spectral fits were obtained, the flux was calculated. These flux values, which were calculated for each of the temporally binned data segments, were then used to calculate a flux-to-rate ratio during that time period. This ratio was then applied to the rate light curve to convert it to a flux light curve, as shown in Figure 1, which assumes that any spectral variability has a negligible impact on the flux-to-rate ratio within a single time bin.
Results
The X-ray light curve is shown in Figure 1. There is clear evidence of strong variability on multiple timescales, with measurable flux doubling on timescales as short as ∼5 days. While shorter timescale variability is certainly possible (within the individual ∼5 ksec observations), it is not measurable within the flux limited error bars provided by these data. Longer timescale variability is also evident, with an extended rise and decay lasting for at least 30 days within the time frame bracketed by MJD 54907 and MJD 54941.
For this work, the hardness ratio is defined as R 2 /R 1 , where R 2 is the rate in the 2-10 keV band and R 1 is the rate in the 0.3-2 keV band. The hardness ratio as a function of time is shown in Figure 2. The mean hardness ratio is 0.96 with a standard deviation of 0.27. Fitting the hardness curve to a constant in time results in a reduced χ 2 equal to 2.01 with 24 degrees of freedom, with a chance probability of 99.8%. While the constant hardness hypothesis does not provide a good fit, it is not poor enough to claim significant evidence of spectral variability.
A simple absorbed power law model, with N H left to vary freely, was fit to the time binned data. The N H was not found to vary significantly. The mean N H was found to be 3.3±0.2×10 21 cm −2 , and all values are consistent, within error, with the value obtained from earlier XMM observations, 3.1 ± 0.3 × 10 21 cm −2 (Hinton et al. 2009). For all subsequent fitting, the N H was fixed to the value obtained from the XMM observations. The photon indices from these fits are shown in Figure 3. The time axis error bars represent the full extent of the time bin used for each model fit, and the vertical error bars are 90% confidence level error bars. We also binned all of the data together to obtain a combined fit of an absorbed power law to the entire data set. The photon index was found to be 1.56±0.06, and the fit of the absorbed power law resulted in a reduced χ 2 of 0.89 with 72 degrees of freedom. The model and data for the overall fit are shown in Figure 4.
A power spectrum periodicity search was performed on the overall light curve shown in Figure 1, using a Lomb normalized periodogram which is applicable to unevenly spaced data (Press et al. 2007). However, no significant periodic signal can be discerned from these data.
Discussion & Conclusions
The X-ray flux from the TeV binary LS I +61 303 (which is potentially a very similar object to HESS J0632+057) varies over a 26.5 day orbital cycle at many wavelengths, including variations by factors of 2 to 6 at both optical and X-ray wavelengths (Paredes et al. 1997;Holder, Falcone, & Morris 2007;Casares et al. 2005), and it has fast timescale flares (Smith et al. 2009;Barthelmy et al. 2008). Therefore, we have probed HESS J0632+057 for binary orbit variability timescales ranging from days to weeks, as well as searching for any large amplitude flaring on short timescales during all observations. The detection of a binary period would clearly allow us to differentiate between X-ray binary models and Be star wind-driven shock models (Townsend et al. 2007), as well as other less likely models such as cosmic rays interacting with a molecular cloud or a remarkably low radio flux blazar on the Galactic plane.
While it is clear that further monitoring observations are needed in order to obtain any conclusions regarding periodicity, it is also clear that the X-ray light curve of this point source is variable on long (> weeks) timescales. The light curve is consistent with periodic variability on timescales longer than those probed in this study, if we assume that there is some additional variability due to flaring, but longer baseline observations are needed to evaluate the reality of any periodic variability. All of the measured fluxes in the epochs during which Swift observed the source are larger (by a factor of ∼two) than the flux measured by XMM in September of 2007. A constant flux hypothesis is ruled out. A simplistic sinusoidal curve is also ruled out. However, it is possible that a more complicated model, involving a long period sinusoidal curve plus flares, could be fit to the light curve. The results of a Lomb-Scargle periodogram analysis show no significant periodicity; however, this analysis does not rule out the possibility of a periodicity at longer timescales beyond the range of these observations (≥54 days) or the possibility of a shorter periodicity timescale with variable or small fractional flux modulation. For example, if the modulation fraction was ≤30%, then Swift-XRT would not be sensitive enough to see a significant deviation from one 5 ksec pointing to the next, given the mean flux from HESS J0632+057. It is also likely that this study would miss periodic flux modulation fractions somewhat larger than this, due to the contamination of the periodic signal by non-periodic flare-like emission. Although the following is not statistically significant, we should also mention that if the two high states measured in these data are due to a binary modulation, then the favored period would be approximately 35-40 days; but, of course, there is no evidence for or against the periodic nature of these two high states since longer timescale observations are necessary for making such a determination. If these two high states are not related to any potential periodicity, then the favored period would be longer than half the timeframe of observations performed thus far (i.e. longer than 54 days).
By plotting N H /N H,gal versus binary orbit for known high mass X-ray binaries, Be-HMXBs and Sg-HMXBs have been found to occupy different regions of parameter space (Bodaghee et al. 2007;Rodriguez & Bodaghee 2008). According to Kalberla et al. (2005), the galactic neutral hydrogen column density, N H,gal , is 5.6 ×10 21 cm −2 in the direction of HESS J0632+057. If we take the orbital period of HESS J0632+057 to be ≥54 days (half the time period of these observations) and the N H to be 3.1 × 10 21 cm −2 (Hinton et al. 2009), then it falls within the middle of the Be-HMXB section of the Bodaghee Diagram. This period, as well as the potential ∼35-40 day period would both be long enough to be consistent with the interpretation of this object as a TeV emitting HMXB associated with the Be star MWC 148.
If MWC 148 was a single star with a wind driven shock, it would be expected to have a reasonable spectral fit with a thermal model such as an absorbed MeKaL model or an absorbed Raymond-Smith model with a temperature of kT∼1 keV (Rauw et al. 2002;Townsend et al. 2007). However, we find that such models do not fit the data well due to the presence of a harder component. The absorbed Mekal model results in a high temperature of kT=17±4 keV, which is too high to be likely for a single star model, with reduced χ 2 =1.36 for 109 degrees of freedom. The absorbed Raymond-Smith model results in a reduced χ 2 =8.15 for 109 degrees of freedom. The spectrum is more naturally fit by an absorbed power law spectrum, which results in a reduced χ 2 of 0.89 and a power law photon index of 1.56±0.06. While not definitive, this evidence favors models involving more than a single star, such as the binary hypothesis.
These monitoring observations also provide the possibility to search for short timescale flaring from either XMMU J063259.3+054801 or from any of the other dimmer X-ray sources in the field of view. X-ray flaring has been observed from other X-ray binaries and from at least one TeV emitting X-ray binary, namely LS I +61 303 (Smith et al. 2009). Short timescale flaring would provide a way to probe the size of the emission region, as well as the power of the engine that must be feeding the associated acceleration site. In principle, short timescale variability could also be used to probe different temporal signatures expected from Be star wind-driven shock models, binary interaction region models, and accretion-driven microquasar jet models. Variability as short as ∼ 5 days was observed, but no flaring at shorter timescales was observed in the flux range probed by these observations. These data are complemented by the simultaneous data that have been obtained by Fermi, as well as radio observations reported by Skilton et al. (2009) and TeV observations reported by Acciari et al. (2009b). In future work, modeling of this composite spectral energy distribution and longer timescale observations will provide strong constraints on potential periodicities and on the acceleration site parameters, such as the B field and the electron distribution, implied by fitting a synchrotron spectrum plus an inverse Compton spectrum to the observations. This work is supported at Pennsylvania State University by NASA contract NAS5-00136 and grant NNX08AV77G.
Fig. 1 .Fig. 2 .Fig. 3 .
123-The X-ray light curve of XMMU J063259.3+054801 from the Swift-XRT observations in the 0.3-10 keV band. The bottom panel shows the flux derived from binning up all photon events into time bins that are illustrated by the horizontal error bars and finding the best fit spectral model with N H fixed to 3.1×10 21 . The flux in the top panel explores temporal variability in more detail by deriving the flux directly from the X-ray rate light curve, with a flux to rate ratio derived for each of the spectral time bins defined by the horizontal error bars in the bottom panel (this assumes that spectral variability is a negligible effect within each time bin). -The X-ray hardness curve of XMMU J063259.3+054801, where hardness is defined as the flux in the 2-10 keV band divided by flux in the 0.3-2 keV band. The mean hardness is shown as a solid line. A fit to this line produces a reduced χ 2 of 2.-Spectral parameters from application of an absorbed power law model to time binned Swift-XRT observations of XMMU J063259.3+054801. The top panel is the power law photon index. The bottom panel is the reduced χ 2 of the model fit. Vertical error bars represent 90% confidence range. Horizontal error bars represent the extent of the time region used in the binning, which is the same temporal binning that was shown inFigure 1.
Fig. 4 .
4-Absorbed power law model fit to sum of all Swift-XRT data analyzed in this paper. N H is fixed to the value obtained previously by XMM, and the photon index is found to be 1.56±0.06, with a reduced χ 2 of 0.89. Residuals are shown in the bottom panel.
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|
[] |
[
"Medical Image Segmentation Using Squeeze-and-Expansion Transformers",
"Medical Image Segmentation Using Squeeze-and-Expansion Transformers"
] |
[
"Shaohua Li \nInstitute of High Performance Computing\nA*STARSingapore\n",
"Xiuchao Sui [email protected] \nInstitute of High Performance Computing\nA*STARSingapore\n",
"Xiangde Luo [email protected] \nUniversity of Electronic Science and Technology of China\nChengduChina\n",
"Xinxing Xu \nInstitute of High Performance Computing\nA*STARSingapore\n",
"Yong Liu \nInstitute of High Performance Computing\nA*STARSingapore\n",
"Rick Goh \nInstitute of High Performance Computing\nA*STARSingapore\n"
] |
[
"Institute of High Performance Computing\nA*STARSingapore",
"Institute of High Performance Computing\nA*STARSingapore",
"University of Electronic Science and Technology of China\nChengduChina",
"Institute of High Performance Computing\nA*STARSingapore",
"Institute of High Performance Computing\nA*STARSingapore",
"Institute of High Performance Computing\nA*STARSingapore"
] |
[] |
Medical image segmentation is important for computer-aided diagnosis. Good segmentation demands the model to see the big picture and fine details simultaneously, i.e., to learn image features that incorporate large context while keep high spatial resolutions. To approach this goal, the most widely used methods -U-Net and variants, extract and fuse multi-scale features. However, the fused features still have small effective receptive fields with a focus on local image cues, limiting their performance. In this work, we propose Segtran, an alternative segmentation framework based on transformers, which have unlimited effective receptive fields even at high feature resolutions. The core of Segtran is a novel Squeeze-and-Expansion transformer: a squeezed attention block regularizes the self attention of transformers, and an expansion block learns diversified representations. Additionally, we propose a new positional encoding scheme for transformers, imposing a continuity inductive bias for images. Experiments were performed on 2D and 3D medical image segmentation tasks: optic disc/cup segmentation in fundus images (REFUGE'20 challenge), polyp segmentation in colonoscopy images, and brain tumor segmentation in MRI scans (BraTS'19 challenge). Compared with representative existing methods, Segtran consistently achieved the highest segmentation accuracy, and exhibited good cross-domain generalization capabilities. The source code of Segtran is released at https://github.com/askerlee/segtran.
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10.24963/ijcai.2021/112
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[
"https://arxiv.org/pdf/2105.09511v3.pdf"
] | 234,790,361 |
2105.09511
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ec561a375a100097f00fcf4924a2c14a9a7735e7
|
Medical Image Segmentation Using Squeeze-and-Expansion Transformers
Shaohua Li
Institute of High Performance Computing
A*STARSingapore
Xiuchao Sui [email protected]
Institute of High Performance Computing
A*STARSingapore
Xiangde Luo [email protected]
University of Electronic Science and Technology of China
ChengduChina
Xinxing Xu
Institute of High Performance Computing
A*STARSingapore
Yong Liu
Institute of High Performance Computing
A*STARSingapore
Rick Goh
Institute of High Performance Computing
A*STARSingapore
Medical Image Segmentation Using Squeeze-and-Expansion Transformers
Medical image segmentation is important for computer-aided diagnosis. Good segmentation demands the model to see the big picture and fine details simultaneously, i.e., to learn image features that incorporate large context while keep high spatial resolutions. To approach this goal, the most widely used methods -U-Net and variants, extract and fuse multi-scale features. However, the fused features still have small effective receptive fields with a focus on local image cues, limiting their performance. In this work, we propose Segtran, an alternative segmentation framework based on transformers, which have unlimited effective receptive fields even at high feature resolutions. The core of Segtran is a novel Squeeze-and-Expansion transformer: a squeezed attention block regularizes the self attention of transformers, and an expansion block learns diversified representations. Additionally, we propose a new positional encoding scheme for transformers, imposing a continuity inductive bias for images. Experiments were performed on 2D and 3D medical image segmentation tasks: optic disc/cup segmentation in fundus images (REFUGE'20 challenge), polyp segmentation in colonoscopy images, and brain tumor segmentation in MRI scans (BraTS'19 challenge). Compared with representative existing methods, Segtran consistently achieved the highest segmentation accuracy, and exhibited good cross-domain generalization capabilities. The source code of Segtran is released at https://github.com/askerlee/segtran.
Introduction
Automated Medical image segmentation, i.e., automated delineation of anatomical structures and other regions of interest (ROIs), is an important step in computer-aided diagnosis; for example it is used to quantify tissue volumes, extract key quantitative measurements, and localize pathology [Schlemper et al., 2019;Orlando et al., 2020]. Good segmentation demands the model to see the big picture and fine details at * Corresponding Author. the same time, i.e., learn image features that incorporate large context while keep high spatial resolutions to output finegrained segmentation masks. However, these two demands pose a dilemma for CNNs, as CNNs often incorporate larger context at the cost of reduced feature resolution. A good measure of how large a model "sees" is the effective receptive field (effective RF) , i.e., the input areas which have non-negligible impacts to the model output.
Since the advent of U-Net [Ronneberger et al., 2015], it has shown excellent performance across medical image segmentation tasks. A U-Net consists of an encoder and a decoder, in which the encoder progressively downsamples the features and generates coarse contextual features that focus on contextual patterns, and the decoder progressively upsamples the contextual features and fuses them with fine-grained local visual features. The integration of multiple scale features enlarges the RF of U-Net, accounting for its good performance. However, as the convolutional layers deepen, the impact from far-away pixels decay quickly. As a results, the effective RF of a U-Net is much smaller than its theoretical RF. As shown in Fig.2, the effective RFs of a standard U-Net and DeepLabV3+ are merely around 90 pixels. This implies that they make decisions mainly based on individual small patches, and have difficulty to model larger context. However, in many tasks, the heights/widths of the ROIs are greater than 200 pixels, far beyond their effective RFs. Without "seeing the bigger picture", U-Net and other models may be misled by local visual cues and make segmentation errors.
Many improvements of U-Net have been proposed. A few typical variants include: U-Net++ [Zhou et al., 2018] and U-Net 3+ [Huang et al., 2020], in which more complicated skip connections are added to better utilize multi-scale contextual features; attention U-Net [Schlemper et al., 2019], which employs attention gates to focus on foreground regions; 3D U-Net [Çiçek et al., 2016] and V-Net [Milletari et al., 2016], which extend U-Net to 3D images, such as MRI volumes; Eff-UNet [Baheti et al., 2020], which replaces the encoder of U-Net with a pretrained EfficientNet [Tan and Le, 2019].
Transformers [Vaswani et al., 2017] are increasingly popular in computer vision tasks. A transformer calculates the pairwise interactions ("self-attention") between all input units, combines their features and generates contextualized features. The contextualization brought by a transformer is analogous to the upsampling path in a U-Net, except that it has unlimited effective receptive field, good at capturing long-range correlations. Thus, it is natural to adopt transformers for image segmentation. In this work, we present Segtran, an alternative segmentation architecture based on transformers. A straightforward incorporation of transformers into segmentation only yields moderate performance. As transformers were originally designed for Natural Language Processing (NLP) tasks, there are several aspects that could be improved to better suit image applications. To this end, we propose a novel transformer design Squeeze-and-Expansion Transformer, in which a squeezed attention block helps regularize the huge attention matrix, and an expansion block learns diversified representations. In addition, we propose a learnable sinusoidal positional encoding that imposes a continuity inductive bias for the transformer. Experiments demonstrate that they lead to improved segmentation performance. We evaluated Segtran on two 2D medical image segmentation tasks: optic disc/cup segmentation in fundus images of the REFUGE'20 challenge, and polyp segmentation in colonoscopy images. Additionally, we also evaluated it on a 3D image segmentation task: brain tumor segmentation in MRI scans of the BraTS'19 challenge. Segtran has consistently shown better performance than U-Net and its variants (UNet++, UNet3+, PraNet, and nnU-Net), as well as DeepLabV3+ [Chen et al., 2018].
Related Work
Our work is largely inspired by DETR [Carion et al., 2020]. DETR uses transformer layers to generate contextualized features that represent objects, and learns a set of object queries to extract the positions and classes of objects in an image. Although DETR is also explored to do panoptic segmentation [Kirillov et al., 2019], it adopts a two-stage approach which is not applicable to medical image segmentation. A followup work of DETR, Cell-DETR [Prangemeier et al., 2020] also employs transformer for biomedical image segmentation, but its architecture is just a simplified DETR, lacking novel components like our Squeeze-and-Expansion transformer. Most recently, SETR [Zheng et al., 2021] and TransU-Net [Chen et al., 2021] were released concurrently or after our paper submission. Both of them employ a Vision Transformer (ViT) [Dosovitskiy et al., 2021] as the encoder to extract image features, which already contain global contextual information. A few convolutional layers are used as the decoder to generate the segmentation mask. In contrast, in Segtran, the transformer layers build global context on top of the local image features extracted from a CNN backbone, and a Feature Pyramid Network (FPN) generates the segmentation mask. [Murase et al., 2020] extends CNNs with positional encoding channels, and evaluates them on segmentation tasks. Mixed results were observed. In contrast, we verified through an ablation study that positional encodings indeed help Seg-tran to do segmentation to a certain degree.
Receptive fields of U-Nets may be enlarged by adding more downsampling layers. However, this increases the number of parameters and adds the risk of overfitting. Another way of increasing receptive fields is using larger stride sizes of the convolutions in the downsampling path. Doing so, however, sacrifices spatial precision of feature maps, which is often disadvantageous for segmentation [Liu and Guo, 2020].
Squeeze-and-Expansion Transformer
The core concept in a transformer is Self Attention, which can be understood as computing an affinity matrix between different units, and using it to aggregate features:
Att_weight(X, X) = f (K(X), Q(X)) ∈ R N ×N , (1) Attention(X) = Att_weight(X, X) · V (X), (2) X out = FFN(Attention(X)),(3)
where K, Q, V are key, query, and value projections, respectively. f is softmax after dot product. Att_weight(X, X) is the pairwise attention matrix between input units, whose i, jth element defines how much the features of unit j contributes to the fused (contextualized) features of unit i. FFN is a feedforward network to further transform the fused features. The basic transformer above is extended to a multi-head attention (MHA) [Vaswani et al., 2017;Voita et al., 2019], aiming to capture different types of associations between input units. Each of the N h heads computes individual attention wights and output features (C/N h -dimensional), and their output features are concatenated along the channel dimension into C-dimensional features. Different heads operate in exclusively different feature subspaces.
We argue that transformers can be improved in four aspects make them better suited for images:
1. In Eq.(2), the intermediate features Attention(X) are obtained by linearly combining the projected input features, where the attention matrix specifies the combination coefficients. As the attention matrix is huge: N ×N , with typically N > 1000, it is inherently vulnerable to noises and overfitting. Reducing the attention matrix to lower rank matrices may help.
2. In traditional transformers, the output features are monomorphic: it has only one set of feature transformations (the multi-head transformer also has one set of transformations after concatenation), which may not have enough capacity to fully model data variations. Just like a mixture of Gaussians almost always better depicts a data distribution than a single Gaussian, data variations can be better captured using a mixture of k transformers.
3. In traditional transformers, the key and query projections are independently learned, enabling them to capture asymmetric relationships between tokens in natural language. However, the relationships between image units are often symmetric, such as whether two pixels belong to the same segmentation class.
4. Pixels in an image have strong locality and semantic continuity. The two mainstream positional encoding
1 2 3 ⋯ (a) 1 ′ ⋮ ′ 1 ⋯ (b) 1 2 3 ⋮ 1 2 3 ⋯ 1 ′ 2 ′ 3 ′ ⋮ ′ 1 ′ 2 ′ 3 ′ ⋮ ′ 1 2 3 ⋮ Figure 3: (a) Full self-attention (N × N ) vs. (b) Squeezed At- tention Block (SAB).
In SAB, first input units x1, · · · , xN attend with a codebook c1, · · · , cM , yielding projected codebook features c 1 , · · · , c M , which then attend back with the input x1, · · · , xN . The two attention matrices are N × M and M × N , respectively. The Squeeze-and-Expansion Transformer aims to improve in all the four aspects. The Squeezed Attention Block computes attention between the input and M inducing points [Lee et al., 2019], and compresses the attention matrices to N ×M . The Expanded Attention Block is a mixture-of-experts model with N m modes ("experts"). In both blocks, the query projections and key projections are tied to make the attention symmetric, for better modeling of the symmetric relationships between image units. In addition, a Learnable Sinusoidal Positional Encoding helps the model capture spatial relationships. [Lee et al., 2019] proposes Induced Set Attention Block (ISAB) by bringing inducing points into the transformer. It was originally designed to learn good features of a set of unordered objects. Here we employ this design to "squeeze" the bloated attention matrix, so as to reduce noises and overfitting in image tasks. We rename ISAB as Squeezed Attention Block (SAB) to highlight its new role in this context 1 .
Squeezed Attention Block
In SAB, inducing points are a set of M learned embeddings c 1 , · · · , c M in an external discrete codebook. Usually M N , the number of input units. The inducing points are first transformed into new embeddings C = c 1 , · · · , c M after attending with the input. The combination of these embeddings form the output features X out = x 1 , · · · , x N (Fig.3):
C = Single-Head(X, C),(4)X out = EAB(C , X),(5)
where Single-Head(·, ·) is a single-head transformer, and EAB(·, ·) is an Expanded Attention Block presented in the next subsection. In each of the two steps, the attention matrix is of N × M , much more compact than vanilla transformers. SAB is conceptually similar to the codebook used for discrete representation learning in [Esser et al., 2020], but the discretized features are further processed by a transformer. SAB can trace its lineage back to low-rank matrix factorization, i.e., approximating a data matrix X n×n ≈ P n×d · Q d×n , which is a traditional regularization technique against data noises and overfitting. Confirmed by an ablation study, SAB helps fight against noises and overfitting as well.
Expanded Attention Block
The Expanded Attention Block (EAB) consists of N m modes, each an individual single-head transformer. They output N m sets of contextualized features, which are then aggregated into one set using dynamic mode attention:
X (k) out = Mode (k) (X),(6)B (k) = Linear (k) (X (k) out ),(7)with k ∈ {1, · · · , N m }, G = softmax B (1) , · · · , B (Nm) ,(8)X out = X (1) out , · · · , X (Nm) out · G ,(9)
where the mode attention G ∈ R Nu×Nm is obtained by doing a linear transformation of each mode features, and taking softmax over all the modes. Eq.(9) takes a weighted sum over the modes to get the final output features X out . This dynamic attention is inspired by the Split Attention of the ResNest model [Zhang et al., 2020b]. EAB is a type of Mixture-of-Experts [Shazeer et al., 2017], an effective way to increase model capacity. Although there is resemblance between multi-head attention (MHA) and EAB, they are essentially different, as shown in Fig.4. In MHA, each head resides in an exclusive feature subspace and provides unique features. In contrast, different modes in EAB share the same feature space, and the representation power largely remains after removing any single mode. The modes join forces to offer more capacity to model diverse data, as shown in an ablation study. In addition, EAB is also different from the Mixture of Softmaxes (MoS) transformer [Zhang et al., 2020a]. Although MoS transformer also uses k sets of queries and keys, it shares one set of value transformation.
Learnable Sinusoidal Positional Encoding
A crucial inductive bias for images is the pixel locality and semantic continuity, which is naturally encoded by convolutional kernels. As the input to transformers is flattened into 1-D sequences, positional encoding (PE) is the only source to inject information about spatial relationships. On the one hand, this makes transformers flexible to model arbitrary shapes of input. On the other hand, the continuity bias of images is non-trivial to fully incorporate. This is a limitation of the two mainstream PE schemes: Fixed Sinusoidal Encoding and Discretely Learned Encoding [Carion et al., 2020;Dosovitskiy et al., 2021]. The former is spatially continuous but lacks adaptability, as the code is predefined. The latter learns a discrete code for each coordinate without enforcing spatial continuity.
We propose Learnable Sinusoidal Positional Encoding, aiming to bring in the continuity bias with adaptability. Given a pixel coordinate (x, y), our positional encoding vector pos(x, y) is a combination of sine and cosine functions of linear transformations of (x, y):
pos i (x, y) = sin(a i x + b i y + c i ) if i < C/2 cos(a i x + b i y + c i ) if i ≥ C/2,(10)
where i indexes the elements in pos, {a i , b i , c i } are learnable weights of a linear layer, and C is the dimensionality of image features. To make the PE behave consistently across different image sizes, we normalize (x, y) into [0, 1] 2 . When the input image is 3D, Eq.(10) is trivially extended by using 3D coordinates (x, y, z). The encoding in Eq.(10) changes smoothly with pixel coordinates, and thus nearby units receive similar positional encodings, pushing the attention weights between them towards larger values, which is the spirit of the continuity bias. The learnable weights and sinusoidal activation functions make the code both adaptable and nonlinear to model complex spatial relationships [Tancik et al., 2020].
Segtran Architecture
As a context-dependent pixel-wise classification task, segmentation faces a conflict between larger context (lower resolution) and localization accuracy (higher resolution). Segtran partly circumvents this conflict by doing pairwise feature contextualization, without sacrificing spatial resolutions. There are five main components in Segtran (Fig.1): 1) a CNN backbone to extract image features, 2) input/output feature pyramids to do upsampling, 3) learnable sinusoidal positional encoding, 4) Squeeze-and-Expansion transformer layers to contextualize features, and 5) a segmentation head.
CNN Backbone
We employ a pretrained CNN backbone to extract features maps with rich semantics. Suppose the input image is X 0 ∈ R H0×W0×D0 , where for a 2D image, D 0 = 1 or 3 is the number of color channels. For a 3D image, D 0 3 is the number of slices in the depth dimension. For 2D and 3D images, the extracted features are CNN(X 0 ) ∈ R C×H×W , and CNN(X 0 ) ∈ R C×H×W ×D , respectively. On 2D images, typically ResNet-101 or EfficientNet-D4 is used as the backbone. For increased spatial resolution, we change the stride of the first convolution from 2 to 1. Then H, W = H 0 /16, W 0 /16. On 3D images, 3D backbones like I3D [Carreira and Zisserman, 2017] could be adopted.
Transformer Layers
Before being fed into the transformer, the visual features and positional encodings of each unit are added up before being fed to the transformer: X spatial = X visual + pos(coordinates(X)). X spatial is flattened across spatial dimensions to a 1-D sequence X 0 ∈ R Nu×C , where N u is the total number of image units, i.e., points in the feature maps.
The transformer consists of a few stacked transformer layers. Each layer takes input features X, computes the pairwise interactions between input units, and outputs contextualized features X out of the same number of units. The transformer layers used are our novel design Squeeze-and-Expansion Transformer (Section 3).
Feature Pyramids and Segmentation Head
Although the spatial resolution of features is not reduced after passing through the transformer layers, for richer semantics, the input features to transformers are usually high-level features from the backbone. They are of a low spatial resolution, however. Hence, we increase their spatial resolution with an input Feature Pyramid Network (FPN) [Liu et al., 2018] and an output FPN, which upsample the feature maps at the transformer input end and output end, respectively.
Without loss of generality, let us assume the EfficientNet is the backbone. The stages 3, 4, 6, and 9 of the network are commonly used to extract multi-scale feature maps. Let us denote the corresponding feature maps as f 1 , f 2 , f 3 , f 4 , respectively. Their shapes are f i ∈ R Ci×Hi×Wi , with H i = H0 2 i , W i = W0 2 i . As described above, f (X 0 ) = f 4 is 1/16 of the original image, which is too coarse for accurate segmentation. Hence, we upsample it with an input FPN, and obtain upsampled feature maps f 34 :
f 34 = upsample ×2 (f 4 ) + conv 34 (f 3 ),(11)
where conv 34 is a 1 × 1 convolution that aligns the channels of f 3 to f 4 , and upsample ×2 (·) is bilinear interpolation. f 34 is 1/8 of the original image, and is used as the input features to the transformer layers. As the transformer layers keep the spatial resolutions unchanged from input to output feature maps, the output feature maps g 34 is also 1/8 of the input image. Still, this spatial resolution is too low for segmentation. Therefore, we adopt an output FPN to upsample the feature maps by a factor of 4 (i.e., 1/2 of the original images). The output FPN consists of two upsampling steps:
f 12 = upsample ×2 (f 2 ) + conv 12 (f 1 ), g 1234 = upsample ×4 (g 34 ) + conv 24 (f 12 ),(12)
where conv 12 and conv 24 are 1 × 1 convolutional layers that align the channels of f 1 to f 2 , and f 2 to f 4 , respectively. This FPN scheme is the bottom-up FPN proposed in [Liu et al., 2018]. Empirically, it performs better than the original top-down FPN [Lin et al., 2017], as richer semantics in top layers are better preserved.
The segmentation head is simply a 1 × 1 convolutional layer, outputting confidence scores of each class in the mask.
Experiments
Three tasks were evaluated in our experiments:
REFUGE20: Optic Disc/Cup Segmentation in Fundus
Images. This task does segmentation of the optic disc and cup in fundus images, which are 2D images of the rear of eyes (Fig. 5). It is a subtask of the REFUGE Challenge 2 [Orlando et al., 2020], MICCAI 2020. 1200 images were provided for training, and 400 for validation. We also used two extra datasets, Drishti-GS dataset [Sivaswamy et al., 2015] and RIM-ONE v3 [Fumero et al., 2011] when training all models. The Disc/Cup dice scores of validation images were obtained from the official evaluation server.
Polyp: Polyp Segmentation in Colonoscopy Images.
Polyps are fleshy growths in the colon lining that may become cancerous. This task does polyp segmentation in colonoscopy images (Fig. 5). Two image datasets [Fan et al., 2020] were used: CVC612 (CVC in short; 612 images) and Kvasir (1000 images). Each was randomly split into 80% training and 20% validation, and the training images were merged into one set.
BraTS19: Tumor Segmentation in MRI Images. This task focuses on the segmentation of gliomas, a common brain tumor in MRI scans. It was a subtask of the BraTS'19 challenge 3 [Menze et al., 2015;Bakas et al., 2017], MICCAI 2019. It involves four classes: the whole tumor (WT), the tumor core (TC), the enhancing tumor (ET) and background. Among them, the tumor core consists of the necrotic regions and non-enhancing tumors (red), as well as the enhancing tumor (yellow). 335 scans were provided for training, and 125 for validation. The dice scores of ET, WT and TC on the validation scans were obtained from the official evaluation server.
Ablation Studies
Two ablation studies were performed on REFUGE20 to compare: 1) the Squeeze-and-Expansion Transformer versus Multi-Head Transformer; and 2) the Learnable Sinusoidal Positional Encoding versus two schemes as well as not using PE.
All the settings were variants of the standard one, which used three layers of Squeeze-and-Expansion transformer with four modes (N m = 4), along with learnable sinusoidal positional encoding. Both ResNet-101 and EfficientNet-B4 were evaluated to reduce random effects from choices of the backbone. We only reported the cup dice scores, as the disc segmentation task was relatively easy, with dice scores only varying ±0.005 across most settings. Type of Transformer Layers. Table 1 shows that Squeezeand-Expansion transformer outperformed the traditional multi-head transformers. Moreover, Both the squeeze attention block and the expansion attention block contributed to improved performance.
Transformer Type
ResNet Positional Encoding. Number of Transformer Layers. Table 3 shows that as the number of transformer layers increased from 1 to 3, the performance improved gradually. However, one more layer caused performance drop, indicating possible overfitting.
Comparison with Baselines
Ten methods were evaluated on the 2D segmentation tasks: [Fan et al., 2020]: The official PyTorch implementation 7 . The pretrained Res2Net-50 [Gao et al., 2020] was recommended to be used as the encoder. • DeepLabV3+ [Chen et al., 2018]: A popular PyTorch implementation 8 , with a pretrained ResNet-101 as the encoder. • Attention based U-Nets [Oktay et al., 2018]: Attention U-Net (AttU-Net) and AttR2U-Net (a combination of AttU-Net and Recurrent Residual U-Net) were evaluated 9 . They learn to focus on important areas by computing element-wise attention weights (as opposed to the pairwise attention of transformers). • nnU-Net [Isensee et al., 2021]: nnU-Net generates a custom U-Net configuration for each dataset based on its statistics. It is primarily designed for 3D tasks, but can also handle 2D images after converting them to pseudo-3D. The original pipeline is time-consuming, and we extracted the generated U-Net configuration and instantiated it in our pipeline to do training and test. • Deformable U-Net [Jin et al., 2019] weights.
• TransU-Net [Chen et al., 2021]: TransU-Net uses a hybrid of ResNet and ViT as the encoder, and a U-Net style decoder. The official implementation 12 was evaluated, by fine-tuning their pretrained weights.
• Segtran: Trained with either a pretrained ResNet-101 or EfficientNet-B4 as the backbone.
Three methods were evaluated on the 3D segmentation task:
• Extension of nnU-Net : An extension of the nnU-Net 13 with two sampling strategies. • Segtran-3D: I3D [Carreira and Zisserman, 2017] was used as the backbone.
Training Protocols
All models were trained on a 24GB Titan RTX GPU with the AdamW optimizer. The learning rate for the three transformer-based models were 0.0002, and 0.001 for the other models. On REFUGE20, all models were trained with a batch size of 4 for 10,000 iterations (27 epochs); on Polyp, the total iterations were 14,000 (31 epochs). On BraTS19, Segtran was trained with a batch size of 4 for 8000 iterations. The training loss was the average of the pixel-wise crossentropy loss and the dice loss. Segtran used 3 transformer layers on 2D images, and 1 layer on 3D images to save RAM. The number of modes in each transformer layer was 4.
Results
Tables 4 and 5 present the evaluation results on the 2D and 3D tasks, respectively. Overall, the three transformer based methods, i.e., SETR, TransU-Net and Segtran achieved best performance across all tasks. With ResNet-101 as the backbone, Segtran performed slightly better than SETR and TransU-Net. With EfficientNet-B4, Segtran exhibited greater advantages.
It is worth noting that, Segtran (eff-B4) was among the top 5 teams in the semifinal and final leaderboards of the REFUGE20 challenge. Among either REFUGE20 or BraTS19 challenge participants, although there were several methods that performed slightly better than Segtran, they usually employed ad-hoc tricks and designs [Orlando et al., 2020;Zhao et al., 2019]. In contrast, Segtran achieved competitive performance with the same architecture and minimal hyperparameter tuning, free of domainspecific strategies.
Cross-Domain Generalization
To explore how well different methods generalize to new domains, we trained three representative methods, U-Net, DeepLabV3+ and Segtran on the 1200 training images of REFUGE20. All the methods used a pretrained ResNet-101 as the encoder/backbone. The trained models were evaluated on both the REFUGE20 training images and the RIM-One dataset [Fumero et al., 2011]. As RIM-One images have drastically different characteristics from REFUGE20, all models suffered severe performance drop, as shown in Table 6. Nevertheless, Segtran had the least performance degradation, showing the best cross-domain generalization. Fig.6 shows a RIM-One image and the corresponding soft segmentation masks (before thresholding) produced by different methods. The mask produced by Segtran contains the fewest artifacts.
Computational Efficiency
Impact of Pretraining
Models for medical image tasks usually benefit from initialization with weights pretrained on natural images (e.g. Ima-geNet [Deng et al., 2009]), as medical image datasets are typically small. To quantitatively study the impact of pretraining, Table 8 compares the performance of using pretrained weights vs. training from scratch of a few methods. Pretraining brought~2.5% increase of average dice scores to the two transformer-based models, and 1% to U-Net (ResNet-101).
Conclusions
In this work, we present Segtran, a transformer-based medical image segmentation framework. It leverages unlimited receptive fields of transformers to contextualize features. Moreover, the transformer is an improved Squeeze-and-Expansion transformer that better fits image tasks. Segtran sees both the global picture and fine details, lending itself good segmentation performance. On two 2D and one 3D medical image segmentation tasks, Segtran consistently outperformed existing methods, and generalizes well to new domains.
Figure 2 :
2Effective receptive fields of 3 models, indicated by nonnegligible gradients in blue blobs and light-colored dots. Gradients are back-propagated from the center of the image. Segtran has nonnegligible gradients dispersed across the whole image (light-colored dots). U-Net and DeepLabV3+ have concentrated gradients. Input image: 576 × 576.
Figure 4 :
4(a) Multi-head attention (MHA) vs. (b) Expanded attention block (EAB). In MHA, each head outputs an exclusive feature subset. In contrast, EAB outputs Nm sets of complete features from Nm modes, and aggregates them with dynamic mode attention.schemes[Carion et al., 2020;Dosovitskiy et al., 2021] do not fully impose such an inductive bias. This bias could be imposed by an improved positional encoding.
Figure 5 :
5Top: Optic disc/cup segmentation in fundus images into 3 classes: disc (grey), cup (white), and background (black). Bottom: Polyp segmentation in colonoscopy images into 2 classes: polyp (white) and background (black).
•
Bag of tricks (2nd place solution of the BraTS'19 challenge) [Zhao et al., 2019]: The winning entry used an ensemble of five models. For fairness, we quoted the best single-model results ("BL+warmup").
Figure 6 :
6Soft segmentation masks produced by different methods on a RIM-One image. The mask by Segtran has the fewest artifacts.
Figure 1: Segtran architecture. It extracts visual features with a CNN backbone, combines them with positional encodings of the pixel coordinates, and flattens them into a sequence of local feature vectors. The local features are contextualized by a few Squeeze-and-Expansion transformer layers. To increase spatial resolution, an input FPN and an output FPN upsamples the features before and after the transformers.Reshaping to
original shape
Visual features
Learnable sinusoidal
positional encoding
…
…
Squeeze-and-Expansion
Transformer layers
Local features
Contextualized features
Output FPN
…
Segmentation
head
,
CNN backbone
Input Encoder
Transformer
Spatially
flattening
Hi-res features
Coordinates
Input FPN
+
Table 1 :
1REFUGE'20 Fundus Optic Cup dice scores change with the type of transformer layers. Single-Mode implies No Expansion.Cell-DETR uses a multi-head transformer and discretely learned PE.
N h : number of attention heads in a MHA. Nm: number of modes
in a Squeeze-and-Expansion transformer.
Table 2
2compares learnable sinu-
soidal positional encoding with the two mainstream PE
schemes and no PE. Surprisingly, without PE, performance
of Segtran only dropped 1~2%. A possible explanation is that
the transformer may manage to extract positional information
from the CNN backbone features [Islam et al., 2020].
Positional Encoding
ResNet-101 Eff-B4
None
0.857
0.853
Discretely learned
0.852
0.860
Fixed Sinusoidal
0.857
0.849
Learnable Sinusoidal
0.862
0.872
Table 2 :
2REFUGE'20 Fundus Optic Cup dice scores change with
the type of positional encoding (PE) schemes.
Table 3 :
3REFUGE20 Optic Cup dice scores change with the num-
ber of transformer layers. Best performers with each backbone are
highlighted.
• U-Net [Ronneberger et al., 2015]: The implementa-
tion in a popular library Segmentation Models.PyTorch
(SMP) was used 4 . The pretrained ResNet-101 was cho-
sen as the encoder. In addition, U-Net implemented in
U-Net++ (below) was evaluated as training from scratch.
• U-Net++ [Zhou et al., 2018]: A popular PyTorch imple-
mentation 5 . It does not provide options to use pretrained
encoders, and thus was only trained from scratch.
• U-Net3+ [Huang et al., 2020]: The official PyTorch im-
plementation 6 . It does not provide options to use pre-
trained encoders.
• PraNet
Table 4 :
4Dice scores on REFUGE20 and Polyp validation sets.
R101: ResNet-101; R50: ResNet-50; eff-B4: EfficientNet-B4.
12 https://github.com/Beckschen/TransUNet 13 https://github.com/woodywff/brats_2019BraTS19
ET
WT
TC
Avg.
Extension of nnU-Net 0.737 0.894 0.807 0.813
Bag of tricks
0.729 0.904 0.802 0.812
Segtran (i3d)
0.740 0.895 0.817 0.817
Table 5 :
5Dice scores on BraTS19 validation set. Only single-model performance is reported.
Table 7
7presents the number of parameters and FLOPs of a few representative methods. In general, transformer-based methods consume more computation and GPU RAM than conventional methods.Our profiling showed that the number of parameters/FLOPs of Segtran are dominated by the output FPN, which vary drastically across different backbones. As the REFUGE RIM-OneDrop
Table 6 :
6Generalization of three methods, measured by drop of Optic Cup dice scores from the REFUGE20 training images to a new test domain RIM-One. The smaller the drop is, the better. All used ResNet-101 as the encoder/backbone. bottom-up FPNs we adopt are somewhat similar to Effi-cientDet[Tan et al., 2020], the model size/FLOPs are optimal when using EfficientNets. With ResNets as the backbone, Segtran has a significantly higher model size/FLOPs, and hence this choice of backbone is not recommended for efficiency-sensitive scenarios.Params (M) FLOPs (G)
nnU-Net
41.2
16.3
AttU-Net
34.9
51.0
SETR (ViT)
307.1
91.1
TransU-Net (R50+ViT)
93.2
32.2
Segtran (R101)
166.7
152.8
Segtran (eff-B4)
93.1
71.3
Table 7 :
7Number of parameters / FLOPs on a 256x256 input image.
Table 8 :
8Impact of using pretrained encoder weights.
We clarify that our contribution is a novel transformer architecture that combines SAB with an Expanded Attention Block.
https://refuge.grand-challenge.org/Home2020/ 3 https://www.med.upenn.edu/cbica/brats-2019/
https://github.com/qubvel/segmentation_models.pytorch/ 5 https://github.com/4uiiurz1/pytorch-nested-unet
AcknowledgementsWe are grateful for the help and support of Wei Jing. This research is supported by A*STAR under its Career Development Award (Grant No. C210112016), and its Human-Robot Collaborative Al for Advanced Manufacturing and Engineering (AME) programme (Grant No. A18A2b0046).
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"https://github.com/askerlee/segtran.",
"https://github.com/Beckschen/TransUNet",
"https://github.com/woodywff/brats_2019BraTS19",
"https://github.com/qubvel/segmentation_models.pytorch/",
"https://github.com/4uiiurz1/pytorch-nested-unet",
"https://github.com/LeeJunHyun/Image_Segmentation",
"https://github.com/RanSuLab/DUNet-retinal-vessel-detection",
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] |
[
"Understanding Leakage Currents through Al 2 O 3 on SrTiO 3",
"Understanding Leakage Currents through Al 2 O 3 on SrTiO 3"
] |
[
"Dror Miron \nDepartment of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n",
"Igor Krylov \nDepartment of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n",
"Maria Baskin \nDepartment of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n",
"Eilam Yalon \nDepartment of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n",
"Lior Kornblum \nDepartment of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n"
] |
[
"Department of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael",
"Department of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael",
"Department of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael",
"Department of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael",
"Department of Electrical Engineering\nThe Andrew & Erna Viterbi\nTechnion -Israel Institute of Technology\n32000HaifaIsrael"
] |
[] |
Leakage currents through insulators received continuous attention for decades, owing to their importance for a wide range of technologies, and interest in their fundamental mechanisms. This work investigates the leakage currents through atomic layer deposited (ALD) Al 2 O 3 , grown on SrTiO 3 . This combination is not only a key building block of oxide electronics, but also a clean system for studying the leakage mechanisms without interfacial layers that form on most of the conventional bottom electrodes. We show how tiny differences in the deposition process can have a dramatic effect on the leakage behavior. Detailed analysis of the leakage behavior rules out Fowler-Nordheim tunneling (FNT) and thermionic emission, and leaves the trap-related mechanisms of trap-assisted tunneling (TAT) and Poole-Frenkel as the likely mechanisms. After annealing the sample in air, the currents are reduced, which is ascribed to transition from trap-based mechanism to FNT, due to the elimination of the traps. The dramatic role of the assumptions regarding the flat-band voltage used for analysis is critically discussed, and the sensitivity of the extracted parameters on this magnitude is quantitatively described. We show that field effect devices based on structures similar to those described here, should be able to modulate >10 13 cm -2 electrons. These results provide general guidelines for reducing and analyzing leakage currents in insulators, and highlight some of the possible approaches and pitfalls in their analysis. _____________________________ a)
|
10.1063/1.5119703
|
[
"https://export.arxiv.org/pdf/1907.05441v1.pdf"
] | 196,470,797 |
1907.05441
|
c171bd763910249cbd145e7854867cb2e8f5fd93
|
Understanding Leakage Currents through Al 2 O 3 on SrTiO 3
Dror Miron
Department of Electrical Engineering
The Andrew & Erna Viterbi
Technion -Israel Institute of Technology
32000HaifaIsrael
Igor Krylov
Department of Electrical Engineering
The Andrew & Erna Viterbi
Technion -Israel Institute of Technology
32000HaifaIsrael
Maria Baskin
Department of Electrical Engineering
The Andrew & Erna Viterbi
Technion -Israel Institute of Technology
32000HaifaIsrael
Eilam Yalon
Department of Electrical Engineering
The Andrew & Erna Viterbi
Technion -Israel Institute of Technology
32000HaifaIsrael
Lior Kornblum
Department of Electrical Engineering
The Andrew & Erna Viterbi
Technion -Israel Institute of Technology
32000HaifaIsrael
Understanding Leakage Currents through Al 2 O 3 on SrTiO 3
1
Leakage currents through insulators received continuous attention for decades, owing to their importance for a wide range of technologies, and interest in their fundamental mechanisms. This work investigates the leakage currents through atomic layer deposited (ALD) Al 2 O 3 , grown on SrTiO 3 . This combination is not only a key building block of oxide electronics, but also a clean system for studying the leakage mechanisms without interfacial layers that form on most of the conventional bottom electrodes. We show how tiny differences in the deposition process can have a dramatic effect on the leakage behavior. Detailed analysis of the leakage behavior rules out Fowler-Nordheim tunneling (FNT) and thermionic emission, and leaves the trap-related mechanisms of trap-assisted tunneling (TAT) and Poole-Frenkel as the likely mechanisms. After annealing the sample in air, the currents are reduced, which is ascribed to transition from trap-based mechanism to FNT, due to the elimination of the traps. The dramatic role of the assumptions regarding the flat-band voltage used for analysis is critically discussed, and the sensitivity of the extracted parameters on this magnitude is quantitatively described. We show that field effect devices based on structures similar to those described here, should be able to modulate >10 13 cm -2 electrons. These results provide general guidelines for reducing and analyzing leakage currents in insulators, and highlight some of the possible approaches and pitfalls in their analysis. _____________________________ a)
I. Introduction
Alumina (Al 2 O 3 ) grown by atomic layer deposition (ALD) is a widespread insulating oxide.
The motivation for studying this material ranges from understanding various fundamental physical aspects of Al 2 O 3 1,2 to its considerable potential for applications in electronics, optics and many other fields. Some two decades ago, ALD Al 2 O 3 received considerable attention as a potential gate insulator for Si technology 3 , owing to its large band gap. Despite the preliminary interest, Al 2 O 3 was eventually sidelined by Hf-based oxides as the high-k gate insulator for ultra-scaled Si logic devices 4 . Nonetheless, Al 2 O 3 has found other uses in Si technology, such as in ultra-thin layers for effective work-function adjustment 5,6 . Beyond Si technologies, ALD Al 2 O 3 emerged as the best passivation layer for Ge 7 , high-Ge content SiGe devices 8 and with III-V based devices as well 9 . Other back-end microelectronics applications of ALD-Al 2 O 3 include metal-insulator-metal (MIM) capacitors for resistive switching random-access memory (RRAM) devices 10 , antifuse devices 11 and others 12 .
More recently, the wide bandgap semiconductor β-Ga 2 O 3 has emerged as a promising candidate for power devices 13 . In such roles, the ability to apply high fields is critical for the performance of power field effect devices, and here the high bandgap of Al 2 O 3 is attractive in reducing the gate leakage currents 14,15 .
Another potential application of ALD-Al 2 O 3 is in oxide electronics, a field greatly invigorated by the discovery of a 2D electron gas (2DEG) at the interface between some insulating oxides 16,17 . One of the promising applications is an oxide field effect transistor (FET) which utilizes the 2DEG as a confined electron channel 18 Many metals form a native oxide surface layer, and when used as the back electrode this layer adds an additional insulator in series; metals that don't have surface oxides, such as Pt, are typically problematic for nucleation of many ALD oxides due to their surface chemistry, which can result in lower quality films.
Conductive oxides are therefore more suitable substrates for studying leakage through ALD-
Al 2 O 3 .
With β-Ga 2 O 3 devices, the oxide substrate is already part of the device, and this issue has been addressed by several works; for example, Hung et al. 34 and Bhuiyan et al. 15 , who both reported trapassisted tunneling as the dominant Al 2 O 3 leakage mechanism and extracted a trap energy of 1.1 eV below the conduction band. Another conductive oxide back electrode, indium tin oxide (ITO), has been employed by Spahr et al. 35 , who reported a thorough investigation of the leakage currents through lowtemperature ALD-Al 2 O 3 , grown at 80 °C. While low temperature processes are crucially important for some applications, their resulting stoichiometry can be less ideal than films grown at 200-300 °C, which is important for leakage reduction.
In this work we address the leakage currents of ALD-Al 2 O 3 grown on conductive STO substrates. We briefly demonstrate that a default ALD recipe is far from ideal for this task, and by comparison to a more optimized process we obtain further insight into the conduction process. The motivation for this study is twofold: to evaluate ALD-Al 2 O 3 and its limits for STO-based oxide electronics, and by employing STO as a conductive back-electrode we aim to understand the leakage mechanisms through Al 2 O 3 , a question that is applicable to many technologies beyond oxide electronics.
II. Experimental
(001) 0.01%(wt) Nb-doped STO (Nb:STO) crystals (CrysTec GmbH) were TiO 2 terminated using the "extended Arkansas" method 36 . This process started with solvent sonication cleaning, capacitance-voltage (CV) characteristics were measured using a Keithely 2450 source meter instrument, and a Keysight E4980A precision LCR meter, respectively, in a shielded light-sealed box with a home-built heating stage. After measuring the sample, it was annealed in air in a tube furnace for 30 min at 500 °C (measured on the outer tube surface).
III. Results and Discussion
The Pt/Al 2 O 3 /Nb:STO structures are treated as metal insulator semiconductor (MIS) capacitors,
where in the general case, the applied voltage on the gate (V g ) can be expressed by the following 37
g FB ox s V V V (1) where FB V isE=V ox /d=(V g -V FB )/d (2)
where d is the oxide thickness. In the absence of Nb:STO depletion, the Al 2 O 3 permittivity can be extracted directly from the CV plot using the parallel plate capacitor expression C ox /A=ε 0 ε r /t ox , where 0 is the vacuum permittivity, r is the relative permittivity of Al 2 O 3 and A is the pad area.
where C s is the series capacitance and R s the series resistance, acquired at low and high frequencies, respectively. At accumulation (+4V), the extracted C s at 5 kHz and R s at 800 kHz before thermal anneal are 0.68 μF/cm 2 and 32 Ω, respectively, while after thermal anneal the extracted C s at 5 kHz and R s at 800 kHz are 0.74 μF/cm 2 and 94 Ω, respectively (slight oxidation of the surface of the Al back contact may account for this small R s increase). Relative permittivity of 7.7 and 8.4 was extracted before and after thermal anneal, respectively, in agreement with previous reports 40,42 . The small increase in the capacitance can be ascribed to slight densification, which could increase the permittivity 43 and reduce the thickness. From this point onward all analysis is done on Recipe B samples. Beyond the measurement voltage range, devices were found to be prone to irreversible damage. The relevant region for positive bias leakage was analyzed at varying temperatures. Figure 2b exhibits distinct temperature dependence and a current increase of ×2 was measured by raising the temperature from 22 to 70 °C. The sample was measured once again after anneal (500 o C for 30 min in air), and showed two main noticeable differences: for any given voltage value the leakage is noticeably smaller, and more importantly, the temperature dependence was diminished, a fact that would be addressed later. The positive bias of the JV curve (5-5.7 V) was fitted by the abovementioned mechanisms ( Figure 3). The relationship between the current density and the electric field in PF emission model is given by 44 :
0 exp / / tr J E q qE kT . (3)
Where E is electric field in the insulator, q is the electron charge, The relationship between the current density and the electric field in TE model is given by 44 : Next, we consider the validity of TAT mechanism, where the relationship between the current density and the electric field, according to Fleischer et. al, 49 can be simplified by the following expression:
2 0 * exp / 4 / Br J A T q qE kT (4) 1 3/2 1 2 3 exp t t t t J C N q E A E (5)
where C t is a slowly varying function of electron energy, 50 qC N P P / (P P ) dx where X 1 =(Vt )/E and P 1 and P 2 are the tunneling probabilities 50 . This simulation is presented in Figure 3h and implies that this simplified model is a good approximation for the TAT integral. TAT provides possible values for the trap levels that vary slowly with V FB , and none can be ruled out. The trap density N t , is seen to vary wildly with the assumed V FB (Figure 3i); changes of ~1 V in the former result in two orders of magnitude difference in the latter. We therefore conclude that extracting the trap density from this model is unreliable in the absence of precise knowledge of the flat-band voltage, and that TAT remains a likely candidate. While the basic TAT models are temperature-independent, some temperature dependence has been observed with TAT 38 and modeled by Yu et al 52 . who accounted for the Fermi-Dirac distribution at the injecting electrode; however, simulation using this model could not yield a satisfying fit for all data as well. We've therefore ruled out TE, and cannot rule out TAT and PF at this point.
Last, we examine FNT, where the relationship between the current density and the electric field is given by the following expression 44 :
1 2 3/2 exp 4 2 * 3 B J E qm E (6)
An FNT linear fit is shown in Figure 3k. Since the slopes of the JV curves before and after anneal are similar, repeating the analysis of Figure 3 on the annealed samples produces nearly identical parameters. However, the disappearance of the temperature dependence following anneal, and the overall leakage reduction, favor FNT. Since our interpretation of TAT or PF in the unannealed samples ascribed the enhanced electron conductivity to For oxide electronics, for example field-effect transistors (FETs) with 2DEG channels, depleting the 2DEG is necessary for closing this "normally-on" device 55 . We therefore estimate the charge modulation ΔQ=C Al2O3 /ΔV for the unannealed structures. We obtain a modulation of ~4×10 12 electrons/cm 2 per 1 V on the gate(0.45 MV/cm using V FB = 1.45 V), or a modulation of ~1.7×10 13 electrons/cm 2 at gate voltages of up to |4| V (2.55 MV/cm), before the onset of detectable leakage currents ( Figure 4). An additional higher-k layer may be considered to create a bilayer structure. This could reduce the field on the Al 2 O 3 without significant reduction of the capacitance 56 . Altogether, our data provides design guidelines for an oxide FET, and particularly for engineering its 2DEG properties, towards achieving low off-state currents and maximizing their I on /I off ratios.
IV. Conclusions
We show that while ALD Al 2 O 3 is a mature, well-established process for gate insulator applications, optimization of small process details can lead to huge benefits in mitigating leakage. The optimized structures can be useful as gate stacks for oxide electronics, owing to their low leakage which enables significant charge modulation of an underlying channel. mathematical agreement to the data, careful consideration of the extracted parameters rules out TE and FNT. Moreover, we show that the effect of the assumed flat-band voltage on the extracted parameters can be huge, necessitating careful handling of this parameter. Altogether, we identify trap-based conduction (TAT or PF) as the most likely leakage mechanism for unannealed samples. These traps can be ascribed to oxygen vacancies in Al 2 O 3 . A moderate anneal reduces the leakage currents and alleviates their temperature dependence. This observation highlights Fowler-Nordheim tunneling (FNT) as the most likely candidate, indicating the possible mitigation of the oxygen vacancies during anneal.
followed by a 3:1 HCl-HNO 3 treatment. A two-step anneal was performed, starting with 1000 o C for 1 hour in air and followed by 650 o C for 30 minutes in flowing O 2 . Nominally 10 nm thick amorphous Al 2 O 3 layer was grown by ALD (Ultratech/Cambridge Nanotech Fiji G2) using trimethyl-aluminum (TMA) and water as the precursors. Two recipes were used and compared: Recipe A, the manufacturer's default for Al 2 O 3 was performed at a substrate temperature of 300 o C and Recipe B was optimized by extending the water pulse by a factor of 5, the TMA purging pulse by ×2.5 and the water purging pulse by ×1.25, at a substrate temperature of 280 o C. Film thickness of 10±0.5 nm, was measured by x-ray reflectivity (XRR, acquired using a Rigaku SmartLab and analyzed with GlobalFit 2.0). 50 nm Pt pads were deposited through a shadow mask using e-beam evaporation and back contact was prepared by ebeam evaporation of 300 nm blanket Al on the back of the wafer. Current density-voltage (JV) and
the flat-band voltage, V ox is the voltage drop across the oxide and s is the band bending, or surface potential of Nb:STO. Since Moon et al. 38 calculated a significant s in 0.7% doped samples, here CV measurements were conducted at a frequency range of 5-800 kHz to assess the possibility of depletion in the Nb:STO. Measurements were conducted in the voltage ranges of -4 V to 4 V where leakage is undetectable (Figure 2a), in order to ensure our accurate interpretation of the capacitance. Figure 1 shows little voltage or frequency dependence of the capacitance. No significant capacitance reduction is observed with voltage, indicating negligible depletion of the Nb:STO in the measured voltage range. We therefore conclude that s is insignificant in Eq. (1), and thus the Nb:STO Fermi level remains near the conduction band edge of the highly-doped STO. The flat-band voltage is taken as
Figure 1 .
1CV measurements for 0.01%(wt) Nb:STO grown with recipe B using Cs-Rs model (inset) in the frequency range of 5-800 kHz, before and after thermal anneal.We start the leakage analysis by comparing ALD Recipe A (instrument default) with Recipe B (an optimized recipe). The JV behavior(Figure 2a)shows that the default Recipe A exhibits considerably higher leakage currents at lower voltages. At the most extreme cases, Recipe A has over 2 orders of magnitude higher leakage currents at the same voltage compared to the optimized Recipe B(e.g. around 4 V). Altogether, our optimized recipe yields undetectable leakage (<10 -8 A/cm 2 ) at fields of under ±4 MV/cm, and no signs of breakdown below 5 MV/cm in the devices tested here. In comparison, Moon et al. demonstrated 4 nm Al 2 O 3 /Nb:STO heterojunction and observed leakage of above 10 -3 A·cm -2 at an approximated electric field of 2.5 MV·cm -1 (Ref. 38 ); however, at these ultrathin dimensions, additional mechanisms may come into play. These results illustrate how seemingly-Cs Rs identical Al 2 O 3 layers grown under similar conditions can vary wildly in their performance as gate insulators, as a result of minute process details.
Figure 2 .
2(a) JV comparison between recipe A and recipe B. The measured noise floor of up to 4.5 pA at 22 o C corresponds to 2·10 -8 A·cm -2 . (b) Varying-temperature JV taken at the positive bias leakage region of Recipe B, before and after anneal. Before anneal the temperature ranges from 22 to 70 o C and after anneal the range is 22 to 50 o C. Four possible leakage mechanism were considered: Fowler-Nordheim tunneling (FNT), trapassisted tunneling (TAT), Poole-Frenkel (PF) emission and Schottky-Richardson thermionic emission (TE). The temperature dependence contradicts the FNT mechanism, where no temperature dependence is expected in its most simplified picture. On the other hand, PF and TE and TAT models qualitatively agree with the measured temperature trend.
is the Al 2 O 3 (high-frequency) relative permittivity and k is the Boltzman constant. The averaged relative permittivity is extracted from the slope of the linear fit of PF (Figure.3b), determined to be 3.4 ± 0.5, which is somewhat higher than the extracted value from optical measurements 45 of 2.2-2.6(46,47) . Furthermore, other works suggested PF has a negligible probability to be a dominant mechanism48 .However, The initial use of a value of 1.45 V as the flat-band voltage is an approximation of the effective work functions difference, neglecting all other potential parasitics such as fixed oxide charges that are likely to affect the flat-band voltage44 . Therefore, a range of flat-band voltages was considered to examine the effect of this elusive magnitude on the parameters extracted from each model.Analyzing the PF conduction mechanism with a broader range of flat-band voltages, yields reasonable flat-band range of voltages of -0.3 to -1.3 V, in which PF emission is possible(Figure 3c) and thus cannot be eliminated.
B
is energy barrier between the conduction band edges of Nb:STO and Al 2 O 3 and A* is the effective Richardson constant. An average energy barrier of 1.7 ± 0.1 eV was extracted from the intercept of the linear fit of TE (Figure 3d), which is in reasonable agreement with the barrier of 2.3 ± 0.3 eV, obtained by spectroscopy 30 . However, the averaged relative permittivity of 0.8 ± 0.1, extracted from the slope, does not agree with the reported value of ~2.4 46 . While good agreement can be obtained for the barrier B , no flat-band voltage in the (arbitrary) range examined could provide a physically-relevant value for the permittivity (Figure 3f), which further validates the exclusion of the TE mechanism.
the electron's effective mass in Al 2 O 3 and is the reduced Planck constant. The value used for m* is 0.23m 0 (Ref.3). This specific TAT model was chosen for its simplicity and its ability to analyze the physical parameters, but we note that TAT modeling has many other variants.By plotting ln JE as a function of E -1 we extract an average trap energy level of 1.6 eV below the conduction band edge of Al 2 O 3(Figure 3h). Theoretical analyses of oxygen vacancies in Al 2 energy levels of 2 eV below the conduction band, in some agreement with this experimental observation. After extracting these parameters, the accuracy of this procedure is examined by simulating the TAT integral
An average barrier height of 1.4 V is extracted from the linear slopes of ln(J·E -2 ) versus E -1 plots, which does not agree with the spectroscopically-measured 2.3 ± 0.3 V 30 . The flat-band analysis (Figure 3l) yields good agreement to the expected barrier value of 2.3 eV at low flat-band voltages. While this may appear a success, the increase of the currents with the temperature does not agree with the basic FNT model. Modifications to FNT can incorporate temperature dependence53 , but FNT should produce the lowest currents from the all the mechanisms discussed above; the observation of lower leakage currents after anneal therefore strongly indicates a trap based conduction mechanism (either PF or TAT) as the leakage mechanism for the unannealed sample. As a result of the large energy barrier between STO and Al 2 O 3 , it can be safely assumed that the first step of the dominant conduction mechanism, whatever it is, is tunneling electrons from STO to traps inside Al 2 O 3 . Subsequently, the electrons either gain enough thermal energy to escape out to the conduction band (PF) as modeled by Jeong et al.54 in their so called 'tunnel assisted PF' (TAPF), or tunnel to the other electrode, as was indicated by Yu et al.52 .
Figure 3 .
3Conduction mechanisms analysis at positive bias. (a-c) Poole-Frenkel (PF), (d-f) Thermionic emission (TE), (g-i) Trap-assisted tunneling (TAT) and (j-l) Fowler-Nordheim tunneling (FNT). The first panel of each row shows a schematic of the conduction mechanism. The second panel presents the temperature-dependent fits to the model represented in that row, assuming VFB = 1.45 V. Symbols represent measured data points and lines represent their linear fits. An additional orange line in panel (h) represents the simulated TAT curves (which coincides with the fits, see text). The third panel of each row illustrates the influence of the flat-band voltage assumption on the parameters extracted from that model.
Figure 4 .
4Q=CV analysis for annealed and unannealed structures, showing the possible degree of charge modulation using the studied MIS structures.
The wide band gap of STO of 3.2 eV 23 makes leakage reduction even more challenging, owing to the relatively low possible barriers with the insulator. We previously addressed this issue growth is the easiest and most widespread ALD process, and it can be robustly performed over wide range of temperatures and other process conditions 33 with excellent results. While many applications are relatively insensitive to the growth conditions, thin gate insulators can be extremely dependent of growth parameters. Otherwise-excellent Al 2 O 3 films may exhibit high leakage currents, poor reproducibility, reliability problems and other issues.. Such devices were demonstrated by epitaxially growing LaAlO 3 (LAO) on
top of single crystal SrTiO 3 (STO) substrates 19,20,21 . The subsequent discovery of oxide 2DEG based on
amorphous oxides grown on STO 22 has quickly paved the way to the application of ALD-Al 2 O 3 for this
purpose as well 23,24 . This concept has been extended by the replacement of STO crystals by thin TiO 2
layers, also grown by ALD 24 . This considerably increases the scalability of oxide electronics, by
circumventing the use of single crystalline STO substrates 25 , available in limited sizes. Similar
Al 2 O 3 /STO and Al 2 O 3 /TiO 2 structures have also been suggested as selectors for memristor crossbar
arrays, 26 for gas sensors 27 and for spintronic devices 28,29 .
spectroscopically, by investigating the band alignment at Al 2 O 3 /STO interfaces, and reported barriers
of 2.0 ± 0.3 eV and 1.4 ± 0.2 eV for electrons and holes, respectively 30 .
One possible explanation to the discrepancy between the large barriers determined by
spectroscopy and the high leakage currents is electron traps or states that may exist inside the Al 2 O 3
band gap, that are typically attributed to oxygen vacancies 31,32 . Therefore, large bandgap and large
barriers are insufficient for mitigating gate leakage.
Al 2 O 3 Considering their importance, leakage currents through ALD Al 2 O 3 have been an integral part
of its development since its early days on Si 3 . Understanding the fundamental properties of the leakage
currents can be done by studying leakage through ALD-Al 2 O 3 grown on a semiconductor, on metal-
coated substrates or on conductive oxides. However, interfacial layers are typically formed at
oxide/semiconductor interfaces, which complicate the analysis of the leakage through the Al 2 O 3 layer 3 .
where Pt is the Pt effective work function that was measured as 5.35 V on Al 2 O 339 , and STO is approximated as the STO electron affinity of 3.9 V 40 . Additional possible contributions to V FB are neglected at this point, an assumption to be revisited later, and since s is small, the electric field in the oxide, E, is taken as1.45
FB
Pt
STO
V
V
the presence of oxygen vacancies,38,51 it is quite likely that anneal in air oxidized the sample enough to decrease the trap density and hence minimize the contribution of the traps, leaving FNT as the dominant current mechanism.Poole-Frenkel
Trap-Assisted Tunneling
Thermionic Emission
Fowler-Nordheim Tunneling
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(j)
(h)
(i)
(k)
(l)
AcknowledgementsThe authors are grateful for the support of the Israeli Science Foundation (ISF Grant 375/17).Partial support in the fabrication and characterization of the samples was provided by the Technion's Micro-Nano Fabrication Unit (MNFU) and the Russell Berrie Nanotechnology Institute (RBNI). We thank Prof. Felix Palumbo for fruitful discussions. Valentina Korchnoy and Arkadi Gavrilov are acknowledged for valuable technical assistance. We acknowledge the contribution of Aviv Haim and Haim Ben-Simhon, whose undergraduate research project inspired parts of this research.
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|
[] |
[
"PRICING WITH COHERENT RISK",
"PRICING WITH COHERENT RISK"
] |
[
"A S Cherny [email protected] \nFaculty of Mechanics and Mathematics\nDepartment of Probability Theory\nMoscow State University\n119992MoscowRussia\n"
] |
[
"Faculty of Mechanics and Mathematics\nDepartment of Probability Theory\nMoscow State University\n119992MoscowRussia"
] |
[] |
This paper deals with applications of coherent risk measures to pricing in incomplete markets. Namely, we study the No Good Deals pricing technique based on coherent risk. Two forms of this technique are presented: one defines a good deal as a trade with negative risk; the other one defines a good deal as a trade with unusually high RAROC. For each technique, the fundamental theorem of asset pricing and the form of the fair price interval are presented. The model considered includes static as well as dynamic models, models with an infinite number of assets, models with transaction costs, and models with portfolio constraints. In particular, we prove that in a model with proportional transaction costs the fair price interval converges to the fair price interval in a frictionless model as the coefficient of transaction costs tends to zero.Moreover, we study some problems in the "pure" theory of risk measures: we present a simple geometric solution of the capital allocation problem and apply it to define the coherent risk contribution.The mathematical tools employed are probability theory, functional analysis, and finite-dimensional convex analysis.
|
10.1137/s0040585x97983158
|
[
"https://export.arxiv.org/pdf/math/0605049v1.pdf"
] | 123,333,256 |
math/0605049
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18b79d744c892bef646801abfbbf4ae85f71644b
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PRICING WITH COHERENT RISK
2 May 2006
A S Cherny [email protected]
Faculty of Mechanics and Mathematics
Department of Probability Theory
Moscow State University
119992MoscowRussia
PRICING WITH COHERENT RISK
2 May 2006and phrases: Capital allocationcoherent risk measuresextreme measuresgenerating setNo Good DealsRAROCrisk contributionrisk-neutral measuressupport functionTail V@Rtransaction costsWeighted V@R
This paper deals with applications of coherent risk measures to pricing in incomplete markets. Namely, we study the No Good Deals pricing technique based on coherent risk. Two forms of this technique are presented: one defines a good deal as a trade with negative risk; the other one defines a good deal as a trade with unusually high RAROC. For each technique, the fundamental theorem of asset pricing and the form of the fair price interval are presented. The model considered includes static as well as dynamic models, models with an infinite number of assets, models with transaction costs, and models with portfolio constraints. In particular, we prove that in a model with proportional transaction costs the fair price interval converges to the fair price interval in a frictionless model as the coefficient of transaction costs tends to zero.Moreover, we study some problems in the "pure" theory of risk measures: we present a simple geometric solution of the capital allocation problem and apply it to define the coherent risk contribution.The mathematical tools employed are probability theory, functional analysis, and finite-dimensional convex analysis.
Introduction
1. Overview. The three basic pillars of finance are:
• optimal investment; • pricing and hedging; • risk measurement and management.
The most well-known financial theories related to the first pillar are the Markowitz meanvariance analysis and Sharpe's CAPM, which are often termed the "first revolution in finance". The most well-known result related to the second pillar is the Black-Scholes-Merton formula, which is often termed the "second revolution in finance". Recently a very important innovation has appeared in connection with the third pillar. In 1997, Artzner, Delbaen, Eber and Heath [4], [5] introduced the concept of a coherent risk measure as a new way of measuring risk. Since 1997, the theory of coherent risk measures has rapidly been evolving and is already termed in some sources the "third revolution in finance" (see [52]). Let us mention, in particular, the papers [1], [3], [20], [31], [32], [38], [40], [53] and the reviews [21], [33;Ch. 4], [47]. Currently, one of the major tasks is the problem of proper risk measurement in the dynamic setting; see, in particular, [12], [25], [36], [45], and [46].
The theory of coherent risk measures is important not only for risk measurement. Indeed, risk (≈ uncertainty) is at the very basis of the whole finance, and therefore, a new way of looking at risk yields new approaches to other problems of finance, in particular, to those related to the first and the second pillars. Nowadays, more and more research is aimed at applications of coherent risk measures to other problems of finance.
One of the major goals of modern financial mathematics is providing adequate price bounds for derivative contracts in incomplete markets. It is known that No Arbitrage price bounds in incomplete markets are typically unacceptably wide, and fundamentally new ideas are required to narrow these bounds. Recently, a promising approach to this problem termed No Good Deals (NGD) pricing has been proposed in [6], [17]. Let us illustrate its idea by an example. Consider a contract that with probability 1/2 yields nothing and with probability 1/2 yields 1000 USD. The No Arbitrage (NA) price interval for this contract is (0, 1000). But if the price of the contract is, for instance, 15 USD, then everyone would be willing to buy it, and the demand would not match the supply. Thus, 15 USD is an unrealistic price because it yields a good deal, i.e. a trade that is attractive to most market participants. The technique of the NGD pricing is based on the assumption that good deals do not exist.
A problem that arises immediately is how to define a good deal. There is no canonical answer, and several approaches have been proposed in the literature. Cochrane and Saá-Requejo [17] defined a good deal as a trade with unusually high Sharpe ratio, Bernardo and Ledoit [6] based their definition on another gain to loss ratio, whileČerný and Hodges [11] proposed a generalization of both definitions (see also the paper [7] by Bjork and Slinko, which extends the results of [17]).
The technique of the NGD pricing can also be motivated as follows. When a trader sells a contract, he/she would charge for it a price, with which he/she will be able to superreplicate the contract. In theory the superreplication is typically understood almost surely, but in practice an agent looks for an offsetting position such that the risk of his/her overall portfolio would stay within the limits prescribed by his/her management (the almost sure superreplication is virtually impossible in practice). These considerations lead to the NGD pricing with a good deal defined as a trade with negative risk. Now, if risk is measured by V@R, this technique leads to the quantile hedging introduced by Föllmer and Leukert [30]. But instead of V@R, one can take a coherent risk measure. The corresponding pricing technique has already been considered in several papers. Carr, Geman, and Madan [9] (see also the review paper [10]) studied this technique in a probabilistic framework (although they do not use the term "good deal"), while Jaschke and Küchler [35] studied this technique in a topological space framework in the spirit of Harrison and Kreps [34] (see also the paper [51] by Staum, which extends the results of [35]). Furthermore, Larsen, Pirvu, Shreve, and Tütüncü [41] considered pricing based on convex risk measures instead of coherent ones (convex risk measures were introduced by Föllmer and Schied [31]). Roorda, Schumacher, and Engwerda [46] studied pricing in the multiperiod model using as a basis dynamic coherent risk measures instead of static ones.
2. Goal of the paper. This is the first of a series of papers dealing with applications of coherent risk measures to the basic problems of finance (the other paper in the series is [15]). The basic idea behind the series is: the whole finance can be built based on coherent risks.
In this paper, we study applications to pricing in incomplete markets. Our approach is similar to that of [9], but [9] assumes an unrealistic world of a finite state space and a finite set of probabilistic scenarios defining a coherent risk measure (most natural coherent risk measures are defined through an infinite set of probabilistic scenarios; see Subsection 2.1). Our model is general in the sense that we consider an arbitrary Ω and a general class of coherent risk measures (satisfying only a sort of compactness condition). Moreover, our approach applies to dynamic models, to models with an infinite number of assets, to models with transaction costs, and to models with convex portfolio constraints. Within this general model, we prove the Fundamental Theorem of Asset Pricing (Theorem 3.4) and provide the form of the fair price interval of a contingent claim (Corollary 3.6). We confine ourselves to static risk measures.
A problem that has attracted attention in several papers is as follows. Consider a model with proportional transaction costs. Is it true that the upper (resp., lower) price of a contingent claim in this model tends to the upper (resp., lower) price of this claim in the frictionless model as the coefficient of transaction costs tends to zero? It was shown in [14], [19], [42], and [50] that, for NA prices, the answer to this question is negative already in the Black-Scholes model (the contingent claim considered in these papers is a European call option). This result might be interpreted as follows: the NA technique is useless in continuous-time models with transaction costs. In this paper (Theorem 3.17), we prove that, for NGD prices, the answer to the above question is positive. This is done within a framework of a general model (the price follows an arbitrary process) with an infinite number of assets and an arbitrary contingent claim (satisfying only some integrability condition). The advantage of the NGD pricing is not only that this result is true, but also that its proof is very short.
Furthermore, we introduce a new variant of pricing based on coherent risk, which we call the RAROC-based NGD pricing. The idea is to define a good deal as a trade with unusually high Risk-Adjusted Return on Capital (RAROC), where RAROC is defined through coherent risk. On the mathematical side, this technique is reduced to the standard NGD pricing (with the original risk measure replaced by another one).
Although this series of papers deals primarily with applications of coherent risk measures to problems of finance, we also establish some results and give several definitions related to "pure" risk measures (these are needed for applications). In particular, we introduce the notion of an extreme measure. The results of this paper and [15] show that this notion is very convenient and important; it appears in the outcomes of several pricing techniques proposed in [15] and in considerations of the equilibrium problem in [15]. In the present paper, we provide a solution of the capital allocation problem in terms of extreme measures (Theorem 2.12). Let us remark that this problem was considered in [21], [24], [28], [39], [44], and [53].
Parallel with the measurement of outstanding risks, a very important problem is measuring the risk contribution of a subportfolio to a "big" portfolio. Based on our solution of the capital allocation problem, we propose several equivalent definitions of the coherent risk contribution.
Another notion we introduce is the notion of a generator. It establishes a bridge between coherent risks and convex analysis, opening the way for geometry. In particular, we provide (see Figure 1) a geometric solution of the capital allocation problem (thus there are two solutions: a probabilistic one is given in terms of extreme measures, while a geometric one is given in terms of generators). We also provide a geometric solution of the pricing and hedging problem (Proposition 3.21) for a model with a finite number of assets. Furthermore, we provide in [15] geometric solutions of several optimization problems, optimality pricing problems, and the equilibrium problem. In fact, for most problems considered in this series of papers, we provide two sorts of results:
• a geometric result applicable to a model with a finite number of assets is given in terms of generators; • a probabilistic result applicable to a general model is typically given in terms of extreme measures.
3. Structure of the paper. Section 2 deals with "pure" risk measures rather than with their applications. Subsection 2.1 recalls some basic definitions related to coherent risks. In Subsection 2.2, we introduce the L 1 -spaces associated with a coherent risk measure (these are employed in the technical conditions in theorems below). Subsection 2.3 presents the definition of an extreme measure. In Subsection 2.4, we provide a solution of the capital allocation problem. Subsection 2.5 deals with equivalent definitions of risk contribution.
Section 3 is related to the NGD pricing. In Subsections 3.1 and 3.2, we study the ordinary and the RAROC-based forms of this technique, respectively. The model considered is a general one, and in Subsections 3.3-3.5, we consider some particular cases of this model: a static model with a finite number of assets (for which fair price intervals admit a simple geometric description; see Figure 3), a continuous-time dynamic model, and a continuous-time dynamic model with transaction costs. Furthermore, in Subsection 3.6, we provide a geometric solution of the hedging problem for a static model with a finite number of assets (see Figure 4).
Acknowledgement. I am thankful to D.B. Madan for valuable discussions and important advice.
Coherent Risk Measures
Basic Definitions
Let (Ω, F , P) be a probability space. The following definition was introduced in [4], [5]. These papers considered only a finite Ω, in which case the continuity axiom (e) is not needed. It was added for a general Ω by Delbaen [20]. Definition 2.1. A coherent utility function on L ∞ is a map u : L ∞ → R with the properties:
(a) (Superadditivity) u(X + Y ) ≥ u(X) + u(Y ); (b) (Monotonicity) If X ≤ Y , then u(X) ≤ u(Y ); (c) (Positive homogeneity) u(λX) = λu(X) for λ ∈ R + ; (d) (Translation invariance) u(X + m) = u(X) + m for m ∈ R;
(e) (Fatou property) If |X n | ≤ 1, X n P − → X , then u(X) ≥ lim sup n u(X n ).
The corresponding coherent risk measure is ρ(X) = −u(X).
Remark. Typically, a coherent risk measure is defined only via conditions (a)-(d), and then one speaks about coherent risk measures with the Fatou property. However, only such risk measures are useful, and for this reason we find it more convenient to add (e) as a basic axiom.
The theorem below was established in [5] for the case of a finite Ω (in this case the axiom (e) is not needed) and in [20] for the general case. We denote by P the set of probability measures on F that are absolutely continuous with respect to P. Throughout the paper, we identify measures from P (these are typically denoted by Q) with their densities with respect to P (these are typically denoted by Z ).
u(X) = inf Q∈D E Q X, X ∈ L ∞ .
(2.1)
So far, a coherent risk measure has been defined on bounded random variables. Let us ask ourselves the following question: Are "financial" random variables like the increment of a price of some asset indeed bounded? The right way to address this question is to split it into two parts:
• Are "financial" random variables bounded in practice?
• Are "financial" random variables bounded in theory?
The answer to the first question is positive (clearly, everything is bounded by the number of the atoms in the universe). The answer to the second question is negative because most distributions used in theory (like the lognormal one) are unbounded. So, as we are dealing with theory, we need to extend coherent risk measures to the space L 0 of all random variables. It is hopeless to axiomatize the notion of a risk measure on L 0 and then to obtain the corresponding representation theorem. Instead, we take representation (2.1) as the basis and extend it to L 0 .
Definition 2.3. A coherent utility function on L 0 is a map u : L 0 → [−∞, ∞] de- fined as u(X) = inf Q∈D E Q X, X ∈ L 0 ,(2.2)
where D ⊆ P and E Q X is understood as E Q X + − E Q X − with the convention ∞ − ∞ = −∞. The corresponding coherent risk measure is ρ(X) = −u(X).
Clearly, a set D , for which representations (2.1) and (2.2) are true, is not unique. However, there exists the largest such set given by {Q ∈ P : E Q X ≥ u(X) for any X}. We introduce the following definition.
Definition 2.4. We will call the largest set, for which (2.1) (resp., (2.2)) is true, the determining set of u.
Remark. Clearly, the determining set is convex. For coherent utility functions on L ∞ , it is also L 1 -closed. However, for coherent utility functions on L 0 , it is not necessarily L 1 -closed. As an example, take a positive unbounded random variable X 0 such that P(X 0 = 0) > 0 and consider D 0 = {Q ∈ P : E Q X 0 = 1}. Clearly, the determining set D of the coherent utility function u(X) = inf Q∈D 0 E Q X satisfies D 0 ⊆ D ⊆ {Q ∈ P : E Q X 0 ≥ 1}. On the other hand, the L 1 -closure of D 0 contains a measure Q 0 concentrated on {X 0 = 0}. Important Remark. Let D be an L 1 -closed convex subset of P . (Let us note that a particularly important case is where D is L 1 -closed, convex, and uniformly integrable; this condition will be needed in a number of places below). Define a coherent utility function u by (2.2). Then D is the determining set of u. Indeed, assume that the determining set D is greater than D , i.e. there exists Q 0 ∈ D \ D . Then, by the Hahn-Banach theorem, we can find X 0 ∈ L ∞ such that E Q 0 X 0 < inf Q∈D E Q X , which is a contradiction. The same argument shows that D is also the determining set of the restriction of u to L ∞ .
In what follows, we will always consider coherent utility functions on L 0 . Example 2.5. (i) Tail V@R (the terms Average V@R, Conditional V@R, and Expected Shortfall are also used) is the risk measure corresponding to the coherent utility function
u λ (X) = inf Q∈D λ E Q X,
where λ ∈ [0, 1] and
D λ = Q ∈ P : dQ dP ≤ λ −1 . (2.3)
In particular, if λ = 0, then the corresponding coherent utility function has the form u(X) = essinf ω X(ω). For more information on Tail V@R, see [3] (ii) Weighted V@R on L ∞ (the term spectral risk measure is also used) is the risk measure corresponding to the coherent utility function
u µ (X) = [0,1] u λ (X)µ(dλ), X ∈ L ∞ , where µ is a probability measure on [0, 1].
Weighted V@R on L 0 is the risk measure corresponding to the coherent utility function
u µ (X) = inf Q∈Dµ E Q X, X ∈ L 0 ,
where D µ is the determining set of u µ on L ∞ . Let us remark that, under some regularity conditions on µ, Weighted V@R possesses some nice properties that are not shared by Tail V@R. In a sense, it is "smoother" than Tail V@R. We consider Weighted V@R as one of the most important classes (or maybe the most important class) of coherent risk measures. For a detailed study of this risk measure, see [1], [2], [26], [40] as well as the paper [16], which is in some sense the continuation of the present paper. ✷
Spaces
L 1 w and L 1 s
For a subset D of P , we introduce the weak and strong L 1 -spaces
L 1 w (D) = {X ∈ L 0 : u(X) > −∞, u(−X) > −∞}, L 1 s (D) = X ∈ L 0 : lim n→∞ sup Q∈D E Q |X|I(|X| > n) = 0 . Clearly, L 1 s (D) ⊆ L 1 w (D). If D = {Q} is a singleton, then L 1 w (D) = L 1 s (D) = L 1 (Q)
, which motivates the notation.
In general, L 1 s (D) might be strictly smaller than L 1 w (D). Indeed, let X 0 be a positive unbounded random variable with P(X 0 = 0) > 0 and let D = {Q ∈ P : E Q X 0 = 1}. Then
X 0 ∈ L 1 w (D), but X 0 / ∈ L 1 s (D)
. (One can also construct a similar counterexample with an L 1 -closed set D ; see Example 2.11). However, as shown by the proposition below, in most natural situations weak and strong L 1 -spaces coincide.
Proposition 2.6. (i) If D λ is the determining set of Tail V@R (see Example 2.5 (i)) with λ ∈ (0, 1], then L 1 w (D λ ) = L 1 s (D λ ). (ii) If D µ is the determining set of Weighted V@R (see Example 2.5 (ii)) with µ concentrated on (0, 1], then L 1 w (D µ ) = L 1 s (D µ ). (iii)
If all the densities from D are bounded by a single constant and P ∈ D , then
L 1 w (D) = L 1 s (D). (iv) If D is a convex combination N n=1 a n D n , where D 1 , . . . , D N are such that L 1 w (D n ) = L 1 s (D n ), then L 1 w (D) = L 1 s (D). (v) If D = conv(D 1 , . . . , D N ), where D 1 , . . . , D N are such that L 1 w (D n ) = L 1 s (D n ), then L 1 w (D) = L 1 s (D).
Lemma 2.7. If µ is a convex combination ∞ n=1 a n δ λn , where λ n ∈ (0, 1], then the determining set D µ of Weighted V@R corresponding to µ has the form ∞ n=1 a n D λn , where D λ is given by (2.3).
Proof. Denote n a n D λn by D . Clearly, D is convex. Fix X ∈ L ∞ . It is easy to see that, for any n, the minimum of expectations of EXZ over Z ∈ D λn is attained (for more details, see [16;Prop. 2.7]). Hence, the minimum of expectations E P XZ over Z ∈ D is attained. By the James theorem (see [29]), D is weakly compact. As it is convex, an application of the Hahn-Banach theorem shows that it is L 1 -closed.
Obviously, u µ (X) = inf Q∈D E Q X for any X ∈ L ∞ . Taking into account the Important Remark following Definition 2.4, we get D µ = D . ✷
Proof of Proposition 2.6. The only nontrivial statement is (ii). In order to prove it, consider the measures µ
= ∞ k=1 a k δ 2 −k ,μ = ∞ k=1 a k δ 2 −k+1 , where a k = µ((2 −k , 2 −k+1 ]). As u µ ≤ u µ ≤ uμ , we have D µ ⊇ D µ ⊇ Dμ . By Lemma 2.7, D µ = ∞ k=1 a k Z k : Z k ∈ D 2 −k , Dμ = ∞ k=1 a k Z k : Z k ∈ D 2 −k+1 . Take X ∈ L 1 w (D µ ). Consider Z k = 2 k−1 I(X < q k ) + c k I(X = q k ),
where q k is the 2 −k+1 -quantile of X and c k is chosen in such a way that E P Z k = 1. Then
E P Z k X = min Z∈D 2 −k+1 E P ZX.
The density Z 0 = ∞ k=1 a k Z k belongs to Dμ and
E P Z 0 X = min Z∈Dμ E P ZX. In view of the inclusion X ∈ L 1 w (D µ ) ⊆ L 1 w (Dμ), the latter quantity is finite. Thus, ∞ k=1 a k min Z∈D 2 −k+1 E P ZX > −∞, which implies that ∞ k=1 a k min Z∈D 2 −k+1 E P Z(−X − ) > −∞.
The same estimate is true for X + , and therefore,
∞ k=1 a k sup Z∈D 2 −k E P Z|X| ≤ 2 ∞ k=1 a k sup Z∈D 2 −k+1 E P Z|X| < ∞. (2.4)
It is clear that X ∈ L 1 , and thus, for each k ,
sup Z∈D 2 −k E P Z|X|I(|X| > n) ≤ 2 k E P |X|I(|X| > n) − −− → n→∞ 0.
This, combined with (2.4), yields
sup Z∈Dµ E P Z|X|I(|X| > n) ≤ sup Z∈D µ E P Z|X|I(|X| > n) = ∞ k=1 a k sup Z∈D 2 −k E P Z|X|I(|X| > n) − −− → n→∞ 0.
Extreme Measures
Definition 2.8. Let u be a coherent utility function with the determining set D . Let X ∈ L 0 . We will call a measure Q ∈ D an extreme measure for X if
E Q X = u(X) ∈ (−∞, ∞).
The set of extreme measures will be denoted by X D (X).
Let us recall some general facts related to the weak topology on L 1 . The weak topology on L 1 is induced by the duality between L 1 and L ∞ and is usually denoted as σ(L 1 , L ∞ ). The Dunford-Pettis criterion states that a set D ⊆ P is weakly compact if and only if it is weakly closed and uniformly integrable. Furthermore, an application of the Hahn-Banach theorem shows that a convex set D ⊆ P is weakly closed if and only if it is L 1 -closed.
Proposition 2.9. If the determining set D is weakly compact and X ∈ L 1 s (D), then X D (X) = ∅.
Proof. It is clear that u(X) ∈ (−∞, ∞). Find a sequence Z n ∈ D such that E P Z n X → u(X). This sequence has a weak limit point Z ∞ ∈ D . Clearly, the map D ∋ Z → E P ZX is weakly continuous. Hence, E P Z ∞ X = u(X), which means that Z ∞ ∈ X D (X). ✷ Example 2.10. (i) If u corresponds to Tail V@R of order λ ∈ (0, 1] (see Example 2.5 (i)) and X has a continuous distribution, then it is easy to see that X D (X) consists of a unique density λ −1 I(X ≤ q λ ), where q λ is a λ-quantile of X .
(ii) If u corresponds to Weighted V@R with the weighting measure µ (see Example 2.5 (ii)) and X has a continuous distribution, then X D (X) consists of a unique density g(X), where g(x) = [F (x),1] λ −1 µ(dλ) and F is the distribution function of X (see [16;Sect. 6]). Note that this density reflects the risk aversion of an agent possessing a portfolio that produces the P&L (Profit&Loss) X . ✷
The condition that D should be weakly compact is very mild and is satisfied for the determining sets of most natural coherent risk measures. For example, the determining set D λ of Tail V@R is weakly compact provided that λ ∈ (0, 1]. The determining set D µ of Weighted V@R is weakly compact provided that µ is concentrated on (0, 1]; this follows from the explicit representation of this set provided in [8] (the proof can also be found in [33;Th. 4.73] or [47; Th. 1.53]); this can also be seen from the representation of D µ provided in [16].
The following example shows that the condition X ∈ L 1 s (D) in Proposition 2.8 cannot be replaced by the condition X ∈ L 1 w (D).
Example 2.11. Let Ω = [0, 1] be endowed with the Lebesgue measure. Consider
Z n = √ nI [0,1/n] + 1 − 1/ √ n, n ∈ N. Then Y n := Z n − 1 L 1
− → 0, and therefore, the set D = 1 + ∞ n=1 a n Y n : a n ≥ 0, ∞ n=1 a n ≤ 1 is convex, L 1 -closed, and uniformly integrable. Thus, D is weakly compact. Now, consider
X(ω) = −1/ √ ω . Then E P Z n X = −4 + 2/ √ n. Thus, inf Q∈D E Q X = −4, while there exists no Q ∈ D such that E Q X = −4. ✷
Capital Allocation
Let (Ω, F , P) be a probability space, u be a coherent utility function with the determining set D , and let X 1 , . . . , X d ∈ L 1 w (D) be the discounted P&Ls produced by different components of a firm (P&L means the Profit&Loss, i.e. the difference between the terminal wealth and the initial wealth). We will use the notation X = (X 1 , . . . , X d ).
Informally, the capital allocation problem is the following. How is the total risk ρ i X i being split between the components 1, . . . , d? In other words, we are looking for a vector (x 1 , . . . , x d ) such that x i means that part of the risk carried by the i-th component. Taking
x i = ρ(X i ) does not work because i ρ(X i ) = ρ i X i .
The following definition of a capital allocation is taken from [21; Sect. 9]. In fact, it is closely connected with the coalitional games (see [24]).
Problem (capital allocation): Find x 1 , . . . , x d ∈ R such that d i=1 x i = u d i=1 X i , (2.5) ∀h 1 , . . . , h d ∈ R + , d i=1 h i x i ≥ u d i=1 h i X i . (2.6)
We will call a solution of this problem a utility allocation between X 1 , . . . , X d . A capital allocation is defined as a utility allocation with the minus sign.
From the financial point of view, −x i is the contribution of the i-th component to the total risk of the firm, or, equivalently, the capital that should be allocated to this component. In order to illustrate the meaning of (2.6), consider the example h i = I(i ∈ J), where J is a subset of {1, . . . , d}. Then (2.6) means that the capital allocated to a part of the firm does not exceed the risk carried by that part. Let us introduce the notation G = cl{E Q X : Q ∈ D}, where "cl" denotes the closure. Note that G is convex and compact. We will call it the generating set or simply the generator for X and u. This term is justified by the line
u( h, X ) = inf Q∈D E Q h, X = inf Q∈D h, E Q X = min x∈G h, x , h ∈ R d .
(2.7)
Note that the last expression is a classical object of convex analysis known as the support function of the convex set G.
Theorem 2.12. The set U of utility allocations between X 1 , . . . , X d has the form
U = argmin x∈G e, x ,(2.
8)
where e = (1, . . . , 1). Furthermore, for any utility allocation x, we have
∀h 1 , . . . , h d ∈ R, d i=1 h i x i ≥ u d i=1 h i X i (2.9)
If moreover X 1 , . . . , X d ∈ L 1 s (D) and D is weakly compact, then
U = E Q X : Q ∈ X D d i=1 X i . (2.10)
Proof. (The proof is illustrated by Figure 1.) For h ∈ R d , we set
L(h) = x ∈ R d : h, x = min y∈G h, y , M(h) = x ∈ R d : h, x ≥ min y∈G h, y .
It is seen from (2.7) that the set of points x ∈ R d that satisfy ( If G is strictly convex (i.e. its interior is nonempty and its border contains no interval), then a utility allocation is unique. However, in general it is not unique as shown by the example below.
Example 2.13. Let d = 2 and X 2 = −X 1 . Then G is the interval with the endpoints (u(X 1 ), −u(X 1 )) and (−u(−X 1 ), u(−X 1 )). In this example, U = G. ✷
Let us now find the solution of the capital allocation problem in the Gaussian case.
Example 2.14. Let X have Gaussian distribution with mean a and covariance matrix C . Let u be a law invariant coherent utility function, i.e. u(X) depends only on the distribution of X ; we also assume that u is finite on Gaussian random variables.
Then there exists γ > 0 such that, for a Gaussian random variable ξ with mean m and variance σ 2 , we have u(ξ) = m − γσ . Let L denote the image of R d under the map x → Cx. Then the inverse C −1 : L → L is correctly defined. It is easy to see that
G = a + {C 1/2 x : x ≤ γ} = a + {y ∈ L : y, C −1 y ≤ γ 2 }.
Let e = (1, . . . , 1) and assume first that Ce = 0. In this case the utility allocation x 0 between X 1 , . . . , X d is determined uniquely. In order to find it, note that, for any y ∈ L such that d dε ε=0
x 0 − a + εy, C −1 (x 0 − a + εy) = 0,
we have e, y = 0. This implies that C −1 (x 0 − a) = α pr L e with some constant α (pr L denotes the orthogonal projection on L). Thus, x 0 = a + αCe. As x 0 should belong to the relative border of G (i.e. the border in the relative topology of a + L), we have x 0 − a, C −1 (x 0 − a) = γ 2 , i.e. α = −γ e, Ce −1/2 . As a result, the utility allocation between X 1 , . . . , X d is a − γ e, Ce −1/2 Ce. Assume now that Ce = 0. This means that e is orthogonal to L, and then the set of utility allocations between X 1 , . . . , X d is G.
Let us remark that in this example the solution of the capital allocation problem depends on u rather weakly, i.e. it depends only on γ . ✷
Risk Contribution
Let (Ω, F , P) be a probability space, u be a coherent utility function with the determining set D , X ∈ L 0 be the discounted P&L produced by a component of some firm, and Y ∈ L 0 be the discounted P&L produced by the whole firm. From the financial point of view, such a firm assesses the risk of X not as ρ(X) but rather as ρ(W + X) − ρ(X). Below we define a risk contribution ρ c (X; W ) in such a way that it is a coherent risk measure as a function of X and ρ c (X; W ) ≈ ρ(W + X) − ρ(W ) provided that X is small as compared to W (the precise statement is Theorem 2.16).
Definition 2.15. The utility contribution of X to Y is
u c (X; W ) = inf Q∈X D (Y ) E Q X.
The risk contribution of X to Y is defined as ρ c (X; Y ) = −u c (X; Y ).
The utility contribution is a coherent utility function provided that X D (Y ) = ∅. If D is weakly compact and X, Y ∈ L 1 s (D) then, by Theorem 2.12,
u c (X; Y ) = inf{x 1 : (x 1 , x 2 ) is a utility allocation between X, Y − X}.
This formula enables one to define risk contribution under a weaker assumption X, Y ∈ L 1 w (D). If D is weakly compact, X 1 , . . . , X d ∈ L 1 s (D), and X D i X i is a singleton, then (in view of Theorem 2.12) the utility allocation between X 1 , . . . , X d is unique and has the form
u c X 1 ; d i=1 X i , . . . , u c X d ; d i=1 X i .
This shows the relevance of the given definition. Another argument supporting this definition is the statement below.
u c (X; Y ) = lim ε↓0 ε −1 (u(Y + εX) − u(Y )).
Proof.
(The proof is illustrated by Figure 2.) Consider the generator G = cl{E Q (X, Y ) : Q ∈ D} and set b = inf{y : ∃x : (x, y) ∈ G}, I = {x : (x, b) ∈ G}, J = {x : ∃y : (x, y) ∈ G}, a = inf{x : x ∈ I}. Note that u c (X; Y ) = a. The minimum min (x,y)∈G (ε, 1), (x, y) is attained at a point (a(ε), b(ε)). We have a (ii) Let X = αY with α ∈ R + . Then u c (X; Y ) = αu(Y ).
(ε) ≤ a, b(ε) ≥ b, and (a(ε), b(ε)) − − → ε↓0 (a, b). Furthermore, εa(ε) + b(ε) ≤ εa + b, which implies that 0 ≤ b(ε) − b ≤ ε(a − a(ε)). As a result, lim ε↓0 ε −1 (u(Y + εX) − u(Y )) = lim ε↓0 ε −1 (εa(ε) + b(ε) − b) = a + lim ε↓0 ε −1 (b(ε) − b) = a = u c (X; Y ).
(iii) Let X, Y have a jointly Gaussian distribution with mean (EX, EY ) and covariance matrix C . Let u be a law invariant coherent utility function that is finite on Gaussian random variables. Then there exists γ > 0 such that, for a Gaussian random variable ξ with mean m and variance σ 2 , we have u(ξ) = m − γσ . Assume that X and Y are not degenerate and corr(X, Y ) = ±1. It follows from Example 2.14 that
u c (X; Y ) = EX − γ e 2 , Ce 2 −1/2 Ce 2 = EX − γ cov(X, Y ) (var Y ) 1/2 = EX + (u(X) − EX) corr(X, Y ),
where e 2 = (0, 1). In particular, if EX = EY = 0, then
u c (X; Y ) u(X) = corr(X, Y ) = V@R c (X; Y ) V@R(X) ,
where var denotes the variance and V@R c denotes the V@R contribution (for the definition, see [43; Sect. 7]). ✷
Good Deals Pricing
Utility-Based Good Deals Pricing
Let (Ω, F , P) be a probability space, u be a coherent utility function with the weakly compact determining set D , and A be a convex subset of L 0 . From the financial point of view, A is the set of various discounted P&Ls that can be obtained in the model under consideration by employing various trading strategies (examples are given in Subsections 3.3-3.5). It will be called the set of attainable P&Ls. We will assume that A is D -consistent (see Definition 3.2 below). It is shown in Subsections 3.3-3.5 that this assumption is automatically satisfied for natural models. First, we give the definition of a risk-neutral measure. Of course, this notion is a classical object of financial mathematics, but the particular definition we need is taken from [13] (it is adapted to the L 0 -case).
Definition 3.1. A risk-neutral measure is a measure Q ∈ P such that E Q X ≤ 0 for any X ∈ A (we use the convention EX = EX + − EX − , ∞ − ∞ = −∞).
The set of risk-neutral measures will be denoted by R or by R(A) if there is a risk of ambiguity. Proof. The "if" part is obvious. Let us prove the "only if" part. Fix X 1 , . . . , X M ∈ A ′ . It follows from the weak continuity of the maps D ∋ Q → E Q X m that the set G = {E Q (X 1 , . . . , X M ) : Q ∈ D} is compact. Clearly, G is convex. Suppose that G ∩ (−∞, 0] M = ∅. Then there exist h ∈ R M and ε > 0 such that h, x ≥ ε for any x ∈ G and h, x ≤ 0 for any x ∈ (−∞, 0] M . Hence, h ∈ R M + . Without loss of generality, m h m = 1. Then X = m h m X m ∈ A and E Q X ≥ ε for any Q ∈ D , so that u(X) > 0.
Definition 3.2. We will say that
A is D -consistent if there exists a set A ′ ⊆ A ∩ L 1 s (D) such that D ∩ R = D ∩ R(A ′ ).
The obtained contradiction shows that, for any X 1 , . . . , X M ∈ A ′ , the set B(X 1 , . . . , X M ) = {Q ∈ D : E Q X m ≤ 0 for any m = 1, . . . , M} is nonempty. As X m ∈ L 1 s (D), the map D ∋ Q → E Q X m is weakly continuous, and therefore, B(X 1 , . . . , X M ) is weakly closed. Furthermore, any finite intersection of sets of this form is nonempty. Consequently, there exists a measure Q that belongs to each B . Then E Q X ≤ 0 for any X ∈ A ′ , which means that Q ∈ D ∩ R(A ′ ). As A is D -consistent, Q ∈ D ∩ R. ✷ Remarks. (i) As opposed to the fundamental theorems of asset pricing dealing with the NA condition and its strengthenings (see [13], [22], [23]), here we need not take any closure of A when defining the NGD. Essentially, this is the compactness of D that yields the fundamental theorem of asset pricing.
(ii) If D = P , then the NGD condition means that there exists no X ∈ A with essinf ω X(ω) > 0. This is very close to the NA condition. However, in this case D is not uniformly integrable and Theorem 3.4 might be violated. Indeed, let A = {hX : h ∈ R}, where X has uniform distribution on [0, 1]. Then the NGD is satisfied, while R = ∅. Now, let F ∈ L 0 be the discounted payoff of a contingent claim. The set of the NGD prices will be denoted by I NGD (F ).
Corollary 3.6 (Fair price interval). For F ∈ L 1 s (D),
I NGD (F ) = {E Q F : Q ∈ D ∩ R}. Proof. Denote {h(F − x) : h ∈ R} by A(x). Clearly, A + A(x)
is D -consistent (in order to prove this, it is sufficient to consider A ′ +A(x)). It follows from Theorem 3.4 that x ∈ I NGD (F ) if and only if D ∩R(A+A(x)) = ∅. It is easy to check that Q ∈ R(A+A(x)) if and only if Q ∈ R and E Q F = x. This completes the proof. ✷
Remark. As opposed to the NA price intervals, the NGD price intervals are closed (this follows from the weak continuity of the map D ∩ R → E Q F ).
To conclude the subsection, we will discuss the origin of D . First of all, D might be the determining set of a coherent utility function like Tail V@R or Weighted V@R. The set D might also correspond to a weighted average or the minimum of several coherent utility functions. It is also possible that D originates from the classical utility maximization as described by the example below.
Example 3.7. Let P 1 , . . . , P N be a family of probability measures, u 1 , . . . , u N be a family of classical utility functions (i.e. smooth concave increasing functions R → R), and W 1 , . . . , W N be a family of random variables. From the financial point of view, P n , u n , and W n are the subjective probability, the utility function, and the future wealth of the n-th market participant, respectively. Consider a measure Q n = c n u ′ n (W n )P n , where c n is the normalizing constant. Then, for any trading opportunity X ∈ L 0 , we have
d dε ε=0 u n (W n + εX) = E Pn u ′ n (W n )X = E Qn c −1 n X (3.1)
(we assume that all the expectations exist and integration is interchangeable with differentiation). Thus, an opportunity εX with a small ε > 0 is attractive to the n-th participant if and only if E Qn X > 0, so that Q n might be called the valuation measure of the n-th participant. Take D = conv(Q 1 , . . . , Q N ) and consider the corresponding coherent utility function u. Then u(X) > 0 if and only if E Qn X > 0 for any n. In view of (3.1), this means that εX with some ε > 0 is attractive to any market participant (this is similar to the notion of a strictly acceptable opportunity introduced in [9]). Thus, in this example the NGD means the absence of a trading opportunity that is attractive to every agent. ✷
RAROC-Based Good Deals Pricing
Let (Ω, F , P) be a probability space, RD ⊂ P be a convex weakly compact set, PD be an L 1 -closed convex subset of RD , and A be a convex subset of L 0 . We will call PD the profit-determining set. Thus, the profit of a position that yields a P&L X is inf Q∈PD E Q X . We will call RD the risk-determining set, so that the risk of a position that yields a P&L X is − inf Q∈D E Q X . A canonical example is: PD = {P} and RD is the determining set of a coherent utility function. We will assume that A is RD -consistent. Finally, we fix a positive number R meaning the upper limit on a possible RAROC.
Definition 3.8. The Risk-Adjusted Return on Capital (RAROC) for X ∈ L 0 is defined as
RAROC(X) = +∞ if inf Q∈PD E Q X > 0 and inf Q∈RD E Q X ≥ 0, inf Q∈PD E Q X − inf Q∈RD E Q X otherwise with the convention 0 0 = 0, ∞ ∞ = 0.
Definition 3.9. A model satisfies the RAROC-based NGD condition if there exists no X ∈ A such that RAROC(X) > R.
1 1 + R PD + R 1 + R RD ∩ R = ∅. (3.2)
Proof. Let us first consider the case R > 0. Then, for any X ∈ L 0 ,
RAROC(X) > R ⇐⇒ inf Q∈PD E Q X + R inf Q∈RD E Q X > 0 ⇐⇒ inf Q∈D E Q X > 0, where D = 1 1+R PD+ R 1+R RD . Clearly, D is weakly compact (note that D ⊆ RD , while L 1 s (D) = L 1 s (RD)
) and A is D -consistent. Now, the statement follows from Theorem 3.4. Let us now consider the case R = 0. Then the "if" part is obvious, and we should check the "only if" part. Take The set of the NGD prices will be denoted by I NGD (F ).
A ′ ⊆ A ∩ L 1 s (RD) such that RD ∩ R = RD ∩ R(A ′ ). For any X ∈ conv A ′ , inf Q∈PD E Q X ≤ 0.
Corollary 3.12 (Fair price interval).
For F ∈ L 1 s (D),
I NGD (F ) = E Q F : Q ∈ 1 1 + R PD + R 1 + R RD ∩ R .
This statement follows from Theorem 3.10.
Static Model with a Finite Number of Assets
We consider the model of the previous subsection with
A = { h, S 1 − S 0 : h ∈ R d },
where S 0 ∈ R d and S 1 1 , . . . , S d 1 ∈ L 1 s (RD). From the financial point of view, S i n is the discounted price of the i-th asset at time n. Clearly, in this model A is RD -consistent and RD ∩ R = RD ∩ M, where M is the set of martingale measures:
M = {Q ∈ P : E Q |S 1 | < ∞ and E Q S 1 = S 0 }.
Remark.
We have M ⊆ R, but the reverse inclusion might be violated. Indeed, let d = 1 and let S 1 be such that E P S + 1 = E P S − 1 = ∞. Then P ∈ R, while P / ∈ M.
Let us now provide a geometric interpretation of Theorems 3.4 and 3.10. For this, we only assume that PD ⊆ RD ⊆ P are convex sets and S 1 ∈ L 1 w (RD). Let us introduce the notation (see Figure 3)
E = cl{E Q S 1 : Q ∈ PD}, G = cl{E Q S 1 : Q ∈ RD}, G R = 1 1 + R E + R 1 + R G, D = conv supp Law P S 1 ,
where "supp" denotes the support, and let D • denote the relative interior of D (i.e. the interior in the relative topology of the smallest affine subspace containing D ). It is easy to see from the equalities
inf Q∈PD E Q h, S 1 − S 0 = inf x∈E h, x − S 0 , inf Q∈RD E Q h, S 1 − S 0 = inf x∈G h, x − S 0
that the following equivalences are true: Example 3.13. Let S 1 have Gaussian distribution with mean a and covariance matrix C . Let PD = {P} and RD be the determining set of a law invariant coherent utility function u that is finite on Gaussian random variables. Let F be such that the vector (S 1 1 , . . . , S d 1 , F ) is Gaussian. Denote c = cov(S 1 , F ) (we use the vector form of notation). There exists b ∈ R d such that Cb = c. We can write F = b, S 1 − a + EF + F . Then E F = 0 and cov( F , S 1 ) = 0, so that F is independent of S 1 . Note that
RAROC-based NGD ⇐⇒ S 0 ∈ G R , utility-based NGD corresponding to u ⇐⇒ S 0 ∈ G, NA ⇐⇒ S 0 ∈ D •σ 2 := var F = var F − var b, S 1 − a = var F − b, Cb = var F − b, c .
Clearly, if σ 2 = 0, then
I NGD (R) (F ) = I NGD (F ) = I NA (F ) = { b, S 0 − a + EF }.
Let us now assume that σ 2 > 0. Figure 3. The geometric representation of price intervals provided by various techniques Obviously, I NA (F ) = R. In order to find I NGD (F ), note that I NGD (F ) = b, S 0 − a + EF + I NGD ( F ). Let L denote the image of R d under the map x → Cx. Then the inverse C −1 : L → L is correctly defined. As u is law invariant, there exists γ > 0 such that, for a Gaussian random variable ξ with mean m and variance σ 2 , we have u(ξ) = m − γσ . From this, it is easy to see that the set G := {E Q (S 1 , F ) : Q ∈ RD} has the form
✲ ✻ R d R S 0 E G R G D • I NGD (R) (F ) I NGD (F ) I NA (F )G = (a, 0) + {(x, y) : x ∈ L, y ∈ R : x, C −1 x + σ −2 y 2 ≤ γ 2 }. (3.3)
Consequently,
I NGD (F ) = b, S 0 − a + EF − α, b, S 0 − a + EF + α ,
where α = (σ 2 γ 2 − σ 2 S 0 − a, C −1 (S 0 − a) ) 1/2 . (In particular, the NGD is satisfied if and only if S 0 − a, C −1 (S 0 − a) ≤ γ 2 .) Similar arguments show that
I NGD (R) (F ) = b, S 0 − a + EF − α(R), b, S 0 − a + EF + α(R) , where α(R) = σ 2 γ 2 R 2 (1+R) 2 −σ 2 S 0 −a, C −1 (S 0 −a) 1/2 . (In particular, the NGD(R) condition is satisfied if and only if S 0 − a, C −1 (S 0 − a) ≤ γ 2 R 2 (1+R) 2 .
) Let us remark that I NGD (F ) and I NGD (R) (F ) depend on u rather weakly, i.e. they depend only on γ . ✷
Dynamic Model with an Infinite Number of Assets
Let (Ω, F , (F t ) t∈[0,T ] , P) be a filtered probability space. We assume that F 0 is trivial. Let D ⊆ P be a convex weakly compact set (in the framework of Subsection 3.1, D is the determining set of u; in the framework of Subsection 3.2, D = 1 1+R PD + R 1+R RD ). Let (S i ), i ∈ I be a family of (F t )-adapted càdlàg processes (the set I is arbitrary and we impose no assumptions on the probabilistic structure of S i like the assumption that S i is a semimartingale). From the financial point of view, S i is the discounted price process of the i-th asset. We assume that S i t ∈ L 1 s (D) for any t ∈ [0, T ], i ∈ I . The set of P&Ls an agent can obtain by piecewise constant trading strategies (and only such strategies can be employed in practice) is naturally defined as
A = N n=1 i∈I H i n (S i un − S i u n−1 ) : N ∈ N, u 0 ≤ · · · ≤ u N , are (F t )-stopping times,
H i n is F u n−1 -measurable, and H i n = 0 for all i, except for a finite set .
A ′ = {H(S i v − S i u ) : u ≤ v ∈ [0, T ], i ∈ I, H is F u -measurableX = N n=1 i∈I H i n (S i un − S i u n−1 ) ∈ A. The process M k = k n=1 i∈I H i n (S i un − S i u n−1 ), k = 0, . . . , N is an (F u k , Q)-local martingale. Suppose that E Q X − < ∞ (otherwise, E Q X = −∞).
Then M is a martingale (see [49; Ch. II, § 1c]), and hence, E Q X = E Q M N = 0. Thus, in any case, E Q X ≤ 0, which proves that Q ∈ R. ✷ Example 3.15. Let us consider the Black-Scholes model in the framework of the RAROC-based pricing. Thus, S t = S 0 e µt+σBt , where B is a Brownian motion; we are given a risk-determining set RD , and we take PD = {P}. Surprisingly enough, in this model sup X∈A RAROC(X) = ∞. Indeed, the set M consists of a unique measure Q 0 and dQ 0 dP is not bounded away from zero, so that condition (3.2) is violated for any R > 0. Let us construct explicitly a sequence X n ∈ A with RAROC(X n ) → ∞. Consider D n = dQ 0 dP < n −1 and set X n = a n I(D n ) − I(Ω \ D n ), where a n is chosen in such a way that E Q 0 X n = 0. Then E P X n → ∞, while inf Q∈RD E Q X ≥ −1, so that RAROC(X n ) → ∞. Actually, X n / ∈ A, but, for each n, there exists a sequence (Y m n ) ∈ A such that −2 ≤ Y m n ≤ a n + 1 and Y m n P −−−→ m→∞ X n (we leave this to the reader as an exer-
cise). Then RAROC(Y m n ) −−−→ m→∞ RAROC(X n ), so that RAROC Y m(n) n → ∞ for some subsequence m(n).
This example shows that complete models are typically inconsistent with the RAROCbased NGD pricing. But this technique is primarily aimed at incomplete models because in complete ones the NA price intervals are already exact.
Let us also remark that the utility-based NGD condition might be naturally satisfied in the Black-Scholes model. ✷
Dynamic Model with Transaction Costs
Let (Ω, F , (F t ) t∈[0,T ] , P) be a filtered probability space. We assume that F 0 is trivial and (F t ) is right-continuous. Let D ⊆ P be a convex weakly compact set. Let S ai , S bi , i ∈ I be two families of (F t )-adapted càdlàg processes. From the financial point of view, S ai (resp., S bi ) is the discounted ask (resp., bid) price process of the i-th asset, so that S a ≥ S b componentwise. We assume that S ai t , S bi t ∈ L 1 s (D) for any t ∈ [0, T ], i ∈ I . The set of P&Ls that can be obtained in this model is naturally defined as Here H i n means the amount of the i-th asset that is bought at time u n (so that n k=0 H i k is the total amount of the i-th asset held at time u n ). Note that if there are no transaction costs, i.e. S ai = S bi = S i for each i, then the set of attainable P&Ls coincides with the set given by (3.4).
A = N n=0 i∈I −H i n I(H i n > 0)S ai un − H i n I(H i n < 0)S bi un : N ∈ N, u 0 ≤ · · · ≤ u N are (F t )-stopping times, H i n is F un -measurable,Lemma 3.16. We have D ∩ R = D ∩ R(A ′ ) = D ∩ M, where A ′ = {G(S bi v − S ai u ) + H(−S ai v + S bi u ) : i ∈ I, u ≤ v are simple (F t )-E Q (S ai v | F u ) ≥ S bi u , E Q (S bi v | F u ) ≤ S ai u . (3.5)
Consider the Snell envelopes
X t = esssup τ ∈Tt E Q (S bi τ | F t ), t ∈ [0, T ], Y t = essinf τ ∈Tt E Q (S ai τ | F t ), t ∈ [0, T ],
where T t denotes the set of simple (F t )-stopping times such that τ ≥ t. (Recall that esssup α ξ α is a random variable ξ such that, for any α, ξ ≥ ξ α a.s. and for any other random variable ξ ′ with this property, we have ξ ≤ ξ ′ a.s.) Then X is an (F t )supermartingale, while Y is an (F t , Q)-submartingale (see [27;Th. 2.12.1]). Let us prove that, for any t ∈ [0, T ], X t ≤ Y t Q-a.s. Assume that there exists t such that P(X t > Y t ) > 0. Then there exist τ, σ ∈ T t such that
Q E Q (S bi τ | F t ) > E Q (S ai σ | F t ) > 0.
This implies that Q(ξ > η) > 0, where ξ = E Q (S bi τ | F τ ∧σ ) and η = E Q (S ai σ | F τ ∧σ ). Assume first that Q({ξ > η} ∩ {τ ≤ σ}) > 0. On the set {τ ≤ σ} we have
ξ = S bi τ = S bi τ ∧σ , η = E Q (S ai σ | F τ ∧σ ) = E Q (S ai τ ∨σ | F τ ∧σ ),
and we obtain a contradiction with (3.5). In a similar way we get a contradiction if we assume that Q({ξ > η} ∩ {τ ≥ σ}) > 0. As a result, X t ≤ Y t Q-a.s. Now, it follows from [37; Lem. 3] that there exists an (F t , Q)-martingale M such that X ≤ M ≤ Y . As a result, Q ∈ M. Let us prove the inclusion D ∩ M ⊆ D ∩ R. Take Q ∈ D ∩ M, so that, for any i, there exists an (F t , Q)-martingale M i such that S bi ≤ M i ≤ S ai . For any
X = N n=0 i∈I −H i n I(H i n > 0)S ai un − H i n I(H i n < 0)S bi un ∈ A, we have X ≤ N n=0 i∈I −H i n I(H i n > 0)M i un − H i n I(H i n < 0)M i un = N n=1 i∈I n−1 k=0 H i k (M i un − M i u n−1 ).
Repeating the arguments used in the proof of Lemma 3.14, we get E Q X ≤ 0. As a result, Q ∈ R. ✷ Consider now a model with proportional transaction costs, i.e. S ai = S i , S bi = (1 − λ i )S i , where each S i is positive, λ i ∈ (0, 1). Denote the interval of the NGD prices in this model by I λ (F ) (the NGD pricing technique might be utilitybased or RAROC-based as the latter one is reduced to the former one by considering D = 1 1+R PD + R 1+R RD ). Let (λ n ) = (λ n ; i ∈ I, n ∈ N) be a sequence such that λ i n − −− → n→∞ 0 for any i. Proof. Let r denote the right endpoint of I 0 (F ). Suppose that the right endpoints of I λn (F ) do not converge to r . Then there exists r ′ > r such that, for each n (possibly, after passing on to a subsequence), there exists Q n ∈ D ∩ R λn with the property: E Q F ≥ r ′ (R λ is the set of risk-neutral measures in the model corresponding to λ). The sequence (Q n ) has a weak limit point Q ∞ ∈ D . Fix i ∈ I , u ≤ v ∈ [0, T ], and a positive bounded F u -measurable function H . For any n, we have
E Qn H((1 − λ i n )S i v − S i u ) ≤ 0. As S i v ∈ L 1 s (D), we have sup Q∈D E Q S i v < ∞, and hence, lim sup n E Qn H(S i v − S i u ) ≤ 0. As the map D ∋ Q → E Q H(S i v − S i u ) is weakly continuous, we get E Q∞ H(S i v − S i u ) ≤ 0.
In a similar way, we prove that
E Q∞ H(−S i v + S i u ) ≤ 0. Thus, S i is an (F t , Q ∞ )-martingale, so that Q ∞ ∈ D ∩ R 0 . As the map D ∋ Q → E Q F is weakly continuous, we should have E Q∞ F ≥ r ′ . But this is a contradiction. ✷
Hedging
Consider the model of Subsection 3.1.
Definition 3.18. The upper and lower NGD prices of a contingent claim F are defined by
V (F ) = inf{x : ∃X ∈ A such that u(X − F + x) ≥ 0}, V (F ) = sup{x : ∃X ∈ A such that u(X + F − x) ≥ 0}.
The problem of finding V (F ) has some similarities with the superreplication problem considered by Cvitanić, Karatzas [18] and by Sekine [48], but the difference is that in those papers the risk is measured not as ρ(X − F + x), but rather as ρ((X − F + x) − ).
Proposition 3.19. If A is a cone and F ∈ L 1 s (D), then V (F ) = sup{E Q F : Q ∈ D ∩ R}, V (F ) = inf{E Q F : Q ∈ D ∩ R}.
Proof. Take x 0 ∈ R and set A(x 0 ) = A + {h(x 0 − F ) : h ∈ R + }. Using Theorem 3.4, we can write
V (F ) ≥ x 0 ⇐⇒ ∃X ∈ A such that u(X − F + x 0 ) > 0 ⇐⇒ ∃X ∈ A(x 0 ) such that u(X) > 0 ⇐⇒ D ∩ R(A(x 0 )) = ∅ ⇐⇒ ∃Q ∈ D ∩ R such that E Q F ≥ x 0 .
This yields the formula for V (F ). The representation of V (F ) is proved similarly. ✷
Remarks. (i) The above theorem is formally true if the NGD is violated. In this case V (F ) = −∞ and V (F ) = ∞.
(ii) The above argument shows that there exist Q, Q ∈ D ∩ R such that E Q F = V (F ), E Q (F ) = V (F ). This is in contrast with the NA technique. Let us now study the sub-and super-replication problem for a particular case of a (frictionless) static model with a finite number of assets. Thus, we are given S 0 ∈ R d and S 1 1 , . . . , S d 1 ∈ L 1 w (D). From the financial point of view, S i n is the discounted price of the i-th asset at time n.
Definition 3.20. The superhedging and subhedging strategies are defined by
H(F ) = {h ∈ R d : u( h, S 1 − S 0 − F + V (F )) ≥ 0}, H(F ) = {h ∈ R d : u( h, S 1 − S 0 + F − V (F )) ≥ 0}.
Below we provide a simple geometric procedure to determine these quantities. Assume that F ∈ L 1 w (D) and let us introduce the notation G = cl{E Q (S 1 , F ) : Q ∈ D}, v = sup{x : (S 0 , x) ∈ G}, v = inf{x : (S 0 , x) ∈ G},
N = {h ∈ R d+1 : ∀x ∈ G, h, x − (S 0 , v) ≥ 0}, N = {h ∈ R d+1 : ∀x ∈ G, h, x − (S 0 , v) ≥ 0},
i.e. G is the generator for (S 1 , F ) and u; N (resp., N ) is the set of inner normals to G at the point (S 0 , v) (resp., (S 0 , v)); see Figure 4. Remark. The statement is true both in the case, where the NGD is satisfied, and in the case, where the NGD is not satisfied (in the latter case S 0 does not belong to the projection of G on R d , v = V (F ) = −∞, v = V (F ) = ∞, N = H = ∅, N = H = ∅). where α = (σ 2 γ 2 − σ 2 S 0 − a, C −1 (S 0 − a) ) 1/2 .
In order to find H and H , we express the upper and lower borders of the set G given by (3.3) as y = ±(σ 2 γ 2 − σ 2 x − a, C −1 (x − a) ) 1/2 . Then by differentiation we get H( F ) = H( F ) = d| x=S 0 (σ 2 γ 2 − σ 2 x − a, C −1 (x − a) ) 1/2 = −σ 2 α −1 C −1 (S 0 − a).
Hence,
H(F ) = b − σ 2 α −1 C −1 (S 0 − a), H(F ) = −b − σ 2 α −1 C −1 (S 0 − a).
We will now provide one more example. Let S 0 ∈ (0, ∞) and S 1 be an integrable random variable such that Law S 1 has no atoms and supp Law S 1 = R + . Let u be the coherent utility function corresponding to Tail V@R of order λ ∈ (0, 1] (see Example 2.5 (i)). We assume that u(S 1 ) < S 0 < −u(−S 1 ). Finally, let F = f (S 1 ), where f : R + → R is a convex function of linear growth. Let us denote Law S 1 by Q and find a, b, c, d such that a + b = λ, d − c = λ, and where q x is the x-quantile of Q.
H(F ) = f (q 1−b ) − f (q a ) q 1−b − q a , H(F ) = − f (q d ) − f (q c ) q d − q c .
Proof. Let us first prove the representation for V (F ) under an additional assumption that f is strictly convex. By Proposition 3.21,
V (F ) = sup Z∈D λ : EZX=S 0
EZf (X) (D λ is given by (2.3)). Take
Z 0 ∈ argmax Z∈D λ : EZX=S 0
EZf (X) (Z 0 exists by a compactness argument). Passing from Z 0 to E(Z 0 | X), we can assume that Z 0 is X measurable, i.e. Z 0 = ϕ(X). Let us prove that Z 0 = λ −1 I(X < q a ) + λ −1 I(X > q 1−b ). (3.6) Assume the contrary. Then there exist 0 < α 1 < α 2 < α 3 < α 4 such that Q({ϕ < λ −1 } ∩ (α 1 , α 2 )) > 0, Q({ϕ > 0} ∩ (α 2 , α 3 )) > 0, Q({ϕ < λ −1 } ∩ (α 3 , α 4 )) > 0.
For h 1 , h 2 , h 3 ∈ [0, λ −1 ], we set
ϕ(x) = ϕ(x), x / ∈ (α 1 , α 4 ), ϕ(x) ∨ h 1 , x ∈ (α 1 , α 2 ), ϕ(x) ∧ h 2 , x ∈ (α 2 , α 3 ), ϕ(x) ∨ h 3 , x ∈ (α 3 , α 4 ).
We can find h 1 , h 2 , h 3 such that Q({ ϕ > ϕ} ∩ (α 1 , α 2 )) > 0, Q({ ϕ < ϕ} ∩ (α 2 , α 3 )) > 0, Q({ ϕ > ϕ} ∩ (α 3 , α 4 )) > 0, Consider the affine function f that coincides with f at α 2 and α 3 . Then Furthermore, as f is strictly convex, f < f on (α 1 , α 2 ), f > f on (α 2 , α 3 ), and f > f on (α 3 , α 4 ). Consequently, ∞ 0 ( ϕ(x) − ϕ(x))f (x)Q(dx) > 0.
Thus, we have found Z 0 = ϕ(X) ∈ D λ such that E Z 0 X = S 0 and E Z 0 f (X) > EZ 0 f (X), which contradicts the choice of Z 0 . As a result, (3.6) is satisfied, which yields the desired representation of V (F ).
Let us now prove the representation for V (F ) in the general case. Take Z 0 given by (3.6). Find a strictly convex function f of linear growth. Then the function f ε = f +ε f is strictly convex and the result proved above shows that EZf ε (X) ≤ EZ 0 f ε (X) for any Z ∈ D λ . Passing on to the limit as ε ↓ 0, we get EZf (X) ≤ EZ 0 f (X) for any Z ∈ D λ . This yields the desired representation of V (F ).
Let us now prove the representation for H(F ). Consider the function
g(x) = sup Z∈D λ : EZX=x EZf (X), x ∈ [u(S 1 ), −u(−S 1 )].
It follows from the reasoning given above that g = g 1 • g −1 2 , where Applying Proposition 3.21, we get
H(F ) = g ′ (S 0 ) f (q 1−b ) − f (q a ) q 1−b − q a .
The representations for V (F ) and H(F ) are proved in a similar way. ✷
representation theorem). A function u satisfies conditions (a)-(e) if and only if there exists a nonempty set D ⊆ P such that
Figure 1 .
12.5) is L(e). The set of points x that satisfy (2.6) is h∈R d+ M(h) = G+R d + .The set of points x that satisfy (2.9) is h∈R d M(h) = G. This proves (2.8) and (2.9). Furthermore, the set {E Q X : Q ∈ D} is closed (the proof is similar to the proof of Proposition 2.9). Now, equality (2.10) follows immediately from (2.8) and the definition of X D .✷ Solution of the capital allocation problem
Figure 2
Theorem 2. 16 .
16If D is weakly compact and X, Y ∈ L 1 s (D), then
Example 2 .
217. (i) Let Y be a constant. In this case X D (Y ) = D , so that u c (X; Y ) = u(X).
Definition 3. 3 .
3A model satisfies the utility-based NGD condition if there exists no X ∈ A such that u(X) > 0.
Theorem 3 . 4 (
34Fundamental Theorem of Asset Pricing). A model satisfies the NGD condition if and only if D ∩ R = ∅.
Definition 3.5. A utility-based NGD price of F is a real number x such that the extended model (Ω, F , P, D, A + {h(F − x) : h ∈ R}) satisfies the NGD condition.
Theorem 3 . 10 (
310Fundamental Theorem of Asset Pricing). A model satisfies the NGD condition if and only if
Repeating the arguments from the proof of Theorem 3.4, we get PD ∩ R = ∅. ✷ Definition 3.11. A RAROC-based NGD price of a contingent claim F is a real number x such that the extended model (Ω, F , P, PD, RD, A + {h(F − x) : h ∈ R}) satisfies the NGD condition.
(
the last equivalence is a well-known result of arbitrage pricing; see [49; Ch. V, § 2e]). Now, let F ∈ L 1 w (RD) be the discounted payoff of a contingent claim. Let E , G, G R , D , and D • denote the versions of the sets E , G, G R , D , and D • defined for S 1 = (S 1 1 , . . . , S d 1 , F ) instead of S 1 . Let I NGD (R) (F ) denote the RAROC-based NGD price interval, I NGD (F ) denote the utility-based NGD price interval (corresponding to u), and I NA (F ) denote the NA price interval. Then I NGD (R) (F ) = {x : (S 0 , x) ∈ G R }, I NGD (F ) = {x : (S 0 , x) ∈ G}, I NA (F ) = {x : (S 0 , x) ∈ D • }.
.
We have D ∩ R = D ∩ R(A ′ ) = D ∩ M, where
and bounded}, M = {Q ∈ P : for any i ∈ I, S i is an (F t , Q)-martingale}. Proof. The inclusions D ∩ R ⊆ D ∩ R(A ′ ) ⊆ D ∩ M are clear. So, it is sufficient to prove the inclusion D ∩ M ⊆ D ∩ R. Let Q ∈ D ∩ M. Take
stopping times, G, H are positive, bounded, F u -measurable}, M = {Q ∈ P : for any i, there exists an (F t , Q)-martingale M i such that S bi ≤ M i ≤ S ai }.(A stopping time is simple if it takes on a finite number of values.) Proof. The inclusion D ∩ R ⊆ D ∩ R(A ′ ) is obvious. Let us prove the inclusion D ∩ R(A ′ ) ⊆ D ∩ M. Take Q ∈ D ∩ R(A ′ ). Fix i ∈ I . For any simple stopping times u ≤ v , we have S ai u , S bi u , S ai v , S bi v ∈ L 1 s (D) and
Theorem 3 . 17 .
317For F ∈ L 1 s (D), we have I λn (F ) − −− → n→∞ I 0 (F )in the sense that the right (resp., left) endpoints of I λn (F ) converge to the right (resp., left) endpoint of I 0 (F ).
✷ (iii) Under the conditions of the above corollary, we have I NGD (F ) = [V (F ), V (F )].
Proposition 3 . 21 .
321We haveV (F ) = v, V (F ) = v, H = {h ∈ N : h d+1 = −1}, H = {h ∈ N : h d+1 = 1}.
Figure 4 .
4Solution of the super-and subhedging problem Example 3.22. Consider the setting of Example 3.13. The results of that example show thatV (F ) = b, S 0 − a + EF + α, V (F ) = b, S 0 − a + EF − α,
xQ(dx) = S 0 ,
(x)Q(dx) = ∞ 0 xϕ(x)Q(dx) = S 0 , ∞ 0 ϕ(x)Q(dx) = ∞ 0 ϕ(x)Q(dx) = 1.
dy), x ∈ [0, λ −1 ].
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|
[] |
[
"OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS",
"OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS",
"OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS",
"OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS"
] |
[
"Yuta Kusakabe ",
"Yuta Kusakabe "
] |
[] |
[] |
We study the Oka properties of complements of closed countable sets in C n (n > 1) which are not necessarily discrete. Our main result states that every tame closed countable set in C n (n > 1) with a discrete derived set has an Oka complement. As an application, we obtain non-elliptic Oka manifolds which negatively answer a long-standing question of Gromov. Moreover, we show that these examples are not even weakly subelliptic. It is also proved that every finite set in a Hopf manifold has an Oka complement and an Oka blowup.
|
10.1090/proc/14832
|
[
"https://export.arxiv.org/pdf/1907.12024v1.pdf"
] | 198,967,926 |
1907.12024
|
4f0a6672f3336d67a294816c7dddf2adbe69fe9d
|
OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS
28 Jul 2019
Yuta Kusakabe
OKA COMPLEMENTS OF COUNTABLE SETS AND NON-ELLIPTIC OKA MANIFOLDS
28 Jul 2019
We study the Oka properties of complements of closed countable sets in C n (n > 1) which are not necessarily discrete. Our main result states that every tame closed countable set in C n (n > 1) with a discrete derived set has an Oka complement. As an application, we obtain non-elliptic Oka manifolds which negatively answer a long-standing question of Gromov. Moreover, we show that these examples are not even weakly subelliptic. It is also proved that every finite set in a Hopf manifold has an Oka complement and an Oka blowup.
Introduction
A complex manifold Y is an Oka manifold if any holomorphic map from an open neighborhood of a compact convex set K ⊂ C n (n ∈ N) to Y can be uniformly approximated on K by holomorphic maps C n → Y . The most basic examples of Oka manifolds are complex Euclidean spaces C n (n ∈ N). This is a consequence of the classical Oka-Weil approximation theorem which generalizes the Runge approximation theorem. For the historical background and the theory of Oka manifolds, we refer the reader to the survey [3] and the comprehensive monograph [4].
It is natural to ask when a closed countable set S in C n has an Oka complement C n \ S. In the case of n = 1, we have a complete answer that the complement C \ S is Oka if and only if S contains at most one point. On the other hand, in the case of n > 1, Forstnerič and Prezelj proved that every tame discrete set in C n has an Oka complement [5,Theorem 1.6]. The following is the definition of tameness. Definition 1.1. A closed countable set S ⊂ C n is tame if there exists a holomorphic automorphism ϕ ∈ Aut C n such that the closure of ϕ(S) in the projective space P n ⊃ C n does not contain the hyperplane at infinity P n \ C n .
For example, every compact countable set S ⊂ C n is tame. Tameness for discrete sets was first introduced by Rosay and Rudin [10] and recently generalized to other complex manifolds [2,11,12]. Rosay and Rudin also constructed a discrete set in C n (n > 1) whose complement is not Oka [10,Theorem 4.5].
In the present paper, we study complements of tame closed countable sets S ⊂ C n which are not necessarily discrete. That is, we do not assume that the derived set S ′ (the set of accumulation points of S) is empty. Our main theorem is the following generalization of the result for tame discrete sets. Theorem 1.2. For any tame closed countable set S ⊂ C n (n > 1) with a discrete derived set, the complement C n \ S is Oka.
In the same manner, we may prove the following theorem for blowups. Theorem 1.3. For any tame closed countable set S ⊂ C n (n > 1) with a discrete derived set S ′ , the blowup Bl S\S ′ (C n \ S ′ ) of C n \ S ′ along S \ S ′ is Oka.
The proofs of Theorem 1.2 and Theorem 1.3 are given in Section 2. These are new applications of the localization principle for Oka manifolds (Theorem 2.1) which was established in our previous paper [9].
In Section 3, we prove the following two corollaries. First, we give an example of a non-elliptic Oka manifold. Ellipticity was introduced by Gromov in his seminal paper [6] in 1989 (see Definition 3.1). It is a consequence of his main result in [6] that ellipticity is a sufficient condition for manifolds to be Oka. In the same paper, he also proved the converse for Stein manifolds [ [1] showed that in a Stein manifold of dimension at least three every compact holomorphically convex 2 set with an infinite derived set has a non-elliptic complement. However, there has been no example of an Oka complement of this type. As an application of Theorem 1.2 and the localization principle for Oka manifolds (Theorem 2.1), we obtain such an example which negatively answers the long-standing question of Gromov.
Corollary 1.4. For any n ≥ 3, the complement C n \ ((N −1 ) 2 × {0} n−2 ) is a non- elliptic Oka manifold where N −1 = {j −1 : j ∈ N} ⊂ C.
There is a weaker variant of ellipticity called weak subellipticity (see Definition 3.1) which is also a sufficient condition to be Oka (cf. [4, Corollary 5.6.14]). In fact, we prove that the complement in Corollary 1.4 is not even weakly subelliptic (Corollary 3.2). It should be mentioned that a still weaker variant, called Condition Ell 1 , characterizes Oka manifolds [9, Theorem 1.3]. The localization principle for Oka manifolds (Theorem 2.1) was proved as a corollary of this characterization.
It is also a fundamental problem whether any point in an Oka manifold of dimension at least two has an Oka complement and an Oka blowup. As another application of our results, we solve this problem for Hopf manifolds (in particular, we solve [3, Problem 2.42]). Recall that a complex manifold Y is a Hopf manifold if it is compact and universally covered by C dim Y \ {0}. The latter condition implies that a Hopf manifold is Oka (cf. [4,Corollary 5.6.11]) and the dimension of a Hopf manifold must be greater than 1.
Corollary 1.5. For any Hopf manifold Y and any finite set S ⊂ Y , the complement Y \ S and the blowup Bl S Y are Oka.
Proofs of the Theorems
Recall the following localization principle for Oka manifolds. Here, a subset of Y is Zariski open if its complement is a closed complex subvariety. We also need the following approximation theorem. Let B n (a, r) denote the closed ball in C n of radius r > 0 centered at a ∈ C n . Lemma 2.2. For any discrete sequence {a j } j in C n there exists a sequence {r j } j of positive numbers such that (1) the closed balls B n (a j , r j ) are mutually disjoint, and (2) for any holomorphic functions f j ∈ O(B n (a j , r j )) and any sequence {ε j } j of positive numbers there exists a holomorphic function f ∈ O(C n ) such that sup B n (a j ,r j ) |f − f j | ≤ ε j for all j.
Proof. We write B R = B n (0, R). Take an increasing sequence {R l } l∈N of positive numbers such that lim l→∞ R l = ∞ and {a j } j ∩ l∈N ∂B R l = ∅. We define B R 0 = ∅ for convenience. Renumbering {a j } j if necessary, we may assume that there exists an increasing sequence 0 = k 0 < k 1 < k 2 < · · · of integers such that {a j } k l+1
j=k l +1 ⊂ B R l+1 \ B R l for all l ≥ 0. Since B R l ∩ {a j } k l+1
j=k l +1 = ∅ for each l, there exists a sequence {r j } j of small positive numbers such that (a) the closed balls B n (a j , r j ) are mutually disjoint,
(b) k l+1 j=k l +1 B n (a j , r j ) ⊂ B R l+1 \ B R l for all l ≥ 0, and (c) B R l ∪ k l+1
j=k l +1 B n (a j , r j ) is polynomially convex for each l ≥ 0. Let us verify the condition (2). Let g 0 ∈ O(B R 0 ) = O(∅) be the unique holomorphic function on ∅. Assume inductively that g l ∈ O(B R l ) has been chosen for some l ≥ 0. By the Oka-Weil approximation theorem, there exists a holomorphic function g l+1 ∈ O(B R l+1 ) such that
(i) sup B R l |g l+1 − g l | ≤ min{ε j } k l j=1 /2 l+1 , and (ii) sup B n (a j ,r j ) |g l+1 − f j | ≤ ε j /2 for all j = k l + 1, . . . , k l+1 .
Then the limit f = lim l→∞ g l exists uniformly on compacts in C n and has the desired property.
Proof of Theorem 1.2. By Theorem 2.1, it suffices to prove that for any fixed point p ∈ C n \ S there exists a Zariski open Oka neighborhood of p. Since S is tame, there exist a holomorphic coordinate system z = (z ′ , z n ) on C n and a constant C > 0 such that p = 0 and S ⊂ {(z ′ , z n ) : |z n | ≤ C(1 + |z ′ |)} by definition. Furthermore, since S is countable, we may also assume that S ⊂ (C n−1 \ {0}) × C and pr ′ | S : S → C n−1 is injective where we denote by pr ′ : C n → C n−1 the projection z → z ′ . Note that the restriction of pr ′ to {(z ′ , z n ) : |z n | ≤ C(1 + |z ′ |)} is proper. Thus pr ′ (S ′ ) is discrete in C n−1 . Let us enumerate pr ′ (S ′ ) = {a j } j . Take a sequence {r j } j of positive numbers which satisfies the conditions (1) and (2) in Lemma 2.2. Note that pr ′ (S) \ j B n−1 (a j , r j ) ⊂ C n−1 is discrete. By the Oka-Cartan extension theorem, there exists g ∈ O(C n−1 ) such that (a) g(0) = 1, and (b) g(z ′ ) = −z n for all (z ′ , z n ) ∈ S ′ ∪ (S \ pr ′−1 ( j B n−1 (a j , r j ))).
By the condition (2) in Lemma 2.2, there exists f ∈ O(C n−1 ) such that on each closed ball B n−1 (a j , r j ) the real part of f satisfies
ℜf ≤ log 1 j · sup z ′ ∈B n−1 (a j ,r j ) (C(1 + |z ′ |) + |g(z ′ )|)
.
It follows that for all (z
′ , z n ) ∈ S ∩ pr ′−1 (B n−1 (a j , r j )) e f (z ′ ) (z n + g(z ′ )) ≤ e ℜf (z ′ ) (C(1 + |z ′ |) + |g(z ′ )|) ≤ 1 j .
Consider the automorphism ϕ ∈ Aut C n defined by ϕ(z ′ , z n ) = (z ′ , e f (z ′ ) (z n + g(z ′ ))). Note that ϕ(p) = (0, e f (0) ) ∈ C n−1 × C * , ϕ(S ′ ) ⊂ C n−1 × {0} and the discrete set D = ϕ(S) ∩ (C n−1 × C * ) in C n−1 × C * is contained in j (B n−1 (a j , r j ) × B 1 (0, 1/j)). Thus pr n (D) is discrete in C * where pr n : C n → C is the n-th projection, and hence pr n ((id C n−1 × exp) −1 (D)) = exp −1 (pr n (D)) is discrete in C. This implies that (id C n−1 × exp) −1 (D) is a tame discrete set in C n (cf. [10, Theorem 3.9]). Therefore (C n−1 × C * ) \ ϕ(S) is a Zariski open Oka neighborhood of ϕ(p) because its universal covering C n \ (id C n−1 × exp) −1 (D) is Oka (cf. [4, Proposition 5.6.3]). It follows that the preimage ϕ −1 (C n−1 × C * ) \ S ⊂ C n \ S is a Zariski open Oka neighborhood of p.
Proof of Theorem 1.3. Let π : Bl S\S ′ (C n \S ′ ) → C n \S ′ denote the blowup of C n \S ′ along S\S ′ . As before, it suffices to prove that for any fixed point p ∈ Bl S\S ′ (C n \S ′ ) there exists a Zariski open Oka neighborhood of p. The argument in the proof of Theorem 1.2 gives a holomorphic automorphism ϕ ∈ Aut C n such that ϕ(π(p)) ∈ C n−1 × C * , ϕ(S ′ ) ⊂ C n−1 × {0} and the discrete set (id C n−1 × exp) −1 (ϕ(S)) ⊂ C n is tame. Since the blowup of C n along a tame discrete set is Oka (cf. [4,Proposition 6.4.12]), the blowup Bl (id C n−1 ×exp) −1 (ϕ(S)) C n is Oka. Therefore the blowup Bl ϕ(S)∩(C n−1 ×C * ) (C n−1 × C * ) covered by Bl (id C n−1 ×exp) −1 (ϕ(S)) C n is also Oka. It follows that
π −1 (ϕ −1 (C n−1 × C * )) = Bl S∩ϕ −1 (C n−1 ×C * ) ϕ −1 (C n−1 × C * ) ⊂ Bl S\S ′ (C n \ S ′ )
is a Zariski open Oka neighborhood of p.
Proofs of the Corollaries
First, let us recall the definitions of ellipticity and its variants. (1) A spray (E, π, s) on Y is a triple (E, π, s) consisting of a holomorphic vector bundle π : E → Y and a holomorphic map s : E → Y such that s(0 y ) = y for each y ∈ Y . (2) Y is elliptic (resp. subelliptic) if there exists a spray (E, π, s) (resp. a family (E j , π j , s j ) (j = 1, . . . , k) of sprays) on Y such that ds 0y (E y ) = T y Y (resp. k j=1 (ds j ) 0y (E j,y ) = T y Y ) for each y ∈ Y . (3) Y is weakly subelliptic if for any compact set K ⊂ Y there exists a family (E j , π j , s j ) (j = 1, . . . , k) of sprays on Y such that k j=1 (ds j ) 0y (E j,y ) = T y Y for each y ∈ K.
As we mentioned, we prove the following stronger result than Corollary 1.4. Proof. To reach a contradiction, we assume that Y \K is weakly subelliptic. Take a relatively compact open neighborhood U ⊂ Y of K. By assumption, there exists a family (E j , π j , s j ) (j = 1, . . . , k) of sprays on Y \ K such that k j=1 (ds j ) 0y (E j,y ) = T y Y for each y ∈ ∂U. By the Hartogs extension theorem for holomorphic vector bundles and sprays [1, Theorem 1.2 and Theorem 4.1], we can extend E j (j = 1, . . . , k) to holomorphic vector bundles π j : E j → Y \ A (j = 1, . . . , k), where A ⊂ K is a finite set, and s j (j = 1, . . . , k) to holomorphic mapss j : E j → Y (j = 1, . . . , k). Note thats j (0 y ) = y for each y ∈ Y \ A by the identity theorem.
Let B ⊂ Y \ A denote the closed complex subvariety of points y ∈ Y \ A such that k j=1 (ds j ) 0y ( E j,y ) = T y Y and set S = A ∪ B. Since U is relatively compact and B ∩ ∂U = ∅, the intersection B ∩ U ⊂ U \ A must be a discrete set. Hence K is not contained in S and thus we may take a point y 0 ∈ ∂K \ S. Let us consider the fiber product and the associated maps E = (e 1 , . . . , e k ) ∈ k j=1 E j \ π −1 j (S) ∪s −1 j (S) :s j (e j ) = π j+1 (e j+1 ), j = 1, . . . , k − 1 , π : E → Y \ S, (e 1 , . . . , e k ) → π 1 (e 1 ), s : E → Y \ S, (e 1 , . . . , e k ) →s k (e k ).
.
Let Y be a complex manifold. Assume that each point of Y has a Zariski open Oka neighborhood. Then Y is an Oka manifold.
Definition 3.1 (cf. [4, Definition 5.6.13]). Let Y be a complex manifold.
Corollary 3 . 2 .
32For any n ≥ 3, the complement C n \ ((N −1 ) 2 × {0} n−2 ) is Oka but not weakly subelliptic. In order to prove Corollary 3.2, we need to improve the result of Andrist, Shcherbina and Wold [1, Theorem 1.1] as follows. The proof is based on their idea and Gromov's method of composed sprays (cf. [4, §6.3]).
Lemma 3 . 3 .
33Let Y be a Stein manifold of dimension at least three and K ⊂ Y be a compact O(Y )-convex set with an infinite derived set. Then Y \ K is not weakly subelliptic.
6, Remark 3.2.A] 1 . Then he asked a question whether the converse holds for all complex manifolds [6, Question 3.2.A ′′ ]. Decades later, Andrist, Shcherbina and Wold
The condition Ell ∞ in [6, Remark 3.2.A] is equivalent to being Oka (cf. [4, §5.15]).2 The holomorphic convexity was not assumed in[1] but used in the proof implicitly.
Acknowledgement I wish to thank my supervisor Katsutoshi Yamanoi for many constructive comments, and Franc Forstnerič for helpful remarks. This work was supported by JSPS KAKENHI Grant Number JP18J20418.
0 y ) : y ∈ Y \ S} onto a neighborhood of the diagonal {(y, y) : y ∈ Y \ S}. weakly subelliptic. In order to prove that the complement is Oka, we use the localization principle for Oka manifolds (Theorem 2.1). Set U j = C j−1 × C * × C n−j (j = 1, . . . , n) definition, π −1 (S) = exp −1 (N −1 ) × (N −1 ) × {0} n−2 . Note that π −1 (S) is tame and its derived set π −1 (S) ′ = exp −1 (N −1 ) ×{0} n−1 is discrete. Y \ S ( Cf, Therefore the fiber preserving map E → (Y \ S) 2 , e → (π(e). s(e)) restricts to a holomorphic submersion from a neighborhood of the zero section {(0 y. Therefore Theorem 1.2 implies that n = dim Y > 1. By Kodaira's argument [8, TheoremY \ S (cf. [4, Lemma 6.3.6]). Therefore the fiber preserving map E → (Y \ S) 2 , e → (π(e), s(e)) restricts to a holomorphic submersion from a neighborhood of the zero section {(0 y , . . . , 0 y ) : y ∈ Y \ S} onto a neighborhood of the diagonal {(y, y) : y ∈ Y \ S}. weakly subelliptic. In order to prove that the complement is Oka, we use the localization principle for Oka manifolds (Theorem 2.1). Set U j = C j−1 × C * × C n−j (j = 1, . . . , n) definition, π −1 (S) = exp −1 (N −1 ) × (N −1 ) × {0} n−2 . Note that π −1 (S) is tame and its derived set π −1 (S) ′ = exp −1 (N −1 ) ×{0} n−1 is discrete. Therefore Theorem 1.2 implies that n = dim Y > 1. By Kodaira's argument [8, Theorem
In this coordinate system, the discrete set π −1 (S) \ B n (0, 1) ⊂ C n projects to a discrete set in the first coordinate, and hence it is tame (cf. lower triangular (cf. [4, p. 117. 10, Theorem 3.9]). Note that (π −1lower triangular (cf. [4, p. 117]). In this coordinate system, the discrete set π −1 (S) \ B n (0, 1) ⊂ C n projects to a discrete set in the first coordinate, and hence it is tame (cf. [10, Theorem 3.9]). Note that (π −1
The Hartogs extension theorem for holomorphic vector bundles and sprays. R B Andrist, N Shcherbina, E F Wold, Ark. Mat. 542R. B. Andrist, N. Shcherbina, and E. F. Wold. The Hartogs extension theorem for holomor- phic vector bundles and sprays. Ark. Mat., 54(2):299-319, 2016.
A new notion of tameness. R B Andrist, R Ugolini, J. Math. Anal. Appl. 4721R. B. Andrist and R. Ugolini. A new notion of tameness. J. Math. Anal. Appl., 472(1):196- 215, 2019.
Oka manifolds: from Oka to Stein and back. F Forstnerič, Ann. Fac. Sci. Toulouse Math. 226With an appendix by Finnur LárussonF. Forstnerič. Oka manifolds: from Oka to Stein and back. Ann. Fac. Sci. Toulouse Math. (6), 22(4):747-809, 2013. With an appendix by Finnur Lárusson.
Stein manifolds and holomorphic mappings. F Forstnerič, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 563rd Series. A Series of Modern Surveys in MathematicsF. Forstnerič. Stein manifolds and holomorphic mappings, volume 56 of Ergebnisse der Math- ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].
The homotopy principle in complex analysis. Cham Springer, second editionSpringer, Cham, second edition, 2017. The homotopy principle in complex analysis.
Oka's principle for holomorphic submersions with sprays. F Forstnerič, J Prezelj, Math. Ann. 3224F. Forstnerič and J. Prezelj. Oka's principle for holomorphic submersions with sprays. Math. Ann., 322(4):633-666, 2002.
Oka's principle for holomorphic sections of elliptic bundles. M Gromov, J. Amer. Math. Soc. 24M. Gromov. Oka's principle for holomorphic sections of elliptic bundles. J. Amer. Math. Soc., 2(4):851-897, 1989.
Deformations and diffeomorphism types of Hopf manifolds. K Hasegawa, Illinois J. Math. 374K. Hasegawa. Deformations and diffeomorphism types of Hopf manifolds. Illinois J. Math., 37(4):643-651, 1993.
On the structure of compact complex analytic surfaces. K Kodaira, II. Amer. J. Math. 88K. Kodaira. On the structure of compact complex analytic surfaces. II. Amer. J. Math., 88:682-721, 1966.
Elliptic characterization and localization of Oka manifolds. Y Kusakabe, Indiana Univ. Math. J. to appearY. Kusakabe. Elliptic characterization and localization of Oka manifolds. Indiana Univ. Math. J., to appear.
Holomorphic maps from C n to C n. J.-P Rosay, W Rudin, Trans. Amer. Math. Soc. 3101J.-P. Rosay and W. Rudin. Holomorphic maps from C n to C n . Trans. Amer. Math. Soc., 310(1):47-86, 1988.
Tame discrete sets in algebraic groups. J Winkelmann, arXiv:1901.08952J. Winkelmann. Tame discrete sets in algebraic groups. arXiv:1901.08952.
Tame discrete subsets in Stein manifolds. J Winkelmann, 560-0043J. Aust. Math. Soc. 1071Department of Mathematics, Graduate School of Science, Osaka UniversityJ. Winkelmann. Tame discrete subsets in Stein manifolds. J. Aust. Math. Soc., 107(1):110- 132, 2019. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan E-mail address: [email protected]
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"Solving Price Per Unit Problem Around the World: Formulating Fact Extraction as Question Answering"
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"Kushal Kumar \nInc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA\n",
"Anoop S \nInc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA\n",
"K K Saladi [email protected] \nInc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA\n",
"Ismail Tutar [email protected] \nInc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA\n"
] |
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"Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA",
"Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA",
"Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA",
"Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA",
"Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle\nNY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA"
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Price Per Unit (PPU) is an essential information for consumers shopping on e-commerce websites when comparing products. Finding total quantity in a product is required for computing PPU, which is not always provided by the sellers. To predict total quantity, all relevant quantities given in a product's attributes such as title, description and image need to be inferred correctly. We formulate this problem as a question-answering (QA) task rather than named entity recognition (NER) task for fact extraction. In our QA approach, we first predict the unit of measure (UoM) type (e.g., volume, weight or count), that formulates the desired question (e.g., "What is the total volume?") and then use this question to find all the relevant answers. Our model architecture consists of two subnetworks for the two subtasks: a classifier to predict UoM type (or the question) and an extractor to extract the relevant quantities. We use a deep character-level CNN architecture for both subtasks, which enables (1) easy expansion to new stores with similar alphabets, (2) multi-span answering due to its span-image architecture and (3) easy deployment by keeping model-inference latency low. Our QA approach outperforms rule-based methods by 34.4% in precision and also BERT-based fact extraction approach in all stores globally, with largest precision lift of 10.6% in the US store.
|
10.48550/arxiv.2204.05555
|
[
"https://arxiv.org/pdf/2204.05555v1.pdf"
] | 243,844,849 |
2204.05555
|
0600e08e8969cb0c0f0a0526cbc3bc9950c16766
|
Solving Price Per Unit Problem Around the World: Formulating Fact Extraction as Question Answering
Tarik Arici [email protected]
Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle
NY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA
Kushal Kumar
Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle
NY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA
Anoop S
Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle
NY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA
K K Saladi [email protected]
Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle
NY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA
Ismail Tutar [email protected]
Inc. New York Inc. Bengaluru Inc. Seattle Inc. Bengaluru Inc. Seattle
NY, KA, WA, KA, WAUSA, India Hayreddin Çeker, USA, India, USA
Solving Price Per Unit Problem Around the World: Formulating Fact Extraction as Question Answering
10.1145/nnnnnnn.nnnnnnnBERTdeep learningmulti-span answernatural language under- standingquestion-answeringSQuADtransformers
Price Per Unit (PPU) is an essential information for consumers shopping on e-commerce websites when comparing products. Finding total quantity in a product is required for computing PPU, which is not always provided by the sellers. To predict total quantity, all relevant quantities given in a product's attributes such as title, description and image need to be inferred correctly. We formulate this problem as a question-answering (QA) task rather than named entity recognition (NER) task for fact extraction. In our QA approach, we first predict the unit of measure (UoM) type (e.g., volume, weight or count), that formulates the desired question (e.g., "What is the total volume?") and then use this question to find all the relevant answers. Our model architecture consists of two subnetworks for the two subtasks: a classifier to predict UoM type (or the question) and an extractor to extract the relevant quantities. We use a deep character-level CNN architecture for both subtasks, which enables (1) easy expansion to new stores with similar alphabets, (2) multi-span answering due to its span-image architecture and (3) easy deployment by keeping model-inference latency low. Our QA approach outperforms rule-based methods by 34.4% in precision and also BERT-based fact extraction approach in all stores globally, with largest precision lift of 10.6% in the US store.
Table 1
: Below, quantity information relevant to PPU task is shown in red for a few example catalog items.
INTRODUCTION
PPU enables consumers to compare same or substitutable products when they come in a variety of packet sizes. Especially in the case of consumable products, PPU can be an important factor in customer purchase decisions. PPU information leads to a better consumer experience, and customers are sensitive to this especially in online retailing. This sensitivity leads to competitive pricing across online retailers regardless of the location of the customer and even moved offline retailers towards competitive pricing [1]. Grocery products with PPU information had higher purchase rate compared to those without PPU information. Also, some U.S. states and some EU stores such as UK have regulations on PPU reporting [5]. To this end, we train models that can correct wrong PPU information supplied by sellers and also fill in missing PPU information.
Worldwide, sellers often do not provide information about the total quantity directly but specify it in free-text fields like product titles, descriptions or even images. Moreover, this information is usually unstructured which renders regex-based extraction unfeasible which is highlighted in Table 1. In the first product title, the relevant quantity is 24 while 8.3 is not relevant in calculating the final quantity. In the second title, which is also a ground coffee product, all quantities are relevant. Even when there is some structure in the title, computing total quantity can be a challenge. For instance, the third and fourth product titles in Table 1 follow similar syntactic structure, yet the context, words like total and each, decides the selection of the most relevant quantity. In the last product title although all quantities are related to weight, 5000 mg is not a relevant quantity as it indicates the concentration of the amino acid. These examples show that complex patterns, which depend on seller conventions, need to be learned, moreover, high-level information such as Unit of Measure (UoM) type is needed to guide quantity extraction. For example in the first product title knowing that the UoM type is count will increase the model's confidence to select quantity 24 and not 8.3 due to its context.
To the best of our knowledge, Named Entity Recognition (NER) is a de facto formulation for fact extraction problem, where the task is to assign tags to words in a sentence that indicates the begin and end of the answer. Traditionally, these named entities represent either a person, location or an organization [4]. We may formulate our quantity extraction problem similarly, in which we can either have a single extraction class (i.e., relevant quantity) or multiple classes corresponding to the relevant quantities in each UoM type. However, NER solutions do not enable the model to couple start and end indices explicitly, and check for their compatibility during training. Constraints such as the end index has to be bigger than the start index, can not be embedded during training. Moreover, they are prone to small variations in the tokens, for e.g., "fluid ounce" or "fl oz", which need to be explicitly tagged. Other span characteristics such as shorter answers are more likely to occur, can not be learned by the model, which can help eliminating the need for post-processing or regularization. Also, UoM type information, which is shown to be important for quantity extraction, cannot be efficiently fed to NER model other than learning different token representations for each UoM type. To overcome these limitations, we introduce a span-image architecture that works at a characterlevel and employ a QA approach to quantity extraction which conditions the extractor model with UoM type information.
QA models such as the ones used in Stanford Question Answering Dataset (SQuAD) competition find an answer span given a context paragraph and a question. A significant advance in answer span prediction is the BiDAF method proposed in [6], which uses bi-directional LSTM on query-to-context and context-to-query sequences and applies a softmax normalization across the sequence dimension. Hence, a single start index and a single end index is predicted to compute a single answer over context given query. BERT is a language model that significantly improved over the BiDAF model by enabling learning from unlabeled text [2] [3]. BERT and its alikes still use a softmax over the sequence as their goal is to find a single answer span. This is optimal since an answer span requires only one span start and end, and likelihood of start and end locations can be maximized separately. However, our task dictates as many quantity values as reported in product attributes (most likely up to three: weight/volume, items per package, and number of packages). Since we have to extract as many "answers" as needed, we couple start and end prediction outputs and predict spans. We achieve this by outputting a vector for each location in a two-dimensional grid (span-image), where each location ( , ) corresponds to a possible quantity span from character location to . All acceptable spans occur in the upper triangular part of the span-image.
Our model uses available free-form textual attributes of a product, optical character recognition (OCR) text extracted from product images, and categorical features derived from product taxonomy. We designed a lightweight model that uses only character-level embeddings and convolutional layers, and is deep and large enough to learn semantic information to predict UoM type. Constructing a word-vocabulary and learning embeddings requires large number of parameters which is exacerbated by the heavy-tail distribution of words in our dataset (many rare words) as can be seen in Figure 1. Unlike word vocabularies, a character vocabulary of size 128 applies to many stores with alphabets sharing characters. This enables sharing character-based models across various international stores and enjoy benefits of warm-start due to shared information such as brand names, measurements units, linguistic similarities, etc. Character-level convolutional networks have been successfully applied before to several text classification tasks and achieve stateof-the-art results for classification [7]. Our model has multiple use cases that can lead to enhanced quality of an e-commerce catalog by improving coverage and consistency of PPU related information in products display pages. These use cases include, but not limited to, defect rate reduction, backfilling missing values and real-time validation of PPU related information entered during registration of a new product (see Section 3).
Our contributions in this work are as follows:
• We propose using a question-answering framework for extraction, where UoM classifier predicts a question that guides the quantity extractor. We employ a two-stage training approach. UoM classifier is trained in the first stage using more data available for this task, and quantity extractor is trained in the second stage by exploiting the predicted question as a latent variable. • We enable more than one possible answer spans in the input text by introducing a span-image architecture. • We propose a character-based model deep enough to construct concepts and words from characters but light enough to satisfy our latency requirements and real-time use case. Figure 2. Character embeddings are not shared between the two subnetworks since vocabulary size is small. Given text attributes { 1 , 2 , · · · }, model input comprises of character sequences = 1 , 2 , · · · , , where ∈ R are -dimensional character embedding corresponding to the ℎ character in the input sequence of length for attribute . We also pass categorical embeddings as inputs for each category the product belongs to in the product taxonomy.
MODEL ARCHITECTURE
UoM Classifier and Quantity Extractor
UoM classifier consists of below stages (see Figure 2):
• Character embedding layer maps each character to a dimensional vector and the resulting vector sequence is fed into convolutional layers. • Convolutional layers consist of multiple layers with filter sizes 3 and 5. We used maxpooling on the activations. Output sequence vectors are batch normalized and dropout is employed. • Attention module computes an attention vector from all input attributes. Each attribute-encoding vector is affine transformed to obtain attention keys, which are element-wise scaled and summed to find the scores for softmax weights. Weighted averaged attribute-encoding vectors constitute a product-description vector. • Categorical embeddings vectors are created by embedding categorical indices into a high dimensional space (1/ √ dim, where is the number of categories). Every product in our catalog belongs to predefined categories.
• Product-description vector and category-embedding vectors are concatenated and passed to classification layers to produce logits for UoM type. Softmax normalization is applied on the logits to predict the UoM type.
Quantity extraction (QE) model consists of below stages:
• Character embedding layer same as above.
• 1D convolutional layers are applied to obtain an encoded sequence y without any strided pooling. Resultant sequence is batch normalized and dropout is used during training. No pooling is applied since sequence length needs to be kept.
• Each vector in y is concatenated with UoM softmax outputs, and fed into two different 1D convolutional layers to compute two vector sequences s and e of length , with a shrunken depth . This permits specialization for start and end index prediction. We also concatenate different positional dimensions to both these vector sequences to segregate them. • s is tiled horizontally and e is tiled vertically to produce two tensors of size × . These two tensors are multiplied element-wise to create a span-image of width and height equal to and depth of . 2D convolutional filters are applied on the span-image to produce an image of size × and depth 2. • Softmax normalization is applied on the depth dimension as opposed to the sequence dimension. Post-processing is done on the extracted quantities above a certain threshold to obtain the final quantity.
Training and Inference
Our model training is performed in two phases. In the first phase, UoM classifier is trained, in the second phase QE is trained while UoM classifier weights are frozen. This two-phase training strategy suits well with our model architecture. We use auditors to decide on UoM type and total quantity given a product. Our auditors specify a UoM type and total quantity value for each product audited. UoM type is a predefined class (i.e, weight, volume, count). Hence, all available audits can be used for training the UoM classifier. However, total quantity is often a multiplication of other quantities such as number of items and/or packages, and item volume or weight. Since our auditors only provide final quantity value and do not explicitly tag parts-of-text within product attributes, we use some high precision heuristics (see Algorithm 1) to create the ground truth span required for quantity extraction model training. This approach of using heuristics to tag ground truth values does not work well for all samples, resulting in a small loss of audited examples. Our training dataset for learning the quantity extractor is about 15% smaller than our UoM dataset for learning the classifier.
To increase our recall, we perform noising on our inputs by adding and deleting random gibberish words and tokens that are Algorithm 1: Part-of-text tagging for quantity values in the attribute: We start searching for qualified candidates by evaluating combinations of upto 3 candidate quantities. This biases towards more factorial understanding of total quantity and favors learning features for number of packages, number of items, volume or weight. The qualified candidates are used for training quantity extractor model. Our part-of-text tagging scheme for quantity extraction dataset is to find quantity candidates that can be processed to compute the final quantity using post-processing rules. At inference time, thresholding is applied on the span-image to obtain these candidate quantity span. Overlapping spans, if any, are handled by choosing the span with a bigger score. If no span has score higher than the threshold and UoM type is Count, quantity is predicted to be one. In all other cases, the model refrains from making a prediction.
We use rules to compute the final quantity from the obtained candidate quantities. Firstly, we identify the UoM type of each quantity span using cues from the immediate units of measure present in the text. If no unit of measure is found then UoM type is taken to be count. We put all the obtained quantities in a stack and remove duplicate quantities of same UoM type. At every iteration for weight or volume UoM types we sum distinct quantities and remove quantities remaining in the stack which are duplicate to the sum obtained. For count UoM type, we multiply the quantities from the stack and remove duplicates in the same way. Finally, we multiply total weight or volume quantity with the total count quantity if the predicted UoM type is weight or volume, else output the total count quantity.
PPU MODEL USE CASES
Our PPU model can be used primarily for the following three tasks on an e-commerce catalog.
Correction
This task involves fixing incorrect UoM and total quantity information provided by sellers. As this information may exist in one or more attributes, they may conflict with each other, e.g., having a different size in product title versus product image. We can use PPU models to predict UoM type and quantity for products at a particular cadence. If the prediction and associated attribute values in the catalog do not match, we can send them for manual correction. This can help remove defects, ensuring consistent information across attributes.
Backfilling
We can also use our model predictions for backfilling quantity information. For certain stores, we found correlations between the quality of catalog information and the popularity of the product. Specifically, the tail products tend to have inconsistent or missing attribute values along with distribution shifts on UoM types for same categories compared to head products. This leads to lower coverage at the time of backfilling. To address such problems, we experimented with active learning technique by obtaining manual audits on a small set of tail products where the model confidence was low. It helped improve recall on tail products by 13% with only 1% drop on the head products in that store.
Validation
As new products are created in an e-commerce catalog everyday, the challenge of fixing incorrect information or backfilling missing values is ever lasting. Instead, we can ensure consistent attribute information during creation itself by using real-time model validation owing to low model latency. Quantity related attributes can be validated using our model predictions, and the merchant can be notified to recheck and correct inconsistent attribute values.
EXPERIMENTS
We evaluate our Quantity Extraction model based on its predicted total quantity and UoM type and not individual quantity spans. This is a stricter metric since a true prediction requires all relevant quantities in the input to be extracted correctly. We compare precision for quantity extraction task as we need to meet a high precision threshold for deployment, while for classification task, we compare F1 scores as both precision and recall are important.
Dataset
Our dataset was created within a 16-month time frame. We first used a rule-based model as a UoM classifier, and flagged items in the top 3 product categories (we will refer to them as A, B and C categories in this paper) when the prediction did not match the catalog values. Initially, we worked with internal audit teams to correct UoM and quantity values manually. After collecting about 40K examples in our dataset, we trained our deep learning-based UoM classifier, which had about 120K parameters. Designing a lightweight model allowed us to use deep learning technqiues early in our project. As we obtained more audits, our model size scaled proportional to our dataset sizes. Currently, our model includes 657K number of parameters. As the model performance improved, we started choosing candidates based on the correctness of quantity values as well as of the UoM type. Table 2 shows the distribution of products with respect to the number of spans. Large proportion of the products in our training dataset did not contain any span, which either meant that the relevant quantities are missing from the text or that the product is of type count and the total quantity is 1.
Similar to US catalog, IN catalog is also in English and shares the same vocabulary, yet there are several distribution differences when compared to US catalog. Fine-tuning US model even on a small training dataset can lift performance by increasing confidence scores on tokens including but not limited to unit words in the metric system for e.g. kilogram, millilitre, etc. IN catalog is also rife with out-of-vocabulary (OOV) words for US model, which are borrowed directly from the regional language, for eg -atta which is Hindi for flour and agarbatti which is Hindi for incense sticks, that carry useful signals for UoM classification task. We also observed that longer text attributes such as product description and bullet points were seldom informative on PPU related information over shorter attributes like title. Moreover, there were some distributional changes as well, like the distribution of UoM types across same product categories varied across the stores. Also, count UoM type was more noisy where it was difficult to predict UoM type by using the product titles alone, and needed signals from product taxonomy. These differences needed to be addressed appropriately when testing US model and improving it further in IN store. . Regex also catered to various composite patterns such as "2 × 200 ml". We also applied guardrails to the per unit quantity value for each UoM type for better precision.
Baselines to our PPU Model
Fine-tuned BERT models.
We fine-tuned BERT models separately for both the UoM classification and quantity extraction tasks as another baseline. We took pre-trained Google BERT base model (bert_uncased_L-12_H-768_A-12) and trained it further on our catalog corpus for both MLM and NSP tasks for English stores like US and IN [2]. This model is then fine-tuned for the UoM classification task in the US store which has the largest share of training examples.
For quantity extraction task, we used BERT model to compare our approach with fact extraction formulation. We used BERT-base (bert-base-uncased) model that is available from Transformers library which is trained on lower-cased English text with 12 layer, 768 hidden dimensions, 12 attention-heads and 110M parameters. We fine-tune it with PPU dataset as a fact extraction problem using a UoM agnostic question "What is the total quantity?". Current implementation of BERT does not support multi-span answer prediction. We modify the last linear layer and use two affine transformations (outer-product) to convert the separate begin and end vectors into a matrix where each pixel corresponds to a potential answer. The probability of each span can be computed by applying sigmoid function on each pixel in the output matrix. Using sigmoid makes no assumption on number of spans. This way, we can set a threshold to find out all plausible answers.
Results
We note that all results reported in this paper are in absolute terms. Table 3 shows performance comparison between rule based versus deep learning based PPU model in US store. Despite handling the most common quantity patterns in the rule-based model, rule-based model fails due to more complex patterns in the text and due to lack of semantic understanding of the product. We see that the deep learning approach significantly outperforms such rule-based models with F1 jump of over 65% in UoM classification task and over 34% in precision for quantity extraction task, where deep learning model crossed the set precision threshold for all UoM types but rule-based model did not. Also, the deep learning-based model has the potential to improve continuously as our corrections process yields more data as a byproduct while rule-based model has limited improvement potential (see Section 3.1). Table 4 compares performance in the UoM type classification task across all three stores between pre-trained BERT model and our UoM Classifier model. We compare F1 score for classification tasks, and as we can see in Table 4, our model outperforms in US store while performs somewhat comparably to BERT in IN store, although BERT model performs better in EU-5 store. Overall, despite any prior knowledge on English language or the e-commerce catalog and with much less parameters, our model comes reasonably close to BERT performance globally. This shows that our CNN architecture is deep enough to learn the semantic information to accurately predict UoM type.
Comparison with rule-based models.
Comparison with BERT models.
As mentioned previously, the task of quantity extraction can also be viewed as a fact extraction problem, where the model directly predicts whether a certain quantity in the text attribute is relevant or not. We compare our Question Prediction and Answering approach on the lightweight PPU model against fact extraction approach using a bulkier BERT based model which was pre-trained for better English language understanding (see Table 5). We see that our model outperforms BERT in US store by 10.6% and in EU-5 store by 0.9% in precision while performing comparably in IN store, despite added advantage to the BERT model. Notably, our model crossed Table 6 shows performance comparison across stores for the three most prominent product categories in our dataset -A, B and C. Across all stores, product category A has a particularly lower F1 score on weight UoM type than the overall performance on that store. This is mainly because of incorrect UoM type classification since a lot of products in A had a weight information in the product title but the audited UoM type was count.
For example a product title may look like -"Patanjali Saundarya Swarn Kanti Fairness Cream(1.75 Oz)" where the model predicts its UoM type as weight but the audited UoM type is count and total quantity is 1. For the same reason, recall on count is higher in general for product category A than the overall average. Also, we found that volume and weight UoM types have better precision than count UoM type across categories. This is intuitive despite comparable classification performance across types as count UoM type products usually have many more count related numbers (thread count, roll count, number of packs, etc.) whose relevancy need to be accurately predicted. Due to the same reason, for EU-5 and IN stores, we are unable to reach the set precision bar for count UoM type across all 3 categories, unlike for weight and volume UoM types, using our PPU model.
Feature Selection in IN.
As seen in Section 4.1, there are data distribution differences between US and IN stores. We found that a simple replacement of OOV words in IN store with US counterpart lifts US model's confidence by upto 10% and even correct the predicted UoM type in some cases. Given our model is built on character-level features, it quickly adapts to the new set of words that are important for UoM classification task when fine-tuned (US fine-tuned (all text) model), giving a substantial lift in recall by 26% with respect to US baseline model. Inferring using only short text attributes such as title with fine-tuned model (US fine-tuned (short text) model) led to no notable drop in recall with improvements in weight and volume UoM types. Thus, the longer text attributes like product description and bullet points rarely contained extra information related to PPU over shorter attributes. Furthermore, we trained a model from scratch using all the text attributes (IN (all text) model) and found that recall only increases by 16% compared to US baseline model which is 10% lower than the lift in US fine-tuned model. This also quantifies the impact of transfer-learning over training afresh, as the model transfers knowledge such as product type and brand names with similar UoM types from US store. We achieved best performance with a model trained from scratch using only short text attributes and categorical attributes (IN (short text & categories) model) with an overall recall lift of 36% on held-out dataset with respect to US baseline. The role of categorical attributes is significant, where in its absence (IN (short text) model) the recall lift is 10% lower on the held-out set for quantity extraction task and 20% lower on hard examples for UoM classification task compared to the best model. We also trained a model from scratch using all text attributes as well as categorical attributes (IN (all text & categories) model) for completeness. This model performed reasonably well on the classification task on hard examples, yet failed on extraction task compared to the best model due to noisy signals from longer text attributes. See Table 7 for results comparison across all the models.
Latency
Model latency is a critical aspect for deployment, especially in the real-time validation use case (see Section 3.3). The validation models are required to have low-latency (less than 50 milliseconds). Given our PPU model is designed to be light-weight and sufficiently deep, when compared to large language models like BERT, our model scales well with latency. On a machine with 2 CPU cores, mean latency for PPU model is 17% better than BERT model and further improves as the number of CPU cores increases. Even with 16 CPU cores, we were unable to achieve less than 50 milliseconds of mean latency for BERT model. Latency improves tremendously if we drop long text attributes like product description & bullet points and use short text and categorical attributes, as in IN store (see Table 8).
Figure 1 :
1Word frequency plot for American English corpus and our dataset for top 5000 most common words. For an easier comparison, word frequency values are normalized in both corpuses so that log of word frequencies corresponding to most common words is one.
Figure 2 :
2Our model architecture consists of two subnetworks: UoM classification and quantity extraction networks in
closely related to UoM classes. This leads to better generalization and introduces robustness to typos.Input
_
: start & end indices of candidate quantities
_
: total quantity determined by auditors
_
: UoM type determined by auditors
Result:
_
: trainable candidate spans
if
_
== 1 and
_
==
then
return
for ∈ [3, 2, 1] do
forall
in ℎ
(
_
, ) do
if (quantities ∈
) ==
_
then
if no weight/volume quantity in
and
_
==
then
return
if only one weight/volume quantity in
and
_
≠
then
return
Table 2 :
2Distribution of products by number of spans# of spans Percentage
0
54.0%
1
34.5%
2
11.3%
3
0.2%
4.2.1 Rule-based models. The rule based model is the first baseline to PPU model, which comprises of regex rules for predicting UoM type and capturing all quantities present in the text. For predicting UoM type, the rule-based model relies on UoM specific keywords like ounce, liquid, pieces, etc. A simple regex rule to capture weight quantities for instance, can be as follows: "[decimal number][space][weight unit]"
Table 3 :
3Performance gains for our PPU model over the rule-based model in US store3 Task
PPU model (Δ)
Volume Weight Count Overall
UoM Classification
ΔP
1.6
26.0
17.9
16.4
ΔR
64.0
56.9
90.0
76.3
ΔF1
51.6
44.1
86.8
65.4
Quantity Extraction
ΔP
11.5
37.4
42.2
34.4
ΔR
19.1
0.2
27.7
19.1
ΔF1
22.4
8.0
40.6
26.1
Table 4 :
4Performance gains across stores over pre-trained Google BERT Base model fine-tuned first on catalog from English
stores for both MLM and NSP tasks and then on UoM Classification task across all stores: EU-5, IN and US
Store
PPU Model Classifier (Δ)
Volume Weight Count Overall
EU-5
ΔP
-11.8
-20.8
-7.2
-12.0
ΔR
-9.0
-7.8
-3.9
-5.4
ΔF1
-10.2
-13.5
-5.6
-8.6
IN
ΔP
0.0
0.0
0.0
0.0
ΔR
-8.0
-4.4
-3.9
-5.0
ΔF1
-4.4
-2.3
-2.1
-2.7
US
ΔP
0.5
-5.0
0.3
-0.7
ΔR
5.8
12.3
3.6
6.8
ΔF1
3.3
3.7
2.0
3.2
Table 5 :
5Problem formulation -Performance gains for Question Prediction and Answering approach using our PPU model
over Fact Extraction approach using BERT across all stores: EU-5, IN and US
Store
Question Prediction and Answering (Δ)
Volume Weight Count
Overall
EU-5
ΔP
1.7
1.2
0.3
0.9
ΔR
-27.3
-20.2
-15.3
-19.6
ΔF1
-30.7
-25.7
-17.2
-22.4
IN
ΔP
0.1
0.1
-0.6
0.1
ΔR
-14.1
-7.4
11.3
-0.5
ΔF1
-8.7
-4.7
10.8
1.8
US
ΔP
-0.3
-0.1
19.8
10.6
ΔR
-8.3
-4.8
8.5
1.6
ΔF1
-7.6
-5.0
12.0
3.8
the set precision threshold for US store and on two out of three
UoM types in both IN and EU-5 stores, while fact extraction using
BERT crossed it only for weight and volume UoM types in IN and
US stores. This reinstates that our approach of question prediction
(in the form of UoM type) and answering is desirable since the UoM
latent variables are critical in disambiguating among candidate
quantity spans. It helps in achieving comparable performance to
BERT with orders of magnitude smaller architecture which is easy
to deploy.
4.3.3 Product Categories.
Table 6 :
6Performance gains on 3 main Product Categories across different UoM types on quantity extraction task with respect to the overall performance in that storeStore
A
B
C
Volume Weight Count
Volume Weight Count
Volume Weight Count
EU-5
ΔP
0.7
-15.1
7.9
-1.3
-3.3
-12.9
-7.0
0.1
8.1
ΔR
-4.0
-9.5
41.2
10.3
45.2
7.4
-2.4
-0.8
19.3
ΔF1
-5.9
-15.8
41
13.2
47.4
7.1
-3.8
-1.2
16.6
IN
ΔP
0.1
0.0
0.9
0.2
-0.1
0.7
0.2
0.0
-0.3
ΔR
-10.3
-28.9
-5.2
-4.4
7.8
-17.4
-11.1
-5.3
-8.0
ΔF1
-7.3
-23.6
-4.6
-2.9
4.9
-17.6
-7.9
-3.7
-7.6
US
ΔP
0.4
0.0
0.4
0.0
-0.1
0.2
1.0
-4.6
0.7
ΔR
-4.7
-0.5
12.0
15.7
36.3
-5.6
-21.3
26.5
9.8
ΔF1
-4.6
-0.8
12.6
15.1
29.9
-6.6
-24.7
21.5
10.5
Table 7 :
7Performance gains on held-out dataset and hard examples for IN across different model variants with respect to US baseline model. On both the held-out dataset and hard examples, we compare recall and F1 score at a fixed high precision point.Model
Held-out Set (QE)
Hard Examples (UoM)
ΔRecall
ΔF1
ΔRecall
ΔF1 4
US fine-tuned (all text)
0.26
0.27
0.14
0.17
US fine-tuned (short text)
0.25
0.26
0.12
0.13
IN (all text)
0.16
0.17
0.14
0.17
IN (all text & categories)
0.11
0.12
0.32
0.31
IN (short text)
0.26
0.26
0.16
0.18
IN (short text & categories)
0.36
0.35
0.36
0.34
ΔP -lift in micro-averaged precision, ΔR -lift in micro-averaged recall
Both ΔRecall and ΔF1 are micro-averaged metrics
CONCLUSIONWe presented a lightweight deep learning model that can i) perform semantic learning and ii) scale well with our dataset sizes and iii) be shared across different stores, thanks to its fully character based architecture. UoM for computing quantity depends on factors such as brand, product type, conventions in a store etc. This can only be learned from domain experts for each store through rigorous audit processes. As human labeled data is limited in size, large language models are not the best fit. A model fully scalable with dataset sizes is desired while low latency is a must for real-time use cases. Also, sharing models across stores is important as brands, product types, etc., have mostly same unit of measure. These restrictions coupled with the need to avoid tokenization errors makes fully character based architectures desirable, while span-image architecture allows multi-answering. Solving quantity extraction as a question prediction and answering task gives better performance over fact extraction formulation even with bulkier pre-trained language models like BERT.
More Amazon Effects: Online Competition and Pricing Behaviors. Alberto Cavallo, 10.3386/w2513825138National Bureau of Economic Research. Working PaperAlberto Cavallo. 2018. More Amazon Effects: Online Competition and Pricing Behaviors. Working Paper 25138. National Bureau of Economic Research. https: //doi.org/10.3386/w25138
BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, arXiv:1810.04805Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. CoRR abs/1810.04805 (2018). arXiv:1810.04805 http://arxiv.org/abs/1810.04805
SpanBERT: Improving Pre-training by Representing and Predicting Spans. Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S Weld, Luke Zettlemoyer, Omer Levy, arXiv:1907.10529Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S. Weld, Luke Zettlemoyer, and Omer Levy. 2019. SpanBERT: Improving Pre-training by Representing and Pre- dicting Spans. CoRR abs/1907.10529 (2019). arXiv:1907.10529 http://arxiv.org/abs/ 1907.10529
Introduction to the CoNLL-2003 Shared Task:Language-Independent Named Entity Recognition. Erik F Tjong, Kim Sang, Fien De Meulder, SIGNLL Proceedings of the Seventh Conference on Natural Language Learning at HLT-NAACL 2003. Erik F. Tjong Kim Sang and Fien De Meulder. 2003. Introduction to the CoNLL- 2003 Shared Task:Language-Independent Named Entity Recognition. SIGNLL Proceedings of the Seventh Conference on Natural Language Learning at HLT- NAACL 2003 (2003), 142-147. https://www.aclweb.org/anthology/W03-0419
Unit Pricing Guide: A Best Practice Approach to Unit Pricing. NIST Special Publication 1181. National Institute of Standards and Technology. David Sefcik, 10.3386/w25138David Sefcik. 2015. Unit Pricing Guide: A Best Practice Approach to Unit Pricing. NIST Special Publication 1181. National Institute of Standards and Technology. https://doi.org/10.3386/w25138
Bidirectional Attention Flow for Machine Comprehension. Min Joon Seo, Aniruddha Kembhavi, Ali Farhadi, Hannaneh Hajishirzi, arXiv:1611.01603Min Joon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. 2016. Bidirectional Attention Flow for Machine Comprehension. CoRR abs/1611.01603 (2016). arXiv:1611.01603 http://arxiv.org/abs/1611.01603
Character-level convolutional networks for text classification. Xiang Zhang, Junbo Zhao, Yann Lecun, Advances in neural information processing systems. Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015. Character-level convolutional networks for text classification. In Advances in neural information processing systems. 649-657.
|
[] |
[
"A System for General In-Hand Object Re-Orientation",
"A System for General In-Hand Object Re-Orientation"
] |
[
"Tao Chen [email protected] \nMassachusetts Institute of Technology\n\n",
"Jie Xu [email protected] \nMassachusetts Institute of Technology\n\n",
"Pulkit Agrawal [email protected] \nMassachusetts Institute of Technology\n\n"
] |
[
"Massachusetts Institute of Technology\n",
"Massachusetts Institute of Technology\n",
"Massachusetts Institute of Technology\n"
] |
[] |
Figure 1: We present a simple framework for learning policies for reorienting a large number of objects in scenarios where the (1) hand faces upward, (2) hand faces downward with a table below the hand and (3) without the support of the table. The object orientation in the rightmost image in each row shows the target orientation.Abstract: In-hand object reorientation has been a challenging problem in robotics due to high dimensional actuation space and the frequent change in contact state between the fingers and the objects. We present a simple model-free framework that can learn to reorient objects with both the hand facing upwards and downwards. We demonstrate the capability of reorienting over 2000 geometrically different objects in both cases. The learned policies show strong zero-shot transfer performance on new objects. We provide evidence that these policies are amenable to realworld operation by distilling them to use observations easily available in the real world. The videos of the learned policies are available at: https://taochenshh. github.io/projects/in-hand-reorientation.
| null |
[
"https://arxiv.org/pdf/2111.03043v1.pdf"
] | 242,757,300 |
2111.03043
|
0ee9b633a0914b51f1eec3ad434752aa58e10149
|
A System for General In-Hand Object Re-Orientation
Tao Chen [email protected]
Massachusetts Institute of Technology
Jie Xu [email protected]
Massachusetts Institute of Technology
Pulkit Agrawal [email protected]
Massachusetts Institute of Technology
A System for General In-Hand Object Re-Orientation
Dexterous manipulationin-hand manipulationobject reorientation
Figure 1: We present a simple framework for learning policies for reorienting a large number of objects in scenarios where the (1) hand faces upward, (2) hand faces downward with a table below the hand and (3) without the support of the table. The object orientation in the rightmost image in each row shows the target orientation.Abstract: In-hand object reorientation has been a challenging problem in robotics due to high dimensional actuation space and the frequent change in contact state between the fingers and the objects. We present a simple model-free framework that can learn to reorient objects with both the hand facing upwards and downwards. We demonstrate the capability of reorienting over 2000 geometrically different objects in both cases. The learned policies show strong zero-shot transfer performance on new objects. We provide evidence that these policies are amenable to realworld operation by distilling them to use observations easily available in the real world. The videos of the learned policies are available at: https://taochenshh. github.io/projects/in-hand-reorientation.
Introduction
A common maneuver in many tasks of daily living is to pick an object, reorient it in hand and then either place it or use it as a tool. Consider three simplified variants of this maneuver shown in Figure 1. The task in the top row requires an upward-facing multi-finger hand to manipulate an arbitrary object in a random orientation to a goal configuration shown in the rightmost column. The next two rows show tasks where the hand is facing downward and is required to reorient the object either using the table as a support or without the aid of any support surface respectively. The last task is the hardest because the object is in an intrinsically unstable configuration owing to the downward gravitational force and lack of support from the palm. Additional challenges in performing such manipulation with a multi-finger robotic hand stem from the control space being high-dimensional and reasoning about the multiple transitions in the contact state between the finger and the object. Due to its practical utility and several unsolved issues, in-hand object reorientation remains an active area of research.
Past work has tackled the in-hand reorientation problem via several approaches: (i) The use of analytical models with powerful trajectory optimization methods [1,2,3]. While these methods demonstrated remarkable performance, the results were largely in simulation with simple object geometries and required detailed knowledge of the object model and physical parameters. As such, it remains unclear how to scale these methods to real-world and generalize to new objects. Another line of work has employed (ii) model-based reinforcement learning [4,5]; or (iii) model-free reinforcement learning with [6,7,8,9] and without expert demonstrations [10,11,12,13]. While some of these works demonstrated learned skills on real robots, it required use of additional sensory apparatus not readily available in the real-world (e.g., motion capture system) to infer the object state, and the learned policies did not generalize to diverse objects. Furthermore, most prior methods operate in the simplified setting of the hand facing upwards. The only exception is pick-and-place, but it does not involve any in-hand re-orientation. A detailed discussion of prior research is provided in Section 5.
In this paper, our goal is to study the object reorientation problem with a multi-finger hand in its general form. We desire (a) manipulation with hand facing upward or downward; (b) the ability of using external surfaces to aid manipulation; (c) the ability to reorient objects of novel shapes to arbitrary orientations; (d) operation from sensory data that can be easily obtained in the real world such as RGBD images and joint positions of the hand. While some of these aspects have been individually demonstrated in prior work, we are unaware of any published method that realizes all four. Our main contribution is building a system that achieves the desiderata. The core of our framework is a model-free reinforcement learning with three key components: teacher-student learning, gravity curriculum, and stable initialization of objects. Our system requires no knowledge of object or manipulator models, contact dynamics or any special pre-processing of sensory observations. We experimentally test our framework using a simulated Shadow hand. Due to the scope of the problem and the ongoing pandemic, we limit our experiments to be in simulation. However, we provide evidence indicating that the learned policies can be transferred to the real world in the future.
A Surprising Finding: While seemingly counterintuitive, we found that policies that have no access to shape information can manipulate a large number of previously unseen objects in all the three settings mentioned above. At the start of the project, we hypothesized that developing visual processing architecture for inferring shape while the robot manipulates the object would be the primary research challenge. On the contrary, our results show that it is possible to learn control strategies for general in-hand object re-orientation that are shape-agnostic. Our results, therefore, suggest that visual perception may be less important for in-hand manipulation than previously thought. Of course, we still believe that the performance of our system can be improved by incorporating shape information. However, our findings suggest a different framework of thinking: a lot can be achieved without vision, and that vision might be the icing on the cake instead of the cake itself.
Method
We found that simple extensions to existing techniques in robot learning can be used to construct a system for general object reorientation. First, to avoid explicit modeling of non-linear and frequent changes in the contact state between the object and the hand, we use model-free reinforcement learning (RL). An added advantage is that model-free RL is amenable to direct operation from raw point cloud observations, which is preferred for real-world deployment. We found that better policies can be trained faster using privileged state information such as the velocities of the object/fingertips that is easily available in the simulator but not in the real world. To demonstrate the possibility of transferring learned policies to the real world in the future, we overcome the need for privileged information using the idea of teacher-student training [14,15]. In this framework, first, an expert or teacher policy (π E ) is trained using privileged information. Next, the teacher policy guides the learning of a student policy (π S ) that only uses sensory inputs available in the real world. Let the state space corresponding to π E and π S be S E and S S respectively. In general, S E = S S .
We first trained the teacher policy to reorient more than two thousand objects of diverse shapes (see Section 2.1). Next, we detail the method for distilling π E to a student policy using a reduced state space consisting of only the joint positions of the hand, the object position, and the difference in orientation from the goal configuration (see Section 2.2.1). However, in the real world, even the object position and relative orientation must be inferred from sensory observation. Not only does this process require substantial engineering effort (e.g., a motion capture or a pose estimation system), but also inferring the pose of a symmetric object is prone to errors. This is because a symmetric object at multiple orientations looks exactly the same in sensory space such as RGBD images.
To mitigate these issues, we further distill π E to operate directly from the point cloud and position of all the hand joints (see Section 2.2.2). We propose a simple modification that generalizes an existing 2D CNN architecture [16] to make this possible.
The procedure described above works well for manipulation with the hand facing upwards and downwards when a table is available as support. However, when the hand faces downward without an underlying support surface, we found it important to initialize the object in a stable configuration. Finally, because gravity presents the primary challenge in learning policies with a downwardfacing hand, we found that training in a curriculum where gravity is slowly introduced (i.e., gravity curriculum) substantially improves performance. These are discussed in Section 4.2.
Learning the teacher policy
We use model-free RL to learn the teacher policy (π E ) for reorienting an object ({O i |i = 1, ..., N }) from an initial orientation α o 0 to a target orientation α g (α o 0 = α g ). At every time step t, the agent observes the state s t , executes the action a t sampled from the policy π E , and receives a reward r t . π E is optimized to maximize the expected discounted return: π = arg max π E E T −1 t=0 γ t r t , where γ is the discount factor. The task is successful if the angle difference ∆θ between the object's current (α o t ) and the goal orientation (α g ) is smaller than the threshold valueθ, i.e., ∆θ ≤θ. To encourage the policy to be smooth, the previous action is appended to the inputs to the policy (i.e., a t = π E (s t , a t−1 )) and large actions are penalized in the reward function. We experiment with two architectures for π E : (1) an MLP policy π M , (2) an RNN policy π R . We use PPO [17] to optimize π E . More details about the training are in Section C.1 and Section C.2 in the appendix.
Observation and action space: We define S E to include joint, fingertip, object, and goal information as detailed in Table B.1 in the appendix. Note that S E does not include object shape or information about friction, damping, contact states between the fingers and the object, etc. We control the joint movements by commanding the relative change in the target joint angle (q target t ) on each actuated joint (action a t ∈ R 20 ): q target t+1 = q target t + a t × ∆t, where ∆t is the control time step. We clamp the action command if necessary to make sure |∆q target t | ≤ 0.33 rad. The control frequency is 60 Hz.
Dynamics randomization: Even though we do not test our policies on a real robot, we train and evaluate policies with domain randomization [18] to provide evidence that our work has the potential to be transferred to a real robotic system in the future. We randomize the object mass, friction coefficient, joint damping and add noise to the state observation s t and the action a t . More details about domain randomization are provided in Table C.4 in the appendix.
Learning the student policy
We distill the teacher π E into the student policy π S using Dagger [19], a learning-from-demonstration method that overcomes the covariate shift problem. We optimize π S by minimizing the KL-divergence between π S and π E : π S = arg min π S D KL π E (s E t , a t−1 )||π S (s S t , a t−1 )) . Based on observation data available in real-world settings, we investigate two different choices of S S .
Training student policy from low-dimensional state
In the first case, we consider a non-vision student policy π S (s S t , a t−1 ). s S t ∈ R 31 includes the joint positions q t ∈ R 24 , object position p o t ∈ R 3 , quaternion difference between the object's current and target orientation β t ∈ R 4 . In this case, S S ⊂ S E , and we assume the availability of object pose information, but do not require velocity information. We use the same MLP and RNN network architectures used for π E on π S except the input dimension changes as the state dimension is different.
Training student policy from vision
In the second case, S S only uses direct observations from RGBD cameras and the joint position (q t ) of the robotic hand. We convert the RGB and Depth data into a colored point cloud using a pinhole camera model [20]. Our vision policy takes as input the voxelized point cloud of the scene W t , q t , and previous action command a t−1 , and outputs the action a t , i.e., a t = π S (W t , q t , a t−1 ).
Goal specification: To avoid manually defining per-object coordinate frame for specifying the goal quaternion, we provide the goal to the policy as an object point cloud rotated to the desired orientation W g , i.e., we only show the policy how the object should look like in the end (see the top left of Figure 2). The input to π S is the point cloud W t = W s t ∪ W g where W s t is the actual point cloud of the current scene obtained from the cameras. Details of obtaining W g are in Section C.2.
Sparse3D-IMPALA-Net: To convert a voxelized point cloud into a lower-dimensional feature representation, we use a sparse convolutional neural network. We extend the IMPALA policy architec- ture [16] for processing RGB images to process colored point cloud data using 3D convolution. Since many voxels are unoccupied, the use of regular 3D convolution substantially increases computation time. Hence, we use Minkowski Engine [21], a PyTorch library for sparse tensors, to design a 3D version of IMPALA-Net with sparse convolutions (Sparse3D-IMPALA-Net) 1 . The Sparse3D-IMPALA network takes as input the point cloud W t , and outputs an embedding vector which is concatenated with the embedding vector of (q t , a t−1 ). Afterward, a recurrent network is used and outputs the action a t . The detailed architecture is illustrated in Figure 2.
Mitigating the object symmetry issue: π E is trained with the the ground-truth state information s E t including the object orientation α o t and goal orientation α g . The vision policy does not take any orientation information as input. If an object is symmetric, the two different orientations of the object may correspond to the same point cloud observation. This makes it problematic to use the difference in orientation angles (∆θ ≤θ) as the stopping and success criterion. To mitigate this issue, we use Chamfer distance [22] to compute the distance between the object point cloud in α o t and the goal point cloud (i.e., the object rotated by α g ) as the evaluation criterion. The Chamfer distance is computed
as d C = a∈W o t min b∈W g a − b 2 2 + b∈W g min a∈W o t a − b 2 2 , where W o
t is the object point cloud in its current orientation. Both W o t and W g are scaled to fit in a unit sphere for computing d C . We check Chamfer distance in each rollout step. If d C ≤d C (d C is a threshold value for d C ), we consider the episode to be successful. Hence, the success criterion is (∆θ ≤θ) ∨ (d C ≤d C ). In training, if the success criterion is satisfied, the episode is terminated and used for updating π S .
Experimental Setup
We use the simulated Shadow Hand [23] in NVIDIA Isaac Gym [24]. Shadow Hand is an anthropomorphic robotic hand with 24 degrees of freedom (DoF). We assume the base of the hand to be fixed. Twenty joints are actuated by agonist-antagonist tendons and the remaining four are under-actuated.
Object datasets: We use the EGAD dataset [25] and YCB dataset [26] that contain objects with diverse shapes (see Figure B.2) for in-hand manipulation experiments. EGAD contains 2282 geometrically diverse textureless object meshes, while the YCB dataset includes textured meshes for objects of daily life with different shapes and textures. We use the 78 YCB object models collected with the Google scanner. Since most YCB objects are too big for in-hand manipulation, we proportionally scale down the YCB meshes. To further increase the diversity of the datasets, we create 5 variants for each object mesh by randomly scaling the mesh. More details of the object datasets are in Section B.2.
Setup for visual observations: For the vision experiments, we trained policies for the scenario of hand facing upwards. We place two RGBD cameras above the hand ( Figure B these cameras is combined to create a point cloud observation of the scene [20]. For downstream computation, the point cloud is voxelized with a resolution of 0.003 m.
Setup with the upward facing hand: We first consider the case where the Shadow Hand faces upward and is required to reorient objects placed in the hand (see Figure 3a). We use the coordinate system where the z-axis points vertically upward and the xy-plane denotes the horizontal plane. The object pose is initialized with the following procedure: xy position of the object's center of mass (COM) p o 0,xy is randomly sampled from a square region of size 0.09 m × 0.09 m. The center of this square is approximately located on the intersection of the middle finger and the palm so that the sampling region covers both the fingers and the palm. The z position of the object is fixed to 0.13 m above the base of the hand to ensure that the object does not collide with the hand at initialization. The initial and goal orientations are randomly sampled from the full SO(3) space.
Setup with the downward facing hand: Next, we consider the cases where the hand faces downward. We experiment with two scenarios: with and without a table below the hand. In the first case, we place a table with the tabletop being 0.12 m below the hand base. We place objects in a random pose between the hand and the table so that the objects will fall onto the table. We will describe the setup for the second case (without a table) in Section 4.2.2.
Evaluation criterion: For non-vision experiments, a policy rollout is considered a success if θ ≤θ. θ = 0.1 rad. For vision experiments, we also check d C ≤d C as another criterion andθ = 0.2 rad,d C = 0.01. The initial and goal orientation are randomly sampled from SO(3) space in all the experiments. We report performance as the percentage of successful episodes when the agent is tasked to reorient each training object 100 times from arbitrary start to goal orientation. We report the mean and standard deviation of success rate from 3 seeds.
Results
We evaluate the performance of reorientation policies with the hand facing upward and downward. Further we analyze the generalization of the learned policies to unseen object shapes.
Reorient objects with the hand facing upward
Train a teacher policy with full-state information We train our teacher MLP and RNN policies using the full state information using all objects in the EGAD and YCB datasets separately. The progression of success rate during training is shown in Figure D.5 in Appendix D.1 . Figure D.5 also shows that using privileged information substantially speeds up policy learning. Results reported in Table 1 indicate that the RNN policies achieve a success rate greater than 90% on the EGAD dataset (entry B1) and greater than 80% on the YCB dataset (entry G1) without any explicit knowledge of the object shape 2 . This result is surprising because apriori one might believe that shape information is important for in-hand reorientation of diverse objects.
The visualization of policy rollout reveals that the agent employs a clever strategy that is invariant to object geometry for re-orienting objects. The agent throws the object in the air with a spin and catches it at the precise time when the object's orientation matches the goal orientation. Throwing the object with a spin is a dexterous skill that automatically emerges! One possibility for the emergence of this skill is that we used very light objects. This is not true because we trained with objects in the range of 50-150g which spans many hand-held objects used by humans (e.g., an egg weighs about 50g, a glass cup weighs around 100g, iPhone 12 weighs 162g, etc.). To further probe this concern, we evaluated zero-shot performance on objects weighing up to 500g 3 and found that the learned policy can successfully reorient them. We provide further analysis in the appendix showing that forces applied by the hand for such manipulation are realistic. While there is still room for the possibility that the learned policy is exploiting the simulator to reorient objects by throwing them in the air, our analysis so far indicates otherwise.
Next, to understand the failure modes, we collected one hundred unsuccessful trajectories on YCB dataset and manually analyzed them. The primary failure is in manipulating long, small, or thin objects, which accounts for 60% of all errors. In such cases, either the object slips through the fingers and falls, or is hard to be manipulated when the objects land on the palm. Another cause of failures (19%) is that objects are reoriented close to the goal orientation but not close enough to satisfy ∆θ <θ. Finally, the performance on YCB is lower than EGAD because objects in the YCB dataset are more diverse in their aspect ratios. Scaling these objects by constraining l max ∈ [0.05, 0.12]m (see Section 3) makes some of these objects either too small, too big, or too thin and consequently results in failure (see Figure D.6). A detailed object-wise quantitative analysis of performance is reported in appendix Figure D.9. Results confirm that sphere-like objects such as tennis balls and orange are easiest to reorient, while long/thin objects such as knives and forks are the hardest to manipulate.
Train a student policy with a reduced state space The student policy state is s S t ∈ R 31 . In Table 1 (entries E1 and J1), we can see that π S R can get similarly high success rates as π E R . The last two columns in Table 1 also show that the policy is more robust to dynamics variations and observation/action noise after being trained with domain randomization.
Reorient objects with the hand facing downward
The results above demonstrate that when the hand faces upwards, RL can be used to train policies for reorienting a diverse set of geometrically different objects. A natural question to ask is, does this still hold true when the hand is flipped upside down? Intuitively, this task is much more challenging because the objects will immediately fall down without specific finger movements that stabilize the object. Because with the hand facing upwards, the object primarily rests on the palm, such specific finger movements are not required. Therefore, the hand facing downwards scenario presents a much harder exploration challenge. To verify this hypothesis, we trained a policy with the downward-facing hand, objects placed underneath the hand (see Figure 3b), and using the same reward function (Equation (1)) as before. Unsurprisingly, the success rate was 0%. The agent's failure can be attributed to policy needing to learn to both stabilize the object under the effect of gravity and simultaneously reorient it. Deploying this policy simply results in an object falling down, confirming the hard-exploration nature of this problem.
Reorient objects on a table
To tackle the hard problem of reorienting objects with the hand facing downward, we started with a simplified task setup that included a table under the hand (see Figure 3c). Table eases exploration by preventing the objects from falling. We train π E M using the same reward function Equation (1) on objects sampled from the EGAD and YCB datasets. The success rate using an MLP policy using full state information for EGAD and YCB is 95.31% ± 0.9% and 81.59% ± 0.7% respectively. Making use of external support for in-hand manipulation has been a challenging problem in robotics. Prior work approach this problem by building analytical models and constructing motion cones [27], which is challenging for objects with complex geometry. Our experiments show that model-free RL provides an effective alternative for learning manipulation strategies capable of using external support surfaces.
Reorient objects in air with hand facing downward
To enable the agent to operate in more general scenarios, we tackled the re-orientation problem with the hand facing downwards and without any external support. In this setup, one might hypothesize that object shape information (e.g., from vision) is critical because finding the strategy in Section 4.1 is not easy when the hand needs to overcome gravity and stabilize the object while reorienting it. We experimentally verify that even in this case, the policies achieve a reasonably high success rate without any knowledge of object shape.
A good pose initialization is what you need: The difficulty of directly training the RL policies when the hand faces downward is mainly because of the hard-exploration issue in learning to catch the objects that are moving downward. However, catching is not necessary for the reorientation. Even for human, we only reorient the object after we grasp it. More specifically, we first train an object-lifting policy to lift objects from the table, collect the ending state (joint positions q T , object position p o T and orientation α o T ) in each successful lifting rollout episode, and reset the hand and objects to these states (velocities are all 0) for the pose initialization in training the reorientation policy. The objects have randomly initialized poses and are dropped onto the table. We trained a separate RNN policy for each dataset using the reward function in Section C.2. The success rate on the EGAD dataset is 97.80%, while the success rate on the YCB dataset is 90.11%. Note that objects need to be grasped first to be lifted. Our high success rates on object lifting also indicate that using an anthropomorphic hand makes object grasping an easy task, while many prior works [28,29] require much more involved training techniques to learn grasping skills with parallel-jaw grippers. After we train the lifting policy, we collect about 250 ending states for each object respectively from the successful lifting episodes. In every reset during the reorientation policy training, ending states are randomly sampled and used as the initial pose of the fingers and objects. With a good pose initialization, policies are able to learn to reorient objects with high success rates. π E R trained on EGAD dataset gets a success rate more than 80% while π E R trained on YCB dataset gets a success rate greater than 50%. More results on the different policies with and without domain randomization are available in Table D.6 in the appendix. This setup is challenging because if at any time step in an episode the fingers take a bad action, the object will fall.
Improving performance using gravity curriculum: Since the difficulty of training the reorientation policy with the hand facing downward is due to the gravity, we propose to build a gravity curriculum to learn the policy π E . Since π E already performs very well on EGAD objects, we apply gravity curriculum to train π E on YCB objects. Our gravity curriculum is constructed as follows: we start the training with g = 1 m/s 2 , then we gradually decrease g in a step-wise fashion if the evaluation success rate (w) is above a threshold value (w) until g = −9.81 m/s 2 . More details about gravity curriculum are in Section C.4. In Table D.6 (Exp Q and T) in the appendix, we can see that adding gravity curriculum (g-curr) significantly boost the success rates on the YCB dataset.
Zero-shot policy transfer across datasets
We have shown the testing performance on the same training dataset so far. How would the policies work on a different dataset? To answer this, we test our policies across datasets: policies trained with EGAD objects are now tested with YCB objects and vice versa. We used the RNN policies trained with full-state information and reduced-state information respectively (without domain randomization) and tested them on the other dataset with the hand facing upward and downward. In the case of the hand facing downward, we tested the RNN policy trained with gravity curriculum. Table 3 shows that policies still perform well on the untrained dataset.
Object Reorientation with RGBD sensors
In this section, we investigate whether we can train a vision policy to reorient objects with the hand facing upward. As vision-based experiments require much more compute resources, we train one vision policy for each object individually on six objects shown in Table 4. We leave training a single vision policy for all objects to future work. We use the expert MLP policy trained in Section 4.1 to supervise the vision policy. We also performed data augmentation on the point cloud input to the policy network at each time step in both training and testing. The data augmentation includes the random translation of the point cloud, random noise on the point positions, random dropout on the points, and random variations on the point color. More details about the data augmentation are in Section D.5. We can see from Table 4 that reorienting the non-symmetric objects including the toy and the mug has high success rates (greater than 80%). While training the policy for symmetric objects is much harder, Table 4 shows that using d C as an episode termination criterion enables the policies to achieve a success rate greater than 50%.
Related Work
Dexterous manipulation has been studied for decades, dating back to [30,31]. In contrast to paralleljaw grasping, pushing, pivoting [32], or pick-and-place, dexterous manipulation typically involves continuously controlling force to the object through the fingertips of a robotic hand [33]. Some prior works used analytical kinematics and dynamics models of the hand and object, and used trajectory optimization to output control policies [1,2,34] or employed kinodynamic planning to find a feasible motion plan [35]. However, due to the large number of active contacts on the hand and the objects, model simplifications such as simple finger and object geometries are usually necessary to make the optimization or planning tractable. Sundaralingam and Hermans [34] moved objects in hand but assumes that there is no contact breaking or making between the fingers and the object. Furukawa et al. [36] achieved a high-speed dynamic regrasping motion on a cylinder using a high-speed robotic hand and a high-speed vision system. Prior works have also explored the use of a vision system for manipulating an object to track a planned path [37], detecting manipulation modes [38], precision manipulation [39] with a limited number of objects with simple shapes using a two-fingered gripper. Recent works have explored the application of reinforcement learning to dexterous manipulation. Model-based RL works learned a linear [4,7] or deep neural network [5] dynamics model from the rollout data, and used online optimal control to rotate a pen or Baoding balls on a Shadow hand. However, when the system is unstable, collecting informative trajectories for training a good dynamics model that generalizes to different objects remains challenging. Another line of works uses model-free RL algorithms to learn a dexterous manipulation policy. For example, OpenAI et al. [11] and OpenAI et al. [12] learned a controller to reorient a block or a Rubik's cube. Van Hoof et al. [40] learned the tactile informed policy via RL for a three-finger manipulator to move an object on the table. To reduce the sample complexity of model-free learning, [9,13,8,41,6] combined imitation learning with RL to learn to reorient a pen, open a door, assemble LEGO blocks, etc. However, collecting expert demonstration data from humans is expensive, time-consuming, and even incredibly difficult for contact-rick tasks [8]. Our method belongs to the category of model-free learning. We use the teacher-student learning paradigm to speed up the deployment policy learning. Our learned policies also generalize to new shapes and show strong zero-shot transfer performance. To the best of our knowledge, our system is the first work that demonstrates the capabilities of reorienting a diverse set of objects that have complex geometries with both the hand facing upward and downward. A recent work [42] (after our CoRL submission) learns a shape-conditioned policy to reorient objects around z-axis with an upward-facing hand. Our work tackles more general tasks (more diverse objects, any goal orientation in SO (3), hand faces upward and downward) and shows that even without knowing any object shape information, the policies can get surprisingly high success rates in these tasks.
Discussion and Conclusion
Our results show that model-free RL with simple deep learning architectures can be used to train policies to reorient a large set of geometrically diverse objects. Further, for learning with the hand facing downwards, we found that a good pose initialization obtained from a lifting policy was necessary, and the gravity curriculum substantially improved performance. The agent also learns to use an external surface (i.e., the table). The most surprising observation is that information about shape is not required despite the fact that we train a single policy to manipulate multiple objects. Perhaps in hindsight, it is not as surprising -after all, humans can close their eyes and easily manipulate novel objects into a specific orientation. Our work can serve a strong baseline for future in-hand object reorientation works that incorporate object shape in the observation space.
While we only present results in simulation, we also provide evidence that our policies can be transferred to the real world. The experiments with domain randomization indicate that learned policies can work with noisy inputs. Analysis of peak torques during manipulation (see Figure D.11 in the appendix) also reveals that generated torque commands are feasible to actuate on an actual robotic hand.
Finally, Figure D.9 and Figure D.10 in the appendix show that the success rate varies substantially with object shape. This suggests that in the future, a training curriculum based on object shapes can improve performance. Another future work is to directly train one vision policy for a diverse set of objects. A major bottleneck in vision-based experiments is the demand for much larger GPU memory. Learning visual representations of point cloud data that can ease the computational bottleneck is, therefore, an important avenue for future research.
Appendix A Evidence indicating transfer to real-world
Due to the scope of the problem and the ongoing pandemic, we limit our experiments to be in simulation. However, we provided evidence indicating that the learned policies can be transferred to the real world in the future in the paper. We summarize this evidence as follows.
Convex decomposition The objects after the convex decomposition still have geometrically different and complex geometries as shown in Figure B.3. The objects in the EGAD dataset are 3D printable. The YCB objects are available in the real world.
Action space We control the finger joints via relative position control as explained in Section 2.1. This suffers less sim-to-real gap compared to using torque control on the joints directly.
Student policies We designed two student policies and both of them use the observation data that can be readily acquired from the real world. The first student policy only requires the joint positions and the object pose. Object pose can be obtained using a pose estimation system or a motion capture system in the real world. Our second student policy only require the point cloud of the scene and the joint positions. We can get the point cloud in the real world by using RGBD cameras such as Realsense D415, Azure Kinect, etc.
Domain randomization
We also trained and tested our policies with domain randomization. We randomized object mass, friction, joint damping, tendon damping, tendon stiffness, etc. The results in Table D.5, Table D.6, and Table 4 show that even after adding randomization/noise, we can still get good success rates with the trained policies. Even though we cannot replicate the true real-world setups in the simulation, our results with domain randomization indicates a high possibility that our policies can be transferred to the real Shadow hand. Prior works [11] have also shown the domain randomization can effectively reduce the sim-to-real gap.
Torque analysis
We also conducted torque analysis as shown in Section D.4. We can see that the peak torque values remain in an reasonable and affordable range for the Shadow hand. This indicates that our learned policies are less likely to cause motor overload on the real Shadow hand. Figure B.2 shows examples from the EGAD and YCB dataset. We can see that these objects are geometrically different and have complex shapes. We also use V-HACD [43] to perform an approximate convex decomposition on the object meshes for fast collision detection in the simulator. Figure B.3 shows the object shapes before and after the decomposition. After the decomposition, the objects are still geometrically different.
B.3 Camera setup
We placed two RGBD cameras above the hand, as shown in Figure B For the non-vision policies, we experimented with two architectures: The MLP policy π M consists of 3 hidden layers with 512, 256, 256 neurons respectively. The RNN policy π R has 3 hidden layers (512 − 256 − 256), followed by a 256-dim GRU layer and one more 256-dim hidden layer. We use the exponential linear unit (ELU) [44] as the activation function.
For our vision policies, we design a sparse convolutional network architecture (Sparse3D-IMPALA-Net). As shown in Figure 2, the point cloud W t is processed by a series of sparse CNN residual modules and projected into an embedding vector. q t and a t−1 are concatenated together and projected into an embedding vector via an MLP. Both embedding vectors from W t and (q t , a t−1 ) are concatenated and passed through a recurrent network to output the action a t .
C.2 Training details
All the experiments in the paper were run on at most 2 GPUs with a 32GB memory. We use PPO [17] to learn π E . Table C.2 lists the hyperparameters for the experiments. We use 40K parallel environments for data collection. We update the policy with the rollout data for 8 epochs after every 8 rollout steps for the MLP policies and 50 rollout steps for the RNN policies. A rollout episode is terminated (reset) if the object is reoriented to the goal orientation successfully, or the object falls, or the maximum episode length is reached. To learn the student policies π S , we use Dagger [19]. While Dagger typically keep all the state-action pairs for training the policy, we do Dagger in an online fashion where π S only learns from the latest rollout data.
For the vision experiments, the number of parallel environments is 360 and we update policy after every 50 rollout steps from all the parallel environments. The batch size is 30. We sample 15000 points from the reconstructed point cloud of the scene from 2 cameras for the scene point cloud W s t and sample 5000 points from the object CAD mesh model for the goal point cloud W g .
We use Horovod [45] for distributed training and Adam [46] optimizer for neural network optimization.
Reward function for reorientation: For training π E for the reorientation task, we modified the reward function proposed in [24] to be:
r(s t , a t ) = c θ1 1 |∆θ t | + θ + c θ2 1(|∆θ t | <θ) + c 3 a t 2 2(1)
where c θ1 > 0, c θ2 > 0 and c 3 < 0 are the coefficients, ∆θ t is the difference between the current object orientation and the target orientation, θ is a constant, 1 is an indicator function that identifies whether the object is in the target orientation. The first two reward terms encourage the policy to reorient the object to the desired orientation while the last term suppresses large action commands.
Reward function for object lifting: To train the lifting policy, we use the following reward function:
r(s t , a t ) = c h1 1 |∆h t | + h + c h2 1(|∆h t | <h) + c 3 a t
Goal specification for vision policies: We obtain W g by sampling 5000 points from the object's CAD mesh using the Barycentric coordinate, rotating the points by the desired orientation, and translating them so that these points are next to the hand. Note that one can also put the object in the desired orientation right next to the hand in the simulator and render the entire scene altogether to remove the need for CAD models. We use CAD models for W g just to save the computational cost of rendering another object while we still use RGBD cameras to get W s t . Comparing the Column 1 and Column 2 in Table D.5, we can see that if we directly deploy the policy trained without domain randomization into an environment with different dynamics, the performance drops significantly. If we train policies with domain randomization (Column 3), the policies are more robust and the performance only drops slightly compared to Column 1 in most cases. The exceptions are on C3 and H3. In these two cases, the π S M policies collapsed in training during the policy distillation along with the randomized dynamics.
C.3 Dynamics randomization
C.4 Gravity curriculum
We found building a curriculum on gravity helps improve the policy learning for YCB objects when the hand faces downward. Algorithm 1 illustrates the process of building the gravity curriculum. In our experiments, we only test on the training objects once (one random initial and goal orientation) to get the surrogate average success rate w on all the objects during training.w = 0.8, g 0 = 1 m/s 2 , ∆g = −0.5 m/s 2 , K = 3, L = 20, ∆T min = 40. and greater than 70% on the YCB dataset (entries F1, G1) without any explicit knowledge of the object shape. This result is surprising because intuitively, one would assume that information about the object shape is important for in-hand reorientation.
Testing performance -Student We experimented with the following three combinations:
(1) distill π E M into π S M , (2) distill π E M into π S R , (3) distill π E R into π S R .
The student policy state is s S t ∈ R 31 . In Table D.5 (entries C1-E1, H1-J1), we can see that π E R → π S R gives the highest success rate on π S , while π E M → π S M leads to much worse performance (36% drop of success rate in EGAD, and 47% drop in YCB). This shows that π S requires temporal information due to reduced state space. The last two columns in Table D.5 also show that the policy is more robust to dynamics variations and observation/action noise after being trained with domain randomization. Figure D.6 shows some example failure cases. If the objects are too small, thin, or big, the objects are likely to fall. If objects are initialized close to the hand border, it is much more difficult for the hand to catch the objects. Another failure mode is that the objects are reoriented close to the goal orientation but not close enough to satisfy ∆θ ≤θ. Example visualization We show an example of reorienting a cup in Figure
Failure cases
D.3 Success rate on each type of YCB objects
We also analyzed the success rates on each object type in the YCB dataset. Using the same evaluation procedure described in Section 3, we get the success rates of each object using π E R . Figure D.9 shows the distribution of the success rates on YCB objects with the hand facing upward while Figure D.10 corresponds to the case of reorienting the objects in the air with the hand facing downward. We can see that sphere-like objects such as tennis balls and orange are easiest to reorient. Long or thin objects such as knives and forks are the hardest ones to manipulate.
D.4 Torque analysis
We randomly sampled 100 objects from the YCB dataset, and use our RNN policy trained without domain randomization with the hand facing downward to reorient each of these objects 200 times. We record the joint torque values for each finger joint at each time step. Let the joint torque value of i th joint at time step j in k th episode be J k ij . We plot the distribution of {max i=[ [1,24]
D.5 Vision experiments with noise
We also trained our vision policies with noise added to the point cloud input. We added the following transformations to the point cloud input.
We applied four types of transformations on the point cloud: • RandomDropout: Randomly dropout points with a dropout ratio uniformly sampled from [0, 0.4].
• JitterColor: Jitter the color of points with the following 3 transformations: (1) jitter the brightness and rgb values, (2) convert the color of 30% of the points into gray, (3) jitter the color contrast. Each of this transformation can be applied independently with a probability of 30% if JitterColor is applied.
Each of these four transformations is applied independently with a probability of 40% for each point cloud at every time step. Table D.7 shows the success rates of the vision policies trained with the aforementioned data augmentations until policy convergence and tested with the same data augmentations. We found that adding the data augmentation in training actually helps improve the data efficiency of the vision policy learning even though the final performance might be a bit lower. As a reference, we show the policy performance trained and tested without data augmentation in Table D.7. For the mug object, adding data augmentation in training improves the final testing performance significantly. Without data augmentation, the learned policy reorients the mug to a pose where the body of the mug matches how the mug should look in the goal orientation, but the cup handle does not match. Adding the data augmentation helps the policy to get out of this local optimum. 053_mini_soccer_ball 017_orange 055_baseball 014_lemon 018_plum 056_tennis_ball 054_softball 063-b_marbles 015_peach 013_apple 058_golf_ball 057_racquetball 012_strawberry 065 We exclude a few outliers in the plot, i.e., we only plot the data distributions up to 99% quantile.
Figure 2 :
2Visual policy architecture. MK stands for Minkowski Engine. q t is the joint positions and a t is the action at time step t.
Figure 3 :
3Examples of initial poses of the hand and object. (a): hand faces upward. (b), (c), (d): hand faces downward. (b): both the hand and the object are initialized with random poses . (c): there is a table below the hand. (d): the hand and the object are initialized from the lifted poses.
. 4 .
4In ISAAC gym, we set the camera pose by setting its position and focus position. The two cameras' positions are shifted from the Shadow hand's base origin by [−0.6, −0.39, 0.8] and [0.45, −0.39, 0.8] respectively. And their focus points are the points shifted from the Shadow hand's base origin by [−0.08, −0.39, 0.15] and [0.045, −0.39, 0.15] respectively.
Figure B. 2 :
2First row: examples of EGAD objects. Second row: examples of YCB objects.
Figure B. 3 :
3Examples of EGAD objects. The first and third row shows the visual mesh of the objects. The second and fourth row show the corresponding collision mesh (after V-HACD decomposition).
Figure
Figure B.4: Camera positions
finger static friction ×U(0.7, 1.3) joint damping ×E(0.3, 3.0) N (µ, σ): Gaussian distribution with mean µ and standard deviation σ. U(a, b): uniform distribution between a and b. E(a, b) = exp U (log(a),log(b)) . +: the sampled value is added to the original value of the variable. ×: the original value is scaled by the sampled value.
Figure D. 5 :
5avg(Q) >w and ∆T > ∆T min then 10: g = max(g − ∆g, Learning curves of the MLP and RNN policies on the EGAD (Left) and YCB datasets (Middle). The Right plot shows that using the full state information speeds up the policy learning compared to only using the reduced state information.
Figure D. 6 :
6Examples of failure cases. (a) and (b): objects are too small. (c): the object is reoriented close to the target orientation, but not close enough. (d): the object is too big and initialized around the palm border. D.2 Hand faces downward (in the air) Testing performance For the case of reorienting objects in the air with the hand facing downward Table D.6 lists the success rates of different policies trained with/without domain randomization, and tested with/without domain randomization.
D.7 and an example of reorienting a sponge in Figure D.8. More examples are available at https://taochenshh.github. io/projects/in-hand-reorientation.
Figure D.11shows that the majority of the maximum torque magnitude is around 0.2 N m.
Figure D. 7 :
7An example of reorienting a cup with the hand facing downward. From left to right, top to bottom, we show the some moments in an episode.
Figure D. 8 :
8An example of reorienting a sponge with the hand facing downward. From left to right, top to bottom, we show the some moments in an episode.
Figure D. 9 :
9Reorientation success rates for each object in the YCB dataset when the hand faces upward.
Figure D. 10 :
10Reorientation success rates for each object in the YCB dataset when the hand faces downward without a table. maximum torque magnitude on all jointsFigure D.11: Distribution of the maximum absolute joint torque values on all joints for all the time steps.
Sparse Conv.-× 3
kernel 3
stride 1
MK. Max Pooling
kernel 3
stride 2
Residual Block
Residual Block
x 4, [16, 32, 32, 32] channels
Global Max Pooling
MK. FC
FC 256
1D Batch Norm
ELU
FC 256
2 !
! !"#
24
20
FC 256
ELU
FC 256
512
GRU 256
concat.
ℎ !"#
512
FC 256
ELU
x 2
FC 256
FC 256
ELU
FC 20
! !
RELU
Sparse Conv.
kernel 3
stride 1
Sparse Conv.
RELU
kernel 3
stride 1
add
Residual Block
Table 1 :
1Success rates (%) of policies tested on different dynamics distribution.θ = 0.1rad. DR: domain randomization and observation/action noise. X→Y: distill policy X into policy Y. The full table is in Table D.5.1
2
3
Table 2 :
2Performance of the student policy when the hand faces upward and downwardDataset
Upward
Downward (air) Downward (air, g-curr)
EGAD 91.96 ± 1.5
74.10 ± 2.3
YCB
81.04 ± 0.5
45.22 ± 2.1
67.33 ± 1.9
Table 3 :
3Zero-shot RNN policy transfer success rates (%) across datasets. U. (D.) means hand faces upward (downward). FS (RS) means using full-state (reduced-state) information.EGAD → YCB YCB → EGAD
U.FS
68.82 ± 1.7
96.41 ± 1.2
U.RS
59.64 ± 1.8
96.38 ± 1.3
D.FS
62.73 ± 2.2
85.45 ± 2.9
D.RS
55.30 ± 1.3
77.91 ± 2.1
Table 4 :
4Vision policy success rate (θ =
0.2 rad,dC = 0.01)
Object
Success rate (%)
025_mug
89.67 ± 1.2
065-d_cups
68.32 ± 1.9
072-b_toy_airplane
84.52 ± 1.4
073-a_lego_duplo
58.16 ± 3.1
073-c_lego_duplo
50.21 ± 3.7
073-e_lego_duplo
66.57 ± 3.1
Table C.4lists all the parameters we randomized in our experiments. We also add noise to the state observation and action commands as shown inTable C.4. For the vision experiments, we also added noise (various ways of data augmentation including point position jittering, color jiterring, dropout, etc.) to the point cloud observation in training and testing as explained in Section D.5.
Table C .
C4 list all the randomized parameters as well the state observation noise and action command
noise.
Table C .
C2: Hyperparameter Setup
Hyperparameter
Value Hyperparameter Value
Hyperparameter
Value
Num. batches
5
Entropy coeff.
0.
Num. pts sampled from W s
t
15000
Actor learning rate 0.0003 GAE parameter
0.95 Num. pts sampled from W g
5000
Critic learning rate 0.001
Discount factor
0.99
Num. envs
40000
Num. epochs
8
Episode length
300
Num. rollout steps per
policy update (MLP/RNN)
8/50
Value loss coeff.
0.0005 PPO clip range
0.1
c θ1
1
c θ2
800
c 3
0.1
θ
0.1
θ
0.1rad
c h1
0.05
h
0.02
h
0.04
c h2
800
Table C.3: Mesh Parameters
Parameter
Range
longest side of the bounding box of EGAD objects [0.05, 0.08]m
longest side of the bounding box of YCB objects
[0.05, 0.12]m
mass of each object
[0.05, 0.15]kg
No. of EGAD meshes
2282
No. of YCB meshes
78
No. of variants per mesh
5
Voxelization resolution
0.003 m
Table C.4: Dynamics Randomization and Noise
Parameter
Range
Parameter
Range
Parameter
Range
state observation
+U(−0.001, 0.001)
action
+N (0, 0.01)
joint stiffness
×E(0.75, 1.5)
object mass
×U(0.5, 1.5)
joint lower range +N (0, 0.01) tendon damping
×E(0.3, 3.0)
object static friction
×U(0.7, 1.3)
joint upper range +N (0, 0.01) tendon stiffness
×E(0.75, 1.5)
Table D .
D5: Success rates (%) of policies tested on different dynamics distribution.θ = 0.1rad. DR
stands for domain randomization and observation/action noise. X→Y: distill policy X into policy Y.
1
2
3
Exp. ID Dataset
State
Policy
Train without DR
Train with DR
Test without DR Test with DR Test with DR
A
EGAD
Full state
MLP
92.55 ± 1.3
78.24 ± 2.4
91.92 ± 0.4
B
RNN
95.95 ± 0.8
84.27 ± 1.0
88.04 ± 0.6
C
Reduced state
MLP→MLP
55.55 ± 0.2
25.09 ± 3.0
23.77 ± 1.8
D
MLP→RNN
85.32 ± 1.2
68.31 ± 2.6
81.05 ± 1.2
E
RNN→RNN
91.96 ± 1.5
78.30 ± 1.2
80.29 ± 0.9
F
YCB
Full state
MLP
73.40 ± 0.2
54.57 ± 0.6
66.00 ± 2.7
G
RNN
80.40 ± 1.6
65.16 ± 1.0
72.34 ± 0.9
H
Reduced state
MLP→MLP
34.08 ± 0.9
12.08 ± 0.4
5.41 ± 0.3
I
MLP→RNN
69.30 ± 0.8
47.38 ± 0.6
53.07 ± 0.9
J
RNN→RNN
81.04 ± 0.5
64.93 ± 0.2
65.86 ± 0.7
Table D .
D6: Success rates (%) of policies trained with hand facing downward and to reorient objects in the air. Due to the large number of environment steps required in this setup, we fine-tune the model trained without DR with randomized dynamics instead of training models with DR from scratch.1
2
3
Exp. ID Dataset
State
Policy
Train without DR
Finetune with DR
Test without DR Test with DR
Test with DR
K
EGAD
Full state
MLP
84.29 ± 0.9
38.42 ± 1.5
71.44 ± 1.3
L
RNN
82.27 ± 3.3
36.55 ± 1.4
67.44 ± 2.1
M
Reduced state
MLP→RNN
77.05 ± 1.6
29.22 ± 2.6
59.23 ± 2.3
N
RNN→RNN
74.10 ± 2.3
37.01 ± 1.5
62.64 ± 2.9
O
YCB
Full state
MLP
58.95 ± 2.0
26.04 ± 1.9
44.84 ± 1.3
P
RNN
52.81 ± 1.7
26.22 ± 1.0
40.44 ± 1.5
Q
RNN + g-curr
74.74 ± 1.2
25.56 ± 2.9
54.24 ± 1.4
R
Reduced state
MLP→RNN
46.76 ± 2.5
25.49 ± 1.4
34.14 ± 1.3
S
RNN→RNN
45.22 ± 2.1
24.45 ± 1.2
31.63 ± 1.6
T
RNN + g-curr→ RNN
67.33 ± 1.9
19.77 ± 2.8
48.58 ± 2.3
] |J k ij | | j ∈ [[1, T ]], k = [[1, 20000]]}, where [[a, b]] represents {x | x ∈ [a, b], x ∈ Z}.
• RandomTrans :
RandomTransTranslate the point cloud by [∆x, ∆y, ∆z] where ∆x, ∆y, ∆z are all uniformly sampled from [−0.04, 0.04]. • JitterPoints: Randomly sample 10% of the points. For each sampled point i, jitter its coordinate by [∆x i , ∆y i , ∆z i ] where ∆x i , ∆y i , ∆z i are all sampled from a Normal distribution N (0, 0.01) (truncated at −0.015m and 0.015m).
Table D .
D7: Vision policy success rates when the policy is trained and tested with/without data augmentation (θ = 0.2 rad,d C = 0.01) Object Without data augmentation (noise) With data augmentation (noise)025_mug
36.51 ± 2.8
89.67 ± 1.2
065-d_cups
79.17 ± 2.3
68.32 ± 1.9
072-b_toy_airplane
90.25 ± 3.7
84.52 ± 1.4
073-a_lego_duplo
62.16 ± 3.7
58.16 ± 3.1
073-c_lego_duplo
58.21 ± 3.9
50.21 ± 3.7
073-e_lego_duplo
76.57 ± 3.6
66.57 ± 3.1
0
20
40
60
80
100
Success rate
056_tennis_ball
053_mini_soccer_ball
055_baseball
057_racquetball
063-b_marbles
017_orange
058_golf_ball
054_softball
012_strawberry
018_plum
015_peach
014_lemon
013_apple
072-a_toy_airplane
016_pear
065-h_cups
065-d_cups
065-b_cups
065-a_cups
065-j_cups
065-e_cups
005_tomato_soup_can
070-a_colored_wood_blocks
063-a_marbles
061_foam_brick
062_dice
065-g_cups
065-i_cups
010_potted_meat_can
070-b_colored_wood_blocks
065-c_cups
002_master_chef_can
065-f_cups
024_bowl
073-b_lego_duplo
007_tuna_fish_can
072-e_toy_airplane
019_pitcher_base
073-e_lego_duplo
072-d_toy_airplane
073-f_lego_duplo
025_mug
073-a_lego_duplo
038_padlock
006_mustard_bottle
073-d_lego_duplo
072-c_toy_airplane
003_cracker_box
036_wood_block
073-c_lego_duplo
035_power_drill
071_nine_hole_peg_test
028_skillet_lid
008_pudding_box
022_windex_bottle
077_rubiks_cube
004_sugar_box
009_gelatin_box
072-b_toy_airplane
050_medium_clamp
021_bleach_cleanser
073-g_lego_duplo
051_large_clamp
026_sponge
011_banana
052_extra_large_clamp
029_plate
059_chain
040_large_marker
048_hammer
044_flat_screwdriver
037_scissors
042_adjustable_wrench
031_spoon
033_spatula
043_phillips_screwdriver
030_fork
032_knife
Object ID
99.91%
99.83%
99.83%
99.83%
99.74%
99.74%
99.57%
99.48%
99.40%
99.40%
99.40%
99.30%
99.22%
98.61%
98.52%
98.36%
97.34%
97.30%
96.78%
96.43%
96.35%
95.92%
95.91%
95.83%
95.33%
95.22%
94.70%
94.43%
94.36%
93.13%
92.33%
91.94%
91.90%
91.65%
91.23%
90.05%
89.79%
89.65%
89.23%
89.04%
89.04%
88.80%
88.64%
88.56%
88.26%
87.66%
86.96%
86.89%
86.70%
86.52%
85.91%
82.04%
81.02%
80.52%
80.08%
79.57%
79.30%
77.55%
77.07%
74.89%
74.26%
69.74%
67.69%
66.96%
61.03%
59.06%
58.09%
50.98%
50.21%
49.73%
40.43%
30.43%
29.10%
28.72%
28.66%
28.09%
25.95%
11.24%
We also experimented with a 3D sparse convolutional network based on ResNet18, and found that 3D IMPALA-Net works better.
More quantitative results on the MLP policies are available inTable D.5 in the appendix.
We change the mass of the YCB objects to be in the range of [0.3, 0.5]kg, and test π E R from the YCB dataset on these new objects. The success rate is around 75%.
2(2) where ∆h t = max(p b,z t − p o,z t , 0) and p b,z t is the height (z coordinate) of the Shadow Hand base frame, p o,z t is the height of the object,h is the threshold of the height difference. The objects have randomly initialized poses and are dropped onto the table.
AcknowledgmentsWe thank the anonymous reviewers for their helpful comments in revising the paper. We thank the members of Improbable AI lab for providing valuable feedback on research idea formulation and manuscript. This research is funded by Toyota Research Institute, Amazon Research Award, and DARPA Machine Common Sense Program. We also acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing HPC resources that have contributed to the research results reported within this paper.Appendix B Environment SetupB.1 State definitionThe full state space S E includes joint, fingertip, object and goal information detailed inTable B.1. To compute the angle difference between two quaternion orientations α 1 and α 2 , we first compute the difference rotation quaternion: β = α 1 α −1 2 . Then the angle difference (distance between two rotations) ∆θ is computed as the angle of β from the axis-angle representation of β.B.2 DatasetWe use two object datasets (EGAD and YCB) in our paper. To further increase the diversity of the datasets, we create 5 variants for each object mesh by randomly scaling the mesh. The scaling ratios are randomly sampled such that the longest side of the objects' bounding boxes l max lies in [0.05, 0.08]m for EGAD objects, and l max ∈ [0.05, 0.12]m for YCB objects. The mass of each object is randomly sampled from [0.05, 0.15]kg. When we randomly scale YCB objects, some objects become very small and/or thin, making the reorientation task even more challenging. In total, we use 11410 EGAD object meshes and 390 YCB object meshes for training.Appendix D Supplementary ResultsD.1 Hand faces upwardLearning curvesFigure D.5 shows the learning curve of the RNN and MLP policies on the EGAD and YCB datasets. Both policies learn well on the EGAD and YCB datasets. The YCB dataset requires much more environment interactions for the policies to learn. We can also see that using the full-state information can speed up the learning and give a higher final performance.Testing performance -Teacher The testing results inTable D.5show that both the MLP and RNN policies are able to achieve a success rate greater than 90% on the EGAD dataset (entries A1, B1)
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|
[] |
[
"Towards Similarity-Aware Time-Series Classification",
"Towards Similarity-Aware Time-Series Classification"
] |
[
"Daochen Zha ",
"Kwei-Herng Lai ",
"Kaixiong Zhou ",
"Xia Hu "
] |
[] |
[] |
We study time-series classification (TSC), a fundamental task of time-series data mining. Prior work has approached TSC from two major directions: (1) similaritybased methods that classify time-series based on the nearest neighbors, and (2) deep learning models that directly learn the representations for classification in a data-driven manner. Motivated by the different working mechanisms within these two research lines, we aim to connect them in such a way as to jointly model time-series similarities and learn the representations. This is a challenging task because it is unclear how we should efficiently leverage similarity information.To tackle the challenge, we propose Similarity-Aware Time-Series Classification (SimTSC), a conceptually simple and general framework that models similarity information with graph neural networks (GNNs). Specifically, we formulate TSC as a node classification problem in graphs, where the nodes correspond to time-series, and the links correspond to pair-wise similarities. We further design a graph construction strategy and a batch training algorithm with negative sampling to improve training efficiency. We instantiate SimTSC with ResNet as the backbone and Dynamic Time Warping (DTW) as the similarity measure. Extensive experiments on the full UCR datasets and several multivariate datasets demonstrate the effectiveness of incorporating similarity information into deep learning models in both supervised and semi-supervised settings. Our code is available at https://github.com/daochenzha/SimTSC.
|
10.1137/1.9781611977172.23
|
[
"https://arxiv.org/pdf/2201.01413v2.pdf"
] | 245,704,672 |
2201.01413
|
107cef3e8a6a6a6f128e87954812bc11df2696db
|
Towards Similarity-Aware Time-Series Classification
Daochen Zha
Kwei-Herng Lai
Kaixiong Zhou
Xia Hu
Towards Similarity-Aware Time-Series Classification
We study time-series classification (TSC), a fundamental task of time-series data mining. Prior work has approached TSC from two major directions: (1) similaritybased methods that classify time-series based on the nearest neighbors, and (2) deep learning models that directly learn the representations for classification in a data-driven manner. Motivated by the different working mechanisms within these two research lines, we aim to connect them in such a way as to jointly model time-series similarities and learn the representations. This is a challenging task because it is unclear how we should efficiently leverage similarity information.To tackle the challenge, we propose Similarity-Aware Time-Series Classification (SimTSC), a conceptually simple and general framework that models similarity information with graph neural networks (GNNs). Specifically, we formulate TSC as a node classification problem in graphs, where the nodes correspond to time-series, and the links correspond to pair-wise similarities. We further design a graph construction strategy and a batch training algorithm with negative sampling to improve training efficiency. We instantiate SimTSC with ResNet as the backbone and Dynamic Time Warping (DTW) as the similarity measure. Extensive experiments on the full UCR datasets and several multivariate datasets demonstrate the effectiveness of incorporating similarity information into deep learning models in both supervised and semi-supervised settings. Our code is available at https://github.com/daochenzha/SimTSC.
Introduction
Time-series classification (TSC) is a fundamental task of time-series data mining. Given a collection of timeseries with the attached labels, TSC aims to train a classifier to classify unseen time-series. With the increasing amount of temporal data available, TSC has broad applications, such as human activity recognition, health care, and cyber security [28].
Many research efforts have been devoted to TSC. Similarity-based (distance-based) methods are widely used [1]. The main idea is to combine a k-NN clas- * Department of Computer Science, Rice University. sifier with a similarity measure for classification. Dynamic Time Warping (DTW) [23,26], which calculates the optimal match between two time-series, is one of the most popular similarity measures. It is shown that DTW plus a 1-NN classifier can achieve reasonably good accuracy [27]. However, the similarity is often obtained in an unsupervised fashion followed by a simple k-NN classifier, which could be sub-optimal. Another promising research line is deep learning. Without any crafting in feature engineering, deep learning methods perform end-to-end training on the raw time-series and learn the representations. Recent studies suggest that convolutional layers, such as ResNet and Fully Convolutional Networks (FCN), significantly outperform DTW and achieve competitive performance to the state-of-the-art TSC algorithms on the UCR benchmarks [30,9].
While deep learning methods are simple and effective, they highly rely on the supervision of training labels for automatic representation learning; they thus often fall short when very few labels are given. Figure 1 compares the average ranks of ResNet and DTW plus 1-NN on the full 128 UCR datasets [7] with different numbers of provided training labels per class. While ResNet dominates DTW with sufficient training labels, it delivers unsatisfactory performance with very few labels. In contrast, DTW classifies the time-series by reasoning with pair-wise similarities instead of directly learning time-series representations, which could be less sensitive to the number of provided labels. Motivated by the different working mechanisms within these two research lines, we explore the possibility of connecting them in such a way as to jointly model time-series similarities and learn the representations.
However, it is non-trivial to achieve this goal due to the following challenges. First, it is unclear how we can incorporate similarity information into representation learning. The commonly used architectures, such as CNN and LSTM, cannot model similarity. While some papers have explored neural networks for time-series similarity learning [2,12], they learn the similarity in the first place and then simply apply the learned similarity to classification, which still relies on k-NN. Second, even though we can enable similarity in deep learning models, it is challenging to balance similarity information and the original representation learning. Incorporating too much similarity information may lead to indistinguishable representations with poor accuracy.
To address these challenges, we propose Similarity-Aware Time-Series Classification (SimTSC), a conceptually simple and general framework for incorporating similarity information into deep learning models. Motivated by graph neural networks (GNNs) in modeling node relationships [8,13,17], we reformulate TSC as a node classification problem. Specifically, we correspond each time-series to a node in a graph and each pair-wise similarity to a link between nodes. A graph convolution layer is then applied on the top of a backbone (e.g., ResNet) to jointly perform feature extraction and model time-series similarities. We instantiate SimTSC with ResNet and DTW, which are the representative deep learning model and similarity measure, respectively. Extensive experiments suggest that this simple design improves accuracy, particularly with very few labels. We make the following contributions:
• Explore the possibility of connecting the research efforts of similarity-based methods and deep learning models for TSC.
• Propose SimTSC, a simple and general framework that can combine any similarity measures with any deep learning models from the view of graphs.
• Design a graph construction strategy that focuses on the top neighbors for efficient aggregation of graph convolution. We also develop a batch training algorithm with negative sampling to enable the training of SimTSC on large datasets.
• Instantiate SimTSC with ResNet and DTW. We conduct extensive experiments on the full 128 UCR datasets and several multivariate datasets. SimTSC outperforms ResNet, DTW, and the stateof-the-art supervised and semi-supervised deep models significantly, particularly when very few labels are given. We also present extensive hyperparameter studies and ablations. Symbol Definition
x ∈ R A real-value in a time-series x ∈ R T A univariate time-series with length T X ∈ R M ×T An M -dimensional time-series with length T X ∈ R N ×M ×T A 3D matrix consisting of N multivariate time-series X train A set of training time-series y train
The labels corresponding to X train X test A set of testing time-series y test
The labels corresponding to X test X unlabeled A set of unlabeled time-series d(X1, X2)
The similarity (distance) of two time-series
D ∈ R N ×N The similarity (distance) matrix A ∈ R N ×N
The normalized adjacency matrix in graph
X ∈ R N ×M
The attribute information matrix in graph α A scaling factor K
The number of neighbors for each node
Preliminaries
We start with a problem description and then provide a background of time-series similarity measure, deep learning for TSC, and graph neural networks.
Problem Statement
We use lowercase alphabet, e.g., x ∈ R, to represent a scalar value, lowercase boldface letter, e.g.,
x = [x 1 , x 2 , ..., x T ] ∈ R T ,
to denote a vector of length T , uppercase boldface alphabet, e.g., Without loss of generality, we unify the above definitions by expanding dimension for univariate time-series. Specifically, we regard a univariate time-series x as a 1dimensional time-series X ∈ R 1×T , that is, a univariate time-series is a special case of multivariate time-series with M = 1. We will use the unified notation X to represent a time-series throughout the paper.
X = [x 1 , x 2 , ..., x M ] ∈ R M ×T ,
We formally describe the problem of TSC. Given some testing time-series X test = [X 1 , X 2 , ..., X N test ] and the labels y test = [y 1 , y 2 , ..., y N test ], where N test is the number of testing time-series, we aim to train a classifier that can predict the labels based on X test under one of the following settings:
• Supervised setting: The classifier is trained based on a training time-series dataset X train = [X 1 , X 2 , ..., X N train ] and its corresponding labels y train = [y 1 , y 2 , ..., y N train ], where N train is the number of training time-series.
• Inductive semi-supervised setting: In addition to X train and y train , the classifier can also access some unlabeled time-series X unlabeled , which does not overlap with X test .
• Transductive semi-supervised setting: In addition to X train , y train and X unlabeled , the classifier is exposed to testing time-series X test . Note that y test is not accessible in training. The above settings differ in how much unlabeled timeseries data the classifier can access. With more unlabeled data, the classifier could better learn the underlying data distributions and often achieve better accuracy.
Time-Series Similarity Measure
Given two time-series X 1 and X 2 , we aim at providing a distance d(X 1 , X 2 ), such that similar time-series tend to have smaller d(X 1 , X 2 ). Dynamic Time Warping (DTW) [23] is one of the most popular ones. The key idea is to calculate the optimal match between two time-series such that the sum of matched series has the smallest values. The troughs and peaks of the same pattern can be matched even if they are not perfectly synced up. DTW is a standard tool with many efficient implementations, such as the UCR Suite [26]. In this work, we adopt DTW as the similarity measure; one can also use other similarity measures under our framework.
Deep Neural Networks for TSC
Numerous deep learning models have been developed for timeseries classification [30,15,16,10]. In this work, we mainly focus on Residual Network (ResNet) since it is shown to have superior performance on the majority of UCR Time Series Classification Archive [9]. The network consists of multiple residual blocks. Each block consists of three 1D convolutional layers followed by batch normalization and a ReLU activation function, with shortcuts to enable a direct flow of the gradient:
H 1 = ReLU(BatchNorm(Conv1d(X))), (2.1) H 2 = ReLU(BatchNorm(Conv1d(H 1 ))), (2.2) H 3 = ReLU(BatchNorm(Conv1d(H 2 ))), (2.3)Ĥ = ReLU(H 3 + X), (2.4)
whereĤ is the output of residual block, Conv1d(·) denotes 1D convolutional, BatchNorm(·) is batch normalization, and ReLU(·) is ReLU activation function.
Graph Neural Networks
Graph neural networks (GNNs) have achieved great success in modeling node dependencies in graph [8,17,37]. To capture the node dependency, Graph Convolution Network (GCN) [17] performs joint learning of feature extraction and aggregation of neighboring nodes. Given a graph G = ( A, X), where A ∈ R N ×N is the normalized adjacency matrix, X ∈ R N ×M is the attribute information matrix, N is the number of nodes, and M is the feature dimension, a GCN layer performs feature aggregation of neighboring nodes with
(2.5)Ẑ = A XW,
where W denotes trainable parameters, andẐ denotes the output of the GCN layer. Similar to ResNet, we can stack multiple GCN layers with activation functions. While there are hundreds of GNNs, we adopt the basic GCN to make our contribution focused; one can also use other GNNs under our framework. Figure 2 shows an overview of Similarity-Aware Time-Series Classification (SimTSC), which consists of three modules: (1) a graph construction module that connects the time-series based on a similarity measure (e.g., DTW), (2) a backbone that performs feature extraction with deep neural networks (e.g., ResNet), and (3) a GNN module that aggregates the features of neighboring time-series (e.g., GCN). The graph construction is unsupervised so that it can flexibly adapt to all the three settings defined in Section 2.1.
Methodology
Graph Construction with Similarity
This subsection describes how we construct a graph based on a similarity measure of time-series, e.g., DTW.
Let X = [X 1 , X 2 , ..., X N ] denote all the accessible time-series. In the supervised setting, X is simply all the training data. In the semi-supervised settings defined in Section 2.1, the X consists of both labeled and unlabeled time-series. Based on the learned similarity measure d(·, ·), we can obtain a similarity matrix for X as (3.6)
D = d(X 1 , X 1 ) d(X 1 , X 2 ) · · · d(X 1 , X N ) d(X 2 , X 1 ) d(X 2 , X 2 ) · · · d(X 2 , X N ) . . . . . . . . . . . . d(X N , X 1 ) d(X N , X 2 ) · · · d(X N , X N ) .
Given the similarity matrix D, we construct the graph as follows. First, we introduce a scaling hyperparameter α ∈ [0, ∞) to control the importance of top neighbors. Specifically, let D ij denote the (i, j) th entry of D. The adjacency matrix A is obtained by
(3.7) A ij = 1 e αDij , ∀i, j,
where A ij denotes the (i, j) th entry of A. A larger α will give more importance to the top neighbors. When α = 0, each node will equally aggregate the features of all the nodes, and all the nodes will have indistinguishable features. When α → ∞, our framework reduces to the backbone itself since A reduces to a diagonal matrix.
Second, to filter out irrelevant neighbors, we sample the top-K neighbors for each node. Specifically, for each row a i in A, we only keep K entries with the largest weights and zero out the others, which leads to a sparse matrix. Finally, we normalize the adjacency
matrix with A ij = Aij j A ij , ∀i, j.
Joint Learning of Backbone and Graph Convolution Layers
This subsection introduces how we optimize the backbone and the GNN module.
The backbone takes all the accessible time-series X as inputs and produces an attribute information matrix X ∈ R N ×M , where N denotes the number of all the accessible time-series, and M denotes the feature dimension. Here, the output of the backbone is assumed to be flattened to be 1-dimensional. Given X and A, the GNN aggregates the node features and produces the final representation followed by a softmax layer for classification.
Let Z ∈ R N ×C denote the final output of the model, where C is the number classes, and Z train ∈ R N train ×C denotes the rows that have labels. The objective is to minimize the cross-entropy over the labeled time-series:
(3.8) L = − N train i=1 C j=1 Y train ij log Z train ij ,
where Y train ij denotes the j th element of one-hot encoded label of the i th labeled time-series. The weights of the backbone and the GNN module can be jointly optimized using gradient descent.
Batch Training with Negative Sampling
This subsection presents how we handle large datasets with batch training and how we use negative sampling to improve efficiency.
A naive training strategy is to put all the timeseries into the GPU memory in the first place and then train the network with backpropagation. However, this strategy has the following limitations. First, it cannot scale to large datasets with many time-series or very long time-series. Second, each time-series can only interact with a fixed number of neighbors since the constructed graph is static. However, the top-K dropping strategy may improperly drop some important connections, which leads to sub-optimal performance.
To overcome the above limitations, we propose a batch training strategy with negative sampling as follows. Given a batch size B, we sample B/2 labeled and B/2 unlabeled time-series, respectively, as a batch for training. The unlabeled time-series are "negatively sampled" in that we can usually access far more unlabeled data in real-world applications. Then, we construct a graph with this batch of time-series and update the model accordingly. The above training procedure is summarized in Algorithm 1. Here, X unlabeled will also cover testing data in the transductive setting 1 .
This design has three nice properties. First, it can scale to large datasets since we only need to put a batch of data into GPU memory in each update step. Second, each node will interact with more neighbors for aggregation since a new graph will be constructed in every randomly sampled batch. Third, compared with random sampling, negative sampling ensures that half of the data in each batch is labeled so that we can have sufficient learning signals to update the weights. In particular, if very few labels are available, random sampling may result in very few or even no labeled data in a sampled batch, which will reduce training efficiency.
At testing time, we similarly sample B/2 testing and B/2 non-testing time-series in each batch to construct the graph. In this way, we can aggregate some of the non-testing representations to enhance the representations of the testing samples.
Algorithm 1 Batch training of SimTSC with negative sampling 1: Input: Labeled time-series X train , unlabeled timeseries X unlabeled , similarity measure d(·, ·), batch size B 2: X ← {X train , X unlabeled } 3: Precompute the similarity matrix D for X based on Eq. 3.6 4: for each epoch do 5: for each batch X train batch of size B/2 in X train do 6:
Sample a batch X unlabeled batch of size B/2 from X unlabeled 7:
X batch ← {X train batch , X unlabeled batch } 8:
Obtain submatrix D batch from D with sampled indices 9:
Construct adjacency matrix A from D batch based on Eq. 3.7
10:
Compute normalized matrix A with A ij = Aij j A ij , ∀i, j 11:
Obtain model output Z with X batch and A
12:
Update the model weights based on Eq. 3.8
13:
end for 14: end for
Experiments
We evaluate SimTSC across various settings to answer the following research questions: RQ1: How does SimTSC compare with existing deep learning and similarity-based methods (Section 4.2)? RQ2: Can SimTSC also enhance other neural architectures, such as MLP and FCN (Section 4.3)? RQ3: How will the number of graph convolutional layers impact the performance (Section 4.4)? RQ4: How will graph structure impact the performance of SimTSC (Section 4.5)? RQ5: Is the proposed negative sampling strategy effective (Section 4.6)? RQ6: Can SimTSC also be applied to multivariate time-series classification tasks (Section 4.7)? RQ7: How does SimTSC learn the representations with similarity information (Section 4.8)?
Experimental Setup
As suggested in [7], we evaluate the performance on the full 128 UCR datasets 2 . We merge the original training and testing data to create new splits for all the datasets to simulate the three settings defined in Section 2.1. First, we randomly split 20% of the data as the hold-out set for testing purposes, denoted as X test . Second, we vary the number of training labels per class from the set {1, 5,10,15,20,25,30,35,40,45, 50} to test different levels of supervision and sample a subset from the remain-2 https://www.cs.ucr.edu/~eamonn/time_series_data_2018/ ing 80% of data as the training data to create few-shot settings, denoted as X train . Third, the time-series out of the above two splits will serve as another split of some unlabeled data, denoted as X unlabeled . We further consider four multivariate datasets: Character Trajectories [3], ECG [25], KickVsPunch [5], and NetFlow [5].
Metric. For UCR datasets, we follow previous work [7,9] and rank the algorithms on each dataset according to the mean accuracy and report the average ranks across the 128 datasets. We perform Wilcoxon signed-rank test with a significance level of 0.05. For the multivariate datasets, we report accuracies.
Baselines. For deep learning models, we consider four supervised architectures, including MLP, Fully Convolutional Network (FCN) [30], ResNet [11], and InceptionTime [10], and a state-of-the-art semisupervised framework TapNet [33]. For similaritybased methods, we include DTW with a 1 Nearest-Neighbor (1-NN) classifier [27]. We train SimTSC in supervised, inductive semi-supervised, and transductive semi-supervised settings, denoted as SimTSC-S, SimTSC-I, and SimTSC-T, respectively. Implementation Details. We use DTW as the similarity measure and ResNet as the backbone. The hyperparamters are set based on the accuracy on the training data, with the scaling factor α as 0.3, the number of neighbors K as 3, one GCN layer, the batch size as 128, and the number of epochs to be 500, across all the experiments. For a fair comparison, the backbones used in SimTSC are exactly the same as the baselines. We use the authors' implementations of InceptionTime 3 and TapNet 4 with the default hyperparameters. We run five times and report the average accuracy. More details of the neural architectures, hardware, and the dataset statistics are provided in supplementary materials.
Performance Comparison on Benchmarks
To study RQ1, we report the average ranks of SimTSC on the UCR datasets in Table 2, separately present TapNet in Table 3 because TapNet does not support batch training and suffers from memory explosion on 45 datasets. We make the following observations.
First, SimTSC outperforms the baselines with very few labels. Given 1, 5, or 10 labels per class, SimTSC trained in all the settings achieve better ranks than ResNet significantly. Given more labels, i.e, 15, 20, 25, 30 , or 35, SimTSC also beats ResNet consistently. An interesting observation is that SimTSC-S performs well even though it only uses very few time-series to construct the graph. A possible explanation is that the graph may serve as a regularizer for ResNet to help alleviate the overfitting issue. Second, similarity information is less effective when we have sufficient labels. Given 40, 45, or 50 labels, ResNet and SimTSC achieve similar ranks. Nevertheless, the results suggest that SimTSC can still deliver competitive performance with enough labels, which shows the flexibility of SimTSC. Third, SimTSC tends to perform better with more unlabeled data. With very few exceptions, SimTSC-T > SimTSCI > SimTSC-S. This is because we can construct a better graph to capture the underlying data distributions with more unlabeled time-series. Fourth, SimTSC consistently outperforms TapNet across all different numbers of labels when both evaluated under transductive semi-supervised setting. The result again verifies the effectiveness of SimTSC.
Results on Other Neural Architectures
To investigate RQ2, we show the results of applying SimTSC on MLP and FCN on the UCR datasets in Table 4. SimTSC can also enhance these two neural architectures. In particular, we observe significant performance gains when applying SimTSC on MLP.
Analysis of Graph Convolutional Layers
To answer RQ3, we report the ranks using more GCN layers. We focus on the transductive setting with 10 and 20 labels per class ( Figure 3). SimTSC achieves the best performance with only one GCN layer. We speculate that this is because the graph is constructed based on the pair-wise similarity, which can fully capture the relationship between each pair of time-series. As such, stacking more GCN layers will not introduce more information but instead makes the model more susceptible to over-smoothing [35,36]. Nevertheless, stacking more GCN layers could help in larger time-series datasets by computing a submatrix and leverage multiple GCN layers to capture the multihop connections, which is deferred as our future work.
Analysis of Graph Structure
For RQ4, we vary the hyperparameters of graph construction. In Figure 4, we vary α from the set {0.1, 0.3, 0.5, 0.7, 0.9, 1.1} with K fixed to be 3. We observe a significant performance drop when α = 0.1. This is because when α → 0, each node will aggregate more information from the neighbors, and the resulting representation will become less distinguishable. There is also a performance drop when α = 1.1. A larger α will reduce the similarity information in the model, which suggests that incorporating similarity information is indeed helpful. In Figure 5, we vary K from the set {1, 2, 3, 4, 5, 6} with α fixed to be 0.3. The best performance is achieved with K = 2. When K = 1, our framework reduces to backbone itself since there will be only self-connections in the graph. When K becomes larger, the performance also drops since aggregating more information from the neighbors makes the representations less distinguishable. Overall, we find that incorporating an appropriate amount of similarity information lead to the best performance.
Ablation Study
For RQ5, we consider two ablations to study the effectiveness of negative sampling. First, we consider a variant that uses random sampling for batch training. Second, we consider a variant that puts all the time-series into GPU memory without batch training. Unfortunately, this variant will explode the GPU memory for 7 of the datasets so that only 121 datasets are reported. We perform a grid-search of K and find that K = 10 achieves the best performance. Table 5 summarizes the results. First, we observe that negative sampling outperforms random sampling significantly. Second, the variant without batch training does not perform well. A possible explanation is that the graph is pre-constructed so that each time-series can only interact with a limited number of connected neighbors, which may lead to sub-optimal performance.
Results on Multivariate Datasets
For RQ6, we evaluate SimTSC on multivariate time-series classification tasks in Table 6. First, SimTSC outperforms ResNet and DTW with very few labels. Given 5, 10, or 15 labels, at least one of the SimTSC variants achieves the best performance. Second, when more labels are given, the similarity information helps on some datasets but worsens some others' performance. SimTSC delivers poor performance on ECG in the transductive setting given 50 labels, while we observe a consistent improvement on CharacterTrajectories and NetFlow.
Visualization of Learned Representations
To answer RQ7, we conduct a case study on the Coffee dataset from UCR Archive. Figure 6 visualizes the learned representations of SimTSC and ResNet as well as the constructed graphs. We observe that there is an overlap between the two classes' representations learned by ResNet. Thus, ResNet cannot distinguish those overlapped time-series and only gives 83% accuracy. Whereas, the representations learned by SimTSC form clear clusters so that SimTSC achieves 100% accuracy. A possible reason is that the overlapped time-series tend to be close to those in the same class in terms of DTW, and thus their representations are corrected by aggregation. SimTSC achieves better accuracy by jointly performing feature extraction and aggregation.
Related Work
Deep Learning for TSC. Prior deep learning models for TSC can be mainly grouped into (1) learning representations of time-series in an unsupervised manner and then applying a classifier to the learned representations [22], and (2) training a classifier in an end-toend fashion [11,4,15]. However, the existing models mainly focus on feature extraction but cannot capture similarity information. Although [32] has introduced GNNs to capture time-series dependencies, they focus on time-series forecasting and can only model the dependencies among multivariate time-series. Whereas, we adopt GNNs to jointly perform feature extraction and model the time-series similarities for TSC. Similarity-Based TSC Methods. Similaritybased methods first obtain time-series similarities and then use a k-NN classifier for classification [24,31]. Recently, some similarity learning approaches have been proposed to learn the similarities [2,12]. However, they often rely on a separate procedure for classification. In contrast, we connect time-series similarity with deep models under a unified framework.
Leveraging Unlabeled Data in Time-Series. Prior work has explored semi-supervised learning [29], domain adaptation [20], and anomaly detection [21,34,18,19] on the unlabeled data, most of which do not target deep models. Recently, [14] proposes a deep TSC method with auxiliary forecasting tasks. [33] augments networks with task-adaptive projection. However, they do not support batch training. Unlike the previous work, our framework brings benefits to not only semisupervised setting but also the supervised setting.
Conclusions and Future Work
This work explores connecting the research efforts of time-series similarity measuring and deep learning for TSC. To jointly model feature extraction and similarity information, we formulate TSC as a node classification problem in graphs and introduce GNNs on the top of a backbone to enable end-to-end training. We instantiate our framework with ResNet and DTW with extensive experiments on the full 128 UCR datasets and several multivariate datasets. Experimental results suggest that incorporating similarity information can improve deep models significantly. In the future, we will investigate differentiable DTW [6] for graph construction. [7]. These datasets are collected from various domains, including Image, Spectro, Sensor, Simulated, Device, Motion, ECG, Traffic, EOG, HRM, Trajectory, and Hemodynamics. These datasets also have diverse data sizes, numbers of classes, and lengths. For example, Elec-tricDevices, one of the largest datasets in the archive, has 16,637 time-series in total, while BeetleFly only has 40 time-series. Similarly, ShapesAll has 60 classes, while many other datasets have only 2 or 3 classes. The timeseries length can also be up to 2,709 and can also be as short as 60. Moreover, 11 of the datasets have varying lengths for different time-series. Thus, the UCR datasets provide a rigorous test for time-series classification.
Multivariate Time-Series Datasets. We focus on four multivariate datasets from different domains with varying characteristics.
• Character Trajectories:
is a handwriting dataset captured at 200Hz by a WACOM tablet. The three dimensions are the positions of the xaxis, the y-axis, and the pen tip force. The data has been numerically differentiated, and Gaussian smoothed, with a sigma value of 2.
• ECG: traces the electrical activity recorded during heartbeats. The two classes are the normal heartbeat and the myocardial infarction.
• KickVsPunch: is a motion capture dataset collected by CMU-MOCAP. Each dimension is a motion marker. The two classes are the two actions, i.e., the kick and the punch.
• NetFlow: is the traffic flow of websites.
We summarize the statistics of the above datasets in Table 4. Third, we separate out the last 20% of the shuffled indices as the hold-out testing time-series. Fourth, given the target number of labels per class, we iterate over the first 80% of the shuffled indices sequentially until we find enough number of labeled data for each class. It is possible that we can not find enough labeled data for some classes even after iterating all the indices. In this case, we simply use as many labels as we can. For example, for a class A and a target number 10, if the number of class A data in the first 80% of the shuffled indices is only eight, we simply use eight training data for class A. The remaining data in the first 80% of the shuffled indices will serve as the third split (unlabeled data). The above three splits are used to simulate the supervised, inductive semi-supervised, and transductive semi-supervised settings defined in Section 2.1. The performance will be evaluated on the hold-out testing set. The above splitting procedure is applied to all the univariate and multivariate time-series data. Since data splitting may significantly affect the performance, particularly when we have very few labels, we run each experiment 5 times on different splits.
A.4 Dynamic Time Warping Dynamic Time
Warping (DTW) is s standard algorithm for measuring the similarity between two time-series. The main idea of DTW is to calculate the optimal match between two time-series such that the sum of matched series has the smallest values. In this work, the DTW is computed based on a Python wrapper of The UCR Suite 6 . This suite provides a highly efficient C++ implementation of DTW via dynamic programming. For all the datasets, we set the size of the warping window to be 100. If either of the two time-series is shorter than 100, we use the shortest time-series length.
For multivariate datasets, we compute independent DTW. Specifically, we first compute the DTW for each pair of univariate time-series and then sum them up to represent the distance between two multivariate timeseries. We have uploaded the pre-computed DTW to Google Drive for reproducibility.
Since most of the datasets are small, it will not take much time to compute the full similarity matrix. For most of the datasets, the computation of DTW can be finished in minutes. For some larger datasets, it takes at most a day using one CPU core.
A.5 Neural Architecture of Backbone We use PyTorch to implement all the neural networks. For most of the experiments, we use ResNet as the backbone. The network consists of three residual blocks. Each residual block consists of three 1-D convolution layers. The kernel sizes of the three convolution layers are 7, 5, and 3, respectively. After each convolution layer, we use a 1-D batch normalization layer to stabilize training, followed by a ReLU activation function. The number of channels is set to be 64 for all the convolution layer. We find that using more channels will lead to unsatisfactory performance with very little training data due to overfitting issue. A skip connection is added in each block to enable direct flow to alleviate the gradient vanishing issue. The three blocks are stacked sequentially to perform feature extraction. To reduce the feature dimension, we add a global average pooling layer to the last residual block's output. For our SimTSC, we directly use the global average pooling layer's output as the extracted features. For the ResNet baseline, these features will be further processed by a fully-connected layer with a softmax activation for classification purposes. The weights of the networks are initialized with the default initializers in PyTorch. Figure 7 summarizes the neural architecture of ResNet. For the MLP baseline, we use four fully-connected layers with a hidden size of 500. We use a ReLU activation function after each fully-connected layer followed by a dropout layer to avoid overfitting. The dropout rates are set to be 0.1, 0.2, 0.2, 0.3 for the four layers. We flatten the time-series into one dimension so that it can be directly used in MLP. Different from convolution layers, MLP can not capture temporal information, which leads to unsatisfactory performance. For the FCN baseline, we use three 1-D convolution layers. Each convolution layer is followed by a batch normalization layer. The kernel sizes are set to be 7, 5, and 3. Similar to ResNet, the number of channels is set to 64. For both MLP and PyTorch, we use the default initializers in PyTorch to initialize the weights.
Note that, for all the experiments, we use exactly the same architecture for the backbone of SimTSC and the baseline. The only difference of SimTSC is adding a GCN layer on the top of the backbone. Thus, the comparison is fair. The performance gain is solely attributed to modeling the similarity information.
A.6 Graph Convolution Layers The graph convolution layers are implemented based on the standard implementation of GCN 7 . Specifically, each GCN layer takes node features and an adjacency matrix as the input, where the adjacency matrix is a sparse tensor. Then it aggregates the neighbors' features by performing matrix multiplication of the input features, weights, and the adjacency matrix. Finally, a bias term is added to the obtained features. For multiple GCN layers, we add a dropout layer after each GCN layer to avoid overfitting.
The adjacency matrix used in the graph convolution layers is constructed as follows. We maintain a full precomputed similarity matrix in the memory. In each update step, we sample a batch of indices for training. We then use the sampled index to obtain a submatrix of the full similarity matrix. This submatrix will only contain the indices in this batch. Further, we rank each row's values in ascending order and only keep the top-K similar neighbors for each row. We finally use the top-K neighbors to construct the graph, which is represented by a sparse matrix. The above constructing procedure is efficient since we only need to take care of a batch of indices instead of all the indices.
A.7 Hyperparameter Settings
We summarize the hyperparmeters of graph, optimizer, and how we train SimTSC and all the baselines as follows.
• Graph Construction: We set the scaling factor α = 0.3 and number neighbors for each node K = 3.
• Graph Convolution: We use one GCN layer for most of the experiments. For multiple GCN layers, the feature dimension is set to be 64, and the dropout rate is set to be 0.5.
• Optimizer: We use Adam optimizer. The learning rate is set to be 0.0001. The is set to be 10 −8 .
• Training Procedure: For SimTSC and all the baselines, we use the model that achieves the best performance on the training data for evaluation. Specifically, we calculate the accuracy based on the training data after each training epoch and store the model's weights with the highest accuracy. Then the stored weights will be reloaded for evaluation purposes. For all the models, we train 500 epochs. While validating on the training data may lead to overfitting, we find in practice that it works better than separating a validation set from the training data. This is because a separated validation set will be too small to perform a meaningful evaluation.
A.8 Hardware and Software Descriptions
We conduct all the experiments on a server with two AMD EPYC 7282 16-Core processors, four GeForce RTX 3090 GPUs, and 252 GB memory. We use Ubuntu 18.04.5 LTS system and PyTorch 1.7.0.
Figure 1 :
1{daochen.zha,khlai,Kaixiong.Zhou,Xia.Hu}@rice.edu Average ranks (↓) of ResNet and DTW on the full 128 UCR datasets, where different numbers of labels per class is given (see Section 4.1 for more details).
Figure 2 :
2An overview of SimTSC framework. The graph is constructed based on the pair-wise similarities (e.g., DTW distances) of the time-series. Each time-series is processed by a backbone (e.g., ResNet) for feature extraction. The GNN module will aggregate the features and produce the final representations for classification.
Figure 3 :
3Impact of the number of GCN layers.
Figure 4 :
4Impact of α.
Figure 5 :
5Impact of K.
Figure 6 :
6Learned representations of ResNet and SimTSC on Coffee with 56 time-series, two classes marked in blue and green, respectively, and only one time-series labeled in each class (circled in red).
Figure 7 :
7Neural architecture of ResNet. Each layer is followed by a batch normalization layer and a ReLU activation.
Table 1 :
1Main symbols and definitions.
to denote a matrix consisting of M vectors, where each vector can have a different length, and calligraphic font, e.g.,X = [X 1 , X 2 , ..., X N ] ∈ R N ×M ×T , to denote a 3D matrix.We summarize the main symbols inTable 1. We first give formal definitions of time-series. series x of length T is represented as a vector [x 1 , x 2 , ..., x T ]. Definition 2. (Multivariate Time-Series) An M -dimensional time-series X consists of M univariate time-series [x 1 , x 2 , ..., x M ].Definition 1. (Univariate Time-Series) A
uni-
variate time-
Table 2 :
2Average ranks (↓) of SimTSC and baselines on the full 128 UCR datasets with different numbers of training labels per class. †, , and to denote the cases where SimTSC-S, SimTSC-I, and SimTSC-T are significantly better than the other algorithms w.r.t. the Wilcoxon signed rank test (p < 0.05), respectively.Algorithm
Labels 1
5
10
15
20
25
30
35
40
45
50
DTW
3.776
4.163
4.465
4.738
4.824 †
5.048 †
4.965 †
5.160 †
5.309 †
5.199 †
5.211 †
MLP
5.504 †
5.496 †
5.438 †
5.309 †
5.316 †
5.256 †
5.367 †
5.477 †
5.195 †
5.402 †
5.348 †
FCN
4.630
4.310
4.383
4.508
4.723 †
4.803 †
4.699 †
4.910 †
4.773 †
4.883 †
4.852 †
ResNet
4.846 †
4.857 †
4.617 †
4.047
4.449
4.039
4.102
4.090
4.086
3.840
3.895
InceptionTime 5.484 †
5.302 †
5.438 †
5.434 †
5.215 †
5.145 †
5.168 †
4.914 †
4.941 †
5.066 †
5.039 †
SimTSC-S
4.224
4.278
4.074
4.277
4.141
4.044
4.148
3.988
3.887
3.918
4.047
SimTSC-I
3.724
3.817
3.793
3.836
3.746
4.031
3.762
3.734
3.852
3.867
3.797
SimTSC-T
3.811
3.778
3.781
3.852
3.586
3.632
3.789
3.727
3.957
3.824
3.812
Table 3 :
3Average ranks of TapNet and SimTSC on 83 datasets, on which TapNet does not suffer from memory explosion. suggests SimTSC is significantly better.Algorithm
Labels
5
10
15
20
25
TapNet
1.645
1.524
1.548
1.536
1.530
SimTSC
1.355
1.476
1.452
1.464
1.470
Table 4 :
4Average ranks of using MLP and FCN as backbones. †, , and suggest SimTSC-S, SimTSC-I, and SimTSC-T are significantly better, respectively.Algorithm
Labels 10
20
30
40
MLP
3.137 †
2.906 †
2.879 †
2.898 †
SimTSC-S with MLP backbone 2.738
2.633
2.618
2.578
SimTSC-I with MLP backbone 2.078
2.258
2.414
2.258
SimTSC-T with MLP backbone 2.047
2.203
2.090
2.266
FCN
2.672
2.695
2.641
2.563
SimTSC-S with FCN backbone 2.523
2.492
2.504
2.438
SimTSC-I with FCN backbone 2.500
2.590
2.539
2.547
SimTSC-T with FCN backbone 2.309
2.223
2.316
2.453
Table 5 :
5Comparison of negative and random sampling on the full UCR datasets (top). Comparison with the variant without batch training on 121 UCR datasets (bottom), on which w/o batch training does not suffer from GPU memory explosion. suggests negative sampling is significantly better than the other algorithms.Algorithm
Labels 10
20
30
40
random sampling
1.637
1.609
1.547
1.582
negative sampling
1.363
1.391
1.453
1.418
random sampling
2.152
1.979
1.868
1.926
w/o batch training
2.068
2.277
2.331
2.322
negative sampling
1.780
1.744
1.802
1.752
Table 6 :
6Classification accuracy on multivariate time-series datasets across different numbers of training labels per class. For KickvsPunch, we only report the performance up to 15 labels since there are less than 15 labels per class. TapNet is not reported on Character Trajectories and Netflow due to memory explosion. 418±.052 .456±.046 .484±.058 .618±.049 .642±.036 .657±.024 .678±.014 .675±.036 .681±.018 .681±.Dataset
Algorithm
Labels
5
10
15
20
25
30
35
40
45
50
Character
Trajectories
DTW
.847±.014 .881±.005 .895±.009 .900±.014 .908±.014 .907±.013 .906±.010 .906±.007 .909±.010 .913±.008
ResNet
.834±.024 .898±.017 .920±.010 .937±.010 .939±.008 .941±.009 .949±.008 .956±.011 .958±.007 .958±.007
InceptionTime .883±.010 .939±.007 .947±.006 .968±.006 .964±.003 .974±.005 .979±.003 .978±.005 .979±.001 .986±.003
TapNet
-
-
-
-
-
-
-
-
-
-
SimTSC-S
.894±.020 .939±.009 .949±.007 .947±.017 .964±.011 .975±.003 .977±.011 .975±.004 .981±.007 .982±.005
SimTSC-I
.914±.012 .944±.009 .951±.015 .953±.012 .969±.011.978±.006.981±.007 .979±.005 .977±.008 .980±.003
SimTSC-T
.903±.014 .946±.005.957±.011 .964±.009 .967±.012 .973±.009 .976±.009 .981±.006 .983±.008 .986±.004
ECG
DTW
.605±.124 .670±.086 .740±.112 .755±.103 .805±.043 .825±.050 .805±.053 .800±.057 .805±.053 .800±.057
ResNet
.745±.048 .795±.037 .805±.058 .800±.079 .860±.030.855±.048 .850±.052 .855±.029 .830±.037 .870±.029
InceptionTime .750±.045 .805±.033 .785±.020 .800±.037 .820±.037 .830±.043 .825±.016 .850±.027 .855±.0010 .850±.016
TapNet
.770±.043 .780±.012 .755±.025 .795±.048 .810±.037 .795±.029 .785±.025 .815±.037 .830±.019 .845±.024
SimTSC-S
.795±.043 .810±.020 .855±.040.840±.051 .830±.056 .840±.020 .860±.041 .825±.047 .830±.071 .860±.025
SimTSC-I
.790±.062 .765±.072 .830±.070 .730±.159 .740±.087 .800±.091 .830±.048 .750±.052 .790±.108 .735±.108
SimTSC-T
.810±.041.815±.046 .770±.108 .815±.115 .730±.118 .745±.075 .745±.099 .780±.051 .775±.071 .710±.101
KickvsPunch
DTW
.433±.082 .433±.082 .433±.082
-
-
-
-
-
-
-
ResNet
.667±.183 .833±.149 .833±.183
-
-
-
-
-
-
-
InceptionTime .667±.000 .533±.125 .567±.226
-
-
-
-
-
-
-
TapNet
.700±.125 .767±.082 .733±.013
-
-
-
-
-
-
-
SimTSC-S
.733±.200 .767±.133 .867±.125
-
-
-
-
-
-
-
SimTSC-I
.700±.125 .833±.105 .800±.125
-
-
-
-
-
-
-
SimTSC-T
.600±.133 .767±.133 .767±.082
-
-
-
-
-
-
-
NetFlow
DTW
.611±.016 .559±.128 .607±.132 .595±.118 .546±.103 .568±.125 .523±.154 .481±.203 .503±.217 .504±.214
ResNet
.613±.074 .714±.063 .749±.022 .763±.038 .739±.058 .767±.050 .769±.054 .767±.049 .787±.026 .797±.039
InceptionTime .015
TapNet
-
-
-
-
-
-
-
-
-
-
SimTSC-S
.519±.108 .720±.071 .705±.055 .709±.089 .738±.082 .786±.036 .765±.091 .790±.045 .784±.063 .799±.047
SimTSC-I
.766±.043 .788±.036 .689±.139 .776±.042 .731±.084 .755±.104 .834±.037 .798±.066 .810±.065 .839±.035
SimTSC-T
.769±.052.805±.035.785±.101 .766±.095 .745±.092.825±.029 .801±.065 .827±.059 .847±.023 .852±.028
Table 7 :
7Statistics of the multivariate time series datasets.Dataset
Attribute Data SizeDimension# Classes Length
Character Trajectories 2858
3
20
109-205
ECG
200
2
2
39-152
KickvsPunch
26
62
2
274-841
NetFlow
1337
2
4
50-997
A Experimental Details
A.1 Datasets We use both univariate datasets and
multivariate datasets in our experiments. We provide
detailed descriptions below.
Univariate Time-Series Datasets. The experi-
ments are conducted on the full 128 datasets from the
UCR Time Series Classification Archive 5
Specifically, we merge the original training and testing data to create new splits as follows. First, we merge the Numpy arrays of training and testing data, where the training data is put before the testing data. Second, we randomly shuffle all the indices.5 https://www.cs.ucr.edu/~eamonn/time_series_data_2018/
A.2 Data Preprocessing For both univariate and
multivariate time-series data, we store them as 3-
dimensional Numpy arrays, where the first dimension is
the number of time-series, the second dimension is the
number of time-series (for univariate time-series, there
is only one time-series), and the third dimension is the
length of the time-series. We impute missing values with
zeros for the datasets with varying lengths to make the
time-series have the same length. All the time-series are
z-normalized before feeding into the models.
A.3 Data Splitting The original splits in UCR
datasets are very diverse. For example, ElectricDevices
has 8,926 training samples, while ECGFiveDays only
has 23 training samples. This makes it difficult to un-
derstand how the models will behave under different
amounts of supervision. Specifically, when an algorithm
performs well, it is hard to tell how it performs when we
have very few labels and how it will perform when we
have enough labels. Motivated by few-shot learning, we
simulate the settings where different numbers of labels
per class are given.
In the supervised setting, we only sample labeled time-series.
https://github.com/hfawaz/InceptionTime 4 https://github.com/xuczhang/tapnet
https://www.cs.ucr.edu/~eamonn/UCRsuite.html Tim e-Ser ies 7x1 conv, 64 5x1 conv, 64 3x1 conv, 64 7x1 conv, 64 5x1 conv, 64 3x1 conv, 64 7x1 conv, 64 5x1 conv, 64 3x1 conv, 64 Global Aver age Pooling
https://github.com/tkipf/pygcn
AcknowledgementsThe work is, in part, supported by NSF (#IIS-1849085, #IIS-1900990, #IIS-1750074). The views and conclusions in this paper are those of the authors and should not be interpreted as representing any funding agencies.
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"Conduction spectroscopy of a proximity induced superconducting topological insulator"
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"M P Stehno \nFaculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands\n",
"N W Hendrickx \nFaculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands\n",
"M Snelder \nFaculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands\n",
"T Scholten \nFaculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands\n",
"Y K Huang \nInstitute of Physics\nWaals -Zeeman Institute\nUniversity of Amsterdam\nThe Netherlands\n",
"A Brinkman [email protected] \nFaculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands\n\nInstitute of Physics\nWaals -Zeeman Institute\nUniversity of Amsterdam\nThe Netherlands\n"
] |
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"Faculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands",
"Faculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands",
"Faculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands",
"Faculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands",
"Institute of Physics\nWaals -Zeeman Institute\nUniversity of Amsterdam\nThe Netherlands",
"Faculty of Science and Technology and MESA+ Institute for Nanotechnology\nUniversity of Twente\nThe Netherlands",
"Institute of Physics\nWaals -Zeeman Institute\nUniversity of Amsterdam\nThe Netherlands"
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The combination of superconductivity and the helical spin-momentum locking at the surface state of a topological insulator (TI) has been predicted to give rise to p-wave superconductivity and Majorana bound states. The superconductivity can be induced by the proximity effect of a an s-wave superconductor (S) into the TI. To probe the superconducting correlations inside the TI, dI/dV spectroscopy has been performed across such S-TI interfaces. Both the alloyed Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 and the stoichiometric BiSbTeSe 2 have been used as three dimensional TI. In the case of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , the presence of disorder induced electron-electron interactions can give rise to an additional zero-bias resistance peak. For the stoichiometric BiSbTeSe 2 with less disorder, tunnel barriers were employed in order to enhance the signal from the interface. The general observations in the spectra of a large variety of samples are conductance dips at the induced gap voltage, combined with an increased subgap conductance, consistent with p-wave predictions. The induced gap voltage is typically smaller than the gap of the Nb superconducting electrode, especially in the presence of an intentional tunnel barrier. Additional uncovered spectroscopic features are oscillations that are linearly spaced in energy, as well as a possible second order parameter component. arXiv:1707.03640v1 [cond-mat.supr-con]
|
10.1088/1361-6641/aa7f88
|
[
"https://arxiv.org/pdf/1707.03640v1.pdf"
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1707.03640
|
d22f639022e5d04e9b6d6e105710fa090cdf1e31
|
Conduction spectroscopy of a proximity induced superconducting topological insulator
M P Stehno
Faculty of Science and Technology and MESA+ Institute for Nanotechnology
University of Twente
The Netherlands
N W Hendrickx
Faculty of Science and Technology and MESA+ Institute for Nanotechnology
University of Twente
The Netherlands
M Snelder
Faculty of Science and Technology and MESA+ Institute for Nanotechnology
University of Twente
The Netherlands
T Scholten
Faculty of Science and Technology and MESA+ Institute for Nanotechnology
University of Twente
The Netherlands
Y K Huang
Institute of Physics
Waals -Zeeman Institute
University of Amsterdam
The Netherlands
A Brinkman [email protected]
Faculty of Science and Technology and MESA+ Institute for Nanotechnology
University of Twente
The Netherlands
Institute of Physics
Waals -Zeeman Institute
University of Amsterdam
The Netherlands
Conduction spectroscopy of a proximity induced superconducting topological insulator
The combination of superconductivity and the helical spin-momentum locking at the surface state of a topological insulator (TI) has been predicted to give rise to p-wave superconductivity and Majorana bound states. The superconductivity can be induced by the proximity effect of a an s-wave superconductor (S) into the TI. To probe the superconducting correlations inside the TI, dI/dV spectroscopy has been performed across such S-TI interfaces. Both the alloyed Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 and the stoichiometric BiSbTeSe 2 have been used as three dimensional TI. In the case of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , the presence of disorder induced electron-electron interactions can give rise to an additional zero-bias resistance peak. For the stoichiometric BiSbTeSe 2 with less disorder, tunnel barriers were employed in order to enhance the signal from the interface. The general observations in the spectra of a large variety of samples are conductance dips at the induced gap voltage, combined with an increased subgap conductance, consistent with p-wave predictions. The induced gap voltage is typically smaller than the gap of the Nb superconducting electrode, especially in the presence of an intentional tunnel barrier. Additional uncovered spectroscopic features are oscillations that are linearly spaced in energy, as well as a possible second order parameter component. arXiv:1707.03640v1 [cond-mat.supr-con]
Introduction
The proximity effect between an s-wave superconductor and a three-dimensional topological insulator (TI) or nanowire with strong spin-orbit coupling has been predicted to create a stable zero-energy mode that may serve as the building block of a quantum computer, the Majorana zero-energy mode [1,2,3,4,5,6,7,8,9,10,11,12,13]. The presence of peaks in the conductance at zero-energy has been interpreted as experimental evidence for the existence of Majorana modes in such systems [14,15,16,17].
In order to create a Majorana zero-energy mode, theory requires a dominant (induced) p-wave component in the superconducting order parameter of the material. It is also known from theoretical studies that a zero-bias conductance peak and conductance dips at the characteristic gap energy are typical features of (dominant) p-wave superconductivity [18,13,19,20,21,22,23,24,25]. Effective s-wave pairing with subdominant p-wave admixture has been predicted [25,26,27] at the interface between a topological insulator and an s-wave superconductor. However, as was shown by Tkachov in Ref. [26], s and p-wave correlations have a different spatial dependence suggesting that device geometry may play a crucial role in observing p-wave features in topological insulator/superconducting devices. It is, therefore, desired to investigate conductance spectra in proximized topological materials.
To study the proximity effect between an s-wave superconductor (Nb) and a 3D TI with dominant surface transport, we performed differential conductance measurements on Nb/TI/Au devices, where the TI is either alloyed Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 or stoichiometric BiSbTeSe 2 . This generation of topological insulators has been shown before to have negligible bulk conductance in thin flakes [28,29]. The large difference in topological surface state mobility between the two types of TI provides different spectroscopic features such as a resistance peak due to electron-electron interactions in the disordered case and conductance oscillations in the cleaner, stoichiometric case. However, we reveal that the underlying spectra are very much alike. The general observations in the spectra of a large variety of samples are conductance dips at the induced gap voltage, combined with an increased sub-gap conductance, consistent with p-wave predictions. The induced gap voltage is typically smaller than the gap of the Nb superconducting electrode, especially in the presence of an intentional tunnel barrier.
2. Proximity effect into disordered Bi 1.5 Sb 0.5 Te 1.7 Se 1.3
Device fabrication
All devices were prepared by exfoliation from a single crystal of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , grown by the Bridgeman method, which was found to exhibit vanishing bulk conductance in earlier experiments [30,31]. The flakes were transferred from the Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 crystal to a Si/SiO 2 substrate by means of mechanical exfoliation. By means of e-beam lithography (EBL) the Au electrode is defined on the flake. Thereafter we perform a low voltage 15 second Ar etching step to avoid large damaging of the surface followed by sputtering in-situ a 3 nm Ti layer and a 60 nm Au layer. The Ti layer is grown for better adhesion of the Au on the flake. Next, a second EBL step is performed to define the Nb part of the device. After the EBL step the structure is etched for 15 seconds again at low voltage followed by in-situ sputtering of 80 nm Nb. BSTS has a small mean free path of 10-40 nm. Therefore we optimized the EBL step so that a spacing of 50 nm between the two electrodes could be realized, in order to minimize the resistance contribution of the Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 interlayer itself. The width of the electrodes is about 300 nm, see also Fig. 1. The thickness of the exfoliated flakes is about 100 nm, which is thick enough to rule out a direct coupling between the top and bottom topological surfaces. Because of the very small electrode spacing at the top surface, we expect the contribution of the bottom surface to the spectra to be small.
Conductance spectroscopy
The conductance was measured by applying a DC current with an additional AC current by means of a lock-in amplifier. We measured the differential voltage and the DC voltage across the devices.
The conductance spectra at 1.7 K of two different devices, but with the same dimensions and geometry (samples D1 and D2) are shown in Fig. 2a. Although the line shapes of the devices look slightly different, two common features are readily identified. First of all, both spectra display dips in the conductance at a voltage that we will show is most likely related to the proximity induced gap value. The corresponding gap values are significantly lower in energy than the gap of the Nb electrode (∆ is about 1.3 meV for our Nb with a T c = 8.4 K). Secondly, the subgap conductance is increased with respect to the normal state resistance.
The characteristic energy scale associated with the induced superconductivity is related to the dwell time in the junction and is known as the Thouless energy. In the diffusive limit it is defined ashD/L 2 where D is the diffusion constant [32,33,34]. The Thouless energy corresponds very well to energy scale attributed to the width of the dip features in all three the devices. This suggest that mainly diffusive transport contributes to the measured conductance spectra and that the observed features can be interpreted as signatures of induced superconductivity in Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 .
P-wave symmetry?
One may raise the question whether or not the conductance dips at the gap value and the increased conductance near zero energy can be explained by the existence of p-wave superconducting correlations in the TI. Note, that a conductance dip near the (induced) gap energy is one of the characteristic signatures of p-wave superconductivity. At first sight, we do not expect a dominant p-wave order parameter to exist, as is noted in Ref. [25,26,27]. Rather, an equal admixture of p and s-wave correlations coexist. Nevertheless, as is shown by Tkachov et al. [26,27] the amplitude of the s and p-wave order parameters have different spatial dependencies. For a dirty TI (i.e. the coherence length ξ N is much larger than the mean free path l mean ), it is shown that the s-wave order parameter is always dominant over the p-wave and both decay to zero on a length scale of the order of the coherence length. However, for the clean limit or intermediate ratios of ξ N and l mean , the p-wave order parameter can become dominant over the s-wave correlations, depending on the distance from the interface. For an estimated mean free path of 10 nm and a T c of 8.4 K we obtain ξ N = hD 2πK B Tc = 15.5 nm ,which corresponds with ξ/l mean = 1.5, or an intermediate regime according to Ref. [26]. Now it depends on the distance over which the induced superconductivity is probed whether or not a dominant p-wave order parameter can be observed.
The existence of a zero-bias conductance increase and conductance dips at gap edges, as observed in our devices, are also known to appear in s-wave diffusive SNN' devices as described by Volkov et al. [32]. It is shown that the total diffusion constant, directly related to the measured differential conductance, is given by
D(E) = 1 + r 1 + r 2 r 1 /M 1 (E) + r 2 /M 2 (E) + m(E) ,(1)
where r 1 and r 2 are the barrier resistances at the Nb/Bi If one of the interfaces is dominating it follows from this diffusion constant that the measured conductance is similar to the properties of this interface through which we can make an estimation of the distance over which the superconducting relations are probed. Device D2 has the largest modulation in the conductance and corresponds to 12% of the total measured conductance. Following Volkov et al. [32] this would correspond to a r 1 + r 2 of the order of one. This implies that the different resistances are comparable to each other, which results in a measured conductance that is a weighted average of the three spectral densities and hence there is no dominant interface. This makes the conductance spectra rather complicated and prevents us from extracting a well-defined position from which one can conclude [26,27] whether p-wave correlations are dominant over the s-wave correlations.
Disorder induced zero-bias resistance peak
In order to distinguish between the two scenarios for the conductance spectra, spectroscopic features can be investigated as function of temperature. For this purpose, another device (D3) was cooled down further, see Fig. 2b. The position of the induced gap values clearly changes as function of temperature, as one would expect. However, the zero-bias conductance increase (dashed line around zero voltage in Fig. 2b) now becomes masked by a zero-bias resistance peak at temperatures below 1.5 K. We attribute the zero-bias resistance peak to disorder induced electron-electron interactions (EEI, also called the Altshuler-Aronov effect) since this feature is also there in the absence of superconducting electrodes (e.g. see our reference junctions in Ref. [35]). Moreover, the interplay between EEI and the related weak antilocalization (WAL) is a well studied phenomenon in topological insulators [36,37]. In our case, the voltage here determines the relevant energy scale instead of temperature. The logarithmic dependence is expected for two-dimensional EEI [36], whereas a square root dependence would have been expected for a three-dimensional situation. The zero-bias conductance peak in a perpendicular magnetic field (see Fig. 2c) in related Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Nb junctions (samples S1 and S2) with the same geometry, can be best fitted (cross section at a field of 1 T in Fig. 2d) with a logarithmic dependence on voltage.
In order to further explore the p-wave symmetry as an explanation for the spectra, one has to get rid of the strong disorder that causes the resistance peak at low temperatures. Therefore, we now turn to stoichiometric BiSbTeSe 2 , with higher surface state mobility. Furthermore, to enhance the spectroscopic signal from the interface with respect to the topological layer, we employ edge-type side contacts with intentional tunnel barriers.
Proximity effect into stoichiometric BiSbTeSe 2
Device fabrication
To make edge-type side contacts, flakes were again transferred from the stoichiometric BiSbTeSe 2 crystal to a Si/SiO 2 substrate by mechanical exfoliation. The big drain contact is defined by optical lithography. Next, a short, low power Ar etch is performed, to enhance the transparency of the contact, followed by the sputter deposition of 5 nm of Pd, a 100 nm Nb layer, and a 5 nm Pd capping layer. Thereafter, the edge is defined by optical lithography and the flake is Argon ion milled down. A short wet etch in nitric acid is performed to remove part of the disordered material, formed by the ion milling. The (crosslinked) photoresist defining the edge is not removed, so the top of the flake remains covered. A third optical lithography step is used to define the edge contacts. The exact alignment is not important, since the crosslinked photoresist protects the top of the flake, as is visible in Fig. 3a. The edge contacts are sputter deposited under an angle of 45 • , to ensure good side wall coverage. First, a layer of aluminium is deposited and oxidised, to form a tunnel barrier, followed by a 100 nm Nb layer and a 5 nm Pd capping layer. Finally, both layers of photoresist are dissolved, causing the metal layer to break at the top of the flake, leaving a side contact, as shown in Fig. 3b and the image shown in Fig. 3c. The exfoliated flake thicknesses range between 50 and 300 nm. Since superconductivity can also be induced from the side into the bottom topological surface, the conductivity spectra could have parallel contributions from the top and bottom surfaces.
Conductance spectra at renormalized voltage
Both the bias voltage as well as the differential resistance are measured between the current sourcing contact and the big drain contact. Therefore, this voltage drops not only over the side contact interface, but also over the flake itself as well as the big drain contact interface. Since the flake is only a thin sheet of material and BSTS is not doped, this resistance can actually have a significant value. In order to estimate the flake resistance, it is possible to solve Poisson's equation for a piece of material shaped like this flake.
By modeling the Poisson equation for the geometry of the sample (see Fig. 3c) and by assuming that the drain contact is transparent (because of the relatively large size and absence of barrier), the potential profile was numerically calculated. An example is given in Fig. 3d, for conduction across the top contact. Using the TI resistivity and by combining the measurement results of local and non-local configurations (i.e. measuring the potential at other contacts then the current injectors), an estimate for the ratio between interface resistance and interlayer resistance was made. Using the estimated flake resistances, it is possible to correct the bias voltage for the voltage drop across the flake, simply by considering the flake and contact to form a voltage divider. The correction factors are found to be f C1 = 0.431 and f C2 = 0.589. Now the dI/dV curves can be depicted as function of the estimated bias voltage across the side contact interface only, as can be seen in Fig. 4a. Note that using these correction factors, the sizes of the small gaps of both devices C1 and C2 match, being approximately 0.2 mV, as well as many of the other peaks as will be discussed later on.
The conductance spectra of samples C1 and C2 resemble the spectra of the more disordered samples D1 and D2. However, the Nb/BiSbTeSe 2 interface now has a stronger weight in the spectra and the mean free path in BiSbTeSe 2 is higher compared to the more disordered Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 . It is, therefore, more plausible now that the spectra can indeed be explained by dominant p-wave correlations [26]. Now, that less disorder is present, we do not expect EEI to become important, and we can study the induced gap values as function of temperature. The results are plotted in Fig. 4b. Upon cooling down, it can be seen that the induced gap opens up around 1.3 K, and then quickly increases in size, reaching its final width at 0.8 K. Note, that upon cooling down even further, another small conductance peak at zero bias seems to arise. The size of the induced gap is plotted versus the temperature in the inset, together with the expected Bardeen-Cooper-Schrieffer gap size dependence of a superconductor with a critical temperature of 1.4 K. There is a good agreement between the two.
Interference effect
The differential resistance spectra show clear oscillations with a period that is constant in bias voltage. Assigning a peak number to the dips in the spectra of sample C1, starting from zero bias, the renormalized bias voltage positions of the oscillation peaks are plotted versus the peak number in Fig. 4c. The same was done for sample C2, but the assumption was made that occasionally a peak is missing for C2, in order to match peaks/dips of different contacts at the same bias voltage. The similarities between the two contacts are striking. The resonances are present throughout the entire range of bias voltages, both within the induced gap, as well as out of the gap. The transition between two regions with linearly spaced peaks is around 0.5 mV, somewhat outside the induced gap value. The difference in the period between these two domains is about a factor of 4. These oscillations cannot be detected on any of the nonlocal contacts, implying that they occur in or near the local, current sourcing, junction.
There can be many causes of oscillations in tunnel junctions. Multiple Andreev reflections (MAR) would cause a subgap interference pattern with maxima at eV = 2∆/n [38]. The oscillations we observed are linearly spaced, excluding MAR as a possible reason. We also rule out de Gennes-Saint James resonances [39]. These would be almost linearly spaced in voltage, but they require the absence of states within the gap (good interface), as indicated in Refs. [40,41]. As the zero bias background conductance in our device does not vanish inside the gap, these kind of oscillations likely cannot be present. Also, self-induced steps from internal coupling to Josephson radiation emission would require a Josephson junction to be formed across the tunnel barrier, which we do not resolve experimentally.
Tomasch oscillations are geometrical resonances on the superconductor side of the interface [42]. When an electron-like (hole-like) quasiparticle Andreev reflects into a hole-like (electron-like) quasiparticle on the interface, it can immediately interfere with itself, due to the mixed nature of the quasiparticle state. This results in a series of oscillations with peaks in the conductance at eV p = ∆ 2 + nhv F 2d S 2 , with n the oscillation index, v F the renormalised Fermi velocity in the superconductor and d S its thickness. This will thus result in an almost linear series of resonances in the differential conductance spectrum. Assuming a Fermi velocity of 6 · 10 5 ms −1 [28] in the BiSbTeSe 2 , the outer set of resonances can only be fitted assuming a superconductor thickness of about 5 µm. This does not match any dimensions present in our device. We thus conclude that Tomasch oscillations do not seem to be the origin of the observed resonances. Similarly we rule out McMillan-Rowell oscillations [43] inside the flake (i.e. between the side contact and the drain), since this would require phase coherence and ballistic transport across at least two times the width of the flake, i.e. > 10 µm, so this is highly unlikely. Note, that the resonances in S-TI-S junctions by Finck et al. [44,45] are in fact attributed to Fabry-Perot resonances in the TI, but there the interlayer thickness is a lot shorter. Furthermore, if the resonances were to occur in our flake, they should also be present in the nonlocal signal. This is not the case in our measurements.
Since the interference does not seem to match any geometric resonance, the origin of the oscillations could perhaps be sought in interference between multiple order parameter contributions. Self-consistent calculations of the order parameter of a TI on top of a superconductor have been performed by Dellabetta et al. [46], and show oscillations in the order parameter in k-space . If the size of the order parameter was to fluctuate, the magnitude of the gap (and thus the DOS) would also show oscillations for different k-vectors. It could be that the oscillations we observed, are caused by similar effects, related to multiple order parameters interfering.
Speculatively, another indication for the presence of two order parameters is provided by the temperature dependence of the spectra at higher temperatures. A different device (C3) was measured above 1.7 K, see Fig. 4d. Note, that in this case, the bias voltage shown is not corrected for the voltage drop over the BiSbTeSe 2 . By comparing the shape of the curve at 1 K with the 1 K data of samples C1 and C2, the zero bias conductance increase can readily be associated with the induced gap (rescaled value) that is plotted as function of temperature in the inset of Fig. 4a. However, at higher temperatures, the spectra of sample C3 show a dominant decrease of the conductance around zero bias, with an associated gap value that is about twice as high in energy. This could perhaps indicate the presence of an additional order parameter component.
Discussion
The observed conductance spectra are consistent with an unconventional order parameter symmetry, such as the predicted p-wave contribution. Also Yang et al. have reported the observation of a conductance increase at low energy in a Sn-Bi 2 Se 3 junction [47], which they attributed to an induced unconventional order parameter. Furthermore, they have also observed resistance spikes at nonzero energy, similar to our observations. Koren et al. also observed a zero-bias conductance peak in a proximity induced superconducting phase in a TI [48]. They find a conductance increase in conductance spectroscopy measurements of a pure bismuth film, which they attribute to induced superconductivity, due to islands of bismuth becoming superconducting. Argon milling is known to disorder the surface of a material extensively, so a similar effect could, in principle, also occur in our junctions. In Ref. [50] Koren et al. show that proximity superconductivity can really be realised in Bi 2 Se 3 , by observing an increase of the conductance of grainy NbN-films when a Bi 2 Se 3 -film is deposited on top. Many publications on proximity induced superconductivity into Bi 2 Se 3 report the proximity induced superconductivity to appear at a temperature below the superconducting electrode T c , similar to our findings [48,49,51]. Recently, also the proximity effect into topological HgTe has been studied by conductance spectroscopy.
In summary, all spectroscopic devices studied here show spectra that are reminiscent of p-wave superconductors, i.e. having conductance dips at the value of the induced gap, as well as an increase in conductance below the gap. The stoichiometric BiSbTeSe 2 is more suited for spectroscopy than the disordered Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , in which case electron-electron interaction induced zero-bias resistance peaks can mask the actual conductance increase. The presence of oscillations as well as the possible observation of a second order parameter component are experimental observations that can, hopefully, provide input to theoretical modeling of the induced proximity effect. The voltage axis has not been corrected for the contribution from the interlayer. Besides the conductance dips and zero-bias conductance enhancement, at higher temperature, a dominant conductance decrease is revealed around zero bias, with a cut-off that is higher in energy.
. 5
5Sb 0.5 Te 1.7 Se 1.3 and Au/Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 interface, respectively, normalized to the resistance of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , M 1 (E) is the spectral density in Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 at the position of the Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 /Nb interface, M 2 (E) the spectral density of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 at the Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 /Au interface and m(E) is the spectral density of Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 integrated over the distance between the two interfaces.
Figure 1 .
1(a) Schematic view of the Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Au spectroscopic devices, indicating the current and voltage contacts. (b) Scanning electron microscopy image of the device. The spacing between the Au and Nb electrodes is about 50 nm and the width of the contacts is about 300 nm. (c) Schematic side view. In this geometry, the Nb induces superconductivity in the topological insulator underneath. The proximity effect extends laterally beyond the Nb electrode.
Figure 3 .
3(a) Schematic drawing of one of the fabrication steps of Nb/Al 2 O 3 -BiSbTeSe 2 -Pd/Nb/Pd devices with stoichiometric interlayers. The Nb electrode deposition is preceded by a thin layer of oxidized Al to form a tunnel barrier, effectively increasing the role of the Nb-BiSbTeSe 2 interface in the conductance spectra. (b) Layout after the final fabrication step. The Nb-BiSbTeSe 2 contact is in the plane of the topological surface state, rather than perpendicular to that. (c) Optical microscopy image of a sample with the BiSbTeSe 2 devices C1 and C2. The geometry used for calculating the potential distribution is overlaid on the image. (d) Calculation result of the electric potential model when the device is biased between the right electrode and the top left electrode. Similarly, for the contacts C1 and C2, the contribution to the resistance of the interlayer was estimated.
Figure 4 .
4(a) Conductance spectra of BiSbTeSe 2 sample C1 and C2 at 15 mK. After correcting for the voltage across the interlayer, the spectroscopic features (gap as well as interference peaks) of the same sample seem to coincide quite well. (b) Conductance spectra of BiSbTeSe 2 device C1 at various temperatures. The bias voltage has been corrected for the potential across the interlayer. Conductance dips at a characteristic gap voltage as well as a conductance enhancement at zero bias are present in these devices too. The position of the gap feature (black circles) has been plotted as function of temperature in the inset. The red solid line shows a fit to the Bardeen-Cooper-Schrieffer gap equation for a superconductor with a critical temperature of 1.4 K. (c) The voltage at which peaks/dips are observed in the conductance spectra of samples C1 and C2 at 15 mK are plotted as function of the peak position. (d) Sample C3 (BiSbTeSe 2 device on a different crystal flake) is measured at temperatures up to 4.2 K.
Figure 2. (a) Conductance spectra of two different Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Au spectroscopic devices, with disordered interlayer, samples D1 and D2, at T = 1.7 K. Generic features are conductance dips at some characteristic gap voltage and an increased conductance around zero bias. (b) Conductance of Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Au device D3 at various temperatures. For clarity, the data have been off-set in steps of 0.1 × 10 −3 Ω −1 with respect to the data at 1.5 K. The black dashed lines are guides to the eye to see how the gap features move to lower voltage for increased temperature. The red lines indicate the onset of a disorder induced electron-electron interaction (EEI)resistance peak, masking the conductance increase at zero bias as observed at higher temperature for samples D1 and D2. (c) Sample S1 is a Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Nb device with electrode spacing of 250 nm. The EEI resistance peak at zero-bias is plotted as function of bias current and the magnetic field applied perpendicular to the sample. (d) The conductance at a field of 1 T of a similar Nb-Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 -Nb device with electrode spacing of 200 nm shows a logarithmic dependence on voltage (best fit: green line), as expected for EEI on a two-dimensional surface. Three-dimensional EEI would have given a square root dependence (best fit: blue line).1.4
1.6
250 mK
400 mK
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1.2 K
dI/dV (10
-3
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ln(V)
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|
[] |
[
"Enhancing Multi-Scale Implicit Learning in Image Super-Resolution with Integrated Positional Encoding",
"Enhancing Multi-Scale Implicit Learning in Image Super-Resolution with Integrated Positional Encoding"
] |
[
"Ying-Tian Liu \nDepartment of Computer Science and Technology\nTsinghua University\n\n",
"Yuan-Chen Guo \nDepartment of Computer Science and Technology\nTsinghua University\n\n",
"Song-Hai Zhang \nDepartment of Computer Science and Technology\nTsinghua University\n\n",
"Bnrist \nDepartment of Computer Science and Technology\nTsinghua University\n\n"
] |
[
"Department of Computer Science and Technology\nTsinghua University\n",
"Department of Computer Science and Technology\nTsinghua University\n",
"Department of Computer Science and Technology\nTsinghua University\n",
"Department of Computer Science and Technology\nTsinghua University\n"
] |
[] |
Is the center position fully capable of representing a pixel? There is nothing wrong to represent pixels with their centers in a discrete image representation, but it makes more sense to consider each pixel as the aggregation of signals from a local area in an image super-resolution (SR) context. Despite the great capability of coordinate-based implicit representation in the field of arbitrary-scale image SR, this area's nature of pixels is not fully considered. To this end, we propose integrated positional encoding (IPE), extending traditional positional encoding by aggregating frequency information over the pixel area. We apply IPE to the state-of-the-art arbitrary-scale image super-resolution method: local implicit image function (LIIF), presenting IPE-LIIF. We show the effectiveness of IPE-LIIF by quantitative and qualitative evaluations, and further demonstrate the generalization ability of IPE to larger image scales and multiple implicit-based methods. Code will be released.
| null |
[
"https://arxiv.org/pdf/2112.05756v1.pdf"
] | 245,124,125 |
2112.05756
|
65b32d135917dee1142f0255d8598ab271c18f30
|
Enhancing Multi-Scale Implicit Learning in Image Super-Resolution with Integrated Positional Encoding
Ying-Tian Liu
Department of Computer Science and Technology
Tsinghua University
Yuan-Chen Guo
Department of Computer Science and Technology
Tsinghua University
Song-Hai Zhang
Department of Computer Science and Technology
Tsinghua University
Bnrist
Department of Computer Science and Technology
Tsinghua University
Enhancing Multi-Scale Implicit Learning in Image Super-Resolution with Integrated Positional Encoding
Is the center position fully capable of representing a pixel? There is nothing wrong to represent pixels with their centers in a discrete image representation, but it makes more sense to consider each pixel as the aggregation of signals from a local area in an image super-resolution (SR) context. Despite the great capability of coordinate-based implicit representation in the field of arbitrary-scale image SR, this area's nature of pixels is not fully considered. To this end, we propose integrated positional encoding (IPE), extending traditional positional encoding by aggregating frequency information over the pixel area. We apply IPE to the state-of-the-art arbitrary-scale image super-resolution method: local implicit image function (LIIF), presenting IPE-LIIF. We show the effectiveness of IPE-LIIF by quantitative and qualitative evaluations, and further demonstrate the generalization ability of IPE to larger image scales and multiple implicit-based methods. Code will be released.
Introduction
Single image super-resolution (SISR) aims to reconstruct a visually-natural high-resolution (HR) image from its lowresolution (LR) counterpart. SISR is an ill-posed problem since there exist many potential reasonable HR images for a single LR input and the down-sampling filters for HR images are not determined. Many methods have been proposed to bridge the gap in signal frequency between LR and HR images by either regressing high-frequency textures [9,13,24,42,52] or generating perceptually reasonable details [16,23,26,34,43]. Although super-resolution (SR) with predefined scale factor has been fully explored with convolution-based architectures [9,19,[22][23][24]52], arbitraryscale SISR is a more attractive and challenging task that is still under exploration.
Recently, implicit neural representations have emerged to represent various kinds of signals. The key idea is to represent a signal as a function parameterized with a multilayer perceptron (MLP) that maps a coordinate input to the signal value at that position. What makes the neural implicit representation appealing is that a signal is often represented continuously or in infinite resolution and contains sufficient detail. Different local signals could form various kinds of entities on the whole, such as density and radiance field for 3D scenes [3,30], signed distance field for 3D shapes [7,8,10,27,31] and RGB values for images/videos [27,36,40].
Based on implicit neural representations, Meta-SR [13] achieved arbitrary-scale image SR by generating variant weights of the up-scaling filters for a position and scale input. Chen et al. [6] proposed the local implicit image function (LIIF), for continuously representing natural and complex images. LIIF learns an implicit decoder across images to predict the color within the continuous neighborhood of a feature vector in a 2D feature map. Therefore, LIIF builds a bridge between the discrete and continuous representation for 2D images. Both Meta-SR [13] and LIIF [6] exploit the convenience of implicit representation.
However, MLPs in implicit representation have difficulty in learning high-frequency functions, which is also known as the spectrum bias [32]. The bias hinders the generation of detail-rich contents in some scenarios, such as image SR and novel view synthesis. NeRF [30], a novel framework for novel view synthesis, adopts positional encoding (PE) to address this issue. PE allows MLPs to spatially distinguish inputs with small differences to recover high-frequency details. But in NeRF's rendering, each pixel emits only one ray to predict its color, which may lead to observable aliasing artifacts. To conform as much as possible to the physical aggregation of light, an approximate integral of PE along the ray cone is used in [3] to make full use of multiscale training images and improve the rendering efficiency of anti-aliased views.
In SISR, the same problem exists. Imagine that we have an HR image and a corresponding LR image of half the size, where the pixels in the LR image are not necessarily the same as the pixels at even positions in the HR image. To address the problem, LIIF [6] directly inputs the pixel size to the implicit decoder to distinguish them, called cell decoding. However, the direct input of coordinates and size makes it difficult for the implicit decoder to fit highfrequency details. Although UltraSR [46] has noticed the problem and integrates PE to improve the SR performance, it still does not consider multi-scale variations simultaneously, which is critical in tasks with the output of different pixel sizes. We believe that each pixel in the SR task should be treated as the aggregation of colors over a small area, instead of being represented exactly by its center location. Both pixel size and position are critical for high-quality arbitrary-scale SR. For this purpose, we propose IPE, which is to compute the integral of PE over the pixel to be predicted. We then present IPE-LIIF, where the IPE instead of the coordinate input is concatenated with the local feature vector as the input of the implicit decoder. Experiments reveal that IPE-LIIF achieves state-of-the-art performance on several popular SR datasets, and multiple methods adopting IPE also give consistently better results than the original version when the SR scales varies over a wider range. The contributions of this paper are threefold:
• We propose integrated positional encoding (IPE), which can provide scale and position-dependent encoding for implicitly represented pixels in image SR.
• The arbitrary scale SR model incorporating IPE has demonstrated its optimal competence in both qualitative and quantitative experiments.
• Further experiments highlight that IPE can be applied to a variety of SR models based on implicit represen-tations, consistently improving their capacity to learn multi-scale outputs.
Related Work
In this section, we will review the SISR task and the implicit representation with a focus on positional encoding (PE).
Single image super-resolution SISR is one of the most popular tasks in the visual community [5,9,11,19,35,39,41,42,48,51]. Traditional methods [5,41] are mainly examplebased. SRCNN [9] is the first end-to-end CNN-based SR model. It implements the process of feature extraction, nonlinear mapping, and image reconstruction with a linear three-layer structure. Recursive strategies [20,38] were proposed to improve the training stability and detail fidelity. VDSR [19] and IRCNN [49] made the network deeper by stacking convolutional layers with residual learning. For the first time, ESPCNN [35] proposed to compute on lowresolution inputs before the up-sampling layer, significantly reducing computational overhead. SRResNet [23] introduced the residual connection [12] into image SR and generated perceptually sound details via adversarial training. EDSR [24] improved the ResNet block by removing the BatchNorm [17] layer. Moreover, it proposed to learn simultaneously the backbone corresponding to various SR scales, forming the fusion model MDSR [24], which allowed multi-scale SR. Similarly, LapSR [22] proposed a progressive up-scaling architecture and a novel hierarchical reconstruction loss to complete SR with scales for powers of 2. With the observation that the dense connection [14] allows feature reuse from all the preceding layers within each block, RDN [52] adopted stacked dense blocks to make a full use of hierarchical features extracted from LR image, thereby achieving relatively-high performance.
Arbitrary scale super-resolution Image SR on an arbitrary scale factor is obviously more attractive and convenient than that on a pre-defined one. Hu et al. [13] proposed to learn a meta-upscale module to dynamically predict the weights of the up-scaling filters on different scales. Meta-SR [13] model could combine the meta-upscale module with the feature learning module from most previous methods [24,51,52], producing convincing SR results on arbitrary scale. LIIF [6] replaced the meta-upscale module with an implicit decoder, representing natural and complex images in a continuous manner. And it is proved to be effective for image SR at out-of-distribution scales, yielding visually smoother results than Meta-SR [13].
Coordinate-based neural representation Neural implicit representations can faithfully reconstruct various signals and have therefore received extensive attention and research [8,[29][30][31]]. An object in implicit representation can be typically modeled as a multi-layer perceptron (MLP), mapping a coordinate input to the signal at that 2D/3D location. Such an idea has shown effectiveness in a variety of scenarios, e.g. 3D shape modeling [2,7,8,10], neural volume rendering [3,25,30], 3D scene modeling [18,37] and image/video representation [6,36]. A recent work [32] has highlighted that there exists a spectral bias in the MLP activated by ReLU, which leads to lack of details in high-frequency signal representation. This problem can be solved by either replacing ReLU with periodic functions [36] or adding an extra PE of the coordinate input [3,25,30,40,46]. The use of proper PE can significantly accelerate the training process and improve the fidelity of signals at high-frequency positions. UltraSR [46] adds spatial encoding into LIIF [6], reducing checkboard artifacts on the large scale SR. Inspired by the pre-computed multi-scale anti-aliasing technique (e.g. mipmap [45]), Mip-NeRF [3] accelerated the anti-alias novel view rendering in NeRF with an approximate integral of PE within the interest region.
The method we propose, named IPE, provides position and shape information in a uniform encoding for each pixel. IPE-LIIF applys IPE to LIIF, achieving the state-of-the-art performance in the arbitrary-scale SR task. We verify the effectiveness and generality of our design with extensive experiments and convincing ablation studies. s
Preliminaries: local implicit image function
LIIF [6] performs the arbitrary-scale image SR by learning a continuous representation for images. In continuous representation, each image I is represented as a 2D feature map M ∈ R H×W ×C , where the size is H × W and the channel is of length C. To predict the RGB value at an arbitrary position, LIIF uses a learnable implicit function, which takes the coordinate and the nearest feature as the input and predicts the RGB value. The implicit function is parameterized by an MLP f θ (with θ as its parameter) and has the following form:
s = f θ (z, x) (1) where x is a continuous 2D coordinate in the range χ = [0, 2W ] × [0, 2H]
and z is a latent vector representing the local feature. The output s is the predicted color at position x. With a pre-defined f θ , f θ (z, ·) becomes a local implicit function representing the local image area. In practice, the H × W features are evenly distributed in χ. The RGB value of the continuous image I at a query coordinate x q is
I(x q ) =f θ (z * , x q − p(z * ))(2)
where z * is the nearest feature vector to position x q in M , p(z) is the position of z. Note that the implicit function f θ here describes the location relatively, which is different
from f θ defined in Eq. (1) with absolute coordinate input . In Eq. (2), the relative coordinate δx = x q − p(z * ) is in the range of [−1, 1] 2 .
Each latent code z in M is responsible for predicting the color in such a local area around p(z). With the shared implicit decoder, LIIF bridges the gap between features at discrete positions and a continuous image. With this procedure for rendering the continuous image at the arbitrary resolution, training LIIF is straightforward: sampling pixels from HR images to build coordinate-color pairs, we minimize the difference between the predicted RGB values and the ground truths with gradient descent. Since LIIF works as a decoder that converts the feature map into an image, it is usually jointly trained with a feature extraction module in CNN models [24,52].
Method
In this section, we first discuss the shortcomings of LIIF [6], propose the solution, and finally present the overall architecture of our model. As described in Sec. 3, LIIF [6] samples only the center to render each pixel. Essentially, LIIF ignores the fact that an image pixel is actually the aggregation of the color in a small area, and considers the color value at the central point as the color of that pixel. Imagine that we have an HR image and a corresponding LR image down-sampled by 2, where pixels in the LR image are not necessarily the same as those at even positions in HR. Therefore the scaling factor or, in other words, the size of the target pixel is crucial in this task. And to address the problem, the original LIIF [6] simply concatenates the pixel size and the central position of pixel area as input to the MLP, called cell decoding. But both our and LIIF's experiments reveal that this cell decoding strategy is not guaranteed to improve the quality of results.
Integrated Positional Encoding
We solve the problem by introducing IPE of each pixel, like [3], enabling the implicit decoder to focus on the local region rather than a center point without area. Moreover, with IPE, the implicit decoder could adapt to a variety of SR scales, which are essentially related to the pixel size to decode.
The commonly used PE [30,40] transforms each position with periodic function in different frequencies:
γ(x) = [sin(x), cos(x), ..., sin(2 L−1 x), cos(2 L−1 x)] (3)
Similar with [3], we construct the featurized representation of the pixel with the expected PE of all the coordinates that lie in it. The main differences among several spatial encoding methods are illustrated in Fig. 2. In [3], the integral of PE in a conical frustum is quite tricky and has no closed form solution, so they use a multivariate Gaussian with the same mean and variance to approximate the area. In the 2D case we are discussing here, the integral is straightforward: within a pixel l whose center is c(c x , c y ) and radius (half of the edge) is r(r W , r H ), the sine encoding with frequency ω has an expectation as:
( * ) " ! ( * , ) ( * ) 2 ! 2 " " ! ( * , , ) * * ( * ) " ! ( * , ( )) * ( * ) " ! ( * , +( , )) * 2 ! 2 " (a) (b) (c) (d)
Similarly, the cosine encoding is as follows:
Hence, the pixel l defined by its center c and radius r has the following IPE: γ(c, r) = 1 r [sin(c) sin(r), cos(c) sin(r), ...,
sin(2 L−1 c) sin(2 L−1 r) 2 L−1 , cos(2 L−1 c) sin(2 L−1 r) 2 L−1 ](6)
Intuitively, the IPE contains both location and size information, which are modulated by sine/cosine and sinc function, respectively. When the SR scale is small, the sinc part makes plain PE smoother, in which case the predicted pixel can inherit enough information from the feature vector. As the SR scale increases, the sinc part will move towards 1, making the IPE closer to plain PE (see Fig. 3 for brief comparison). In that case, each feature vector is responsible for a large number of pixels, which is roughly the square of the scale. Therefore, the implicit decoder cannot reconstruct a detail-rich image solely from the features. The IPE will provide high-frequency encoding of the location to compensate for the lack of information here. The choice of hyperparameter L will further be discussed in Sec. 5.3.
Architecture details
Given a pixel-based image I ∈ R H×W ×3 , we extract the 2D feature map M ∈ R H×W ×C with a feature extraction module E. The module can inherit from EDSR-baseline [24] and RDN [52]. Feature unfolding in [6] is preserved. Each latent vector in the new feature mapM ∈ R H×W ×9C is the concatenation of the 3 × 3 local neighboring vectors from zero-padded M . The feature extraction process is for-
M = E(I) M ij = Concat({M i+k,j+m , k, m ∈ {−1, 0, 1}})(7)
To make the predicted RGB value smoother, we apply the local ensemble as [6]. That is, given a SR scale s and a query coordinate x q , we extend Eq. (2) to
I s (x q ) = t∈{00,01,10,11} S t S ·f θ (z * t ,γ(x q − p(z * t ), ( 1 s , 1 s ))) (8) where {z *
t } are four feature vectors surrounding x q inM , and they are on the vertices of a square. S is the area of this square and S t is the area of the rectangle inside this square that is opposite to p(z * t ) (Fig. 4). The query pixel radius is ( 1 s , 1 s ) here if we specify the range of the xy-axis of the per-pixel local coordinate system in the original input LR image to [−1, 1].
The implicit decoderf θ is implemented as an MLP with skip connections [30,31] from the input to all hidden layers. The MLP has 4 hidden layers, each with a width of 256. The bandwidth parameter L of IPE is set to 10. To fuse the spatial encodingγ(c, r) and the feature vector z * , we concatenate them as the input off θ (Fig. 2 (d)).
Experiments
In this section, we introduce the datasets and metrics in the experiments, the implementation details, and compared methods. Then we compare IPE-LIIF with some baseline alternatives both quantitatively and qualitatively. We further design experiments to discuss the generalization ability of IPE and validate some of the design choices via ablation studies.
Datasets and metrics
We train all models using the DIV2K dataset [1] from NTIRE 2017 Challenge for all the experiments. This dataset consists of 1,000 HR images in 2K resolution and their corresponding LR images of different down-sampling scales (×2 − ×4). We follow the original split setting, i.e. 800 images for training, 100 images for validation. Consistent with previous works [6,13,24,52], we also test our model on 4 popular benchmarks, i.e. Set5 [4], Set14 [47], B100 [28], and Urban100 [15], and report the results. We compare our method with baseline methods on several discrete scales (×2, ×3, ×4). To demonstrate that our method can be applied for arbitrary-scale SR, we also compare the SR results on a variety of large scales (×6 − ×30) that do not appear in training time. We use the Peak Signal-to-Noise Ratio (PSNR) and SSIM [44] as our evaluation metrics. Consistent with most SR works, the metrics are computed on RGB channels for DIV2K [1] and Y channel (luminance)for other benchmarks [4,15,28,47].
Training and inference details During training, we feed patches of size 48 × 48 into the network. To enable the network to perform arbitrary-scale SR on the input patches, we sample the SR scale s from U(1, 4). We then randomly crop a patch of size 48s × 48s from the training HR image. The cropped HR patch is then down-sampled to 48 × 48 with bi-cubic interpolation as the LR input. For the implicit decoder training, we randomly select 48 2 pixels from each HR patch. All the modules are trained from scratch with L1 loss [24,52] for better convergence. We use Adam [21] optimizer with β = (0.9, 0.999) to train the whole network for 1000 epochs. Each epoch contains 1000 iterations. In each iteration, we feed a batch of 16 sampled patches. The learning rate is initialized to 10 −4 and decays to half every 200 epochs. For evaluation on scales within the distribution of training (×2 ∼ ×4), we use the LR images provided in the DIV2K and different benchmark datasets. For out-ofdistribution validation (×6 ∼ ×30), we generate the LR inputs by resizing the HR images with bi-cubic interpolation.
Compared methods The methods involved in the experiments include: 1. models with CNN architectures targeted for a specific scale: EDSR-baseline [24], RDN [52], 2. the state-of-the-art arbitrary-scale SR methods at present: Meta-SR [13], LIIF [ Table 2. Quantitative comparison on 4 benchmarks [4,15,28,47] with PSNR (dB) / SSIM on different scales. LIIF [6], IPE-LIIF (ours) use a uniform model for all the scales. RDN [52] uses different models for different scales and cannot be evaluated on out-of-distribution scales. The best results on different settings are bolded. two groups, both the feature extraction modules in EDSRbaseline and RDN could be adopted. In Sec. 5.1, we evaluate all the models except IPE-MetaSR on benchmarks with various scales to verify the effectiveness of IPE on improving performance for arbitrary-scale SR methods. In Sec. 5.2, we evaluate LIIF, MetaSR and their IPE-version after training on a wider range of SR scales to highlight the generality of IPE on multiple methods and larger training scales.
Results and comparison
We compare IPE-LIIF with five baseline methods: bicubic interpolation, EDSR-baseline [24], RDN [52], MetaSR [13] and LIIF [6]. Note that EDSR-baseline and RDN train different models for different scales, thus cannot be applied to out-of-distribution scales. The remaining parametric methods are designed for arbitrary-scale image SR and their feature extraction modules are exchangeable. The feature extraction module used by the three models is indicated by the prefix of the name. All the parametric methods are trained on the DIV2K [1] training set.
Quantitative comparison The quantitative results on the 5 benchmark datasets [1,5,15,28,47] are listed in Tab. 1 and Tab. 2. We observe that our method surpasses existing methods under almost all the test settings, especially on large datasets [15,28]. For SR on out-of-distribution scales, we achieve consistently better results than existing methods on all the benchmarks. In summary, the use of IPE enhances the existing SR model based on implicit representation, allowing them to go beyond previous scale-specific ones [52] to support multiple scale factors. We argue that the power of IPE-LIIF comes from the combination of IPE and implicit representation, which effectively fuse information from different scales and spatial locations.
Qualitative comparison In Fig. 1 and Fig. 5, we further show visual image comparisons of the previous work and our method on different SR scales. For areas containing scale variant stripes, the previous method [6] will produce checkboard artifacts and could not faithfully recover the HR contents, while IPE-LIIF can distinguish texture details by encoding the query pixel spatial information. In these cases, IPE is able to better identify the position and size of complex textures so that they can be clearly distinguished.
General enhancement for multi-scale implicit learning
Both Meta-SR [6] and LIIF [6] take the advantage of the implicit representation. The former predicts an up-scaling filter for each output pixel and the latter predicts the color directly with a coordinate input via implicit decoding. The IPE strategy can also be easily applied to improve Meta-SR. That is, for a given pixel (i, j) on the SR image output upscaled by s, we redefine the original input to Meta-Upscale Module [13]: Table 3. Validation results of LIIF [6] and Meta-SR [13] on DIV2K [1] combined with EDSR-baseline and different coordinate encodings, including none, plain positional encoding, and integrated positional encoding. The best results are bolded.
v ij = ( i s − i s , j s − j s , 1 s ) (9) as v ij =γ(( i s − i s , j s − j s ), ( 1 s , 1 s ))(10)
By comparing the results on large scales presented in Tab. 3 and Tab. 1, it can be seen that LIIF fails to exceed the performance of our method without these training samples even if the data on ×6 − ×12 is provided. This experiment illustrates the generality of IPE on multiple implicit models and a wider range of SR scales.
Ablation studies
We discuss the effectiveness of some design choices of IPE-LIIF in this section, including: 1. the bandwidth L of integrated positional encoding, 2. whether to use the integrated version of PE, 3. whether to use the original cell decoding, 4. whether to use skip connections in the implicit decoder. EDSR-baseline is used as the feature extraction module for all the ablation studies. The quantitative results are listed in Tab. 4 and green indicates a decrease in results.
Bandwidth L of integrated positional encoding In terms of bandwidth L, we choose L = 10 in IPE-LIIF and set L = 4, 16 for competitors. Comparing the two variants (L = 4, 16) with the original model, we observe that the effect of the bandwidth is minor. Taking the results and computational overhead into account, we choose a mild value L = 10. The intuitive explanation of how a large L does not improve the SR quality is that the additional terms in IPE (Eq. (6)) have large denominators and remain almost constant near 0 as the query pixel region changes. That is, a larger L does not necessarily provide valid spatial information, which is also highlighted in mip-NeRF [3].
Integrated positional encoding An obvious possible improvement on LIIF [6] is to add plain PE to the coordinate input. To verify the effectiveness of IPE, we perform plain PE on the input and form LIIF with PE. For a fair comparison, we also set the bandwidth L to 10. It can be seen that with PE, the network focuses more on high-frequency information and performs poorly on relatively small scale Table 4. Ablation of design choices of IPE-LIIF on DIV2K [1] validation set. EDSR-baseline is used as the feature extraction module for all the settings. The results with degraded performance are marked in green.
factors, sometimes even worse than the original LIIF.
Other design choices In Tab. 4, we show the ablation results on some other design choices, including the use of cell decoding and skip connections in the implicit decoder. We concatenate the cell size with the IPE input and form IPE-LIIF + cell. It can be seen that cell decoding makes the results on large scaling factors worse, which is because it is not able to generalize to arbitrary scale factor. For the out-of-distribution scales, the cell code is relatively small compared with those seen in training time. We also remove skip connections and form IPE-LIIF-skip. The results verify that SR on almost all scales benefits from the skip connections, as it provides a more flexible network depth. From Tab. 4, we find that IPE performs good aggregation of the spatial information in the query region. It combines the advantages of PE and cell decoding and provides a universal solution to implicit learning tasks with multi-scale samples.
Limitations
Although we have improved the existing arbitrary-scale image SR methods with IPE, regression-based methods still cannot fill in the missing details when the scale factor is large. In contrast, generative methods [23,33,43] could induct reasonable semantic fill-in and produce visually clear results with a poor PSNR but high perceptual similarity [50]. In addition, in IPE-LIIF, the wider layers in skip-MLP will increase the training time by 10% to 20%.
Conclusion
In this paper, we propose to use IPE for arbitrary-scale image SR, and apply it to the state-of-the-art method LIIF. With the enhancement provided by IPE, IPE-LIIF generates consistently high-fidelity SR results on various scales, which is verified by quantitative and qualitative experiments. Further experiments on the generalization ability suggest that IPE can be seamlessly integrated to multiple architectures to increase the capacity of implicit learning on multi-scale samples.
Figure 1 .
1×4 SR results of methods using the integrated positional encoding (IPE) compared with those without IPE.
Figure 2 .
2Illustrations of different spatial encodings on this 3 × 3 feature map: (a) no encoding, (b), cell decoding used in[6], (c) plain PE used in[30,46], (d) IPE (ours). xq denotes the center of the query pixel. The query pixel is marked with a red box in (b) and (d), with the radius (rW , rH ). The differences in the expressions are marked in green.
Figure 3 .
3Comparison between the sine encoding of IPE (ours) and that of plain PE with the frequency ω = 8.
ωc) sin(ωr) ωr = sin(ωc) sinc(ωr)
Figure 4 .
4Local ensemble in IPE-LIIF. The query pixel along with its center is marked in red. The RGB value of the query pixel is obtained by bilinear interpolation of the results predicted by its surrounding feature vectors.
Table 1. Quantitative comparison on DIV2K[1] validation set with PSNR (dB) / SSIM on different scales. Bicubic, MetaSR[13], LIIF[6] uses a uniform model/algorithm for all the scales. EDSR[24] uses different models for different scales and cannot be evaluated on out-of-distribution scales. The best results under different settings are bolded.Method
In-distribution
Out-of-distribution
×2
×3
×4
×6
×12
×18
×24
×30
Bicubic [24] 31.01/0.9393 28.22/0.8906 26.66/0.8521 24.82/0.8014 22.27/0.7354 20.50/0.6986 19.79/0.6817 19.43/0.6711
EDSR-baseline [24] 34.55/0.9671 30.90/0.9298 28.94/0.8962
-
-
-
-
-
EDSR-baseline-MetaSR [13] 34.64/0.9674 30.93/0.9301 28.92/0.8962 26.61/0.8442 23.55/0.7652 22.03/0.7285 21.06/0.7058 20.37/0.6903
EDSR-baseline-LIIF [6] 34.67/0.9675 30.96/0.9306 29.00/0.8974 26.75/0.8477 23.71/0.7724 22.17/0.7369 21.18/0.7137 20.48/0.6977
EDSR-baseline-IPE-LIIF (ours) 34.72/0.9678 31.01/0.9310 29.04/0.8979 26.79/0.8483 23.75/0.7731 22.21/0.7374 21.22/0.7143 20.51/0.6979
RDN-MetaSR [13] 35.00/0.9692 31.27/0.9340 29.25/0.9016 26.88/0.8508 23.73/0.7721 22.18/0.7337 21.17/0.7104 20.47/0.6943
RDN-LIIF [6] 34.99/0.9691 31.26/0.9339 29.27/0.9017 26.99/0.8528 23.89/0.7771 22.34/0.7412 21.31/0.7181 20.59/0.7013
RDN-IPE-LIIF (ours) 35.04/0.9694 31.32/0.9343 29.32/0.9025 27.04/0.8537 23.93/0.7778 22.38/0.7415 21.34/0.7177 20.63/0.7011
mulated as:
MetaSR(ours) 34.51 30.86 28.94 26.74 23.71and form IPE-MetaSR. To verify the effectiveness of
IPE on a relatively large range of scale variations, e.g.
U(1, 8), U(1, 16), we train several candidate models and
evaluate them on DIV2K [1] over the same scale distribu-
tion. From Tab. 3, we can observe that with IPE, both mod-
els obtain consistent improvements on different SR scales.
barbara from Set14 [47] (×2)
Scale
Method
×2
×3
×4
×6
×12
U(1, 8)
LIIF [6]
34.60 30.94 29.01 26.79
-
LIIF w/ PE
34.47 30.87 29.01 26.81
-
ipe-LIIF(ours)
34.65 30.98 29.04 26.81
-
MetaSR [13]
34.57 30.91 28.97 26.72
-
MetaSR w/ PE
34.60 30.93 28.99 26.74
-
ipe-MetaSR(ours) 34.61 30.93 28.99 26.74
-
U(1, 16)
LIIF [6]
34.50 30.89 28.97 26.76 23.75
LIIF w/ PE
34.36 30.79 28.95 26.78 23.79
IPE-LIIF(ours)
34.54 30.92 29.00 26.79 23.76
MetaSR [13]
34.48 30.86 28.94 26.72 23.70
MetaSR w/ PE
34.50 30.85 28.93 26.72 23.70
IPE-
IPE-LIIF (ours)34.72 31.01 29.04 26.79 22.21 20.51 IPE-LIIF (L = 16) 34.72 31.01 29.04 26.79 22.21 20.52 IPE-LIIF (L = 4) 34.71 31.00 29.03 26.78 22.21 20.52 LIIF w/ PE (L = 10) 34.59 30.92 29.00 26.76 22.21 20.52 IPE-LIIF + cell 34.72 31.01 29.04 26.79 22.18 20.49 IPE-LIIF-skip 34.67 30.97 29.01 26.76 22.19 20.50Method
In-distribution
Out-of-distribution
×2
×3
×4
×6
×18
×30
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"Ingo Roth \nQuantum research centre\nTechnology Innovation Institute\nAbu DhabiUnited Arab Emirates\n",
"Martin Kliesch \nInstitute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany\n\nInstitute for Quantum-Inspired and Quantum Optimization\nHamburg University of Technology\nGermany\n",
"Mirko Arienzo \nInstitute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany\n",
"Markus Heinrich \nInstitute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany\n",
"Ingo Roth \nQuantum research centre\nTechnology Innovation Institute\nAbu DhabiUnited Arab Emirates\n",
"Martin Kliesch \nInstitute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany\n\nInstitute for Quantum-Inspired and Quantum Optimization\nHamburg University of Technology\nGermany\n"
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"Institute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany",
"Institute for Quantum-Inspired and Quantum Optimization\nHamburg University of Technology\nGermany",
"Institute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany",
"Institute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany",
"Quantum research centre\nTechnology Innovation Institute\nAbu DhabiUnited Arab Emirates",
"Institute for Theoretical Physics\nHeinrich Heine University Düsseldorf\nGermany",
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Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haarrandom (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables. We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.Mirko Arienzo: [email protected] Markus
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[
"https://export.arxiv.org/pdf/2211.09835v1.pdf"
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2211.09835
|
e741a1b8a46f2552d57b3b4907885bf4f3937d35
|
Closed-form analytic expressions for shadow estima- tion with brickwork circuits
Mirko Arienzo
Institute for Theoretical Physics
Heinrich Heine University Düsseldorf
Germany
Markus Heinrich
Institute for Theoretical Physics
Heinrich Heine University Düsseldorf
Germany
Ingo Roth
Quantum research centre
Technology Innovation Institute
Abu DhabiUnited Arab Emirates
Martin Kliesch
Institute for Theoretical Physics
Heinrich Heine University Düsseldorf
Germany
Institute for Quantum-Inspired and Quantum Optimization
Hamburg University of Technology
Germany
Closed-form analytic expressions for shadow estima- tion with brickwork circuits
Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haarrandom (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables. We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.Mirko Arienzo: [email protected] Markus
Introduction
Retrieving information about the state of a quantum system is a long-standing problem in quantum information processing and of central practical importance in quantum technologies. Full quantum state tomography can recover a complete, precise classical description of the state but requires a large number of state copies [1][2][3][4][5], making the protocol feasible only for a very moderate number of qubits. Nevertheless, for many concrete tasks, complete knowledge of the quantum state is often unnecessary [6], and estimation schemes for specific properties are often scalable.
A particularly attractive estimation primitive is nowadays referred to as shadow estimation [7,8]. Here, an approximation of a repeatedly prepared unknown quantum state, the so-called classical shadow, is constructed from measurements in randomly selected bases. In the limit of many bases, this approach allows, in principle, for full state tomography. For this reason, classical shadows can be further post-processed to construct estimators for the expectation value of arbitrary sets of observables. Importantly, for certain random measurement ensembles, rigorous analytical guarantees ensure that precise estimates of expectation values can be evaluated long before one has collected enough measurement statistics for full quantum state tomography.
The original examples with strict guarantees on the sample complexity are, in a sense, two "extreme" scenarios: The first one is characterized by evolving the state with a global random Clifford unitary before performing a basis measurement. It is particularly suited for predicting global properties; for instance, fidelity estimation requires a constant number of samples with this setting. The second scheme is built on local Clifford unitaries and effectively amounts to perform measurements in random local Pauli bases. In this case, local properties can often be efficiently estimated [9][10][11]. Moreover, biasing the distribution of local Clifford unitaries to the estimation task at hand can yield further improvements in sample complexity [12].
An accurate estimation requires a precise experimental implementation of the random unitaries. Although more robust variants of shadow estimation exist [13,14], the implementation of global multi-qubit Clifford unitaries on near-term hardware will typically introduce too much noise to be useful for estimation.
Experimentally feasible alternatives, naturally interpolating between the two extreme cases and potentially lowering the sample complexity over local Clifford unitaries, are short Clifford circuits [15]. However, finding expressions for classical shadows for random low-depth Clifford circuits is a challenging task. For instance, the construction of Ref. [15] involves numerically solving a large system of equations.
In this work, we derive closed-form analytic expressions for the arguably simplest non-trivial circuit construction of classical shadows: One round of a brickwork circuit consisting of two layers of products of random unitaries. Besides providing a more direct construction of the classical shadow, these analytic expressions allow us to compare the sample complexity of the circuit construction to the one with local Clifford unitaries. In particular, we first observe that for Pauli observables, one shall look at pairs of adjacent qubits in the support of such observables and their relative position in the circuit. Then, we find that the (very short) brickwork shadows outperform the local Clifford ones for Pauli observables supported on sufficiently many qubits of a brickwork circuit. Conversely, we also observe that local Clifford unitaries yield a lower sample complexity in the case of Pauli observables supported on sparsely distributed qubits in the sense of the brickwork circuit.
The remainder is structured as follows: Following the observation that the associated measurement channel can be interpreted as a frame (super-)operator [16] in Section 2.2, we work out its matrix representation in the Pauli basis in Section 3. In particular, using well-known expressions for the second-moment operator of sufficiently uniform probability measures over the unitary group, we derive recurrence relations for subcircuits that can be analytically solved. In Section 4, we identify the regime where the resulting sample complexity outperforms the shadow estimation protocol with the local Cliffords ensemble, and in Section 4.1 we compare numerically the performance of brickwork and local Cliffords shadows.
Related works.
During the completion of this work, two other papers on brickwork circuits were published [17,18]. Both describe shadows associated with brickwork circuits of arbitrary depth and numerically study the measurement channels associated with such circuits using tensor network techniques. In particular, Ref. [17] applies the formalism based on entanglement features introduced in Ref. [19] and discusses bounds on sample complexity based on the locally scrambled shadow norm [15]. A similar discussion, following a probabilistic interpretation of the eigenvalues of the measurement channels, is done in Ref. [18], which also discusses bounds based on the shadow norm.
Preliminaries
Notation
We denote the Hilbert-Schmidt inner product by a braket-like notation, namely
Tr(A † B) ≡ (A|B) A, B ∈ C d×d .(1)
Likewise, the outer product |A)(B| denotes the superoperator C → (B |C)A. We parametrize single-qubit Pauli operators by binary vectors v = (z, x) ∈ F 2 2 as
W (0, 0) := 1, W (0, 1) := X, W (1, 0) := Z, W (1, 1) := Y,(2)
where X, Y, Z ∈ C 2×2 are the usual Pauli matrices. Then, we define the n-qubit Pauli operators as tensor products of the single-qubit Pauli operators, indexed by vectors v = v 1 ⊕ · · · ⊕ v n ∈ F 2n 2 :
W (v) := W (v 1 ) ⊗ · · · ⊗ W (v n ) .(3)
For a given vector v = v 1 ⊕ · · · ⊕ v n ∈ F 2n 2 , we define its weight vector as the binary vector wt(v) ∈ F n 2 such that wt(v) i = 0 if v i = (0, 0) and wt(v) i = 1 else. In other words, wt(v) has a zero in the ith position if and only if W (v) is the identity on the ith qubit. We use the shorthand
notation |v) ≡ 1 √ d W (v) (4)
for the normalized Pauli operators. Hence, the set {|v)} denotes the orthonormal Pauli basis in C d×d , where d = 2 n denotes the dimension of the Hilbert space of n qubits from now on. Finally, for any k ∈ N, we set [k] := {1, . . . , k}.
Classical shadows formalism
In this section, we review the shadow estimation protocol [7] in the language of frame theory (see Ref. [20] for an introduction to frame theory [1,22], and the associated measurement channel
ρ = σ, i |U ρU † |i = i |U σU † |i . This ensures that {E i,U } is a frameS(ρ) := i∈[d] E U ∼ν |E i,U )(E i,U |ρ) = i∈[d] E U ∼ν i |U ρU † |i U † |i i|U (5)
has the interpretation as a frame operator. In particular, S is positive definite, and thus invertible. Then, {Ẽ i,U := S −1 (E i,U )} is the so-called canonical dual frame, and we have the following relation
Tr(Oρ) = Tr(OS −1 S(ρ)) = i E U ∼ν (O|Ẽ i,U )(E i,U |ρ) .(6)
Therefore, the last expression can be interpreted as the expected value of f O (i, U ) := (O|Ẽ i,U ) when sampling U ∼ ν and i ∼ (E i,U |ρ) and is, thus, the limit of the empirical average over many experimental snapshots. However, the computation of the canonical dual frame is in general a highly non-trivial task. Analytical inversion of S is often only possible in special cases where the probability measure ν is very structured. For instance, if ν is the Haar measure on U(d), or a unitary 2-design, then the POVM {E i,U } is a complex projective (state) 2-design and, thus, forms a tight frame on the subspace of traceless Hermitian matrices. As a consequence, S is a depolarizing channel and can be readily inverted. A similar argument can be applied when the unitaries U are drawn Haar-randomly from a subgroup G ⊂ U(d) [23]. More generally, one has to rely on numerical methods which are not only expensive, but may also be numerically unstable since there are no general guarantees on the condition number of S. In principle, the condition number can even be exponentially large [23].
Under certain conditions, the inversion of S is however drastically simplified: For instance, if the measure ν is right-invariant under multiplication with Pauli operators, then S is diagonal in the Pauli basis [19]. This follows from the observation that, in this case, we have
W(v) † SW(v) = S, where W(v) := W (v)( · )W (v) † ,
and hence S is invariant under the channel twirl over the Pauli group. Thus, it is a Pauli channel and in particular diagonal in the Pauli basis, which means S −1 can be computed via entrywise inversion of the diagonal elements (v |S |v).
Finally, if O = W (v) is a Pauli observable (we call this task Pauli estimation), the sample complexity of shadow tomography can be bounded for simple circuits. In particular, if S is diagonal in the Pauli basis, we simply have , and the second layer is shifted by one position. On the left, the bricks in the second layer acts on qubits (2i, 2i + 1) , i ∈ [5], with the periodic identification n + 1 = 1. On the right, the second layer acts on qubits (2i, 2i + 1) , i ∈ [4], leaving the first and the nth qubit untouched.
f W (v) (i, U ) = 1 (v |S |v) (W (v)|E i,U ) .(7)
Note that this expression features only a single diagonal element of the frame operator independent of i and U . The sample complexity of the corresponding mean estimatorŵ(v) can be controlled using the variance of f W (v) (i, U ) which can be shown to be dominated by (v |S |v) −1 [19]. Chebyshev's inequality then ensures that the mean estimator is -precise using O((v |S |v) −1 −2 δ −1 ) many snapshots with probability 1 − δ. Note that a Hoeffding bound here yields a worse bound scaling as (v |S |v) −2 . If the expectation values of 'many' observables are to be estimated at once, it may be beneficial to use the median-of-mean estimator with sample complexity depending only logarithmically on δ [7]. In general, however, it is not easy to find strict guarantees for the sample complexity, since it is hard to analytically bound the variance, even for different classes of Pauli invariant measures. In these cases, one can rely on the weaker notion of locally scrambled shadow norm [15,17,18], which can be interpreted as the average variance over all states. In particular, since the variance is linear in the state ρ, the locally scrambled shadow norm thus quantifies the performance when ρ is the completely mixed state.
The brickwork circuit: analytical results
We assume for simplicity that the number of qubits is even and consider one round of a onedimensional brickwork (BW) circuit built in the following way: a first layer of n/2 two-local Haar random unitaries is applied to qubits (2i − 1, 2i) for i ∈ [n/2]. The second layer, built in the same way but shifted by one position, applies Haar random unitaries to qubits (2i, 2i + 1). Here, we consider two cases, see also Figure 1. First, the second layer has periodic boundary conditions such that qubits n + 1 and 1 are identified, and consequently, the n/2th random unitary acts on the qubit pair (n, 1). Second, we treat the case of open boundary conditions, where the second layer does not act on the first and the nth qubit.
In the following, we derive analytical results for the frame operator of random brickwork circuits with open and periodic boundary conditions. Both BW circuit ensembles are clearly (left and right) invariant under tensor products of single-qubit unitaries, in particular they are right-invariant under Pauli operators. By the preceding discussion in Section 2.2, we thus know that the frame operator S is diagonal in the Pauli basis. It is thus sufficient to compute the matrix elements (v |S |v) for all v ∈ F 2n 2 . Moreover, both BW circuit ensembles are also invariant under local Clifford unitaries, i.e. tensor products of single-qubit Clifford gates. This implies that (v |S |v) is invariant under the exchange of X, Y , and Z operators, and hence depends only on the weight vector wt(v). As we show shortly, (v |S |v) is in fact determined by non-vanishing pairs of elements in wt(v) corresponding to a brick in the second layer, and by their positions in the circuit. To make this precise, we have to introduce some definitions.
For a given v = v 1 ⊕ · · · ⊕ v n ∈ F 2n 2 , we define the vector of supported bricks as Figure 2: Example how the vector of supported bricks is computed from a 10-qubit Pauli operator. The structure of the shaded bricks is the one of the second layer of the circuit. First, the Pauli operator is transformed into its binary representation v ∈ F 20 2 . We apply a logical or (∨) per qubit to compute the weight vector wt(v). Subsequently, this procedure is repeated for qubit pairs (2i, 2i + 1) and yields the vector of supported bricks v. For periodic boundary conditions, the last entry ofṽ is computed between the last and first entry of wt(v) (here depicted by appending the first entry at the end in parentheses). The brickwork support of this example
v = (ṽ 1 , . . . ,ṽ n/2 ) ∈ F n/2 2 ,ṽ i := wt(v) 2i ∨ wt(v) 2i+1 , i ∈ [n/2] ,(8)W (v) = Y ⊗ X ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ Z ⊗ 1 ⊗ 1 ⊗ Y v = 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ wt(v) = 1 1 0 0 0 0 1 0 0 1 (1) ∨ ∨ ∨ ∨ ∨ v = 1 0 1 0 1is supp BW (v) = {1, 3, 5}, while its partition into local factors is part BW (v) = (1, 2).
where x ∨ y is the logical or between two bits x, y ∈ F 2 , i.e. x ∨ y = 1 if x = 1 or y = 1, and 0 else. The last entry ofṽ is defined according to the boundary conditions of the second layer, in particularṽ n/2 = wt(v) n ∨ wt(v) 1 for periodic boundary conditions, andṽ n/2 ≡ 0 for open boundary conditions, see also Fig. 2 for an explicit example howṽ is computed. We say that the ith brick in the second layer,
with i ∈ [n/2], is in the support of v ifṽ i = 1. Then, one can define the brickwork support of v ∈ F 2n 2 as supp BW (v) := {i |ṽ i = 0}.
In the following, however, it will be equally important to keep track of sequences of consecutive supported bricks in the circuit. Hence, we introduce the following notation: A one-component ofṽ is a maximal tuple of consecutive ones inṽ, where "consecutive" is again meant w.r.t. the boundary conditions of the BW circuit. Then, we define the partition of the brickwork support part BW (v) to be the integer sequence given by the (non-unique) sizes of the one-components ofṽ. For instance, if we have periodic boundary conditions andṽ = (1, 0, 1, 0, 1) as in Fig. 2, then part BW (v) = (1, 2). Note that the maximal number of consecutive ones is n/2 − 1 and n/2 for open and periodic boundary conditions, respectively.
We can now state our main result:
Theorem 1. Let S be the frame operator associated with one round of a two-local brickwork circuit with open or periodic boundary conditions in the second layer. Then, S is diagonal in the Pauli basis, and for
v ∈ F 2n 2 (v |S |v) = Σ pb (n) , part BW (v) = (n/2) , l∈part BW (v) Σ ob (2l + 2) , otherwise ,(9)
where Σ pb (m) =
√ 41 + 5 m/2 + (−1) m/2 √ 41 − 5 m/2 (5 √ 2) m ,(10)Σ ob (m) = 5 2 √ 41 25 − 3 √ 41 √ 41 + 5 m/2 + (−1) m/2+1 25 + 3 √ 41 √ 41 − 5 m/2 5 √ 2 m .(11)
We provide a proof for the theorem in Section 3.1. Let us briefly comment on the interpretation of the matrix elements of S. These values, determined by the elements of part BW (v), are associated with different topologies of the effective BW circuit.
First, notice that the case part BW (v) = (n/2) can occur for periodic boundary conditions only and corresponds to all bricks being in the support of v. In particular, for open boundary conditions, the second case in Eq. (9) always applies.
Next, let us motivate the second case in Eq. (9). Concretely, let us first assume part BW (v) = (n/2 − 1). In the case of open boundary conditions, this assumption corresponds to all bricks being On the left, the effective BW circuit associated with a v ∈ F 20 2 such thatṽ2 = 0 and part BW (v) = (4). In this case, open boundary conditions apply and the circuit is topologically equivalent to the right one in Figure 1. On the right, the effective BW circuit associated with v ∈ F 20 2 such thatṽ2,ṽ5 = 0, which implies supp BW (v) = {1, 3, 4} and part BW (v) = (1,2). In this case, the circuit is the product of two smaller subcircuits with open boundary conditions. In particular, the subcircuits are defined on 4 and 6 qubits, respectively.
in the support of v, sinceṽ n/2 = 0 by definition. Likewise, this situation occurs in the BW circuit with periodic boundary conditions whenever there exists exactly one i ∈ [n/2] such thatṽ i = 0, see Figure 3. Then, we can make two observations: First, the topology of the effective circuit changes from periodic to open boundary conditions. Second, the effective circuit is equivalent -up to reordering of qubits on which it acts-to the fully supported circuit with open boundary conditions described before, which is depicted in Figure 1. In general, the effective circuit splits into as many independent BW circuits with open boundary conditions as the number of elements in part BW (v), and the diagonal elements of S are given by products of different contributions as in Eq. (9). These elements also determine the number of qubits on which these subcircuits act, see Figure 3 for an example with |part BW (v)| = 2.
The frame operator provides, as proved in Ref. [19], a bound on the variance of the Pauli estimation task. Here, the variance of a single sample is given as
σ 2 BW (v, ρ) := i E U ∼µ f W (v) (i, U ) 2 (E i,U |ρ) − (W (v)|ρ) 2 ,(12)
where f W (v) is defined as in Eq. (7). Then, the following result holds:
σ 2 BW (v, ρ) ≡ σ 2 BW (v) ≤ 1 (v |S |v) .(13)
We remark that the latter holds in general for any ensemble invariant under Pauli multiplication. In Appendix C we provide an alternative proof for circuits with periodic boundary conditions which holds for unitary 3-designs.
Proof of Theorem 1
In the following, given an operator A ∈ L(C 2 ⊗ C 2 ), we denote by A (2)
≡ A ⊗ A ∈ L(C 4 ⊗ C 4 )
the operator acting on two copies of two qubit "sites". If it is clear from the context, we will also simply write A (2) ≡ A. In particular, we will extensively use the operator F (2) ≡ F ⊗ F , where F is the flip operator which swaps tensor factors of C 2 ⊗ C 2 .
As we observed in Section 2.2, the frame operator is diagonal in the Pauli basis [19] for Pauli invariant measures. However, since unitaries in the BW circuit are Haar random, we can characterize the matrix elements of the frame operator exploiting some known results about the second moment operator [1], which we summarize in Lemma 6 in Appendix A. As a side note, this also implies that we can draw unitaries from any unitary 2-design instead [24,25]. In particular, it is more practical to draw unitaries from the Clifford group, which, for qubits systems, is even a 3-design [26,27]. In fact, on the one hand, implementing Haar-random unitaries is very hard [28] and, on the other hand, employing Clifford unitaries ensures one can classically post-process shadows efficiently [29].
Then, the following holds:
(v |S |v) = 1 (10 √ 3) n t pb (n) , part BW (v) = (n/2) 1 15 |ṽ| l∈part BW (v) t ob (2l+2) (2 √ 5) 2l+2 , otherwise ,(14)
where |ṽ| is the Hamming weight ofṽ, and
t pb (n) := Tr (1 + F (2) ) ⊗n/2 D 4F (2) − 1 ⊗n/2 D −1 ,(15)t ob (n) := Tr (1 + F (2) ) ⊗n/2 1 4 ⊗ (4F (2) − 1) ⊗(n/2−1) ⊗ 1 4 ,(16)
with D|ψ 1 ⊗ · · · ⊗ |ψ n := |ψ 2 ⊗ · · · ⊗ |ψ n ⊗ |ψ 1 being a cyclic shift operator.
Proof. Let µ be a probability measure on the BW circuit. Hence, µ is the product of probability measures µ ij , where i = 1, . . . , n/2, and j = 1, 2. In other terms, the operator corresponding to the i j th brick is sampled independently from all the others. Then, given u, v ∈ F 2n 2 , we have
(u |S |v) = 1 d i E U ∼µ (u|E i,U )(E i,U |v) = 1 d i E U ∼µ Tr[W (u) † U |i i|U † ] Tr[U |i i|U † W (v)] = E U ∼µ 0 |U ⊗2 † W (u) ⊗ W (v)U ⊗2 |0 .(17)
Consider now the following factorization of W (v):
W (v) = W (v 2,3 ) ⊗ · · · ⊗ W (v n,1 ) ,(18)
where, for
each i ∈ [n/2], v 2i,2i+1 ≡ v 2i ⊕ v 2i+1 ∈ F 2 2 ⊕ F 2 2 ,(19)
and each W (v 2i,2i+1 ) is as in Eq. (3). Moreover, writing U = DU 2 D † U 1 , where U i is the tensor product of two-local Haar random unitaries, it follows
(u |S |v) = Tr E U1 U ⊗2 1 |0 0|U ⊗2 † 1 D E U2 U ⊗2 † 2 D † W (u) ⊗ W (v)DU ⊗2 2 D † .(20)
Hence, by Lemma 6 in Appendix A, we have
E U1 U ⊗2 1 |0 0| ⊗2 U ⊗2 † 1 = 1 10 n/2 P ⊗n/2 sym 2 ,(21)E U2 U ⊗2 2 D † W (v) ⊗ W (u)DU ⊗2 † 2 = δ u,v i∈[n/2] Qṽ i ,(22)
where
Qṽ i := 1 ifṽ i = 0 , 1 15 4F (2) − 1 otherwise .(23)
Therefore, writing P sym 2 = 1 2 1 + F (2) ,
(v |S |v) = 1 (2 √ 5) n Tr (1 + F (2) ) ⊗n/2 D i∈[n/2] Qṽ i D −1 .(24)
Finally, we distinguish cases for the latter according to part BW (v). In particular, if part BW (v) = (n/2), then Qṽ i = 1 15 4F (2) − 1 ∀i = 1, . . . , n/2, and Eq. (15) reads immediately from Eq. (24). Next, assume part BW (v) = (n/2 − 1). In particular, due to invariance under translations of bricks, we can assume without loss of generality thatṽ n/2 = 0, meaning Qṽ n/2 = 1. This yields
(v |S |v) = 1 (2 √ 5) n 1 15 n/2−1 Tr (1 + F ) ⊗n/2 1 ⊗ (4F − 1) ⊗n/2−1 ⊗ 1 .(25)
Consider now there exists i ∈ [n/2 − 1] such thatṽ i = 0. Then, we distinguish between two cases. If i = 1 or i = n/2 − 1, then (v |S |v) is still given by an expression that is morally equivalent to Eq. (25) up to obvious modifications determined by part BW (v) = (n/2 − 2). More specifically, we have
(v |S |v) = 1 (2 √ 5) n 1 15 n/2−2 Tr [1 + F ] Tr (1 + F ) ⊗n/2−1 1 ⊗ (4F − 1) ⊗n/2−2 ⊗ 1 = 1 (2 √ 5) n−2 1 15 n/2−2 t ob (n − 2) .(26)
On the other hand, if i ∈ {2, . . . , n/2 − 2}, we have part BW (v) = (i − 1, n/2 − 1 − i), and the circuit splits into two subcircuits, yielding
(v |S |v) = 1 (2 √ 5) n Tr (1 + F ) ⊗i 1 ⊗ (4F − 1) ⊗(i−1) ⊗ 1 × Tr (1 + F ) ⊗n/2−i 1 ⊗ (4F − 1) ⊗n/2−1−i ⊗ 1 = 1 (2 √ 5) n t ob (2i) t ob (n − 2i) .(27)
All other cases follow from analogous considerations.
Note that the traces in the latter expression have two main contributions. The first one, which is proportional to the projector on the symmetric subspace P sym 2 , comes from scrambling E i with the first layer of the BW circuit, and it is independent of v. The second layer, on the other hand, acts on W (u) ⊗ W (v), and the result of the scrambling for each pair of qubits is an operator that depends on v. This means that effectively the second layer determines whether the circuit factorizes at a given position, and the number of qubits on which each subcircuit is defined is determined by the corresponding first layer of random unitaries.
The next couple of technical results will give a way to evaluate the traces appearing in the previous lemma. The core steps of the proofs are most conveniently presented in terms of tensor network diagrams and deferred to Appendix B. Lemma 4. Let t 1 (n), t 2 (n), t 3 (n) defined as follows:
t 1 (n) := Tr 1 ⊗ (1 + F ) ⊗n/2−1 ⊗ 1 (4F − 1) ⊗n/2 ,(28)t 2 (n) := Tr F ⊗ (1 + F ) ⊗n/2−1 ⊗ F (4F − 1) ⊗n/2 ,(29)t 3 (n) := Tr 1 ⊗ (1 + F ) ⊗n/2−1 ⊗ F (4F − 1) ⊗n/2 .(30)
Then, t pb (n) = t 1 (n) + t 2 (n) and the following system of recursive relations hold true:
t 1 (n) = 24 t 3 (n − 2) t 2 (n) = 24 t 3 (n − 2) + 60 t 2 (n − 2) t 3 (n) = 24 t 1 (n − 2) + 60 t 3 (n − 2)
, n ≥ 2 , n = 0 mod 2 ,
with the following base conditions:
t 1 (2) = 0 t 2 (2) = 60 t 3 (2) = 24 .(32)
Proof. The fact that t pb (n) = t 1 (n) + t 2 (n) is clear from the definition of t 1 and t 2 . Relations (31) and (32) are proved in Appendix B.
Lemma 5.
Let t 1 (n), t 2 (n) be defined as follows:
t 1 (n) := Tr 1 4 ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 4 ,(33)t 2 (n) := Tr F ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 4 .(34)
Then, t ob (n) = 4t 1 (n) + 2t 2 (n), and the following recursive relations hold true:
t 1 (n) = 24 t 2 (n − 2) t 2 (n) = 24 t 1 (n − 2) + 60 t 2 (n − 2) n ≥ 4 , n = 0 mod 2 ,(35)
with the following base conditions:
t 1 (4) = 48 t 2 (4) = 216 .(36)
Proof. First, observe that
t ob (n) := Tr (1 + F (2) ) ⊗n/2 1 ⊗ 4F (2) − 1 ⊗(n/2−1) ⊗ 1 = Tr Tr 1 (1 + F (2) ) ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 = Tr (41 + 2F ) ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 = 4 t 1 (n) + 2 t 2 (n) .(37)
Relations (35) and (36) are proved in Appendix B.
Proof of Theorem 1. As discussed before, S is diagonal in the Pauli basis, and we only need to characterize its diagonal elements (v |S |v), which, by Lemma 3, are determined by part BW (v). In the first case, when part BW (v) = (n/2), the circuit retains periodic boundary conditions, and (v |S |v) is proportional to t pb (n) according to Eq. (14). By Lemma 4, t pb (n) can be expressed in terms of the solution of the system of recurrence relations (31), whose closed form yields Eq. (10). Otherwise, part BW (v) determines the factorization into (possibly many) subcircuits with open boundary conditions. In particular, each entry l ∈ part BW (v) determines a (factorized) subcircuit acting on 2l+2 qubits. Each such subcircuit evaluates up to a multiplicative constant to t ob (2l+2), that, by Lemma 5 fulfills the recurrence relations (35). The latter relations also admit a closed solution, which leads to Eq. (11) for each subcircuit.
As a final remark, observe that the proof of the theorem can be generalized to systems of arbitrary prime or power of prime local dimension. In particular, redefiningṽ according to the local dimension, Eq. (24) holds true with obvious modifications for any prime p, and the same holds for the traces Eqs. (15) and (16).
Finally, one may wonder whether it is possible to find analytical expressions for the frame operator associated with circuits with more layers. However, in this case, splitting the scrambling over multiple layers is more involved, since non-trivial 'intertwinings' between layers occur. This implies that the analytical contraction of the corresponding tensor network is more difficult compared to the calculations of Appendix B, and one might only resort to numerical methods to evaluate the frame operator [17,18].
Discussion and comparison with local Clifford circuits
Given the closed analytic expressions for the frame operator associated with the BW circuit, we can now compare the performance with the local Cliffords (LCs) ensemble.
For LCs, the variance is exponential in the weight of the Pauli observable [7]. More precisely, for any v ∈ F 2n 2 we define the local Cliffords support as the set of weighted sites of v, namely
supp LC (v) := {i | wt(v) i = 0}.
Then, since the LCs ensemble is clearly invariant under multiplication with Pauli operators, one can apply Proposition 2 to get a bound on the variance. In particular,
σ 2 LC (v) ≤ 1 (v |S LC |v) = 3 |supp LC (v)| ,
where S LC is the frame operator associated with LCs shadows (see Appendix A.1 for the derivation of its matrix elements). Notice also that, although this bound corresponds to the shadow norm [7], one does not need to maximize over all the states. The inequality solely originates in disregarding the square of the first-moment (E i,U f W (v) (i, U )) 2 which agrees by construction with Tr(W (v)ρ) 2 for any ensemble. Therefore, we are most of the time comparing the exact expressions for the second moment E i,U [f W (v) (i, U ) 2 ] allowing us to formally deduce lower and upper bounds. In the following, all our expressions for the variances are understood as being up to first-moment terms and we write, e.g. σ 2 LC = 3 n . As derived in the previous section, the variance for the brickwork circuit depends on the partitioning of the brick support into local factors.
We obtain the simplest expression when part BW (v) = (n/2). In this case, Theorem 1 together with Proposition 2 implies that 0.8 · 2 n < σ 2 BW < 2.1 n+1 ; see Appendix D for details. To compare the scaling of σ 2 BW to the one of σ 2 LC , we introduce some notation to distinguish different regimes. First, recall that we say that v ∈ F 2n 2 is supported on the ith brick ifṽ i = 1, and, by definition,ṽ i = 1 if at least one of wt(v) 2i and wt(v) 2i+1 is non-zero. A supported brick can further be of two types. If wt(v) 2i ∧ wt(v) 2i+1 = 1, namely the logical and between the two local weights is non-trivial, the ith brick is said to be fully supported. Otherwise, the ith brick is said to be half supported if the logical xor between the two local weights is non-trivial, or more formally wt(v) 2i wt(v) 2i+1 = 1.
Still assuming part BW (v) = (n/2), we have two extreme cases:
• If each brick is fully supported, then σ 2 LC = 3 n > 2.1 n+1 > σ 2 BW for all n ≥ 2. Thus, the brickwork circuits have an improved sample complexity compared to single qubit random Clifford unitaries. The number of samples is reduced by one order of magnitude for n ≥ 8 and by a factor of about 0.5 · 10 −4 for n = 20.
• If each brick is half-supported, then σ 2 LC = 3 n/2 < 1.75 n < 0.8 · 2 n < σ 2 BW for all n ≥ 2. In this case, the BW circuit retains its periodic structure, while LCs shadow sees the 'correct' number of qubits in the support leading to a smaller sample complexity.
Similar considerations apply if part BW (v) = (n/2 − 1) i.e. when (v |S |v) is given by a single term with open boundary condition. Evaluations of the expressions for both cases are summarized in Figure 4a. We observe that the scaling for both cases only differ in a constant factor as we also explain analytically in Appendix D.
The two extreme cases suggest that shadows with BW circuits outperform the LCs ones when the number of fully supported bricks reaches a certain threshold. More specifically, based on our bounds, we can guarantee a lower sample complexity with BW circuits if |supp LC (v)| ≥ 0.68(n + 1) for the cases part BW (v) = (n/2) and part BW (v) = (n/2 − 1), see Appendix D. Furthermore, the additional constant term can be decreased for larger number of qubits. Evaluations of the threshold for up to 100 qubits are summarized in Figure 4b.
Relaxing the restriction on part BW , we can ensure that σ 2 BW ≤ σ 2 LC provided that
|supp LC (v)| ≥ 0.8| part BW (v)| + 1.4 |ṽ| ,(38)
where | part BW (v)| is the number of entries in part BW (v) and |ṽ| the Hamming weight ofṽ or, in other words, the sum of all the entries of part BW (v). The derivation of this criterion is given in Appendix D.
The threshold criterion is likely to hold for random Pauli observables, since, for a fixed n, few additional qubits are needed to reach the threshold. On the other hand, for any random Pauli string v ∈ F 2n 2 , the probability of the ith brick to be fully supported is strictly larger than the probability of being half supported. Indeed, evaluating the bounds for random Pauli strings we observe that the brickwork circuit gives better performance with high probability drawing random Pauli strings, and p(σ 2 BW ≤ σ 2 LC ) n big − −− → 1, see Figure 5.
Numerical experiments
We now compare numerically performances of BW and LCs estimation procedures. We fix n = 10 as the number of qubits, and consider for simplicity ρ = |0 0| as the input state. Then, we collect numerical data for three different Z-type operators, that we assume to be supported on each brick. In particular, we consider the following Pauli strings: v full , which is supported on each qubit, v half , where each brick is half supported, and v thres , which is supported on 8 qubits, ensuring it satisfy the threshold criterion discussed in the last section. Notice that it does not matter where the two half supported bricks are located in v thres , since all of them are supported. Finally, drawing unitaries from the Clifford group, we can classically simulate the whole procedure efficiently using standard techniques [29][30][31]. More details on the algorithms are provided in Appendix E and at the following link: https://github.com/MirkoArienzo/shadow_short_circuits. Then, we fix m as the number of samples, and compute the empirical average over all samples as described in Section 2.2, which yields an estimator for the given observables and ρ = |0 0|. We run this procedure 100 times, and evaluate the average of the estimators over all runs. The latter has standard deviation given by σ/ √ 100m, with σ = σ BW , σ LC . Finally, the task is repeated for different values of m.
The results of the simulations, shown in Figure 6, agree with the previous discussion. In particular, for circuits that are fully supported or over the threshold, the convergence to the expected values is faster using BW circuits, see Figure 6a and Figure 6b, while the converse
Conclusions
Shadow estimation with short circuits can interpolate between the originally proposed constructions with local and global Clifford unitaries. We derived closed-form analytic expressions for the frame operator (and its inverse) associated with the arguably simplest circuit construction: one round of a brickwork circuit. In particular, we observed how the 2-design property of bricks can be used to determine systems of recurrence relations for the contributions of subcircuits with effectively periodic and open boundary conditions. The recurrence relations admit closed-form solutions and can be used to calculate the classical shadow and the corresponding linear estimators. Furthermore, and in contrast to numeric approaches, we explicitly worked out and analyse the sample complexity of Pauli estimation with one round of a brickwork circuit. This gave rise to a simple criterion for the structure of the support of the Pauli observable in order to have a scaling advantage compared to using local Clifford unitaries. Going beyond the worked out example, our results provide clear evidence for the potential of using short circuits for shadow estimation but also indicate limitations and the need for careful comparison in specific applications. Besides shadow estimation, the analytic expression for the frame operator can potentially also unlock the usage of short depth circuits in related tasks involving randomized measurements such as benchmarking and mitigation.
We expect that generalizing our analytic approach will become considerably more involved for deeper circuits, especially for deriving the exact frame operator required for constructing the classic shadow. To this end, we regard numerical methods as a considerably more flexible approach. Nonetheless, it might be possible to derive sample complexity bounds for deeper circuits and other estimation tasks following and generalizing the argument presented here. Altogether, our work provides both the analytical results and the motivation for implementing short depth quantum circuits in an actual shadow estimation experiment. After all, the merits of the approach have to be evaluated in practice.
Appendices
A Auxiliary results
Lemma 6. Let ν be the normalized Haar measure on U(d). Then, for any integer
k ≥ 1 E U ∼ν U ⊗k |0 0|U ⊗k † = k + d − 1 d − 1 −1 P sym k ,(39)
which evaluates to
2 d(d+1) (1 + F ) for k = 2,
where P sym k is the projector onto the completely symmetric subspace and F ∈ L(C d ⊗ C d ) the flip operator. Moreover, given u, v ∈ F 2n 2 ,
E U ∼ν U ⊗2 W (u) ⊗ W (v)U ⊗2 † = 1 , if u = 0 and v = 0 δu,v d 2 −1 (d F − 1) , otherwise ,(40)
where δ u,v is the Kronecker δ over F 2n 2 .
Proof. See e.g. [1, Section II.I].
A.1 Frame operator for Local Cliffords shadows
Here, we include the calculation for the frame operator of LC shadows. It follows the same steps as the calculation given in Section 3.1 for BW shadows. However, dealing with only one layer of Clifford unitaries makes things more straightforward. In particular, given u, v ∈ F 2n 2 , and denoting by S LC the frame operator associated with LC shadows, we have
(u |S LC |v) = 1 d i E U ∼µ (u|E i,U )(E i,U |v) = 1 d i E U ∼µ Tr[W (u) † U |i i|U † ] Tr[U |i i|U † W (v)] = E U ∼µ 0 |U ⊗2 † W (u) ⊗ W (v)U ⊗2 |0 .(41)
Considering the factorization
W (v) = W (v 1 ) ⊗ · · · ⊗ W (v n ) ,(42)
by Lemma 6, we have
E U ∼µ U ⊗2 † W (u) ⊗ W (v)U ⊗2 = δ u,v 1 3 |supp LC (v)| (2 F − 1) ⊗|supp LC (v)| ,(43)
where
supp LC (v) := {i | wt(v) i = 0}.
Therefore, it follows
(v |S LC |v) = 1 3 |supp LC (v)| 0 |2F − 1|0 |supp LC (v)| = 1 3 |supp LC (v)| .(44)
B Tensor networks for Lemma 4 and 5
In this section, we show how t 1 , t 2 , t 3 (respectively, t 1 , t 2 ) appearing in Lemma 4 (respectively, Lemma 5) can be written as a system of recurrence relations using tensor networks.
First, notice that each operator in the traces Eqs. (15) and (16) acts on two copies of 2 qubits. That means that each brick is represented by two overlapping copies, see Fig. 7. Next, given (the two copies of) a brick, we set for notational purpose ≡ .
Now notice that each brick is made up of identities and flips, the latter is usually represented as
F = .(45)
Finally, t pb (n) and t ob (n) will be simplified exploiting linearity and separability of bricks. For this purpose, and to simplify the notation, we rewrite each pair of lines corresponding to the same qubit as a single one. In particular, if two lines are straight (the identity operator is applied), we summarize them as a single black line, otherwise as a red line when the flip operator is applied. For instance, for a brick 1 + F in the first layer of the circuit, we have
1 + F = + ≡ + .
B.1 Proof of relations (31) and (32) We show how to derive recurrence relations in Lemma 4. First, recall the following definitions:
t 1 (n) := Tr 1 ⊗ (1 + F ) ⊗n/2−1 ⊗ 1 (4F − 1) ⊗n/2 (46) = . . . ,(47)t 2 (n) := Tr F ⊗ (1 + F ) ⊗n/2−1 ⊗ F (4F − 1) ⊗n/2 (48) = . . . ,(49)
B.2 Proof of relations (35) and (36)
Recall that, in this case,
t 1 (n) := Tr 1 4 ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 4 (52) = . . . ,(53)t 2 (n) := Tr F ⊗ (1 + F (2) ) ⊗(n/2−1) (4F (2) − 1) ⊗(n/2−1) ⊗ 1 4 (54) = . . . .(55)
For the first trace we have Moreover,
t 1 (n) = . . . = 8 . . .t 1 (4) = 1 + F 4F − 1 = 8 1 + F − 4 1 + F = 48 , t 2 (4) = 1 + F 4F − 1 = 16 1 + F − 2 1 + F = 216 .
C Another take on the variance bound
In this section, we provide an alternative proof of Proposition 2 in the case of BW circuits with periodic boundary conditions that relies on the 3-design property of the multiqubits Clifford group.
In particular, it is based on the following result:
Lemma 7. Let n ∈ N and let d = 2 n be the dimension of a n-qubits system. If 0 = v ∈ F 2n 2 , then
E U ∈Cln(2) U ⊗3 W (v) ⊗2 ⊗ W (u) (U † ) ⊗3 = 1 d 2 − 1 δ u,0 (d F − 1 d 2 ) ⊗ 1 d ∀u ∈ F 2n 2 .(56)
Moreover,
E U ∈Cln(2) U ⊗3 1 ⊗2 d ⊗ W (u) (U † ) ⊗3 = δ u,0 1 d 3 ∀u ∈ F 2n 2 .(57)
Proof. First, we fix notations for the phase space representation of Clifford operators. Let [·, ·] : F 2n 2 × F 2n 2 → F 2 be the standard symplectic product over F 2n 2 . Letα g : F 2n 2 → Z 4 be a centre fixing automorphism of the associated Heisenberg-Weyl group, where g ∈ Sp 2n (2). One can prove thatα g = α g + [w, ·], where α g : F 2n 2 → F 2 is a suitable function satisfying the compatibility condition α g (0) = 0 [32,Sec. 3.3]. Notice also that |Cl n (2)| = d 2 |Sp 2n (2)|.
With these notations, the action of U ∈ Cl n (2) on Weyl operators can be written as
U W (v)U † := χ([a, v] + α g (v))W (g(v)) ,(58)
where χ(v) := i −vz·vx denotes the character of W (v).
Hence,
E U ∈Cln(2) U ⊗3 W (v) ⊗2 ⊗ W (u) (U † ) ⊗3 = 1 |Cl n (2)| a∈F 2n 2 χ([a, 2v + u]) g∈Sp 2n (2) χ(2α g (v) + α g (u))W (g(v)) ⊗2 ⊗ W (g(u)) = 1 2 2n a∈F 2n 2 χ([a, u]) 1 |Sp 2n (2)| g∈Sp 2n (2) χ(α g (u))W (g(v)) ⊗2 ⊗ W (g(u)) = 1 |Sp 2n (2)| δ u,0 g∈Sp 2n (2) W (g(v)) ⊗2 ⊗ W (g(u)) = 1 |Sp 2n (2) · v| δ u,0 w∈Sp 2n (2)·v W (w) ⊗ W (w) ⊗ 1 2 .(59)
Going from the third to the fourth line, we used a∈F 2n 2 χ([a, u]) = d 2 δ u,0 . In the last step, we wrote the average over Sp 2n (2) as an average over the orbit of v under Sp 2n (2). Notice that, since v = 0, Sp 2n (2) acts transitively on F 2n 2 \ 0 [32], the average over such orbit can be rewritten as an average over F 2n 2 \ 0. Moreover, recalling that the flip operator has the following Pauli expansion:
F = 1 d w∈F 2n 2 W (w) ⊗ W (w) ,(60)
it holds that
E U ∈Cln(2) U ⊗3 W (v) ⊗2 ⊗ W (u) (U † ) ⊗3 = 1 d 2 − 1 δ u,0 (dF − 1 d 2 ) ⊗ 1 d .(61)
If v = 0, the assertion follows trivially from previous considerations.
We will also need the following calculation:
Lemma 8. Given two copies of a system of 3 qubits, it holds
Tr 3 P sym 3 = d + 2 6 (1 + F ) .(62)
Proof. We prove the latter using tensor network diagrams. First, let us consider the decomposition P sym 3 = 1 6 1 + P (1,2) + P (1,3) + P (2,3) + P (1,2,3) + P (1,3,2) , P (·) are unitary operators associated with elements of the permutation group S 3 , and each a ∈ S 3 is represented in cyclic notation. Then,
6 P sym = + + + + + = (d + 2)(1 + F ) .
Proposition 9.
For any state ρ, estimate W (v) using BW shadows with periodic boundary conditions. Then, the variance of the estimator depends only on v ∈ F 2n 2 , and
σ 2 BW (v) ≤ 1 (v |S |v) .(63)
Proof. In the following, we will denote by D the cyclic shift operator between Hilbert spaces as before, so that a random brickwork unitary is given by U = DU 2 D † U 1 , where U i , i = 1, 2, is the tensor product of two-local Haar random unitaries. For a given operator A ∈ L(C 2 ⊗ C 2 ), we will also consider the operator acting on three copies of two qubit sites A (3) := A ⊗ A ⊗ A ∈ L(C 8 ⊗ C 8 ), and simply write A (3) ≡ A if it is clear from the context. According to the shadow estimation protocol, we estimate the expectation value Tr[W (v)ρ] of some Pauli observable W (v) by measuring ρ many times in the computational basis after having applied U ∼ µ, where µ is a probability measure on the ensemble of BW operators. Then, a single such sample has a variance σ 2 BW (v, ρ) bounded as
σ 2 BW (v, ρ) := i E U ∼µ (W (v)|Ẽ i,U ) 2 (E i,U |ρ) − Tr[W (v)ρ] 2 ≤ i E U ∼µ Tr S −1 (W (v))U † E i U 2 Tr U † E i U ρ = d E U ∼µ Tr U ⊗3 † |0 0| ⊗3 U ⊗3 S −1 (W (v)) ⊗2 ⊗ ρ = d (v |S |v) 2 E U ∼µ Tr U ⊗3 † |0 0| ⊗3 U ⊗3 W (v) ⊗2 ⊗ ρ = 1 (2 √ 5) n d (v |S |v) 2 E U2 Tr P ⊗n/2 sym 3 D U ⊗3 2 D † W (v) ⊗2 ⊗ ρ D U ⊗3 † 2 D † ,(64)
where in the last step we applied again Eq. (39) from Lemma 6 and U = DU 2 D † U 1 . Notice that P sym 3 acts on triples of two neighboring qubit sites. c u W (u), where c u := 1 d Tr(W (u)ρ), and for any w ∈ F 2n 2 consider the decomposition
D † W (w)D = W (w 2,3 ) ⊗ · · · ⊗ W (w n,1 ) ,(65)
which agrees with the structure of the second layer of the BW circuit. Then, by Lemma 7, we have
E U2 U ⊗3 2 D † W (v) ⊗2 ⊗ W (u) D U ⊗3 † 2 = δ u,0 i∈[n/2] Qv i ⊗ 1 d ,(66)
from which it follows
σ 2 (v, ρ) ≤ 1 (2 √ 5) n d (v |S |v) 2 c 0 Tr P ⊗n/2 sym 3 D i∈[n/2] Qṽ i D −1 ⊗ 1 d ,(67)
where Qṽ i is defined in Eq. (23). Hence, σ 2 (v, ρ) ≡ σ 2 (v), since c 0 = 1/d. According to Lemma 7, this means that each Clifford unitary in the second layer depolarizes any dependency from the corresponding two-qubits Weyl operator appearing in the decomposition of ρ in the Pauli basis; periodic boundary conditions ensure that this applies to each pair of qubits. Finally, by Lemma 8,
σ 2 (v) ≤ 1 (2 √ 5) n 1 (v |S |v) 2 Tr Tr 3 (P ⊗n/2 sym 3 ) D i∈[n/2] Qṽ i D −1 = 1 (2 √ 5) 2n 1 (v |S |v) 2 Tr (1 + F ) ⊗n/2 D i∈[n/2] Qṽ i D −1 = 1 (v |S |v) ,(68)
Finally, notice that Lemma 7 does not hold for arbitrary values of the local dimension. Indeed, in odd dimensions, the flip operator has the Weyl expansion
F = 1 d v∈F 2 p W (v) ⊗ W (−v) ,(69)
meaning the operator 1
d v∈F 2n 2 W (v) ⊗ W (v)
admits a nice expression for fields of characteristic 2 only, and the proof of Proposition 9 holds for qubit systems only.
D Numeric bounds on the variance
For the construction of the classical shadow, the exact expressions stated in Theorem 1 are required. We here derive simpler bounds for controlling the variance.
Let us start with the case relevant for part BW (v) = (n/2). Set a := (
√ 41 + 5) 1/2 /(5 √ 2) and b := i( √ 41 − 5) 1/2 /(5 √ 2)
. We have 2 < 1/a < 2.1, 1 1−|b/a| 2 ≤ 1.2 and 1 1+|b/a| 2 ≥ .89. Further, since |b/a| < 1 and assuming n ≥ 2 and n even, 1 Σ pb (n) = 1 a n + b n = 1 a n
1 1 + (b/a) n ≤ 1 a n 1 1 − |b/a| 2 ≤ 1.2 · 2.1 n ,(70)
and analogously 1 Σ pb (n) ≥ 1 a n 1 1 + |b/a| 2 ≥ 0.8 · 2 n .
(71)
In the same way, we can bound 1/ Σ ob (n). (73)
Note that Eq. (72) goes to c −1 a −n ≈ 0.44 · 2.1 n for large n as the second fraction becomes 1 asymptotically. Similarly, 1 Σ pb (n) asymptotically becomes a −n . The deviation from this asymptotic scaling is small already for small n. E.g., the relative error of the asymptotic approximation is smaller than 10 −2 for n ≥ 6. Asymptotically the frame operator elements for periodic and open boundary conditions only differ by a constant factor c −1 ≈ 0.44.
A bound of the form 1/ Σ ob (n) ≤ Γ∆ n , implies that the variance is dominated by
σ 2 BW ≤ l∈part BW (v) Σ ob −1 (2l + 2) ≤ (Γ∆ 2 ) | part BW (v)| ∆ 2Σ(part BW (v)) ,(74)
where | part BW (v)| denotes the length of the tuple part BW (v), i.e. the number of parts in the partition, and Σ(part BW (v)) := l∈| part BW (v)| l = |ṽ| is the cumulative length of all parts. Inserting the previous bounds for Γ and ∆ (without intermediate rounding), we conclude that
σ 2 BW ≤ 2.2 | part BW (v)| 4.4 Σ(part BW (v)) .(75)
When comparing to the variance of LC circuits in the case where part BW = (n/2) or (n/2 − 1), we are interested in ensuring that Γ∆ n < 3 |supp LC (v)| . This is the case when
|supp LC (v)| ≥ n log 3 ∆ + log 3 Γ ,(76)
E More details on numerical experiments
In this section, we describe in detail the procedure used for our numerical experiments, which are implemented in the following repository: https://github.com/MirkoArienzo/shadow_short_ circuits. First, given ρ = |0 0| and a Pauli observable W (v), Eq. (6) becomes
0 |W (v)|0 = 1 (v |S |v) i E U ∼µ i |U W (v)U † |i | i |U |0 | 2 .(79)
As discussed above, U is chosen to be a Clifford operator, which is represented by a pair (g, a), with g ∈ Sp 2n (2), and a ∈ F 2n 2 . Then, writing U =
n/2 i=1 n/2 j=1 U (i) 1 U (j)
2 , each local symplectic matrix is sampled using König-Smolin's algorithm [31], and a is a uniformly distributed vector in F 2n 2 . Then, samples {(U j , i j )} m j=1 are drawn according to standard stabilizer simulation techniques [29,30], and the estimator is given by the following empirical average:
w(v) = 1 m(v |S |v) m j=1 i j |U j W (v)U † j |i j .(80)
A single estimate requires the calculation of the phase function appearing in Eq. (58), which can be done in time O(n 3 ) [32]. However, when the observable is of Z-type, we can avoid this calculation, and speed up the simulation. To prove this fact, let us consider the decomposition F 2n 2 = Z n ⊕ X n , and label the computational basis by binary vectors i ∈ F n 2 . Then, i |U W (v)U † |i = (−1) αg(v)+[a,gv]+(gv)z·i 1 Zn (gv) ,
where (gv) z ∈ Z n F n 2 is the Z part of the vector gv ∈ F 2n 2 , and 1 Zn is the indicator function on Z n . Then, suppose the outcome of the latter is non-zero, so (gv) x = 0. Hence, since v x = 0 by assumption, we find:
[a, gv] + (gv) z · i = (gv) z · i 0 = (gv) z · j i j (ge j ) x = j i j [gv, ge j ] = j i j [v, e j ] = 0 ,(82)
where we wrote i = i 0 + a x for some i 0 ∈ F n 2 , and then we considered the decomposition i 0 = j i j (ge j ) z , where {e j } ⊂ Z n is the canonical basis. Then, from Eq. (79) we get
(v |S |v) = i E U ∼µ i |U W (v)U † |i | i |U |0 | 2 = i E U ∼µ (−1) αg(v) | i |U |0 | 2 .(83)
Define now p ± := 1 |Sp 2n (2)| |{g ∈ Sp 2n (2) | gv ∈ Z n , (−1) αg(v) = ±1}| .
Then,
p + + p − = 1 |Sp 2n (2)| |{g ∈ Sp 2n (2) | gv ∈ Z n }| = 1 |Sp 2n (2)| |STAB(v)| · |Z n \ 0| = 1 |Sp 2n (2)| |Sp 2n (2)| |Sp 2n (2) · v| · |Z n \ 0| = 2 n − 1 2 2n − 1 = 1 2 n + 1 ,(85)
where STAB(v) denotes the set of stabilizers of v.
On the other hand, p + and p − also have the interpretation of frequencies of ±1 outcomes in Eq. (83) respectively. This means
(v |S |v) = p + − p − = |Z n | 2 2n − 1 = 1 2 n + 1 ,(86)
from which it follows p − = 0.
In conclusion, whenever v x = 0, we only need to check if (gv) x is trivial.
Figure 1 :
1Brickwork circuits acting on n = 10 qubits. The left and right figures show periodic and open boundary conditions, respectively, and the arrow indicates the direction in which the circuit acts on quantum states. For both of them, the first layer is composed of n/2 two-qubit Haar random unitaries acting on qubits (2i − 1, 2i), i ∈ [5]
Figure 3 :
3Effective brickwork circuits associated with non-fully supported Pauli operators in the case n = 10.
Suppose now we have a BW circuit with open boundary conditions, and there exists another index i such thatṽ i = 0. Then, two cases can occur: Either, (a), i = 1 or i = n/2−1, which implies part BW (v) = (n/2 − 2), and we simply obtain a BW circuit with open boundary conditions on n − 4 qubits. Otherwise, (b), part BW (v) = (i − 1, n/2 − i − 1) and the BW circuit again factorizes into two independent BW circuits with open boundary conditions, acting on 2i and n − 2i qubits respectively.
Lemma 3 .
3Let S be the frame operator associated with one round of a two-local brickwork circuit with periodic or open boundary conditions in the second layer. Then, S is diagonal in the Pauli basis, and
Figure 4 :
4Figure 4a: Comparison of variances (exact evaluation of second moments) calculated for brickworks and local Cliffords. For magenta and green triangles, we assumed supp LC (v) = n and supp LC (v) = n/2, respectively. For blue and red dots, we assumed part BW (v) = (n/2) and part BW (v) = (n/2 − 1), respectively.Figure 4b: Numerical evaluation of the threshold, here indicated by s, that determines the BW sample advantage over LCs shadows in the case part BW (v) = (n/2). Notice that the threshold criterion gives a close estimate of the number of qubits necessary to reach the threshold. For instance, for n = 100 we numerically observe 68 qubits are needed for the BW supremacy, while ≈ 69 qubits are expected using the criterion bound.
Figure 5 :
5Probability of σ 2 BW ≤ σ 2 LC , evaluated on 2 16 random bitstrings for each value of n.
a) v full plot: 10 qubits in the support of v. (b) v thres plot: 8 qubits in the support of v. (c) v half plot: 5 qubits in the support of v.
Figure 6 :
6Convergence of the estimatorsŵ(v full ),ŵ(v thres ), andŵ(v half ) as defined in Section 2.2. We consider a system of 10 qubits with input state ρ = |0 0|, ensuring we can classically simulate the whole procedure efficiently. For each fixed m, 100 runs have been performed and then the average over all of them, with the respective standard deviation, has been plotted. BW estimator is converging faster for v full and v half . happens in the case of half supported circuits, seeFigure 6c.
Figure 7 :
7Each local operator corresponds to two overlapping copies of a brick in the BW circuit.
= 24 t 3
3(n − 2) + 60 t 2 (n − 2) .
= 24 t 1
1(n − 2) + 60 t 2 (n − 2) .
which for open and periodic boundary conditions translates to the sufficient condition |supp LC (v)| > 0.68n + 0.12 (77) (the constant term for open boundary conditions is actually negative). More generally, Eq. (75) is smaller than 3 |supp LC (v)| if |supp LC (v)| ≥ 0.8| part BW (v)| + 1.4Σ(part BW (v)) .(78)
). The procedure works as follows: draw unitaries U ∼ ν according to some probability measure ν on the unitary group U(d), apply U to the (unknown) state ρ, and finally measure in the computational basis{E i := |i i|} i∈[d] . Having obtained outcome i, store the classical snapshot (i, U ). Repeating this primitive yields multiple snapshots {i k , U k } m k=1 . Finally, given an observable O, one evaluates a scalar function f O (i, U ) for each snapshot and takes the empirical averageô = m k=1 f O (i k , U k ). Constructing f O (i, U ) as follows ensures thatô is an unbiased estimator for the expectation value Tr(Oρ): First, one shall require that {E i,U := U † E i U } is a tomographically complete, positive operator valued measure (POVM) [21], i.e. there exists a pair (i, U ) such that, for all states
Proposition 2 ([19, Prop. 3]). For any state ρ, estimate W (v) using BW shadows. Then, σ 2BW
depends only on v ∈ F 2n
2 , and
We have, up to adjusting the phase of b, and, thus, for n ≥ 4 we have 0.3 · 2 n < 1 ca n 1 1 + (d/c)|b/a| 4 ≤ − (d/c)|b/a| 4 < 0.6 · 2.1 n .To this end, further set c = 5(25 − 3
√
41)/(2
√
41)
and d = 5(25 + 3
√
41)/(2
√
41). 1
Σ ob (n)
=
1
ca n + db n =
1
ca n
1
1 + (d/c)(b/a) n
(72)
1
Σ ob (n)
≤
1
ca n
1
1
AcknowledgementsWe thank Leandro Aolita and Renato Mello for fruitful discussions on shadow estimation and Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan for discussions on relations to their work[18]. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Emmy Noether program (grant number 441423094) and by the German Federal Ministry of Education and Research (BMBF) within the funding program "quantum technologies -from basic research to market" via the joint projects MIQRO (grant number 13N15522) and MANIQU (grant number 13N15578).
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|
[
"https://github.com/MirkoArienzo/shadow_short_circuits.",
"https://github.com/MirkoArienzo/shadow_short_"
] |
[
"WEAK CONVERGENCE RATES OF SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION",
"WEAK CONVERGENCE RATES OF SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION"
] |
[
"Charles-Edouard ",
"Ludovic Goudenège "
] |
[] |
[] |
This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work[8], for the temporal discretization of the stochastic Allen-Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to 1 2 , like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity.To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.1991 Mathematics Subject Classification. 60H15;65C30;60H35.Nonlinearities. For all t ≥ 0 and allNote that for all t ≥ 0, z ∈ R, Φ t (z) = z + tΨ t (z). Moreover, Φ t (·) t≥0 is the flow map associated with the ODEż = z − z 3 = Ψ 0 (z).Lemma 2.1 below states the properties of Φ ∆t and Ψ ∆t , and their derivatives, which are used in order to prove well-posedness and moment estimates, and to derive error estimates.We refer to [8] for a detailed proof (except for the inequality concerning the second order derivative, which is not considered there but is obtained using similar arguments).Lemma 2.1. For every ∆t 0 ∈ (0, 1], there exists C(∆t 0 ) ∈ (0, ∞) such that for all ∆t ∈ [0, ∆t 0 ], and all z ∈ R, 7
|
10.1007/s10543-019-00788-x
|
[
"https://arxiv.org/pdf/1804.04061v2.pdf"
] | 119,688,834 |
1804.04061
|
a62de9e685a6ba3458608e0238176a6c75d217f9
|
WEAK CONVERGENCE RATES OF SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION
18 Apr 2018
Charles-Edouard
Ludovic Goudenège
WEAK CONVERGENCE RATES OF SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION
18 Apr 2018arXiv:1804.04061v2 [math.NA]
This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work[8], for the temporal discretization of the stochastic Allen-Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to 1 2 , like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity.To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.1991 Mathematics Subject Classification. 60H15;65C30;60H35.Nonlinearities. For all t ≥ 0 and allNote that for all t ≥ 0, z ∈ R, Φ t (z) = z + tΨ t (z). Moreover, Φ t (·) t≥0 is the flow map associated with the ODEż = z − z 3 = Ψ 0 (z).Lemma 2.1 below states the properties of Φ ∆t and Ψ ∆t , and their derivatives, which are used in order to prove well-posedness and moment estimates, and to derive error estimates.We refer to [8] for a detailed proof (except for the inequality concerning the second order derivative, which is not considered there but is obtained using similar arguments).Lemma 2.1. For every ∆t 0 ∈ (0, 1], there exists C(∆t 0 ) ∈ (0, ∞) such that for all ∆t ∈ [0, ∆t 0 ], and all z ∈ R, 7
Introduction
In this article, we study numerical schemes introduced in by the authors in [8], for the temporal discretization of the stochastic Allen-Cahn equation, ∂X(t, ξ) ∂t = ∂ 2 X(t, ξ) ∂ξ 2 + X(t, ξ) − X(t, ξ) 3 +Ẇ (t, ξ), t ≥ 0, ξ ∈ (0, 1), driven by Gaussian space-time white noise, with homogeneous Dirichlet boundary conditions. This Stochastic Partial Differential Equation (SPDE) has been introduced in [1] as a model for a two-phase system driven by the Ginzburg-Landau energy
E(X) = |∇X| 2 + V (X),
where X is the ratio of the two species densities, and V (X) = (X 2 − 1) 2 is a double well potential. The first term in the energy models the diffusion of the interface between the two pure phases, and the second one pushes the solution to two possible stable states ±1 (named the pure phases, i.e. minima of V ). The stochastic version of the Allen-Cahn equation models the effect of thermal perturbations by an additional noise term.
The objective of this article is to study weak rates of convergence for two examples of splitting schemes schemes introduced in [8]. Let the SPDE be rewritten in the framework of [12], with X(t) = X(t, ·): dX(t) = AX(t)dt + X(t) − X(t) 3 dt + dW (t), 1 where W (t) t≥0 is a cylindrical Wiener process. If ∆t > 0 denotes the time-step size of the integrator, the numerical schemes are defined as (1) X exp n+1 = e ∆tA Φ ∆t (X exp n ) + (n+1)∆t n∆t e ((n+1)∆t−t)A dW (t), X imp n+1 = S ∆t Φ ∆t (X imp n ) + S ∆t W ((n + 1)∆t) − W (n∆t) . In both schemes, Φ t t≥0 is the flow map associated with the ODEż = z −z 3 , which is known exactly, see (5), and at each time step, one solves first the equation dX(t) = X(t)−X(t) 3 dt, second compute an approximation for the equation dX(t) = AX(t)dt + dW (t). In the first scheme in (1), the second step is also solved exactly, using an exponential integrator and exact sampling of the stochastic integral. In the second scheme, the second step is solved using a linear implicit Euler scheme, with S ∆t = (I − ∆tA) −1 .
The schemes given by (1) have been introduced by the authors in [8], where two preliminary results were established: the existence of moment estimates for the numerical solution, with bounds uniform in the time-step size parameter ∆t, and the mean-square convergence of the scheme, with no order of convergence. In [6], it was established that the first scheme in (1) has strong order of convergence 1 4 , based on a nice decomposition of the error. The contribution of the present article is to prove that both schemes in (1) have a weak order of convergence 1 2 . Numerical schemes for SPDEs have been extensively studied in the last two decades, see for instance the monographs [16,21,24]. We recall that two notions of convergence are usually studied: strong convergence refers to convergence in mean-square sense, whereas weak convergence refers to convergence of distributions. If sufficiently regular test functions are considered, it is usually the case that the weak order of convergence is twice the strong order. For one-dimensional parabolic semilinear SPDEs driven by space-time white noise, the solutions are only Hölder continuous in time with exponent α < 1/4, hence one expects a strong order of convergence equal to 1 4 and a weak order equal to 1 2 . In the case of globally Lipschitz continuous nonlinearities, this result has been proved in recent years for a variety of numerical schemes, exploiting different strategies for the analysis of the weak error: analysis of the error using the associated Kolmogorov equation, see [3,7,13,14,30], using the mild Itô formula approach, see [10,15,17], or other techniques, see [2,28]. In the case of SPDEs with non globally Lipschitz continuous nonlinearities, standard integrators, which treat explictly the nonlinearity, cannot be used. Design and analysis of appropriate schemes, is an active field of research, in particular concerning the stochastic Allen-Cahn equation, see [4,6,8,19,20,22,23,25,29]. Several strategies may be employed: splitting, taming, split-step methods have been proposed and studied in those references. Only strong convergence results have been obtained so far. Up to our knowledge, only the preliminary results in the PhD Thesis [18] deal with the analysis of the weak error, for split-step methods using an implicit discretization of the nonlinearity. Hence, the present article is to present detailed analysis of the weak error for a class of SPDEs with non globally Lipschitz continuous nonlinearity.
The main results of this article are Theorems 3.2 and 3.3, which may be stated as follows: for sufficiently smooth test functions ϕ (see Assumption 3.1), and all α ∈ [0, 1 2
), T ∈ (0, ∞), E ϕ(X(N∆t)) − E ϕ(X N ) ≤ C α (T )∆t α . 2
In the case of the second scheme in (1), numerical experiments were reported in [8] to motivate and illustrate this convergence result.
As in the case of SPDEs with globally Lipschitz continuous nonlinearities, driven by a cylindrical Wiener process, the order of convergence is 1 2 , whereas the strong order of convergence is only 1 4 in general. However, several points in the analysis are original and need to be emphasized. First, moment estimates for the numerical solution, which are non trivial in the case of numerical schemes for equations with non globally Lipschitz nonlinearities, are required. They were proved by the authors in [8] for (1). The analysis of the error is based on the decomposition of the error using the solution of related Kolmogorov equations, see (11) below and Section 3 for a description of the method. Appropriate regularity properties need to be proved for the first and second order spatial derivatives: this is presented in Theorems 4.1 and 4.2, which are auxiliary results in the analysis, but may have a more general interest. The proofs of our main results use different strategies. The proof of Theorem 3.2 is shorter, due to the use of a nice auxiliary continuous-time process, and of appropriate temporal and spatial regularity properties of the process, following [28] and [29]. The proof of Theorem 3.3 follows essentially the same steps as in [13] and related references, in particular a duality formula from Malliavin calculus is used. The estimate of the Malliavin derivative given in Lemma 6.4 uses in an essential way the structure as a splitting scheme, and is one of the original results used in this work.
This article is organized as follows. Assumptions, equations and numerical schemes are given in Section 2. Our main results, Theorems 3.2 and 3.3 are stated in Section 3. The regularity properties for solutions of Kolmogorov equations are stated and proved in Section 4. Section 5 is devoted to the proof of Theorem 3.2, whereas Section 6 is devoted to the proof of Theorem 3.3.
Setting
We work in the standard framework of stochastic evolution equations with values in infinite dimensional separable Hilbert and Banach spaces. We refer for instance to [9,12] for details. Let H = L 2 (0, 1), and E = C([0, 1]). We use the following notation for inner product and norms: for x 1 , x 2 ∈ H, x ∈ E,
x 1 , x 2 = 1 0 x 1 (ξ)x 2 (ξ)dξ , |x 1 | H = x 1 , x 1 1 2 , |x| E = max ξ∈[0,1] |x(ξ)|.
For p ∈ [1, ∞], we also use the notation L p = L p (0, 1) and | · | L p for the associated norm.
D(A) = H 2 (0, 1) ∩ H 1 0 (0, 1), Ax = ∂ 2 ξ x, ∀ x ∈ D(A).
Let e n = √ 2 sin(nπ·) and λ n = n 2 π 2 , for n ∈ N. Note that Ae n = −λ n e n , and that e n n∈N is a complete orthonormal system of H. In addition, for all n ∈ N, |e n | E ≤ √ 2.
3
The linear operator A generates an analytic semi-group e tA t≥0 , on L p for p ∈ [2, ∞) and on E. For α ∈ (0, 1), the linear operators (−A) −α and (−A) α are constructed in a standard way, see for instance [26]:
(−A) −α = sin(πα) π ∞ 0 t −α (tI − A) −1 dt , (−A) α = sin(πα) π ∞ 0 t α−1 (−A)(tI − A) −1 dt,
where (−A) α is defined as an unbounded linear operator on L p . In the case p = 2, note that
(−A) −α x = i∈N ⋆ λ −α i x, e i e i , x ∈ H, (−A) α x = i∈N ⋆ λ α i x, e i e i , x ∈ D 2 (−A) α = x ∈ H; ∞ i=1 λ 2α i x, e i 2 < ∞ .
We denote by L(H) the space of bounded linear operators from H to H , with associated norm denoted by · L(H) . The space of Hilbert-Schmidt operators on H is denoted by L 2 (H), and the associated norm is denoted by · L 2 (H) .
To conclude this section, we state several useful functional inequalities. Inequality (2) is a consequence of the Sobolev embedding W 2η,2 (0, 1) ⊂ C([0, 1]) when 2η > 1 2 , and of the equivalence of the norms W 2η,2 (0, 1) and |(−A) η · | L 2 . For inequalities (3) and (4), we refer to [27] for the general theory, and to the arguments detailed in [7].
• For every η > 1 4 , there exists C η ∈ (0, ∞) such that (2) |(−A) −η · | L 2 ≤ C η | · | L 1 .
• For every α ∈ (0, 1 2 ), ǫ > 0, with α + ǫ < 1 2 , there exists C α,ǫ ∈ (0, ∞) such that
(3) |(−A) −α−ǫ (xy)| L 1 ≤ C α,ǫ |(−A) α+ǫ x| L 2 |(−A) −α y| L 2 .
• For every α ∈ (0, 1 2 ), ǫ > 0, with α + 2ǫ < 1 2 , there exists C α,ǫ ∈ (0, ∞) such that, if ψ : R → R is Lipschitz continuous,
(4) |(−A) α+ǫ ψ(·)| L 2 ≤ C α,ǫ [ψ] Lip |(−A) α+2ǫ · | L 2 , where [ψ] Lip = sup z 1 =z 2 |ψ(z 2 )−ψ(z 1 )| |z 2 −z 1 | .
These inequalities are used in an essential way to prove Lemma 2.2 below. In addition, inequality (2) is also used in the proof of Theorem 4.2.
Wiener process.
Let Ω, F , P denote a probability space, and consider a family β n n∈N of independent standard real-valued Wiener processes. Then set
W (t) = n∈N β n (t)e n .
This series does not converge in H. However, ifH is an Hilbert space, and L ∈ L 2 (H,H) is a linear, Hilbert-Schmidt, operator, then LW (t) is a Wiener process onH, centered and with covariance operator LL ⋆ .
|Φ ′ ∆t (z)| ≤ e ∆t 0 , Ψ ′ ∆t (z) ≤ e ∆t 0 , |Ψ ′ ∆t (z)| ≤ C(∆t 0 )(1 + |z| 2 ), |Ψ ′′ ∆t (z)| ≤ C(∆t 0 )(1 + |z|), |Ψ ∆t (z) − Ψ 0 (z)| ≤ C(∆t 0 )∆t(1 + |z| 5 ).
In particular, the mapping Ψ ∆t satisfies the following one-sided Lipschitz condition: for all z 1 , z 2 ∈ R,
Ψ ∆t (z 2 ) − Ψ ∆t (z 1 ) z 2 − z 1 ≤ e ∆t |z 2 − z 1 | 2 .
Observe also that for ∆t > 0, the mapping Ψ ∆t is of class C ∞ , and admits bounded first and second order derivatives. However, such bounds are not uniform with respect to ∆t > 0.
We conclude this section with an auxiliary result, see [29] for a similar statement.
Lemma 2.2. For every ∆t 0 ∈ (0, 1], η ∈ ( 1 4 , 1), α ∈ (0, 1 2 ), and ǫ > 0 such that α + 2ǫ < 1 2 , there exists C(∆t 0 , η, α, ǫ) ∈ (0, ∞) such that for all ∆t ∈ [0, ∆t 0 ] for all x, y ∈ H, with |(−A) α 2 +ǫ x| H < ∞,
|(−A) −η− α+ǫ 2 Ψ ′ ∆t (x)y | H ≤ C(∆t 0 , η, α, ǫ)(1 + |x| E )|(−A) α 2 +ǫ x| H |(−A) − α 2 y| H .
Proof. Using successively the inequalities (2) and (3),
|(−A) −η− α+ǫ 2 Ψ ′ ∆t (x)y | L 2 ≤ C η |(−A) − α+ǫ 2 Ψ ′ ∆t (x)y | L 1 ≤ C η,α,ǫ |(−A) α+ǫ 2 Ψ ′ ∆t (x)| L 2 |(−A) − α 2 y| L 2 .
The function Ψ ′ ∆t is not globally Lipschitz continuous when ∆t = 0, and lim ∆t→0 [Ψ ′ ∆t ] Lip = ∞. Thus inequality (4) cannot be used directly.
However, the derivative Ψ ′′ ∆t has at most linear growth, uniformly in ∆t ∈ [0, ∆t 0 ]: |Ψ ′′ ∆t (z)| ≤ C(∆t 0 )(1+|z|) for all z ∈ R. By a straightforward truncation argument, applying inequality (4) yields
|(−A) α+ǫ 2 Ψ ′ ∆t (x)| L 2 ≤ C α,ǫ C(∆t 0 )(1 + |x| E )|(−A) α+2ǫ 2 x| L 2 .
This concludes the proof of Lemma 2.2.
2.2.
Stochastic Partial Differential Equations. The stochastic Allen-Cahn equation with additive space-time white noise perturbation, is
(6) dX(t) = AX(t)dt + X(t) − X(t) 3 dt + dW (t) , X(0) = x.
More generally, for ∆t ∈ [0, 1] introduce the auxiliary equation
(7) dX (∆t) = AX (∆t) (t)dt + Ψ ∆t (X (∆t) (t))dt + dW (t) , X (∆t) = x.
If ∆t > 0, since Ψ ∆t is globally Lipschitz continuous, standard fixed point arguments (see for instance [12]) imply that, for any initial condition x ∈ H, the SPDE (7) admits a unique global mild solution X (∆t) (t, x) t≥0 , i.e. which satisfy
X (∆t) (t, x) = e tA x + t 0 e (t−s)A Ψ ∆t (X (∆t) (s, x))ds + t 0 e (t−s)A dW (s), t ≥ 0.
If ∆t = 0, proving global well-posedness requires more refined arguments, in particular the use of the one-sided Lipschitz condition (see for instance [9]). For any initial condition x ∈ E, there exists a unique mild solution X(t, x) t≥0 of Equation (6), and X (0) = X solves Equation (7) with ∆t = 0:
X (0) (t, x) = e tA x + t 0 e (t−s)A Ψ 0 (X (0) (s, x))ds + t 0 e (t−s)A dW (s), t ≥ 0.
To simplify notation, we often write X(t) and X (∆t) (t) and omit the initial condition x. Let
(8) W A (t) = t 0 e (t−s)A dW (s).
Then (see [9, Lemma 6.1.2]) for every T ∈ (0, ∞) and M ∈ N, there exists C(T, M) ∈ (0, ∞) such that
(9) E[ sup t∈[0,T ] |W A (t)| M E ] ≤ C(T, M).
Combined the one-sided Lipschitz condition for Ψ ∆t , see Lemma 2.1, (9) yields moment estimates for X (∆t) .
Lemma 2.3. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and M ∈ N. There exists C(T, ∆t 0 , M) ∈ (0, ∞) such that, for all ∆t ∈ [0, ∆t 0 ] and x ∈ E, E[ sup t∈[0,T ] |X (∆t) (t, x)| M E ] ≤ C(T, ∆t 0 , M)(1 + |x| M E ).
We refer to [8] for a proof.
Kolmogorov equations.
In this section, we introduce functions u (∆t) which play a key role in the weak error analysis, and which are solutions of infinite dimensional Kolmogorov equations associated with (7). Let ϕ : H → R be a function of class C 2 , with bounded first and second order derivatives. Let ∆t ∈ (0, 1]. Note that we do not consider the case ∆t = 0 in this section, the reason why will be clear below.
For every t ≥ 0, set
(10) u (∆t) (t, x) = E ϕ(X (∆t) (t,
x)) . 6 Formally, u (∆t) is solution of the Kolmogorov equation associated with (7):
(11) ∂u (∆t) (t, x) ∂t = L (∆t) u (∆t) (t, x) = Ax + Ψ ∆t (x), Du (∆t) (t, x) + 1 2 j∈N D 2 u (∆t) (t, x).(e j , e j ),
where the first order spatial derivative is identified as an element of H thanks to Riesz Theorem. A rigorous meaning can be given using an appropriate regularization procedure. Since Ψ ∆t is globally Lipschitz continuous for fixed ∆t > 0, this may be performed by a standard spectral Galerkin approximation. However, this choice would not allow us to pass to the limit ∆t → 0 in estimates and keep bounds uniform in the regularization parameter. Instead, we propose to replace the noise dW (t) in (7) by e δA dW (t), with the regularization parameter δ > 0. Rigorous computations are performed with fixed δ > 0. Section 4 deals with regularity properties for Du (∆t) (t, x) and D 2 u (∆t) (t, x), which allow us to obtain bounds which are uniform with respect to δ, hence passing to the limit δ → 0 is allowed.
To simplify notation, we do not mention the regularization parameter δ in the computations and statements below.
Splitting schemes.
We are now in position to rigorously define the numerical schemes which are studied in this article.
A first scheme is defined as:
(12) X n+1 = e ∆tA Φ ∆t (X n ) + (n+1)∆t n∆t e (n∆t−t)A dW (t).
A second scheme is defined as:
(13) X n+1 = S ∆t Φ ∆t (X n ) + S ∆t W ((n + 1)∆t) − W (n∆t) , where S ∆t = (I − ∆tA) −1 .
The schemes are constructed following a Lie-Trotter splitting strategy for the SPDE (6). Firstly, the equation dX(t) = Ψ 0 (X(t))dt is solved explictly using the flow map at time t = ∆t, namely Φ ∆t . Secondly, the equation dX(t) = AX(t)dt + dW (t) is either solved exactly in scheme (12), or using a linear implicit Euler scheme in (13).
As already emphasized in [6,8], observe that (12) and (13) can be interpreted as integrators for the auxiliary equation (7) with nonlinear coefficient Ψ ∆t : respectively
X n+1 = e ∆tA X n + ∆te ∆tA Ψ ∆t (X n ) + (n+1)∆t n∆t e (n∆t−t)A dW (t), X n+1 = S ∆t X n + ∆tS ∆t Ψ ∆t (X n ) + S ∆t W ((n + 1)∆t) − W (n∆t) ,
where in (12), an exponential Euler integrator is used, whereas in (13) a semi-implicit integrator is used.
Moment estimates are available. We refer to [8] for a proof.
Lemma 2.4. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and M ∈ N. There exists C(T, ∆t 0 , M) ∈ (0, ∞) such that, for all ∆t ∈ [0, ∆t 0 ] and x ∈ E, E sup n∈N,n∆t≤T |X n | M E ≤ C(T, ∆t 0 , M)(1 + |x| M E ).
Weak convergence results
This section is devoted to the statement of the main result of this article: the numerical schemes (12) and (13) have a weak convergence rate equal to 1 2 , see Theorems 3.2 and 3.3 below respectively.
The main difficulty and novelty of this contribution is the treatment of SPDEs with non globally Lipschitz continuous nonlinear coefficient. Up to our knowledge, except in the PhD Thesis [18] (where split-step schemes based on an implicit discretization of the nonlinear term, for more general polynomial coefficients, are considered), there is no analysis of weak rates of convergence for that situation in the literature.
Strong convergence of numerical schemes (12) and (13) is proved in [8], without rate. In [6], the strong rate of convergence 1 4 is proved for the scheme (12). Test functions satisfy the following condition.
Assumption 3.1. The function ϕ : H → R is of class C 2 , and has bounded first and second order derivatives:
ϕ 1,∞ = sup x∈H,h∈H,|h| H =1 |Dϕ(x).h| < ∞, ϕ 2,∞ = ϕ 1,∞ + sup x∈H,h,k∈H,|h| H =|k| H =1 |D 2 ϕ(x).(h, k)| < ∞.
We are now in position to state our main results.
Theorem 3.2. Let (X n ) n∈N be defined by the scheme (12). Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and x ∈ E. For all α ∈ [0, 1 2 ), there exists C α (T, ∆t 0 , |x| E ) ∈ (0, ∞)
such that the following holds true.
Let ϕ satisfy Assumption 3.1. For all ∆t ∈ (0, ∆t 0 ] and N ∈ N, such that T = N∆t,
(14) E ϕ(X(N∆t)) − E ϕ(X N ) ≤ C α (T, ∆t 0 , |x| E ) ϕ 1,∞ ∆t α , Theorem 3.3. Let (X n ) n∈N be defined by the scheme (13). Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and x ∈ E. For all α ∈ [0, 1 2 )
, there exists C α (T, ∆t 0 , |x| E ) ∈ (0, ∞) such that the following holds true.
Let ϕ satisfy Assumption 3.1. For all ∆t ∈ (0, ∆t 0 ] and N ∈ N, such that T = N∆t,
(15) E ϕ(X(N∆t)) − E ϕ(X N ) ≤ C α (T, ∆t 0 , |x| E ) ϕ 2,∞ ∆t α .
Theorems 3.2 and 3.3 are natural generalizations of the results obtained for SPDEs with globally Lipschitz continuous nonlinear coefficient, see for instance [13]. We obtain the same weak order of convergence 1 2 , which is twice the strong order of convergence. Remark 3.4. The regularity of the function ϕ is essential to get a weak order of convergence which is twice the strong order, as proved in [5]. If one wants to replace ϕ 2,∞ by ϕ 1,∞ in the right-hand side of (15), the order of convergence has to be replaced by α 2 , even in the absence of nonlinear coefficient.
In the right-hand side of (14), it is sufficient to control only ϕ 1,∞ . This is due to an appropriate decomposition of the weak error. This is not in contradiction with [5]: in the absence of nonlinear coefficient, the weak error is equal to zero.
Proving Theorem 3.2 and 3.3 is the aim of the remainder of the article. The strategy consists in decomposing the weak error as follows:
E ϕ(X(N∆t)) − E ϕ(X N ) ≤ E ϕ(X(N∆t)) − E ϕ(X (∆t) (N∆t)) + E ϕ(X (∆t) (N∆t)) − E ϕ(X N ) .
The first error term is estimated using the following result, quoted from [8], combined with globally Lipschitz continuity of ϕ induced by Assumption 3.1.
Proposition 3.5. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1) and x ∈ E. There exists C(T, ∆t 0 , |x| E ) ∈ (0, ∞) such that for all ∆t ∈ (0, ∆t 0 ], sup t∈[0,T ] E X(t) − X (∆t) (t)) H ≤ C(T, ∆t 0 , |x| E )∆t.
The treatment of the second error term requires more subtle arguments. First, thanks to (10), and a telescoping sum argument,
E ϕ(X (∆t) (n∆t)) − E ϕ(X n ) = u (∆t) (n∆t, x) − E u (∆t) (0, X n ) = n−1 k=0 E u (∆t) ((n − k)∆t, X k ) − u (∆t) ((n − k − 1)∆t, X k+1 ) .
The details then depend on the numerical scheme. First, an auxiliary continuous-time processX is introduced, see (32) and (33). It satisfiesX(k∆t) = X k for all k ∈ N. Second, Itô formula is applied, and the Kolmogorov equation (11) is used. Theorem 3.2 (numerical scheme given by (12)) is proved in Section 5. Theorem 3.3 (numerical scheme given by (13)) is proved in Section 6.
Spatial derivatives Du (∆t) (t, x) and D 2 u (∆t) (t, x) appear in the expansion of the error obtained following this standard strategy. In infinite dimension, see [3,7,13,30], appropriate regularity properties are required to obtain the weak order of convergence 1 2 . They are studied in Section 4.
Regularity properties for solutions of Kolmogorov equations
This section is devoted to state and prove regularity properties of the function u (∆t) , defined by (10), solution of the Kolmogorov equation (11) associated to the auxiliary equation (7). The main difficulty and novelty is due to the poor regularity property of Ψ ∆t : even if for fixed ∆t, Ψ ∆t is globally Lipschitz continuous, there is no bound which is uniform in ∆t > 0, since Ψ 0 is polynomial of degree 3.
Theorems 4.1 and 4.2 below are the main results of this section, and they are of interest beyond the analysis of weak convergence rates. They are natural generalizations in a nonglobally Lipschitz framework of the estimates provided in [13], and extended in [7] with nonlinear diffusion coefficients. We emphasize that the right-hand sides in the estimates (16) and (17) do not depend on ∆t.
Theorem 4.1. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1]. For every α ∈ [0, 1), there exists C α (T, ∆t 0 ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ), x ∈ E, h ∈ H and t ∈ (0, T ],
(16) |Du (∆t) (t, x).h| ≤ C α (T, ∆t 0 )(1 + |x| 2 E ) ϕ 1,∞ t α |(−A) −α h| H .
This may be interpreted as a regularization property: the assumption Du (∆t) (0, x) ∈ H implies that, for positive t, (−A) α Du (∆t) (t, x) ∈ H with α ∈ (0, 1). Theorem 4.2. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1]. For every β, γ ∈ [0, 1), with the condition β +γ < 1, there exists C β,γ (T, ∆t 0 ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ), x ∈ E, h, k ∈ H and t ∈ (0, T ],
(17) |D 2 u (∆t) (t, x).(h, k)| ≤ C β,γ (T, ∆t 0 )(1 + |x| 7 E ) ϕ 2,∞ t β+γ |(−A) −β h| H |(−A) −γ k| H . Remark 4.3.
Results similar to Theorems 4.1 and 4.2 are studied in [18], with different techniques. Note that we obtain stronger results. In Theorem 4.1, one may choose α ∈ [0, 1) instead of α ∈ [0, 1 2 ). In Theorem 4.2, one may choose β, γ ∈ [0, 1) such that β + γ < 1, instead of β, γ ∈ [0, 1 2 ). The stronger statements are useful below, to simplify the treatments of several error terms.
Expressions of Du (∆t) (t, x) and D 2 u (∆t) (t, x) are given below: for h, k ∈ H, x ∈ H, and t ≥ 0,
Du (∆t) (t, x).h = E Dϕ(X (∆t) (t, x)).η h (t, x) , D 2 u (∆t) (t, x).(h, k) = E Dϕ(X (∆t) (t, x)).ζ h,k (t, x) + E D 2 ϕ(X (∆t) (t, x)).(η h (t, x), η k (t, x)) ,(18)
where the processes η h (·, x) and ζ h,k (·, x) are the solutions the random PDEs
(19) dη h (t, x) dt = Aη h (t, x) + Ψ ′ ∆t (X (∆t) (t, x))η h (t, x),
with initial condition η h (0, x) = h, and
(20) dζ h,k (t, x) dt = Aζ h,k (t, x) + Ψ ′ ∆t (X (∆t) (t, x))ζ h,k (t, x) + Ψ ′′ ∆t (X (∆t) (t, x))η h (t, x)η k (t, x),
with initial condition ζ h,k (0, x) = 0.
To simplify notation, the parameter ∆t is omitted in the notation for η h (t, x) and ζ h,k (t, x).
4.1.
Proof of Theorem 4.1. Following the strategy in [7], introduce the auxiliary process
η h (t, x) = η h (t, x) − e tA h.
Then, thanks to (18),
Du (∆t) (t, x).h = E Dϕ(X (∆t) (t, x)).(e tA h) + E Dϕ(X (∆t) (t, x)).η h (t, x) ,
and thanks to (19),
dη h (t, x) dt = Aη h (t, x) + Ψ ′ ∆t (X (∆t) (t, x))η h (t, x) + Ψ ′ ∆t (X (∆t) (t, x))e tA h ,η h (0) = 0.
On the one hand, for any α ∈ [0, 1), h ∈ H and t > 0,
(21) E Dϕ(X (∆t) (t, x)).(e tA h) ≤ ϕ 1,∞ |e tA h| H ≤ C α t α ϕ 1,∞ |(−A) −α h| H .
On the other hand, the processη h may be expressed as
(22)η h (t, x) = t 0 U(t, s) Ψ ′ ∆t (X (∆t) (s, x))e sA h ds,
where U(t, s)h t≥s solves, for every h ∈ H,
(23) dU(t, s)h dt = A + Ψ ′ ∆t (X (∆t) (t, x)) U(t, s)h , U(s, s)h = h.
A straightforward energy estimate, using the one-sided Lipschitz condition for Ψ ′ ∆t , yields 1 2
d|U(t, s)h| 2 dt ≤ (e ∆t 0 − λ 1 )|U(t, s)h| 2 ,
thus by Gronwall Lemma |U(t, s)h| 2 ≤ C(T, ∆t 0 )|h| 2 , for all s ≤ t ≤ T . Thus, thanks to (22), for α ∈ [0, 1),
|η h (t, x)| H ≤ C(T, ∆t 0 ) t 0 |Ψ ′ ∆t (X (∆t) (s, x))e sA h| H ds ≤ C(T, ∆t 0 ) sup s∈[0,T ] |Ψ ′ ∆t (X (∆t) (s, x))| E t 0 C α s α ds|(−A) −α h| H .
Thanks to Lemmas 2.1 and 2.3,
(24) E Dϕ(X (∆t) (t, x)).η h (t, x) ≤ C α (T, ∆t 0 ) t α (1 + |x| 2 E ) ϕ 1,∞ |(−A) −α h| H .
Combining (21) and (24) concludes the proof of (16). For future reference, note that for t ∈ (0, T ],
(25) |η h (t, x)| H ≤ C α (T, ∆t 0 ) t α |(−A) −α h| H (1 + sup s∈[0,T ] |Ψ ′ ∆t (X (∆t) (s, x))| E ),
4.2.
Proof of Theorem 4.2. Thanks to (18),
|D 2 u (∆t) (t, x).(h, k)| ≤ ϕ 1,∞ E[|ζ h,k (t, x)| H + ϕ 2,∞ E[|η h (t, x)| H |η k (t, x)| H ].
On the one hand, thanks to (25), using Lemma 2.1 and Lemma 2.3, we obtain
(26) E[|η h (t, x)| H |η k (t, x)| H ] ≤ C β,γ (T, ∆t 0 )(1 + |x| 4 E ) t β+γ |(−A) −β h| H |(−A) −γ | H , for all β, γ ∈ [0, 1).
On the other hand, using (23), from the equation (20), the process ζ h,k may be expressed as
(27) ζ h,k (t, x) = t 0 U(t, s) Ψ ′′ ∆t (X (∆t) (s, x))η h (s, x)η k (s, x) ds.
Thanks to (2),
(−A) − 1 2 Ψ ′′ ∆t (X (∆t) (s, x))η h (s, x)η k (s, x) H ≤ C Ψ ′′ ∆t (X (∆t) (s, x)) E |η h (s, x)| H |η k (s, x)| H .
The following result allows us to use this inequality in (27). Lemma 4.4. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). There exists C(T, ∆t 0 ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ), x ∈ E, h ∈ H, 0 ≤ s < t ≤ T ,
|U(t, s)h| H ≤ C(T, ∆t 0 ) (t − s) 1 2 1 + sup 0≤r≤T |Ψ ′ ∆t (X (∆t) (r, x))| E |(−A) −1/2 h| H .
The proof of Lemma 4.4 is postponed to Section 4.3. We refer to [11] for a similar result and the idea of the proof.
Thanks to Lemma 4.4 and to (27), we get
E|ζ h,k (t, x)| H ≤ t 0 C(T, ∆t 0 ) (t − s) 1 2 1 + sup 0≤r≤T |Ψ ′ ∆t (X (∆t) (r, x))| E Ψ ′′ ∆t (X (∆t) (s, x)) E |η h (s, x)| H |η k (s, x)| H ds.
Thanks to (25), and Lemmas 2.1 and 2.3, we obtain
E|ζ h,k (t, x)| H ≤ C β,γ (T, ∆t 0 )(1 + |x| 7 E ) t 0 1 (t − s) 1 2 s β+γ ds |(−A) β h| H |(−A) γ k| H , hence (28) E|ζ h,k (t, x)| H ≤ C β,γ (T, ∆t 0 )(1 + |x| 7 E ) t β+γ |(−A) −β h| H |(−A) −γ k| H .
Combining (26) and (28), thanks to (18), then concludes the proof of (17).
Proof of Lemma 4.4.
We claim that for all 0 ≤ s ≤ T and h ∈ H,
T s |U(r, s)h| 2 H dr ≤ C(T, ∆t 0 ) 1 + sup 0≤r≤T |Ψ ′ ∆t (X (∆t) (r, x))| E |(−A) −1/2 h| 2 H .(29)
Lemma (4.4) is then a straightforward consequence of (29), using the mild formulation
U(t, s)h = e (t−s)A h + t s e (t−r)A Ψ ′ ∆t (X (∆t) (s, x))U(r, s)hds.
It remains to prove (29). Let s ∈ [0, T ] be fixed, and define U s : h ∈ H → U(t, s)h s≤t≤T ∈ L 2 (s, T ; H).
Introduce U ⋆ s : L 2 (s, T ; H) → H the adjoint of U s . Then, by a duality argument, the claim (29) is a straightforward consequence of the following estimate: for all F ∈ L 2 (s, T ; H),
(30) |(−A) 1 2 U ⋆ s F | 2 ≤ C(T, ∆t 0 ) 1 + sup 0≤r≤T |Ψ ′ ∆t (X (∆t) (r, x))| E T s |F (r)| 2 H dr.
To prove (30), let F ∈ L 2 (s, T ; H), and observe that U ⋆ s F = ξ s (s), where ξ s (t) s≤t≤T is the solution of the backward evolution equation To obtain the required estimate of |(−A)
dξ s (t) dt = −Aξ s (t) − Ψ ′ ∆t (X (∆t) (t, x))ξ s (t) − F (t) , ξ s (T ) = 0.1 2 U ⋆ s F | 2 = |(−A) 1 2 ξ s (s)| 2 , compute 1 2 d dt |(−A) 1 2 ξ s (t)| 2 H = |(−A)ξ s (t)| 2 H − Ψ ′ ∆t (X (∆t) (t, x))ξ s (t), (−A)ξ s (t) − F (t), (−A)ξ s (t) ≥ |(−A)ξ s (t)| 2 H − |Ψ ′ ∆t (X (∆t) (t, x))| E |ξ s (t)| H |(−A)ξ s (t)| H − |F (t)| H |(−A)ξ s (t)| H ≥ − 1 2 |Ψ ′ ∆t (X (∆t) (t, x))| 2 E |ξ s (t)| 2 H − 1 2 |F (t)| 2
H , thanks to Young inequality. Integrating from t = s to t = T , and using ξ s (T ) = 0, we have
(31) |(−A) 1 2 ξ s (s)| 2 ≤ sup 0≤t≤T |Ψ ′ ∆t (X(t, x))| 2 E T s |ξ s (t)| 2 dt + 1 2 T s |F (t)| 2 H dt.
Moreover, using −Ψ ′ ∆t (·) ≥ −e ∆t 0 , see Lemma 2.1, thanks to Young inequality, we have
1 2 d dt |ξ s (t)| 2 H = |(−A) 1/2 ξ s (t)| 2 H − e ∆t 0 |ξ s (t)| 2 H − F (t), ξ s (t) ≥ |(−A) 1/2 ξ s (t)| 2 H − e ∆t 0 |ξ s (t)| 2 H − |(−A) − 1 2 F (t)| H |(−A) 1 2 ξ s (t)| H ≥ −e ∆t 0 |ξ s (t)| 2 H − 1 2 |(−A) − 1 2 F (t)| 2 H , then by Gronwall Lemma there exists C(T, ∆t 0 ) ∈ (0, ∞) such that for all s ≤ t ≤ T , |ξ s (t)| 2 ≤ C(T, ∆t 0 ) T s |(−A) −1/2 F (r)| 2 dr.
Hence, with the inequality |(−A) − 1 2 · | H ≤ C| · | H and Equation (31), we obtain Equation (30), leading to estimate (29), concluding the proof of Lemma 4.4.
Proof of Theorem 3.2
The aim of this section is to prove Theorem 3.2. Let the numerical scheme X n n∈N be given by (12).
The section is organized as follows. An auxiliary processX and an appropriate decomposition of the error are given in Section 5.1. Error terms are estimated in Section 5.2. Auxiliary results are proved in Section 5.3.
5.1.
Decomposition of the error. As explained in Section 3, the strategy for the weak error analysis requires to apply Itô formula, hence the definition of an appropriate continuoustime processX.
Set, for every n ∈ N, and every t ∈ [n∆t, (n + 1)∆t],
(32)X(t) = e (t−n∆t)A Φ t−n∆t (X n ) + t n∆t e (t−s)A dW (s).
By construction,X(n∆t) = X n for all n ∈ N. Moreover, dX(t) = AX(t)dt + dW (t) + e (t−n∆t)A Ψ 0 (Φ t−n∆t (X n ))dt , t ∈ [n∆t, (n + 1)∆t], n ∈ N.
Recall that Φ t−n∆t and Ψ 0 are defined by (5). 13 The following result gives moment estimates. Proof is postponed to Section 5.3.
Lemma 5.1. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1) and M ∈ N. There exists C(T, ∆t 0 , M) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ] and x ∈ E,
sup t∈[0,T ] E[|X(t)| M E ] ≤ C(T, ∆t 0 , M)(1 + |x| E ) M .
The error is then decomposed as follows, using Itô formula, and the Kolmogorov equation (11)
, with T = N∆t E u (∆t) (T, x) − E u (∆t) (0, X N ) = N −1 k=0 E u (∆t) ((N − k)∆t, X k ) − E u (∆t) ((N − k − 1)∆t, X k+1 ) = N −1 k=0 E (k+1)∆t k∆t Du (∆t) (T − t,X(t)), Ψ ∆t (X(t)) − e (t−k∆t)A Ψ 0 (Φ t−k∆t (X k )) dt = N −1 k=0 d 1 k + d 2 k + d 3 k + d 4 k where d 1 k = E (k+1)∆t k∆t Du (∆t) (T − t,X(t)), Ψ ∆t (X(t)) − Ψ ∆t (X k ) dt, d 2 k = E (k+1)∆t k∆t Du (∆t) (T − t,X(t)), I − e (t−k∆t)A Ψ ∆t (X k ) dt, d 3 k = E (k+1)∆t k∆t Du (∆t) (T − t,X(t)), e (t−k∆t)A Ψ ∆t (X k ) − Ψ ∆t (Φ t−k∆t (X k )) dt, d 4 k = E (k+1)∆t k∆t Du (∆t) (T − t,X(t)), e (t−k∆t)A Ψ ∆t (Φ t−k∆t (X k )) − Ψ 0 (Φ t−k∆t (X k )) dt.
Estimates of error terms.
Treatment of d 1 k . Let η > 1 4 , α ∈ (0, 1 2 ), and ǫ > 0, such that α + 3ǫ < 1 2 . Assume k = 0, then, thanks to Theorem 4.1 and Lemmas 2.1 and Lemma 5.1
|d 1 0 | ≤ C(T, ∆t 0 ) ϕ 1,∞ ∆t(1 + |x| 5 E ). Assume that 1 ≤ k ≤ N − 1. Thanks to Theorem 4.1, |d 1 k | ≤ (k+1)∆t k∆t E | Du (∆t) (T − t,X(t)), Ψ ∆t (X(t)) − Ψ ∆t (X k ) | dt ≤ (k+1)∆t k∆t C η+ α+ǫ 2 (T, ∆t 0 ) (T − t) η+ α+ǫ 2 ϕ 1,∞ E (1 + |X(t)| 2 E ) (−A) −η− α+ǫ 2 Ψ ∆t (X(t)) − Ψ ∆t (X k ) H dt
Thanks to Lemma 2.2 and Taylor formula,
(−A) −η− α+ǫ 2 Ψ ∆t (X(t)) − Ψ ∆t (X k ) H ≤ M(X k ,X(t)) (−A) − α 2 (X(t) − X k ) H , with M(X k ,X(t)) ≤ C α,ǫ (∆t 0 )(1 + |X(t)| 2 E + |(−A) α+2ǫ 2X (t)| 2 H + |X k | 2 E + |(−A) α+2ǫ 2 X k | 2 H ).
Using Hölder inequality, we need to estimate EM(X k ,X(t)) 4 1 4 and
E (−A) − α 2 (X(t) − X k ) 2 H 1 2 .
The following auxiliary results give the required estimates. 1 2 ) and M ∈ N. There exists C α (T, ∆t 0 , M) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ], x ∈ E, and n ∈ N with n∆t ≤ T , 1 2 ). There exists C α (T, ∆t 0 ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ] and x ∈ E, for all t ∈ [0, T ] ∩ [n∆t, (n + 1)∆t], n ∈ N,
Lemma 5.2. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1), α ∈ [0,E[|(−A) α 2 W A (n∆t)| M H ] 1 M + E[|(−A) α 2 X n | M H ] 1 M ≤ C α (T, ∆t 0 , M)(1 + |x| E + (n∆t) − α 2 |x| H ). Lemma 5.3. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1), α ∈ [0,E|(−A) − α 2 (X(t) − X n )| 2 H 1 2 ≤ C α (T, ∆t 0 )∆t α (1 + |x| 3 E + (n∆t) − α 2 |x| H ).
Thanks to Lemma 2.1, Φ ∆t is globally Lipschitz continuous, hence applying (3) yields
|(−A) α+2ǫ 2X (t)| H ≤ |(−A) α+2ǫ 2 Φ t−n∆t (X n )| H + |(−A) α+2ǫ 2 W A (t) − e (t−n∆t)A W A (n∆t) | H ≤ Ce ∆t |(−A) α+3ǫ 2 X n | H + |(−A) α+2ǫ 2 W A (t)| + |(−A) α+2ǫ 2 W A (n∆t)| H .
With the condition α + 3ǫ < 1 2 , Lemma 5.2 above then implies
EM(X k ,X(t)) 4 1 4 ≤ C(1 + |x| 2 E + (k∆t) −α−3ǫ |x| 2 H )
. Finally, using Lemmas 5.1 and 5.3, for 1 ≤ k ≤ n − 1,
|d 1 k | ≤ ∆t α (k+1)∆t k∆t C α,ǫ (T, ∆t 0 ) (T − t) η+ α+ǫ 2 ϕ 1,∞ dt 1 + 1 (k∆t) 2α+3ǫ (1 + |x| 5 E )
Observe that 2α + 3ǫ < 1.
Treatment of d 2 k . Let α ∈ (0, 1 2 ). Then, thanks to Theorem 4.1, for all k ∈ {0, . . . , N − 1},
|d 2 k | ≤ (k+1)∆t k∆t E | Du (∆t) (T − t,X(t)), I − e (t−k∆t)A Ψ ∆t (X k ) | dt ≤ (k+1)∆t k∆t C 2α (T, ∆t 0 ) ϕ 1,∞ (T − t) 2α (−A) −2α (I − e (t−k∆t)A ) L(H) E (1 + |X(t)| 2 E )|Ψ ∆t (X k )| H dt ≤ ∆t 2α (k+1)∆t k∆t C 2α (T, ∆t 0 ) ϕ 1,∞ (T − t) 2α (1 + |x| 5 E )dt, using (−A) −2α (I − e (t−k∆t)A ) L(H) ≤ C 2α (t − k∆t) 2α ≤ C 2α ∆t 2α when 0 ≤ t − k∆t ≤ ∆t.
Treatment of d 3 k . Thanks to Lemma 2.1, for all z ∈ R, and all 0 ≤ τ ≤ ∆t ≤ ∆t 0 ,
|Ψ ∆t (z) − Ψ ∆t (Φ τ (z))| ≤ C(∆t 0 )|z − Φ τ (z)|(1 + |z| 2 ) ≤ C(∆t 0 )τ |Ψ τ (z)|(1 + |z| 2 )
≤ C(∆t 0 )∆t(1 + |z| 5 ). 15 Thanks to Theorem 4.1, for all k ∈ {0, . . . , N − 1},
|d 3 k | ≤ (k+1)∆t k∆t E | Du (∆t) (T − t,X(t)), e (t−k∆t)A Ψ ∆t (X k ) − Ψ ∆t (Φ t−k∆t (X k )) | dt ≤ (k+1)∆t k∆t C 0 (T, ∆t 0 ) ϕ 1,∞ E Ψ ∆t (X k ) − Ψ ∆t (Φ t−k∆t (X k )) H dt ≤ ∆t (k+1)∆t k∆t C 0 (T, ∆t 0 ) ϕ 1,∞ E (1 + |X k | 5 E ) dt ≤ ∆t 2 C(T, ∆t 0 ) ϕ 1,∞ (1 + |x| 5 E ).
Treatment of d 4 k . Thanks to Theorem 4.1 and Lemma 2.1, for all k ∈ {0, . . . , N − 1},
|d 4 k | ≤ (k+1)∆t k∆t E | Du (∆t) (T − t,X(t)), e (t−k∆t)A Ψ ∆t (Φ t−k∆t (X k )) − Ψ 0 (Φ t−k∆t (X k )) | dt ≤ (k+1)∆t k∆t C 0 (T, ∆t 0 ) ϕ 1,∞ E Ψ ∆t (Φ t−k∆t (X k )) − Ψ 0 (Φ t−k∆t (X k )) H dt ≤ ∆t (k+1)∆t k∆t C 0 (T, ∆t 0 ) ϕ 1,∞ E (1 + |X k | 5 E ) dt ≤ ∆t 2 C(T, ∆t 0 ) ϕ 1,∞ (1 + |x| 5 E ).
Conclusion. In conclusion,
E u (∆t) (T, x) − E u (∆t) (0, X N ) ≤ C(T, ∆t 0 , |x| E ) ϕ 1,∞ ∆t + ∆t α N −1 k=1 (k+1)∆t k∆t C α,ǫ (T, ∆t 0 ) (T − t) η+ α+ǫ 2 t 2α+3ǫ dt + ∆t 2α N −1 k=0 (k+1)∆t k∆t C 2α (T, ∆t 0 ) ϕ 1,∞ (T − t) 2α dt + ∆t(N∆t) ≤ C(T, ∆t 0 , |x| E ) ϕ 1,∞ ∆t α ,
with α < 1 2 , and ǫ > 0 such that α + 3ǫ < 1 2 . Combined with the arguments of Section 3, this concludes the proof of Theorem 3.2. Proof of Lemma 5.1. For any n ∈ N, and t ∈ [n∆t, (n + 1)∆t], the definition (32) ofX(t) gives
|X(t)| E ≤ |Φ t−n∆t (X n )| E + |W A (t) − e (t−n∆t)A W A (n∆t)| E ≤ e ∆t |X n | E + |W A (t)| E + |W A (n∆t)| E ,
thanks to Lemma 2.1. Using (9) and Lemma 2.4 then concludes the proof.
Proof of Lemma 5.2. Note that, for all n ∈ N, such that n∆t ≤ T ,
E|(−A) α 2 X n | M H | 1 M ≤ |(−A) α 2 e n∆tA x| H + C(T )∆t n−1 k=0 E|(−A) α 2 e (n−k)∆t Ψ ∆t (X k )| M H 1 M + E|(−A) α 2 W A (n∆t)| M H ) 1 M ≤ (n∆t) − α 2 |x| E + C(T )∆t n−1 k=0 1 (n − k)∆t) α 2 E|Ψ ∆t (X k )| M H 1 M + C α (T, M), since, for all t ∈ [0, T ], E|(−A) α 2 W A (t)| M H ≤ C t 0 (−A) α 2 e (t−s)A 2 L 2 (H) ds M 2 ≤ C t 0 (t − s) −α− 1 2 −ǫ ds M 2 ≤ C α (T, M),
where ǫ ∈ (0, 1 2 − α). Thanks to Lemma 5.1, then
E|(−A) α 2 X n | M H | 1 M ≤ C α (T, ∆t 0 , M)(1 + |x| E + (n∆t) − α 2 |x| H ).
Proof of Lemma 5.3. For t ∈ [n∆t, (n + 1)∆t], t ≤ T ,
X(t) − X n = e (t−n∆t)A X n − X n + (t − n∆t)e (t−n∆t)A Ψ t−n∆t (X n ) + t n∆t e (t−s)A dW (s).
First,
E|(−A) − α 2 (e (t−n∆t)A − I)X n | 2 H 1 2 ≤ (−A) −α (e (t−n∆t)A − I) L(H) E|(−A) α 2 X n | 2 H 1 2 ≤ C α (T, ∆t 0 )∆t α (1 + |x| E + (n∆t) − α 2 |x| H ),
thanks to Lemma 5.2. Second,
E (−A) − α 2 (t − n∆t)e (t−n∆t)A Ψ t−n∆t (X n ) 2 H 1 2 ≤ ∆t E|Ψ t−n∆t (X n )| 2 E 1 2 ≤ C(T, ∆t 0 )∆t(1 + |x| 3 E ),
thanks to Lemma 5.1. Third, by Itô formula, for ǫ = 1 2 − α > 0,
E|(−A) − α 2 t n∆t e (t−s)A dW (s)| 2 H = t n∆t (−A) − α 2 e (t−s)A 2 L 2 (H) ds ≤ C α t n∆t (t − s) − 1 2 −ǫ+α ds ≤ C α ∆t 1 2 −ǫ+α = C α ∆t 2α .
This concludes the proof of Lemma 5.3.
Proof of Theorem 3.3
The aim of this section is to prove Theorem 3.3. Let the numerical scheme X n n∈N be given by (13).
The section is organized as follows. An auxiliary processX and an appropriate decomposition of the error are given in Section 6.1. Error terms are estimated in Section 6.2. Auxiliary results are proved in Section 6.3.
Assume that ϕ satisfies Assumption 3.1, and to simplify notation, without loss of generality, assume that ϕ 2,∞ ≤ 1.
6.1. Decomposition of the error. As explained in Section 3, the strategy for the weak error analysis requires to apply Itô formula, hence the definition of an appropriate continuoustime processX.
Set, for every n ∈ N, and every t ∈ [n∆t, (n + 1)∆t],
(33)X(t) = X n + (t − n∆t)AS ∆t X n + (t − n∆t)S ∆t Ψ ∆t (X n ) + S ∆t W (t) − W (n∆t) ,
where we recall that S ∆t = (I − ∆tA) −1 . By construction,X(n∆t) = X n for all n ∈ N. Moreover,
dX(t) = AS ∆t X n dt + S ∆t Ψ ∆t (X n )dt + S ∆t dW (t) , t ∈ [n∆t, (n + 1)∆t], n ∈ N.
The following result gives moment estimates. Proof is postponed to Section 6.1.
E[|X(t)| M E ] ≤ C(T, ∆t 0 , M)(1 + |x| 3 E ) M .
The error is then decomposed as follows, using Itô formula, and the Kolmogorov equation (11), with T = N∆t,
E[u (∆t) (T, x)]−E[u (∆t) (0, X N )] = E[u (∆t) (T, x)] − E[u (∆t) (T − ∆t, X 1 )] + N −1 k=1 E u (∆t) ((N − k)∆t, X k ) − E u (∆t) ((N − k − 1)∆t, X k+1 ) = E[u (∆t) (T − ∆t, X (∆t) (∆t))] − E[u (∆t) (T − ∆t, X 1 )] + N −1 k=1 a k + b k + c k , 18 where a k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), AX(t) − AS ∆t X k dt b k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), Ψ ∆t (X(t)) − S ∆t Ψ ∆t (X k ) dt c k = 1 2 (k+1)∆t k∆t E j∈N D 2 u (∆t) (T − t,X(t)).(e j , e j ) 1 − 1 (1 + λ j ∆t) 2 dt.
Section 6.2 is devoted to the proof of Lemmas 6.2 and 6.3 below. Theorem 3.3 is a straightforward consequence of these results, thanks to the decomposition of the error above.
Lemma 6.2. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and x ∈ E. For all α ∈ [0, 1 2 ), there exists C α (T, ∆t 0 , |x| E ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ), E[u (∆t) (T − ∆t, X (∆t) (∆t))] − E[u (∆t) (T − ∆t, X 1 )] ≤ C α (T, ∆t 0 , |x| E )∆t α . Lemma 6.3. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1] and x ∈ E. For all α ∈ [0, 1 2 ), there exists C α (T, ∆t 0 , |x| E ) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ), n−1 k=1 (|a k | + |b k | + |c k |) ≤ C α (T, ∆t 0 , |x| E )∆t α .
Compared with Section 5, there are more error tems, and the analysis is more technical. The proof essentially follows the same strategy as in [13]. In particular, Malliavin calculus techniques are employed. For completeness, details are given.
We emphasize that the most important new result is the estimate on the Malliavin derivative, see Lemma 6.4. This result is non trivial, since Ψ ∆t is not globally Lipschitz continuous, and it is obtained thanks to the structure of the numerical scheme, based on a splitting approach.
Note that an alternative approach to treat the term b k below, would be to adapt the strategy used in Section 5.2 to treat the term d 1 k , using Lemma 2.2 in particular. Appropriate versions of Lemmas 5.2 and 5.3 would be required. It seems that this alternative approach does not considerably shortens the proof.
We recall the Malliavin calculus duality formula in some Hilbert space K. Let D 1,2 be the closure of smooth random variables (with respect to Malliavin derivative) for the topology defined by the norm
F D 1,2 = E[|F | 2 K ] + E T 0 |D s F | 2 K ds .
where D s F denotes the Malliavin derivative of F . For F ∈ D 1,2 and Ξ ∈ L 2 (Ω×[0, T ]; K) such that Ξ(t) ∈ D 1,2 for all t ∈ [0, T ] and T 0 T 0 |D s Ξ(t)| 2 dsdt < +∞, we have the integration by part formula: 19 In our context, we will use another form of this integration by parts formula: for u ∈ C 2 b (H), and any adapted process Ξ ∈ L 2 (Ω × [0, T ]; L 2 (H)),
E F T 0 (Ξ(s), dW s ) = E T 0 D s F, Ξ(s) ds .(34) E Du (∆t) (F ), T 0 Ξ(s)dW (s) = E j∈N T 0 D 2 u (∆t) (FE[u (∆t) (T − ∆t, X (∆t) (∆t))]−E[u (∆t) (T − ∆t, X 1 )] ≤ C α (T − ∆t) 2α E|(−A) −2α (X (∆t) (∆t) − X 1 )| H . Note that E|(−A) −2α (X (∆t) (∆t) − X 1 )| H ≤ |(−A) −2α e ∆tA − S ∆t x| H + ∆t 0 E|(−A) −2α e (∆t−t)A Ψ ∆t (X (∆t) (t))| H dt + ∆t|(−A) −2α S ∆t Ψ ∆t (x)| H + E| ∆t 0 (−A) −2α e (∆t−t)A dW (t)| H + E| ∆t 0 (−A) −2α S ∆t dW (t)| H .E|(−A) −α e (∆t−t)A Ψ ∆t (X (∆t) (t))| H ≤ E|Ψ ∆t (X (∆t) (t))| H ≤ C(1 + |x| 3 E )
and
|(−A) −α S ∆t Ψ ∆t (x)| H ≤ |Ψ ∆t (x)| H ≤ C(1 + |x| 3 E )
. Finally, for the stochastic integral terms,
E| ∆t 0 (−A) −2α e (∆t−t)A dW (t)| 2 H + E| ∆t 0 (−A) −2α S ∆t dW (t)| 2 H ≤ 2∆t (−A) −2α 2 L 2 (H)
and (−A) −2α 2 L 2 (H) < ∞ when 2α > 1 4 . It is then straightforward to conclude that, for α ∈ [0, 1 2 ),
E[u (∆t) (T − ∆t, X (∆t) (∆t))] − E[u (∆t) (T − ∆t, X 1 )] ≤ C α (T, |x| E )∆t α .
This concludes the proof of Lemma 6.2. 20 6.2.2. Proof of Lemma 6.3, Part 1. The aim of this section is to prove, for α ∈ [0, 1 2 ), that
N −1 k=1 |a k | ≤ C α (T, ∆t 0 , |x| E )∆t α .
For that purpose, decompose a k as follows:
a k = a 1 k + a 2 k , where a 1 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), A(I − S ∆t )X k dt, a 2 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), A(X(t) − X k ) dt.
Using the formulation
X k = S k ∆t x + ∆t k−1 ℓ=0 S k−ℓ ∆t Ψ ∆t (X ℓ ) + k−1 ℓ=0 (ℓ+1)∆t ℓ∆t S k−ℓ ∆t dW (t),
and the identity I − S ∆t = −∆tS ∆t A, the expression a 1 k is decomposed as
a 1 k = a 1,1 k + a 1,2 k + a 1,3 k , where a 1,1 k = −∆t (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), A 2 S k+1 ∆t x dt, a 1,2 k = −∆t (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), ∆t k−1 ℓ=0 A 2 S k−ℓ+1 ∆t Ψ ∆t (X ℓ ) dt, a 1,3 k = −∆t (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), k−1 ℓ=0 (ℓ+1)∆t ℓ∆t A 2 S k−ℓ+1 ∆t dW (t) dt.
Using (33), the expression a 2 k is decomposed as
a 2 k = a 2,1 k + a 2,2 k + a 2,3 k , where a 2,1 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), (t − t k )A 2 S ∆t X k dt, a 2,2 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), (t − t k )AS ∆t Ψ ∆t (X k ) dt, a 2,3 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), t k∆t AS ∆t dW (s) dt.
Treatment of a 1,1 k . Let α ∈ [0, 1 2 ) and ǫ ∈ (0, 1−2α). Thanks to Theorem 4.1 and Lemma 6.1,
|a 1,1 k | ≤ C 2α+ǫ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ dt|(−A) 2−2α−ǫ S k+1 ∆t x| H ≤ C α,ǫ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ dt (−A) 1−ǫ S k ∆t L(H) (−A) 1−2α S ∆t L(H) ≤ C α,ǫ (T, ∆t 0 , |x| E )∆t 2α (k+1)∆t k∆t 1 (k∆t) 1−ǫ (T − t) 2α+ǫ dt, thanks to the standard inequalities (−A) 1−ǫ S k ∆t L(H) ≤ C ǫ (k∆t) −1+ǫ and (−A) 1−2α S ∆t L(H) ≤ C α ∆t 2α−1 .
Treatment of a 1,2 k . Let α ∈ [0, 1 2 ) and ǫ ∈ (0, 1 − 2α). Thanks to Theorem 4.1, Lemma 6.1, and Cauchy-Schwarz inequality,
|a 1,2 k | ≤ C 2α+ǫ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ dt∆t k−1 ℓ=0 E (−A) 2−2α−ǫ S k−ℓ+1 ∆t Ψ ∆t (X ℓ ) 2 H 1 2 dt.
Thanks to Lemma 6.1,
∆t k−1 ℓ=0 E (−A) 2−2α−ǫ S k−ℓ+1 ∆t Ψ ∆t (X ℓ ) 2 H 1 2 ≤ ∆t k−1 ℓ=0 (−A) 1−ǫ S k−ℓ ∆t L(H) (−A) 1−2α S ∆t L(H) E|Ψ ∆t (X ℓ )| 2 H 1 2 ≤ C α,ǫ (T, ∆t 0 , |x| E )∆t 2α−1 , using ∆t k−1 ℓ=0 1 ((k−ℓ)∆t) 1−ǫ ≤ C ǫ < ∞. Thus |a 1,2 k | ≤ C α,ǫ (T, ∆t 0 , |x| E ) (k+1)∆t k∆t 1 (T − t) 2α+ǫ dt ∆t 2α .
Treatment of a 1,3 k . The Malliavin calculus duality formula (34) is applied, for fixed t, with u = u (∆t) (T − t, ·), F =X(t), and Ξ(s) = A 2 S k−ℓ+1 ∆t for ℓ∆t ≤ s ≤ (ℓ + 1)∆t. This yields the following alternative expression for a 1,3 k :
a 1,3 k = −∆t (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), k−1 ℓ=0 (ℓ+1)∆t ℓ∆t A 2 S k−ℓ+1 ∆t dW (s) dt = −∆t (k+1)∆t k∆t k−1 ℓ=0 (ℓ+1)∆t ℓ∆t j∈N E D 2 u (∆t) (T − t,X(t)).(D e j sX (t), A 2 S k−ℓ+1 ∆t e j ) dsdt.
Lemma 6.4 below provides the required estimate. Its proof is postponed to Section 6.3.
D s X k L(H) ≤ e T .
In addition, D s X k = 0 if k∆t ≤ s. 22 Moreover, for all 0 ≤ s < k∆t ≤ t ≤ (k + 1)∆t ≤ T ,
D sX (t) L(H) ≤ (3 + ∆t|Ψ ′ ∆t (X k )| E )e T .
Let α ∈ [0, 1 2 ), and let κ ∈ ( 1 2 − α, 1 − 2α) and ǫ ∈ (0, 1 − 2α − κ) be two auxiliary parameters. Then 2α + κ + ǫ < 1, and α + κ > 1 2 . Thanks to Theorem 4.2, Lemma 6.4, and the moment estimates, by Lemma 6.1,
|a 1,3 k | ≤ C 0,2α+ǫ+κ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ+κ dt k−1 ℓ=0 ∆t j∈N |(−A) 2−2α−ǫ−κ S k−ℓ+1 ∆t e j | H ≤ C α,ǫ,κ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ+κ dt j∈N λ 1−2α−κ j (1 + λ j ∆t) ≤ C α,ǫ,κ (T, ∆t 0 , |x| E )∆t (k+1)∆t k∆t 1 (T − t) 2α+ǫ+κ dt j∈N (∆tλ j ) 1−α (1 + λ j ∆t) ∆t α−1 λ α+κ j ≤ C α,ǫ (T, ∆t 0 , |x| E ) (k+1)∆t k∆t 1 (T − t) 2α+ǫ+κ dt ∆t α , using ∆t k−1 ℓ=0 1 (k−ℓ)∆t 1−ǫ ≤ C ǫ < ∞, and j∈N 1 λ α+κ j < ∞. Treatment of a 2,1 k . Note that (t − t k )AS ∆t = (t−t k ) ∆t (S ∆t − I).
As a consequence, it is sufficient to repeat the treatment of a 1 k above, and to use t − t k ≤ ∆t, to get the required estimate for a 2,1 k :
|a 2,1 k | ≤ C α,ǫ (T, ∆t 0 , |x| E )∆t α (k+1)∆t k∆t 1 (k∆t) 1−ǫ (T − t) 2α+ǫ+κ dt.
Treatment of a 2,2 k . Let α ∈ [0, 1 2 ). Thanks to Theorem 4.1 and Lemma 6.1, Observe that D sX (t) = S ∆t for k∆t ≤ s ≤ t ≤ (k + 1)∆t. Let α ∈ [0, 1 2 ), and let κ ∈ ( 1 2 − α, 1 − 2α) be an auxiliary parameter. Then 2α + κ < 1 and α + κ > 1 2 . Thanks to 23 Theorem 4.2 and Lemma 6.1,
|a 2,2 k | ≤ C 2α (T, ∆t 0 , |x| E ) (k+1)∆t k∆t |t − t k | (T − t) 2α dt E|(−A) 1−2α S ∆t Ψ ∆t (X k )| 2 H 1 2 ≤ C 2α (T, ∆t 0 , |x| E ) (k+1)∆t k∆t 1 (T − t) 2α dt ∆t 2α .|a 2,3 k | ≤ (k+1)∆t k∆t t k∆t E ∞ j=1
λ j (1 + λ j ∆t) 2 |D 2 u (∆t) (T − t,X(t)).(e j , e j )|dsdt
≤ C 0,2α+κ (T, ∆t 0 , |x| E ) ∞ j=1 ∆tλ j (1 + λ j ∆t) 2 λ 2α+κ j (k+1)∆t k∆t 1 (T − t) 2α+κ dt ≤ C α,ǫ (T, ∆t 0 , |x| E ) (k+1)∆t k∆t 1 (T − t) 2α+κ dt j∈N (λ j ∆t) 1−α (1 + λ j ∆t) 2 ∆t α λ α+κ j ≤ C α,κ (T, ∆t 0 , |x| E ) (k+1)∆t k∆t 1 (T − t) 2α+κ dt∆t α , using j∈N 1 λ α+κ j < ∞.
Conclusion. Gathering estimates above, for all α ∈ [0, 1 2
), N −1 k=1 |a k | ≤ C α (T, ∆t 0 , |x| E )∆t α N −1 k=1 (k+1)∆t k∆t 1 t β 1 (α) (T − t) β 2 (α) dt,
with two parameters β 1 (α), β 2 (α) ∈ [0, 1). Therefore This concludes the first part of the proof of Lemma 6.3. 6.2.3. Proof of Lemma 6.3, Part 2. The aim of this section is to prove, for α ∈ [0, 1 2 ), that
N −1 k=1 |b k | ≤ C α (T, ∆t 0 , |x| E )∆t α .
For that purpose, decompose b k as follows:
b k = b 1 k + b 2 k , with b 1 k = (k+1)∆t
k∆t E Du (∆t) (T − t,X(t)), (I − S ∆t )Ψ ∆t (X k ) dt, b 2 k = (k+1)∆t k∆t E Du (∆t) (T − t,X(t)), Ψ ∆t (X(t)) − Ψ ∆t (X k ) dt.
Introduce real-valued functions Ψ j ∆t (·) = Ψ ∆t (·), e j , for all j ∈ N. Using Itô formula, for t ∈ [k∆t, (k + 1)∆t], Ψ j ∆t (X(t)) − Ψ j ∆t (X k ) = t k∆t 1 2 i∈N D 2 Ψ j ∆t (S ∆t e i , S ∆t e i )ds + t k∆t DΨ j ∆t (X(s)), S ∆t AX k ds + t k∆t DΨ j ∆t (X(s)), S ∆t Ψ ∆t (X k ) ds + t k∆t DΨ j ∆t (X(s)), S ∆t dW (s) .
This expansion gives the decomposition Du (∆t) (T − t,X(t)), e j t k∆t DΨ j ∆t (X(s)), S ∆t dW (s) dt.
b 2 k = b 2,1 k + b 2,2 k + b 2,3 k + b 2,4 k , with b 2,1 k = (k+1)∆t
Treatment of b 1 k . Thanks to Theorem 4.1 and Lemma 6.1,
|b 1 k | ≤ C 2α (T, ∆t 0 , x| E ) (k+1)∆t k∆t 1 (T − t) 2α dt E|(−A) −2α (I − S ∆t )Ψ ∆t (X k )| 2 H 1 2 ≤ C 2α (T, ∆t 0 , |x| E ) (k+1)∆t
U
(t, s)h, F (t) dt = − T s d dt U(t, s)h, ξ s (t) dt = h, ξ s (s) , using the conditions ξ s (T ) = 0 and U(s, s)h = h.
5. 3 .
3Proof of Lemmas 5.1, 5.2 and 5.3.
Lemma 6. 1 .
1Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1) and M ∈ N. There exists C(T, ∆t 0 , M) ∈ (0, ∞) such that, for all ∆t ∈ (0, ∆t 0 ] and x ∈ E, sup t∈[0,T ]
−A) −2α e ∆tA − S ∆t x| H ≤ (−A) −2α (e ∆tA − I) L(H) + (−A) −2α (S ∆t − I) L(H) |x| H ≤ C α ∆t 2α |x| H . Second, since (−A) −2α L(H) < ∞ for α ≥ 0, then using Lemma 2.3 gives
Lemma 6. 4 .
4Let T ∈ (0, ∞). For all k ∈ N, such that k∆t ≤ T , and all s ∈ [0, T ], almost surely,
E
D 2 u (∆t) (T − t,X(t)). D e j sX (t), AS ∆t e j dsdt.
1 (α) (T − t) β 2 (α) dt < ∞.
EE
λ i ∆t) 2 E Du (∆t) (T − t,X(t)), D 2 Ψ ∆t (X(s)).(e i , e i ) Du (∆t) (T − t,X(t)), DΨ ∆t (X(s)).(S ∆t AX k ) Du (∆t) (T − t,X(t)), DΨ ∆t (X(s)).(S ∆t Ψ ∆t (X k ))
EE≤
− t) 2α dt ∆t 2α .Treatment of b 2,1 k . Let α ∈ [0,1 2 ). Thanks to Theorem 4.1 and Lemma 6.1, and to the inequality|D 2 Ψ ∆t (x).(e i , e i )| H ≤ C(1 + |x| E ) M |e i | 2 E , |b 2,1 k | ≤ C 0 (T, ∆t 0 , |x| E ) i∈N ∆t 2 (1 + ∆tλ i ) 2 ≤ C α (T, ∆t 0 , |x| E ) Du (∆t) (T − t,X(t)), DΨ ∆t (X(Du (∆t) (T − t,X(t)), DΨ ∆t (X(s)). C 0 (T, ∆t 0 , |x| E )∆t 2 A 2α S k ∆t L(H) A 1−2α S ∆t L(H) |x| H ≤ C α (T, ∆t 0 , |x| E )∆t 1+2α 1 (k∆t) 2α . C 0 (T, ∆t 0 , |x| E )∆t 2 ∆t k−1 ℓ=0 A 2α S k ∆t L(H) A 1−2α S ∆t L(H) ≤ C(T, ∆t 0 , |x| E )∆t 1+2α , ℓ)∆t) 2α ≤ C α (T ) < ∞ for α ∈ [0,1 2 ). It remains to treat b 2,2,3 k . Using the Malliavin calculus duality formula (34) and the chain rule,
2.1. Assumptions. 2.1.1. Linear operator. Let A denote the unbounded linear operator on H, with
b 2,2,3 k = t k+1 t k t t k k−1 ℓ=0 t ℓ+1 t ℓ j∈N E D 2 u (∆t) (T − t,X(t)). DΨ ∆t (X(s)).(AS k−ℓ+1 ∆t e j ), D e j rX (t) drdsdt + t k+1 t k t t k k−1 ℓ=0 t ℓ+1 t ℓ j∈N
− α, 1 − 2α) be an auxiliary parameter. Thanks to Theorem 4.2, and to Lemma 6.1, using (−A) −β e j = λ −β j e j ,|c k | ≤ C 2α+ǫ (T, ∆t 0 , |x| E ) (k+1)∆t k∆t
AcknowledgmentsThe authors want to thank Arnaud Debussche for discussions, and for suggesting the approach to prove Lemma 4.4. They also wish to thank Jialin Hong and Jianbao Cui for helpful comments and suggestions to improve the presentation of the manuscript.E Du (∆t) (T − t,X(t)), D 2 Ψ ∆t (X(s)).(AS k−ℓ+1 ∆t e j , D e j rX (s)) drdsdt.Let η ∈ ( 1 4 , 1), which allows us to use inequality(2). Thanks to Theorems 4.1 and 4.2, Lemmas 6.1 and 6.4,26Finally,k . Applying Theorem 4.1 and Lemma 6.1 directly givesTreatment of b 2,4 k . Using the Malliavin calculus duality formula (34) and the chain rule,indeed D sX (t) = S ∆t for k∆t ≤ s ≤ t ≤ (k + 1)∆t. Let α ∈ (0, 1 2 ). Thanks to Theorem 4.2 and Lemma 6.1,Conclusion. Gathering estimates above, for all α ∈ [0, 1 2 ),with two parameters β 1 (α), β 2 (α) ∈ [0, 1). ThereforeThis concludes the second part of the proof of Lemma 6.3. If ℓ s ≥ k, then s ≥ k∆t, and by definition of the Malliavin derivative, D s X k = 0.Note that X ℓs+1 = S ∆t Φ ∆t (X ℓs ) + S ∆t (W ((ℓ s + 1)∆t) − W (ℓ s ∆t)), thus D s X ℓs+1 = S ∆t .28Finally, if k ≥ ℓ s + 1, using the chain rule, for all h ∈ H,Finally, for 0 ≤ s < k∆t ≤ t ≤ (k + 1)∆t ≤ T , and h ∈ H, using the chain rule and (33),∆t (X k )| E )|D h s X k | H , and using the estimate above concludes the proof of Lemma 6.4.
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|
[] |
[
"SUBCRITICAL POLARISATIONS OF SYMPLECTIC MANIFOLDS HAVE DEGREE ONE",
"SUBCRITICAL POLARISATIONS OF SYMPLECTIC MANIFOLDS HAVE DEGREE ONE"
] |
[
"Hansjörg Geiges ",
"ANDKevin Sporbeck ",
"Kai Zehmisch "
] |
[] |
[] |
We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni-Wood.
|
10.1007/s00013-021-01605-0
|
[
"https://arxiv.org/pdf/2101.02068v1.pdf"
] | 230,770,296 |
2101.02068
|
068aab1fb9cddc8841f8eb3f805061ed66698e06
|
SUBCRITICAL POLARISATIONS OF SYMPLECTIC MANIFOLDS HAVE DEGREE ONE
6 Jan 2021
Hansjörg Geiges
ANDKevin Sporbeck
Kai Zehmisch
SUBCRITICAL POLARISATIONS OF SYMPLECTIC MANIFOLDS HAVE DEGREE ONE
6 Jan 2021
We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni-Wood.
Donaldson hypersurfaces and symplectic polarisations
Let (M, ω) be a closed, connected, integral symplectic manifold, that is, the de Rham cohomology class [ω] [4] has established the existence of such hypersurfaces for any sufficiently large d.
The pair (M, Σ) is called a polarisation of (M, ω), and the number d ∈ N, the degree of the polarisation. Biran and Cieliebak [2] studied these polarisations in the Kähler case, where ω admits a compatible integrable almost complex structure J. In that setting, the complement (M \ Σ, J) admits in a natural way the structure of a Stein manifold.
As shown recently by Giroux [7], building on work of Cieliebak-Eliashberg, even in the non-Kähler case the complement of a symplectic hypersurface Σ ⊂ M found by Donaldson's construction admits the structure of a Stein manifold. Here, of course, the complex structure on M \ Σ does not, in general, extend over Σ. Complements of Donaldson hypersurfaces are also studied in [3].
Subcritical polarisations
The focus of Biran and Cieliebak [2] lay on subcritical polarisations of Kähler manifolds, which means that (M \ Σ, J) admits a plurisubharmonic Morse function ϕ all of whose critical points have, for dim M = 2n, index less than n. (They also assumed that ϕ coincides with the plurisubharmonic function defining the natural Stein structure outside a compact set containing all critical points of ϕ.) [10, p. 504] propose the study of polarisations (M, Σ) where the complement M \ Σ is homotopy equivalent to a subcritical Stein manifold (of finite type). We relax this condition a little further and call (M, Σ) homologically subcritical if M \ Σ has the homology of a subcritical Stein manifold, that is, of a CW-complex containing finitely many cells up to dimension at most n−1. This means that there is some
More generally, McDuff and Salamon
ℓ ≤ n−1 such that H k (M \Σ) vanishes for k ≥ ℓ + 1 and H ℓ (M \ Σ) is torsion-free.
Motivated by the many examples they could construct, Biran and Cieliebak [2, p. 751] conjectured that subcritical polarisations necessarily have degree 1. They suggested an approach to this conjecture using either symplectic or contact homology. A rough sketch of a proof along these lines, in the language of symplectic field theory, was given by Eliashberg-Givental-Hofer [5, p. 661]. A missing assumption c 1 (M \ Σ) = 0 of that argument and a few more details -still short of a complete proof -were added by J. He [8,Proposition 4.2], who appeals to Gromov-Witten theory and polyfolds.
Here is our main result, which entails the conjecture of Biran-Cieliebak. Our proof is devoid of any sophisticated machinery. The assumption on (M, Σ) to be homologically subcritical guarantees the surjectivity of a certain homomorphism in homology described by Kulkarni and Wood [9]; this implies the claimed indivisibility.
The Kulkarni-Wood homomorphism
We consider a pair (M, Σ) consisting of a closed, connected, oriented manifold M of dimension 2n, and a compact, oriented hypersurface Σ ⊂ M of codimension 2. No symplectic assumptions are required in this section.
Write i : Σ → M for the inclusion map. The Poincaré duality isomorphisms on M and Σ from cohomology to homology, given by capping with the fundamental class, are denoted by PD M and PD Σ , respectively.
In their study of the topology of complex hypersurfaces, Kulkarni and Wood [9] used the following composition, which we call the Kulkarni-Wood homomorphism:
Φ KW : H k (M ) i * −→ H k (Σ) PDΣ −→ H 2n−2−k (Σ) i * −→ H 2n−2−k (M ) PD −1 M −→ H k+2 (M ).
Lemma 2. The Kulkarni-Wood homomorphism equals the cup product with the cohomology class σ :
= PD −1 M (i * [Σ]) ∈ H 2 (M ). Proof. For α ∈ H k (M ) we compute Φ KW (α) = PD −1 M i * PD Σ i * α = PD −1 M i * i * α ∩ [Σ] = PD −1 M α ∩ i * [Σ] = PD −1 M α ∩ PD M (σ) = PD −1 M α ∩ (σ ∩ [M ]) = PD −1 M (α ∪ σ) ∩ [M ] = α ∪ σ.
Lemma 3. If the complement M \ Σ has the homology type of a CW-complex of dimension ℓ for some ℓ ≤ n − 1, then Φ KW :
H k (M ) → H k+2 (M ) is surjective in the range ℓ − 1 ≤ k ≤ 2n − ℓ − 2.
Proof. Write νΣ for an open tubular neighbourhood of Σ in M . By homotopy, excision, duality, and the universal coefficient theorem we have
H k (M, Σ) ∼ = H k (M, νΣ) ∼ = H k (M \ νΣ, ∂(νΣ)) ∼ = H 2n−k (M \ νΣ) ∼ = F H 2n−k (M \ Σ) ⊕ T H 2n−k−1 (M \ Σ),
where F and T denotes the free and the torsion part, respectively. This vanishes for 2n − k − 1 ≥ ℓ, that is, for k ≤ 2n − ℓ − 1. It follows that the homomorphism
i * : H 2n−2−k (Σ) → H 2n−2−k (M ) is surjective for 2n−2−k ≤ 2n−ℓ−1, or k ≥ ℓ−1.
Similarly (or directly by Poincaré-Lefschetz duality) we have
H k (M, Σ) ∼ = H 2n−k (M \ Σ),
which vanishes for 2n − k ≥ ℓ + 1, that is, for k ≤ 2n − ℓ − 1. Hence, the homo-
morphism i * : H k (M ) → H k (Σ) is surjective for k + 1 ≤ 2n − ℓ − 1, that is, for k ≤ 2n − ℓ − 2.
Proof of Theorem 1
Under the assumptions of Theorem 1, the homomorphism Φ KW : H k (M ) → H k+2 (M ) is surjective at least in the range n − 2 ≤ k ≤ n − 1; simply set ℓ = n − 1 in Lemma 3. Thus, we can pick an even number k = 2m in this range. The free part of H 2m+2 (M ) is non-trivial, since this cohomology group contains the element [ω] m+1 of infinite order.
On the other hand, Φ KW is given by the cup product with d[ω], as shown in Lemma 2. If d[ω]/torsion were divisible, so would be all elements in the image of Φ KW in H 2m+2 (M )/torsion, and Φ KW would not be surjective.
dR lies in the image of the homomorphism H 2 (M ) → H 2 dR (M ) = H 2 (M ; R) induced by the inclusion Z → R. The cohomology classes in H 2 (M ) mapping to [ω] dR are called integral lifts, and by abuse of notation we shall write [ω] for any such lift. Following McDuff and Salamon [10, Section 14.5], we call a codimension 2 symplectic submanifold Σ ⊂ M a Donaldson hypersurface if it is Poincaré dual to d[ω] ∈ H 2 (M ) for some integral lift [ω] and some (necessarily positive) integer d. Donaldson
2020
Mathematics Subject Classification. 53D35, 57R17, 57R19, 57R95. This research is part of a project in the SFB/TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics, funded by the DFG (Project-ID 281071066 -TRR 191).
Theorem 1 .
1Let (M, ω) be a closed, integral symplectic manifold, and Σ ⊂ M a compact symplectic submanifold of codimension 2, Poincaré dual to the integral cohomology class d[ω] for some (positive) integer d. If (M, Σ) is homologically subcritical, then d[ω]/torsion is indivisible in H 2 (M )/torsion. In particular, d = 1.
The diffeomorphism type of symplectic fillings. K Barth, H Geiges, K Zehmisch, J. Symplectic Geom. 17K. Barth, H. Geiges and K. Zehmisch, The diffeomorphism type of symplectic fillings, J. Symplectic Geom. 17 (2019), 929-971.
Symplectic topology on subcritical manifolds. P Biran, K Cieliebak, Comment. Math. Helv. 76P. Biran and K. Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv. 76 (2001), 712-753.
Symplectic homology of complements of smooth divisors. L Diogo, S T Lisi, J. Topol. 12L. Diogo and S. T. Lisi, Symplectic homology of complements of smooth divisors, J. Topol. 12 (2019), 967-1030.
Symplectic submanifolds and almost-complex geometry. S K Donaldson, J. Differential Geom. 44S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), 666-705.
Introduction to symplectic field theory. Ya, A Eliashberg, H Givental, Hofer, Geom. Funct. Anal. IIYa. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. 2000, Special Volume, Part II, 560-673.
Contact structures on product five-manifolds and fibre sums along circles. H Geiges, A I Stipsicz, Math. Ann. 348H. Geiges and A. I. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Math. Ann. 348 (2010), 195-210.
Remarks on Donaldson's symplectic submanifolds. E Giroux, Pure Appl. Math. Q. 13E. Giroux, Remarks on Donaldson's symplectic submanifolds, Pure Appl. Math. Q. 13 (2017), 369-388.
Correlators and descendants of subcritical Stein manifolds. J He, Internat. J. Math. 2438J. He, Correlators and descendants of subcritical Stein manifolds, Internat. J. Math. 24 (2013), 1350004, 38 pp.
Topology of nonsingular complex hypersurfaces. R S Kulkarni, J W Wood, Adv. in Math. 35R. S. Kulkarni and J. W. Wood, Topology of nonsingular complex hypersurfaces, Adv. in Math. 35 (1980), 239-263.
D Mcduff, D Salamon, Oxford Graduate Texts in Mathematics. OxfordOxford University PressIntroduction to Symplectic TopologyD. McDuff and D. Salamon, Introduction to Symplectic Topology (3rd edn), Oxford Grad- uate Texts in Mathematics, Oxford University Press, Oxford (2017).
Weyertal 86-90, 50931 Köln, Germany Email address: [email protected]. Mathematisches Institut, deUniversität zu KölnMathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Ger- many Email address: [email protected]
. Germany Bochum, Email, [email protected], [email protected], Germany Email address: [email protected], [email protected]
|
[] |
[
"Asymptotic theory in network models with covariates and a growing number of node parameters",
"Asymptotic theory in network models with covariates and a growing number of node parameters"
] |
[
"Qiuping Wang \nZhaoqing University\n\n\nThe Ohio State University\n‡ Central China\n\nNormal University\n\n",
"Yuan Zhang \nZhaoqing University\n\n",
"Ting Yan "
] |
[
"Zhaoqing University\n",
"The Ohio State University\n‡ Central China",
"Normal University\n",
"Zhaoqing University\n"
] |
[] |
We propose a general model that jointly characterizes degree heterogeneity and homophily in weighted, undirected networks. We present a moment estimation method using node degrees and homophily statistics. We establish consistency and asymptotic normality of our estimator using novel analysis. We apply our general framework to three applications, including both exponential family and nonexponential family models. Comprehensive numerical studies and a data example also demonstrate the usefulness of our method.
|
10.1007/s10463-022-00848-0
|
[
"https://arxiv.org/pdf/2207.09861v1.pdf"
] | 250,699,135 |
2207.09861
|
d39b0ca5be1ad99f7bffa2b051fad00a77342333
|
Asymptotic theory in network models with covariates and a growing number of node parameters
20 Jul 2022
Qiuping Wang
Zhaoqing University
The Ohio State University
‡ Central China
Normal University
Yuan Zhang
Zhaoqing University
Ting Yan
Asymptotic theory in network models with covariates and a growing number of node parameters
20 Jul 2022arXiv:2207.09861v1 [math.ST]β-modeldegree heterogeneitynetwork homophilynetwork method of moments Mathematics Subject Classification: 62F12, 91D30
We propose a general model that jointly characterizes degree heterogeneity and homophily in weighted, undirected networks. We present a moment estimation method using node degrees and homophily statistics. We establish consistency and asymptotic normality of our estimator using novel analysis. We apply our general framework to three applications, including both exponential family and nonexponential family models. Comprehensive numerical studies and a data example also demonstrate the usefulness of our method.
Introduction
Jointly modeling a network and nodal or edge-wise coviarates has long been an interesting problem. One natural idea is to extend a widely-used network model to incorporate covariates. For example, (Yang et al., 2013;Zhang et al., 2016;Binkiewicz et al., 2017) introduce nodal covariates into a stochastic block model (SBM), which captures the clustering structure in networks. In this paper, we will study the extension of another network model, called β-model (Rinaldo et al., 2013;Yan and Xu, 2013;Yan et al., 2016bYan et al., , 2019Chen et al., 2021;Zhang et al., 2021) that characterizes a different important aspect of network data, namely, degree heterogeneity (Cho et al., 2011). The degree (total number of connections) of a node provides important profiling information about its structural role in the network (Borgatti and Everett, 2000;Zhang and Xia, 2022;Maugis, 2020). A famous example is that Babai et al. (1980) shows that efficient graph matching can usually succeed with high probability between two shuffled random graphs, using a degree-based algorithm. The β-model, named by Chatterjee et al. (2011), is an undirected, binary network: P(a ij = 1) = e β i +β j 1 + e β i +β j , a ij = a ji , 1 ≤ i < j ≤ n
where A = (a ij ), 1 ≤ {i, j} ≤ n is a binary adjacency matrix. Later, Yan et al. (2016b); Fan et al. (2022) extend this model to weighted networks with edge distributions including Poisson, geometric, exponential and so on. This paper will generalize Yan et al. (2016b) rather than the original binary edge β-model. What we incorporate into a weighted β-model are edge-wise covariates. We notice that this set up also accommodates nodal covariates (e.g. immutable characteristics such as gender, race and genetic features; and/or mutable ones, including location, occupation and hobbies) since they can be easily transformed into edge-wise similarity/dissimilarity measures. According to Graham (2017), this part of the data encodes the homophily effect in network formation. As a quick illustration, consider two node pairs (i 1 , i 2 ) and (j 1 , j 2 ). Even if {β i 1 , β i 2 } is very different from {β j 1 , β j 2 }, their edge expectations E[a i 1 ,i 2 ] and E[a j 1 ,j 2 ] might not differ too much, if they have similar edge covariates z i 1 ,i 2 ≈ z j 1 ,j 2 .
Jointly modeling both degree heterogeneity and homophily, as well as developing effective estimation and inference methods along with supporting theory, is an interesting challenge. As aforementioned, covariate-assisted stochastic block models have been comparatively well-studied, whereas few works exist to extend the β-model. Among notable exceptions, Graham (2017) generalizes (1) by extending β i + β j to β i + β j + z T ij γ and devises a likelihood-based method for estimation and inference. Recently, independent works Stein andLeng (2020, 2021) further introduce ℓ 1 regularization to the joint model for undirected and directed networks, respectively. Both research groups Graham (2017) and Stein andLeng (2020, 2021) focus exclusively on binary edges for cleanness. On the other hand, many networks, such as communications, co-authorship, brain activities and others, have weighted edges.
In this paper, we develop a general joint model for weighted edges. Different from the likelihood-based approaches in Graham (2017) and Stein andLeng (2020, 2021), we propose and analyze a method-of-moment parameter estimation. As discussed later in the paper, moment method has its unique advantage in addressing slightly dependent network edge formation -despite this paper exclusively focuses on independent edge generation, we understand that a comprehensive study here paves the road towards successfully handling the very challenging problem of dependent edges. We develop a two-stage Newton method that first finds an error bound for β−β ∞ for a fixed γ via establishing the convergence rate of the Newton iterative sequence and then derives γ−γ ∞ based on a profiled equation under some conditions. When all parameters are bounded, the ℓ ∞ norm error for β is in the order of O p (n −1/2 ) while the ℓ ∞ norm error for γ is in the order of O p (n −1 ), both up to a logarithm factor. When the parameters diverge, the error bounds depend on additional factors involved with the ranges of β and γ. Further, we derive an asymptotic representation of the moment estimator, based on which, we derive their asymptotic normal distributions under classical CLT conditions. To illustrate the unified results, we present three applications, along with comprehensive numerical simulations and a real data example.
The rest of the paper is organized as follows. In Section 2, we present our general model. In Section 3, we propose our moment estimation equations. In Section 4, we establish consistency and asymptotic normality of our estimator under mild conditions. Section 5 illustrates the application of our general framework to weighted networks with logistic, Poisson and probit edge formation schemes. Section 6 contains summary and discussion. Due to limited space, simulation results and the real data application are relegated to Supplementary Material.
Covariate-assisted β-model
We shall jointly analyze data from two sources: network and edge-wise covariates. The network data is represented by an adjacency matrix A = (a ij ) n×n , 1 ≤ i < j ≤ n. We study undirected networks without self-loops, i.e. A is symmetric a ij = a ji and a ii = 0. In this paper, each entry a ij may be binary or weighted (such as collaboration counts in a co-authorship network and phone call lengths). Let d i = j =i a ij be the degree of node i and d = (d 1 , . . . , d n ) T be the degree sequence. In addition to network data, we also observe a covariate vector z ij ∈ R p on each edge. This setting also covers the scenario when we observe nodal attributes x i : simply define a similarity/dissimilarity measure g(·, ·) that converts these attributes to edge-wise covariates via g(x i , x j ). Examples including Euclidean distance for continuous x i 's and Hamming distance for binary x i 's.
Our goal is to jointly model degree heterogeneity and homophily. Degree heterogeneity is captured by a latent parameter β i ∈ R on each node. Homophily is driven by edge-wise covariates under our framework. Specifically, it is accounted for by z T ij γ, where the exogenous parameter γ ∈ R p can be understood analogously like a regression coefficient in a generalized linear model. Now we present our model. Given Z = (z ij ), the network entries a ij are generated independently by the following model, which we call "covariate-assisted β-model":
a ij |{z, β, γ} ∼ f a ij β i + β j + z T ij γ .(2)
where f is a known probability density/mass function, β i is the degree parameter of node i and γ is a p-dimensional regression coefficient for the covariate z ij . Our model (2) generalizes the semi-parametric models in econometrics literature (Fernández-Vál and Weidner, 2016) with binary and exponential responses for undirected networks. We focus on these additive models for computational tractability. It would be an interesting future work to generalize the method of our analysis to address the more general case where β, z and γ enter the model as non-additive effects. The model (2) extends not only the well-known β-model, but also many of its variants (Yan et al., 2016b). In many examples, such as β-model, f (·) is an increasing function of β i . Consequently, nodes having relatively large degree parameters will have more links than those nodes with low degree parameters, without considering homophily. To further illustrate the usefulness of (2), we consider two running examples.
Example 1. (Binary edges) In statistical network analysis, a long-studied problem is to jointly model network data with additional covariates. For example, Yang et al. (2013);Zhang et al. (2016) incorporate nodal covariates into a stochastic block model; in contrast our model provides a flexible tool for incorporating nodal and/or edge-wise covariates into a β-model. Here, we consider binary edges, i.e. a ij ∈ {0, 1}. Let F be some properly chosen transformation: F : R → [0, 1]. The probability of a ij is
P(a ij = a) = {F (β i + β j + z T ij γ)} a {1 − F (β i + β j + z T ij γ)} 1−a , a ∈ {0, 1}.
Two popular choices of F (·) are sigmoid transformation F (x) = e x /(1 + e x ) (Graham, 2017) and probit transformation
F (x) = Φ(x), where Φ(x) is the CDF of N(0, 1i + β j + z T ij γ). That is, log P(a ij = a) = a(β i + β j + z T ij γ) − exp(β i + β j + z T ij γ) − log a!.
Parameter estimation
To motivate the estimation method for our covariate-assisted β-model (2), it is helpful to very briefly recall that for the classical β-model, i.e. γ ≡ 0, there are two mainstreams of estimation methods: MLE (Chatterjee et al., 2011;Rinaldo et al., 2013) and method of moments (Yan et al., 2016b). While MLE is a widely-recognized method for model parameter estimation, as aforementioned in the introduction, method-of-moments may enjoy an easier extension to dependent-edge scenarios in the future (see also Section 6). Therefore, we propose and study a moment estimator in this paper. Notice that E[a ij ] depends on the model parameters only through
π ij := β i + β j + z T ij γ.(3)
There exists a function µ(·) such that E[a ij ] = µ(π ij ). In some future texts, we find it more convenient to emphasize that µ(π ij ) can be viewed as the (i, j) element of an n × n matrix. Therefore, we slightly abuse notation and might sometimes use a different notation µ ij (β, γ) to represent µ(π ij ). Now we are ready to present our method of moments parameter estimation. To motivate our formulation, consider a special case when the edge distribution belongs to an exponential family, namely, L(a|β, z, γ) = C(β, z, γ) · e 1≤i<j≤n a ij ·(β i +β j +z T ij γ) · h(a, z).
Examples of (4) include logistic model and Poisson model with covariates, as in Section 5. Sending the partial derivatives of the log-likelihood function of (4) to zero, we obtain the following moment equations.
d i = j:j =i µ ij (β, γ), i ∈ [n] (5) 1≤i<j≤n z ij a ij = 1≤i<j≤n z ij µ ij (β, γ)(6)
where [n] = {1, . . . , n}. Denote the solution to (5) and (6) by ( β, γ). We shall address the natural questions of existence and uniqueness of ( β, γ) in Theorem 1. Now we discuss some computational issues. When the number of nodes n is small and f is the binomial, Probit, or Poisson probability function or Gamma density function, we can simply use the package "glm" in the R language to solve (5) and (6). For relatively large n, it might not have large enough memory to store the design matrix for β required by the R package "glm". In this case, we recommend the use of a two-step iterative algorithm by alternating between solving the first equation in (5) via the fixed point method [Chatterjee et al. (2011)] or the gradient descent algorithm [Bubeck (2015)] and solving the second equation in (6).
Asymptotic properties
In this section, we present the consistency and asymptotic normality of the moment estimator. We start with notation. For any C ⊂ R n , let C 0 and C denote the interior and closure of C, respectively. For a vector x = (x 1 , . . . , x n ) T ∈ R n , let x be a generic notation for vector norm. Specifically, inherit the notion of x p to denote ℓ p norm from functional analysis. Let B ∞ (x, ǫ) = {y : x − y ∞ ≤ ǫ} be an ǫ-neighborhood of x under ℓ ∞ metric. For an n × n matrix J = (J i,j ), let J ∞ denote the matrix norm induced by the ℓ ∞ -norm on vectors in R n , i.e.,
J ∞ = sup x =0 Jx ∞ x ∞ = max 1≤i≤n n j=1 |J i,j |, J max = max i,j |J ij |,
and let J be a generic notion for matrix norm. We use a superscript "*" to mark the true parameter. But in early sections of this paper, without causing ambiguity, we might omit it when stating the model.
Next, we set up regularity conditions for our main theorems. Assume µ(·) has continuous third derivative. Recall π ij = β i + β j + z T ij γ as defined in (3).
Suppose there exist b n0 , b n1 , b n2 , b n3 > 0 such that min i,j µ ′ (π ij ) · max i,j µ ′ (π ij ) > 0, (7a) b n0 ≤ min i,j |µ ′ (π ij )| ≤ max i,j |µ ′ (π ij )| ≤ b n1 ,(7b)max i,j |µ ′′ (π ij )| ≤ b n2 ,(7c)max i,j |µ ′′′ (π ij )| ≤ b n3 .(7d)
hold for all β ∈ B ∞ (β * , ǫ n1 ), γ ∈ B ∞ (γ * , ǫ n2 ), where ǫ n1 , ǫ n2 > 0 are two diminishing numbers as n → ∞. Condition (7a) is mild, requiring that the derivative of the expectation function µ(x) is positive for all x ∈ R or negative. The conditions (34a)-(34c) may seem quite technical and abstract for readers. To help with intuitive understanding, let us illustrate them using Example 1 with a logistic link function. In this case, µ(x) = e x /(1 + e x ). Straight calculations show
µ ′ (x) = e x (1 + e x ) 2 , µ ′′ (x) = e x (1 − e x ) (1 + e x ) 3 , µ ′′′ (x) = e x (1 − 4e x + e 2x ) (1 + e x ) 4 .
It is easy to verify that max |µ ′ (x)|, |µ ′′ (x)|, |µ ′′′ (x)| ≤ 1/4, where we used
|µ ′′ (x)| ≤ e x (1 + e x ) 2 (1 − e x ) (1 + e x ) and |µ ′′′ (x)| = e x (1 + e x ) 2 (1 − e x ) 2 (1 + e x ) 2 − 2e x (1 + e x ) 2 .
Therefore, in this example, we can set b n1 = b n2 = b n3 = 1/4 and
b n0 = min i,j e π ij (1 + e π ij ) 2 ≥ e 2 β * ∞+ γ * 1 z * +2ǫ n1 +pǫ n2 (1 + e 2 β * ∞ + γ * 1 z * +2ǫ n1 +pǫ n2 ) 2 ,(8)
where z * := max i,j z ij ∞ . In addition to the regularity conditions (7a)-(34c) this section, we shall also need the following assumptions:
Assumption 1. (Bounded covariates) Suppose max i,j z ij ∞ ≤ C z holds for some uni- versal constant C z . Assumption 2. (Sub-exponential edge distribution) The distribution of a ij − Ea ij is sub-exponential, with parameter h ij . Denote h n := max i,j h ij .
Assumption 1 is naturally satisfied by some popular dissimilarity measures between nodal covariates, such as Hamming distance. If the observed z ij 's or nodal covariates seem to vary wildly, we can simply apply a transformation such as sigmoid or probit functions to tame them into universally bounded edge covariates, see Section 6 of Zhang and Xia (2022). Assumption 2 is satisfied by many popular edge distributions, such as those in Yan et al. (2016b). We make this assumption mostly to make our narration succinct -it can be replaced by any other conditions that guarantee |d i − Ed i | = O p (n 1/2 ) and | i<j (a ij − Ea ij )| = O p (n), respectively.
Consistency
The asymptotic behavior of the estimator ( β, γ) critically depends on the curvature of µ(β, γ). To study this curvature, we start with setting up some notation. Define
F i (β, γ) = n j=1,j =i µ ij (β, γ) − d i , i = 1, . . . , n.(9)
For simplicity, we write F (β, γ) = (F 1 (β, γ), . . . , F n (β, γ)) T . Also denote F γ,i (β) to be F i (β, γ) for an arbitrary given γ, denote F γ (β) = (F γ,1 (β), . . . , F γ,n (β)) T and define β γ to be the solution to F γ (β) = 0. Set
Q(β, γ) = i<j z ij (µ ij (β, γ) − a ij ),(10)Q c (γ) = i<j z ij (µ ij ( β γ , γ) − a ij ).(11)
By definition, we have the following relationships:
F ( β, γ) = 0, F γ ( β γ ) = 0, Q( β, γ) = 0, Q c ( γ) = 0.
Similar to Chatterjee et al. (2011);Yan et al. (2016a), we define a notion called "L class matrices" for narration convenience. Given some M ≥ m > 0, we say an n × n matrix V = (v ij ) belongs to the matrix class L n (m, M) if V is a diagonally balanced matrix with positive elements bounded by m and M, i.e.,
v ii = n j=1,j =i v ij , i = 1, . . . , n, m ≤ v ij ≤ M, i, j = 1, . . . , n; i = j.(12)
Since π is linear in β, for any 1 ≤ {i = j} ≤ n, we have
∂F i (β, γ) ∂β i = j =i µ ′ (π ij ), ∂F i (β, γ) ∂β j = µ ′ (π ij ).(13)
It is easy to verify that (13) yields that when µ ′ (x) > 0, β ∈ B ∞ (β * , ǫ n1 ) and γ ∈ B ∞ (γ * , ǫ n2 ), we have F ′ γ (β) ∈ L n (b n0 , b n1 ). For simplicity, we assume
F ′ γ (β) ∈ L(b n0 , b n1 ) hereafter (if −F ′ γ (β) ∈ L(b n0 , b n1 ), we could rewrite F γ (β) := −F γ (β)). Define a conve- nient shorthand V (β, γ) := F ′ γ (β),
and define an abbreviation V = V (β * , γ * ). We will establish the consistency of the estimator β γ using the theorems of Newton method, for which we shall need an explicity formulation of F ′ γ (β). This inverse does not have a closed form, but fortunately, by mimicking Simons et al. (1999), Yan et al. (2015 proposed a convenient approximate inversion formula V ∈ L n (m, M) by
S = diag(1/v 11 , . . . , 1/v nn )(14)
at approximate error of O (M 2 /(n 2 m 3 )) under the matrix maximum norm (i.e., the maximum of all absolute elements of a matrix).
With the above notation preparations, now we commence the asymptotic analysis of the estimator. First, we have
∂F γ ( β γ ) ∂γ T = ∂F ( β γ , γ) ∂β T ∂ β γ γ T + ∂F ( β γ , γ) ∂γ T = 0,(15)∂Q c (γ) ∂γ T = ∂Q( β γ , γ) ∂β T ∂ β γ γ T + ∂Q( β γ , γ) ∂γ T .(16)
Combining (15) and (16), the Jacobian matrix Q ′ c (γ) = ∂Q ′ c (γ)/∂γ has the following formulation:
∂Q c (γ) ∂γ T = ∂Q( β γ , γ) ∂γ T − ∂Q( β γ , γ) ∂β T ∂F ( β γ , γ) ∂β T −1 ∂F ( β γ , γ) ∂γ T .(17)
The asymptotic behavior of γ crucially depends on Q ′ c (γ). But the β γ that appears in the definition of Q ′ c (γ) does not have a closed form. To facilitate the quantitative study of the curvature of Q c (γ), define
H(β, γ) = ∂Q(β, γ) ∂γ T − ∂Q(β, γ) ∂β T ∂F (β, γ) ∂β T −1 ∂F (β, γ) ∂γ T ,(18)
which can be viewed as a relaxed version of ∂Q c (γ)/∂γ. When β ∈ B ∞ (β * , ǫ n1 ), by Section 10 of Supplemental Material, we have
1 n 2 H(β, γ * ) ij = 1 n 2 H(β * , γ * ) ij + o(1),(19)
for each given (i, j) : 1 ≤ i < j ≤ n, where recall that the entries of H are sums n(n−1)/2 of terms, thus n −2 would be a proper rescaling factor. We assume H(β, γ) is positively definite. When a ij belongs to exponential family of distributions, H(β, γ) is the Fisher information matrix of the concentrated likelihood function on γ (e.g. page 126 of Amemiya (1985)) and is thus positive definite. See also Section 5. In fact, the asymptotic variance of γ is H −1 (β, γ) when f (·) in (2) is an exponential-family distribution; see the applications in Section 5. Thus, the asymptotic behavior of γ will be ill-posed without this assumption. Define
κ n := sup β∈B∞(β * ,ǫ n1 ) n 2 · H −1 (β, γ * ) ∞(20)
Now we formally state the consistency result.
Theorem 1. Let σ 2 n = n 2 (V −1 − S)Cov(F (β * , γ * ))(V −1 − S) max . Suppose Assumptions 1 and 2 and conditions (7a)-(34c) hold, and
κ 2 n b 4 n1 b n2 b 3 n0 b n2 h 2 n b 3 n0 + σ n = o n log n .(21)
Then the moment estimator γ exists with high probability, and we further have
γ − γ * ∞ = O p κ n b n1 log n n h 2 n b n2 b 3 n0 + σ n = o p (1) β − β * ∞ = O p h n b n0 log n n = o p (1).
Our proof of Theorem 1 analyzes a two-stage Newton method and is thus different from Graham (2017) that uses a convergence rate analysis of the fixed point method in Chatterjee et al. (2011).
When f (·) in model (2) is an exponential family distribution, then V = Cov(F (β * , γ * )). In this case, the expression inside the norm of σ 2 n simplifies into
(V −1 − S)V (V −1 − S) = V −1 − S + v ij (1 − δ ij ) v ii v jj , By Lemma 1, V −1 − S max = O(b 2 n1 b −3 n0 n −2 ). Thus, σ 2 n = O(b 2 n1 /b 3 n0 ).
We have the following corollary.
Corollary 1. Assume V = Cov(F (β * , γ * )) and the conditions of Theorem 1 hold. If
κ 2 n h 2 n b 5 n1 b n2 b 6 n0 = o n log n , then γ − γ * ∞ = O p κ n b 2 n1 h 2 n b n2 log n nb 3 n0 = o p (1) β − β * ∞ = O p h n b n0 log n n = o p (1).
When f (·) in model (2) belongs to exponential-family distributions and β * ∞ and γ * ∞ are universally bounded, then b n0 , b n1 , b n2 and σ n are constants. Further, if all covariates are bounded, H(β * , γ * )/n 2 is approximately a constant matrix such that κ n is also a constant. In this case, the conditions in Theorem 1 easily hold. Further, if b n0 , b n1 , b n2 , κ n , h n are constants, then the convergence rates of β and γ are O p ((log n/n) 1/2 ) and O p (log n/n), respectively. This reproduces the error bound in Chatterjee et al. (2011). This convergence rate matches the minimax optimal upper bound β − β ∞ = O p ((log p/n) 1/2 ) for the Lasso estimator in the linear model with a p-dimensional parameter vector β and the sample size n (Lounici, 2008). The convergence rate O p (log n/n) for γ is very close to the square root rate N −1/2 in the classical large sample theory, where N = n(n − 1)/2.
Asymptotic normality of β
We derive the asymptotic expansion format of β by applying a second order Taylor expansion to F ( β, γ) and showing that various remainder terms are asymptotically negligible.
Theorem 2. Assume the conditions of Theorem 1 hold. If
κ 2 n b 2 n1 h 2 n b n2 b 3 n0 + σ 2 n 2 = o n log n ,
then for any fixed i,
β i − β i = v −1 ii (d i − Ed i ) + O p κ n b n1 log n nb n0 b n2 h 2 n b 3 n0 + σ n . Let u ii = j =i Var(a ij ). If j =i E(a ij − Ea ij ) 3 /v 3/2 ii → 0, then, by the Lyapunov's central limit theorem, u −1/2 ii {d i − E(d i )} converges in distribution to the standard normal distribution.
When considering the asymptotic behaviors of the vector (d 1 , . . . , d r ) with a fixed r, one could replace the degrees d 1 , . . . , d r by the independent random variables d i = a i,r+1 + · · · + a in , i = 1, . . . , r. Therefore, we have the following lemma.
Proposition 1. Under the conditions of Theorem 2, if u −3/2 ii j:j =i E(a ij − Ea ij ) 3 → 0, then we have: (1)For any fixed r ≥ 1, (d 1 − E(d 1 ), . . . , d r − E(d r )
) are asymptotically independent and normally distributed with mean zero and marginal variances u 11 , . . . , u rr , respectively.
(
2)More generally, n i=1 c i (d i − E(d i ))/ √ u ii is asymptotically normally distributed with mean zero and variance ∞ i=1 c 2 i whenever c 1 , c 2 , . . . are fixed constants, and ∞ i=1 c 2 i < ∞.
Part (2) follows from part (1) and the fact that
lim r→∞ lim sup t→∞ Var n k=r+1 c i d i − E(d i ) √ u ii = 0(22)
by Theorem 4.2 of Billingsley (1995). To see (22), it suffices to show that the eigenvalues of the covariance matrix of (d i − E(d i ))/u 1/2
ii , i = r + 1, . . . , n are bounded by 2 for all r < n, which is implied by the well-known Perron-Frobenius theorem: if A is a symmetric positive definite matrix with diagonal elements equaling to 1, with nonnegative off-diagonal elements, then its largest eigenvalue is less than 2. In view of Proposition 1, we immediately have the following corollary.
Corollary 2. Assume that conditions in Theorem 2 hold. If u
−3/2 ii j =i E(a ij − Ea ij ) 3 → 0, then for fixed k the vector (u −1/2 11 v 11 ( β 1 − β * ), . . . , u −1/2 kk v kk ( β k − β * k ) converges in distri- bution to the k-dimensional multivariate standard normal distribution.
Asymptotic normality of γ
Let T ij = e i + e j , where e i ∈ R n is all zero except its ith element equals 1. Define V (β, γ) = ∂F (β,γ) ∂β T , V Qβ (β, γ) = ∂Q(β,γ) ∂β T , s ij (β, γ) = (a ij − Ea ij )(z ij − V Qβ (β, γ)[V (β, γ)] −1 T ij ). When evaluating H(β, γ), Q(β, γ), V (β, γ) and V Qβ (β, γ) at their true values (β * , γ * ), we omit the arguments β * , γ * , i.e., V = V (β * , γ * ). Recall we earlier defined N = n(n − 1). Also defineH = lim n→∞ 1 N H(β * , γ * ),
where we recall the definition of H(β, γ) from (18). We have
Theorem 3. Let U = Var(d). Assume the conditions in Theorem 1 hold. If b n3 h 3 n b −3 n0 = o(n 1/2 /(log n) 3/2 ), then we have √ N( γ − γ * ) =H −1 B * +H −1 × 1 √ N i<j s ij (β * , γ * ) + o p (1), where B * = lim n→∞ 1 2 √ N n k=1 ∂ 2 Q(β * , γ * ) ∂β k ∂β T V −1 UV −1 e k .(23)
Note that s ij (β, γ), i < j, are independent vectors. By Lyapunov's central limit theorem, we have
Proposition 2. Let λ ij = Var(a ij ) and z ij = z ij − V Qβ V −1 T ij . For any nonzero vector c = (c 1 , . . . c p ) T , if i<j (c T z ij ) 3 λ 3 ij [ i<j (c T z ij ) 2 λ ij ] 3/2 = o(1),(24)then (c T Σc) −1/2 i<j s γ ij (β * , γ * ) converges in distribution to the standard normal distri- bution, where Σ = Cov(Q − V Qβ V −1 H).
In view of Proposition 2 and Theorem 3, we immediately have Corollary 3. Assume the conditions in Theorem 3 and (24) hold. Then
√ N c T ( γ − γ) d → N H −1 B * , c TH T ΣHc(25)
When the edge distribution (2) belongs to exponential family, we have V = U. Consequently, ∂F (β * , γ * )/∂β = Var(d), B * and Σ can be simplified as follows:
B * = 1 √ N n k=1 j =k z kj µ ′′ kj (π * ij ) j =k µ ′ kj (π * ij ) ,(26)
and
Σ = i<j z ij z T ij µ ′ ij − n i=1 ( j =i z ij µ ′ ij )( j =i z T ij µ ′ ij ) v ii .
Note that asymptotic normality of γ contains a bias term and needs to be corrected when constructing confidence interval and hypothesis testing. Here, we employ the analytical bias correction formula in Dzemski (2019):
γ bc = γ − N −1/2 H −1 ( β, γ) B,
where B is a plug-in estimator for B * using β and γ. Other bias-corrections include Graham (2017) and Fernández-Vál and Weidner (2016).
Applications
In this section, we illustrate the theoretical result by two applications: the logistic distribution and Poisson distribution for f (·). Moreover, any other distributions such as the geometric distribution that lead to the well-defined moment estimator could also be used, besides the logistic distribution and the Poisson distribution.
The logistic model
We consider the generalized β-model in Graham (2017) with the logistic distribution:
P(a ij = 1) = e β i +β j +z T ij γ 1 + e β i +β j +z T ij γ .
Graham (2017) derived the consistency and asymptotic normality of the restricted MLE. The aim of this application is to show that these properties of the unrestricted MLE continue to hold. In this model, the MLE is the same as the moment estimator. The numbers involved with the conditions in theorems are as follows. Because a ij 's are Bernoulli random variables, they are sub-exponential with h n = 1. The numbers b n0 , b n1 , b n2 and b n3 are as defined in (8) and the paragraph right above it. The condition (21) in Theorem 1 becomes that
κ 2 n ω 3 n = o n log n ,(27)
where ω n = e 2 β * ∞+ γ * ∞ . By Theorem 1, we have the following corollary.
Corollary 4. If (27) holds, then γ − γ * ∞ = O p κ n ω 3 n log n n , β − β * ∞ = O p ω n log n n .
We discuss the condition and convergence rates related to the graph density. The expectation of the graph density is
ρ n := 1 N 1≤i<j≤n Ea ij = 1 N 1≤i<j≤n e β i +β j +z T ij γ 1 + e β i +β j +z T ij γ , where N = n(n − 1)/2.
To see what is κ n , let us consider the case of that z ij is one dimension. By using S in (14) to approximate V −1 (β, γ), one can get
H(β, γ) = 1≤i<j≤n z 2 ij µ ′ (π ij ) − n i=1 1 v ii n j=1,j =i z ij µ ′ (π ij ) 2 .
In this case, κ n is approximately the inverse of n −2 H(β * , γ * ), which depends on the covariates, the configuration of parameters, and the derivative of the mean function µ(·).
Since the relationship between (κ n , b n0 ) and ρ n depends on the configuration of the parameters β and γ, where recall the definition of b n0 in (8), it is not possible to express κ n and b n0 as a function of ρ n for a general β and γ. Therefore, we consider one special case that β 1 = · · · = β n ≤ c for illustration, where c is a constant, and assume that z ij is independently drawn from the standard normality. In this case, by large sample theory,
we have 1 N 1≤i<j≤n z 2 ij µ ′ (π ij ) p. → e 2β 1 (1 + e 2β 1 ) 2 , 1 n n j=1,j =i z ij µ ′ (π ij ) p.
→ 0, such that κ n ≍ 1/ρ n , where a n ≍ b n means c 1 a n ≤ b n ≤ c 2 a n with two constants c 1 and c 2 for sufficiently large n. Further, b n0 = O(ρ n ). Then the condition in Corollary 1 becomes ρ n (log n/n) 1/8 → ∞, and, the convergence rates are
γ − γ * ∞ = O p log n nρ 4 n , β − β * ∞ = O p 1 ρ n log n n .
Here, estimation consistency requires a strong assumption ρ n ≫ (n/ log n) 1/8 . It would be of interest to relax it. Since a ij 's (j < i) are independent, it is easy to show the central limit theorem for d i and N −1/2 j<i s ij (β, γ) as given in Su et al. (2018) and Graham (2017) respectively. So by Theorems 2 and 3, the central limit theorem holds for β and γ. See Su et al. (2018) and Graham (2017) for details.
The Poisson model
We now consider the Poisson model in Example 2. Recall that the expectation of a ij is λ ij = e z T ij γ+β i +β j . In this case, µ(x) = e x . The likelihood function is
P(A) ∝ exp n i=1 β i d i + 1≤i<j≤n a ij (z T ij γ) .
It is a special case of the general exponential random graph model, where (d T , i<j a ij z T ij ) T is the sufficient statistic for the parameter vector (β T , γ T ) T . Therefore, the maximum likelihood equations are identical to the moment equations defined in (5) and (6). Define
q n := sup β∈B∞(β * ,ǫ n1 ),γ∈B∞(γ * ,ǫ n2 ) max i,j |β i + β j + z T ij γ|. So b ni 's (i = 0, . . . , 3) in inequalities (34a), (34b) and (34c) are b n0 = e −qn , b n1 = e qn , b n2 = e qn , b n3 = e qn .
Clearly, Poisson(λ) is sub-exponential with parameter cλ, where c is a constant; see Example 4.6 in Zhang and Chen (2021). Thus, h n in Assumption 1 is ce 2qn . By Theorem 1, we have the following corollary.
Corollary 5. If κ n e 7qn = o((n/ log n) 1/2 ), then then
γ − γ * ∞ = O p ( κ n e 8qn log n n ) = o p (1), β − β * ∞ = O p ( e 2qn (log n) 1/2 n 1/2 ) = o p (1).
We discuss the condition and convergence rates related to the average weight. The expectation of the average weight is
λ n := 1 N i Ed i = 1 n i j =i e β i +β j +z T ij γ .
As in the first application, b n0 , b n1 and b n2 can not be represented as functions on λ n for general parameters β and γ. To get some intuitive understandings, let us consider a simple special case where β 1 = · · · = β n < c with a constant c, γ is a constant and z ij independently follows from a symmetric continuous distribution with a bounded support and the unit variance. In this case,
κ n ≍ λ −1 n , λ n = e 2β 1 +O(1) , c 1 λ n ≤ b n0 , b n1 , b n2 ≤ c 2 λ n , h n = c 3 λ 2 n ,
where c 1 , c 2 and c 3 are positive constants. Then, the condition in Theorem 1 becomes
λ n = o n log n 1/4
, and the convergence rates are
γ − γ * ∞ = O p λ 2 n log n n , β − β * ∞ = O p λ n log n n .
Note that d i = j =i a ij is a sum of n−1 independent Poisson random variables.
Since
v ij = Ea ij = λ ij , we have e −qn ≤ v ij = e β i +β j +z T ij γ ≤ e qn , 1 ≤ i < j ≤ n.
By using the Stein-Chen identity [Stein (1972); Chen (1975)] for the Poisson distribution, it is easy to verify that
E(a 3 ij ) = λ 3 ij + 3λ 2 ij + λ ij .(28)It follows j =i E(a 3 ij ) v 3/2 ii ≤ (n − 1)e qn (n − 1) 3/2 e −qn = O( e 4qn n 1/2 ).
If e 4qn = o(n 1/2 ), then the above expression goes to zero. For any nonzero vector c =
(c 1 , . . . c p ) T , if j<i (c T z ij ) 3 λ 3 ij [ j<i (c T z ij ) 2 λ ij ] 3/2 = o(1),(29)
This verifies the condition (24). Consequently, by Corollaries 2 and 3, we have the fol-lowing result.
Corollary 6. If (29) holds and λ 2 n κ 6 n e 28qn = o(n 1/2 /(log n) 3/2 ), then: (1) N 1/2 Σ −1/2 ( γ − γ * ) converges in distribution to multivariate normal distribution with mean Σ −1/2H −1 B * and covariance I p , where I p is the identity matrix, whereΣ = N −1H −1 ΣH −1 ;
(2) for a fixed r, the vector (v
1/2 11 ( β 1 − β * 1 ), . . . , v 1/2 rr ( β r − β * r )
converges in distribution to the r-dimensional standard normal distribution.
The probit model
The two examples above are exponential family of distributions. Here, we pay attention to the probit distribution, which is not exponential. Let φ(x) = (2π) 1/2 e −x 2 /2 be the standard normal density function and Φ(x) = x −∞ φ(x)dx be its the distribution function. The probit model assumes
P(a ij = 1) = Φ 1 σ (β i + β j + z ⊤ ij γ) ,
where σ is the standard derivation. Since the parameters are scale invariable, we simply set σ = 1. Then,
µ ′ (x) = φ(x), µ ′′ (x) = x √ 2π e −x 2 /2 .
Since φ(x) = (2π) 1/2 e −x 2 /2 is an decreasing function on |x|, we have when |x| ≤ Q n , 1 2π
e −Q 2 n /2 ≤ φ(x) ≤ 1 2π .
Let h(x) = xe −x 2 /2 . Then h ′ (x) = (1 − x 2 )e −x 2 /2 . Therefore, when x ∈ (0, 1), h(x) is an increasing function on its argument x; when x ∈ (1, ∞), h(x) is an decreasing function on x. As a result, h(x) attains its maximum value at x = 1 when x > 0. Since h(x) is a symmetric function, we have |h(x)| ≤ e −1/2 ≈ 0.6. So
b n0 ≍ 1 2π e −(max i,j π * ij ) 2 /2 , b n1 = 1 2π , b n2 = (2πe) −1/2 .
We only consider conditions for consistency here and those for central limit theorem are similar and omitted. By (17), it is not difficult to verify
σ 2 n = O(n 4 V −1 − S 2 max ) = O( b 4 n1 b 6 n0
).
The parameter h n in Assumption 2 for a bounded random variable is a constant. In view of Theorem 1, we have the following corollary.
Corollary 7. If κ n e 3(max i,j π * ij ) 2 = o n log n , then the moment estimator ( β, γ) exists with high probability, and we further have
γ − γ * ∞ = O p κ n e 3(max i,j π * ij ) 2 /2 log n n , β − β * ∞ = O p e (max i,j π * ij ) 2 /2
log n n .
Discussion
In this paper, we present a moment estimation for inferring the degree parameter β and homophily parameter γ in model (2). We establish consistency of the moment estimator ( β, γ) under several conditions and also derive its asymptotic normality. The convergence rates of β and γ are nearly optimal when all parameters are bounded by a constant; but may not be optimal when the numbers b n0 , b n1 , b n2 and κ n diverge. Theorems 2 and 3 require stronger assumptions than consistency, but this is a widely-observed phenomenon in existing literature (Yan et al., 2016a(Yan et al., , 2019Zhang et al., 2021). Whether it is possible to establish consistency and asymptotic normality under even weaker conditions will be an interesting future work. For cleanness, in this work, we assume that max i,j z ij ∞ < c is universally bounded. In fact, our theory can be extended to allow it to slowly diverge. It is another interesting future research to investigate how fast it can diverge while preserving consistency.
The independent edge assumption leads to convenient characterization of the orders of d − Ed ∞ and i<j z ij (a ij − Ea ij ) ∞ , based on which, we establish the central limit theorems of d and i<j z ij a ij . For sub-exponential a ij , the orders of d − Ed ∞ and i<j z ij (a ij − Ea ij ) ∞ are O((n log n) 1/2 ) and O(n log n), respectively, up to a factor determined by the sub-exponential parameter h n . Going forward, we can introduce slight dependency between edges. Under such setting, we can still use some Hoeffding-type inequalities for dependent random variables to establish tail bounds similar to those in this paper (Delyon, 2009), as long as edge dependency is sufficiently light. Remarkably, our method-of-moments estimation remains effective, since it only requires specification of the marginal distributions of a ij 's, not the joint distribution of A. Certainly, quantitative study along this direction would require highly-nontrivial future efforts.
Computation for covariate-assisted β-models is challenging in general. The GLM package we use, which was also employed by Chen et al. (2021); Stein andLeng (2020, 2021), do not scale well. Directly programming the Newton method seems more promising, but still might encounter difficulty when the network is Ω(10 5 ). Unfortunately, the reduction method invented by Zhang et al. (2021) only works for the classical and some general-ized β-models without covariates, not for covariate-assisted β-models. Exploring efficient computational methods is an interesting open challenge for future research.
Appendix
Preliminaries
In this section, we present three results that will be used in the proofs. The first is on the approximation error of using S to approximate the inverse of V belonging to the matrix class L n (b n0 , b n1 ), where V = (v ij ) n×n and S = diag(1/v 11 , . . . , 1/v nn ). Yan et al. (2015) obtained the upper bound of the approximation error, which has an order n −2 . Hillar et al. (2012) gave a tight bound of V −1 ∞ . These results are stated below as lemmas.
Lemma 1 (Proposition 1 in Yan et al. (2015)). If V ∈ L n (b n0 , b n1 ), then the following holds:
V −1 − S max = O b 2 n1 n 2 b 3 n0 .(30)
Lemma 2 (Hillar et al. (2012)). For V ∈ L n (b n0 , b n1 ), we have
1 2b n1 (n − 1) ≤ V −1 ∞ ≤ 3n − 4 2b n0 (n − 1)(n − 2)
.
Let F (x) : R n → R n be a function vector on x ∈ R n . We say that a Jacobian matrix F ′ (x) with x ∈ R n is Lipschitz continuous on a convex set D ⊂ R n if for any x, y ∈ D, there exists a constant λ > 0 such that for any vector v ∈ R n the inequality
[F ′ (x)]v − [F ′ (y)]v ∞ ≤ λ x − y ∞ v ∞
holds. We will use the Newton iterative sequence to establish the existence and consistency of the moment estimator. Gragg and Tapia (1974) gave the optimal error bound for the Newton method under the Kantovorich conditions [Kantorovich (1948)].
Lemma 3 (Gragg and Tapia (1974)). Let D be an open convex set of R n and F : D → R n a differential function with a Jacobian F ′ (x) that is Lipschitz continuous on D with Lipschitz coefficient λ. Assume that x 0 ∈ D is such that [F ′ (x 0 )] −1 exists,
[F ′ (x 0 )] −1 ∞ ≤ ℵ, [F ′ (x 0 )] −1 F (x 0 ) ∞ ≤ δ, ρ = 2ℵλδ ≤ 1, B ∞ (x 0 , t * ) ⊂ D, t * = 2 ρ (1 − 1 − ρ)δ = 2δ 1 + √ 1 − ρ ≤ 2δ.
Then: (1) The Newton iterations
x k+1 = x k −[F ′ (x k )] −1 F (x k ) exist and x k ∈ B ∞ (x 0 , t * ) ⊂ D for k ≥ 0. (2) x * = lim x k exists, x * ∈ B ∞ (x 0 , t * ) ⊂ D and F (x * ) = 0.
Error bound between β γ and β *
The lemma below shows that F γ (β) is Lipschitz continuous. The proofs of all the lemmas in this section are given in the supplementary material.
Lemma 4. Let D = B ∞ (β * , ǫ n1 )(⊂ R n ) be an open convex set containing the true point β * . For γ ∈ B ∞ (γ * , ǫ n2 ), if inequality (34c) holds, then the Jacobian matrix F ′ γ (x) of F γ (x) on x is Lipschitz continuous on D with the Lipschitz coefficient 4b n2 (n − 1).
Since a ij , 1 ≤ i < j ≤ n, are independent and sub-exponential with parameters h ij (≤ h n ), by the concentration inequality for sub-exponential random variables [e.g., Corollary 5.17 in Vershynin (2012) (2012)], we have the following lemma.
In view of Lemmas 4 and 5, we obtain the upper bound of the error between β γ and β * by using the Newton method.
Lemma 6. Let ǫ n1 be a positive number and ǫ n2 = o(b −1 n0 (log n) 1/2 n −1/2 ). Assume that (34a), (34b) and (34c) hold. If
b n2 h n b 2 n0 = o n log n ,(32)
then with probability at least 1 − O(n −1 ), for γ ∈ B ∞ (γ * , ǫ n2 ), β γ exists and satisfies
β γ − β * ∞ = O p h n b n0
log n n = o p (1).
Proofs for Theorem 1
To show Theorem 1, we need three lemmas below.
Lemma 7. Let D = B ∞ (γ * , ǫ n2 )(⊂ R p )
be an open convex set containing the true point γ * . Assume that (34a), (34b), (34c) and (32)
hold. If F (β * , γ * ) ∞ = O(h n (n log n) 1/2 ), then Q c (γ) is Lipschitz continuous on D with the Lipschitz coefficient n 2 b n2 b 3 n1 b −3 n0 .
Lemma 8. Write β * as β γ * and V = ∂F (β * , γ * )/∂β T . β * has the following expansion:
β * − β * = V −1 F (β * , γ * ) + V −1 R,(33)
where R = (R 1 , . . . , R n ) T is the remainder term and
V −1 R ∞ = O p ( b n2 h 2 n log n nb 3 n0
).
Lemma 9.
Let Ω = Cov(F (β * , γ * )). Let σ 2
n = n 2 (V −1 − S)Ω(V −1 − S) max . For any β ∈ B ∞ (β * , ǫ n1 ) and γ ∈ B ∞ (γ * , ǫ n2 ), we have ∂Q(β, γ) ∂β T ( β * − β * ) ∞ = O p nb n1 log n( b n2 h 2 n b 3 n0 + σ n ) .
Further, when Ω = −∂F (β * , γ * )/∂β T , we have
∂Q(β, γ) ∂β T ( β * − β * ) ∞ = O p h 2 n b n1 b n2 n log n b 3 n0
. Now we are ready to prove Theorem 1.
Proof of Theorem 1. We construct the Newton iterative sequence to show the consistency. In view of Lemma 3, it is sufficient to demonstrate the Newton-Kantovorich conditions. We set γ * as the initial point γ (0) and γ (k+1
) = γ (k) − [Q ′ c (γ (k) )] −1 Q c (γ (k)
). By Lemma 6, with probability at least 1 − O(n −1 ), we have we have
β γ − β * ∞ = O p h n b n0
log n n .
This shows that β γ (0) exists such that Q c (γ (0) ) and Q ′ c (γ (0) ) are well defined. Recall the definition of Q c (γ) and Q(β, γ) in (10) and (11). By Lemmas 5 and 9, we
have Q c (γ * ) ∞ ≤ Q(β * , γ * ) ∞ + Q( β γ * , γ * ) − Q(β * , γ * ) ∞ = O p nb n1 log n( h 2 n b n2 b 3 n0 + σ n ) . By Lemma 7, λ = n 2 b 3 n1 b n2 b −3 n0 . Note that ℵ = [Q ′ c (γ * )] −1 ∞ = O(κ n n −2 ). Thus, δ = [Q ′ c (γ * )] −1 Q c (γ * ) ∞ = O p κ n b n1 log n n ( h 2 n b n2 b 3 n0 + σ n ) .
As a result, if equation (21) holds, then
ρ = 2ℵλδ = O p κ 2 n b 4 n1 b n2 log n nb 3 n0 ( h 2 n b n2 b 3 n0 + σ n ) = o p (1).
By Theorem 3, the limiting point of the sequence {γ (k) } ∞ k=1 exists, denoted by γ, and satisfies γ − γ * ∞ = O p (δ).
By Lemma 6, β γ exists, denoted by β, and ( β, γ) is the moment estimator. It completes the proof. Supplementary material for "Asymptotic theory in network models with covariates and a growing number of node parameters"
Section 8 contains details of simulation studies and the application to the Enran email data. The proofs of Lemmas 4, 5 and 6 are given in Sections 9, 10 and 11, respectively. Sections 12, 13 and 14 contain the proofs of Lemmas 7, 8 and 9, respectively. We present the proofs of Theorems 2 and 3 in Sections 15 and 16, respectively. The proof of equation (15) is in Section 17. Section 18 contains the detailed simplification calculations of the bias term B * in equation (19). The following inequalities in the main text are restated here, which will be used in the proofs repeatedly.
b n0 ≤ min i,j |µ ′ (π ij )| ≤ max i,j |µ ′ (π ij )| ≤ b n1 , (34a) max i,j |µ ′′ (π ij )| ≤ b n2 , (34b) max i,j |µ ′′′ (π ij )| ≤ b n3 . (34c)
Simulation studies
We set the parameter values to be a linear form, i.e., α * i = (i − 1)L/(n − 1) for i = 1, . . . , n. We considered four different values for L as L ∈ {0, log(log n), (log n) 1/2 , log n}. By allowing α * to grow with n, we intended to assess the asymptotic properties under different asymptotic regimes. Each node had two covariates X i1 and X i2 . Specifically, X i1 took values positive one or negative one with equal probability and X i2 came from a Beta(2, 2) distribution. All covariates were independently generated. The edge-level covariate z ij between nodes i and j took the form: z ij = (x i1 * x j1 , |x i2 − x j2 |) ⊤ . For the homophily parameter, we set γ * = (0.5, 1) ⊤ . Thus, the homophily effect of the network is determined by a weighted sum of the similarity measures of the two covariates between two nodes.
By Corollary 5, given any pair (i, j), γ). Therefore, we assessed the asymptotic normality of ξ i,j using the quantile-quantile (QQ) plot. Further, we also recorded the coverage probability of the 95% confidence interval and the length of the confidence interval. The coverage probability and the length of the confidence interval of γ were also reported. Each simulation was repeated 10, 000 times.
ξ i,j = [ β i − β j − (β * i − β * j )]/(1/ v i,i + 1/ v j,j ) 1/2 converges in distribution to the standard normality, where v i,i is the estimate of v i,i by replacing (β * , γ * ) with ( β,
We did simulations with network sizes n = 100 and n = 200 and found that the QQplots for these two network sizes were similar. Therefore, we only show the QQ-plots for n = 100 to save space. Further, the QQ-plots for L = 0 and L = log(log n) are similar. Also, for L = (log n) 1/2 and L = log n, they are similar. Therefore we only show those for L = log(log n) and L = log n in Figure 1. In this figure, the horizontal and vertical axes are the theoretical and empirical quantiles, respectively, and the straight lines correspond to the reference line y = x. In Figure 1, when L = log(log n), the empirical quantiles coincide well with the theoretical ones. When L = (log n) 1/2 , the empirical quantiles have a little derivation from the theoretical ones in the upper tail of the right bottom subgraph. These figures show that there may be large space for improvement on the growing rate of β ∞ in the conditions in Corollary 5. Table 1 reports the coverage probability of the 95% confidence interval for β i − β j and the length of the confidence interval. As we can see, the length of the confidence interval decreases as n increases, which qualitatively agrees with the theory. The coverage frequencies are all close to the nominal level 95%. On the other hand, the length of the confidence interval decreases as L increases. It seems a little unreasonable. Actually, the theoretical length of the 95% confidence interval is (1/v ii + v jj ) 1/2 multiple by a constant factor. Since v ii is a sum of a set of exponential items, it becomes quickly larger as L increases. As a result, the length of confidence interval decreases as long as the estimates are close to the true values. The simulated coverage probability results shows that the estimates are very good. So, this phenomenon that the length of confidence interval decreases in Table 1, also agrees with the theory. Table 2 reports the coverage frequencies for the estimate γ and bias corrected estimate γ bc at the nominal level 95%, and the standard error. As we can see, the differences between the coverage frequencies with uncorrected estimates and bias corrected estimates
A real data example
We use the Enron email dataset as an example analysis [Cohen (2004)], available from https://www.cs.cmu.edu/~enron/. The Enron email data was acquired and made public by the Federal Energy Regulatory Commission during its investigation into fraudulent accounting practices. The raw data is messy and needs to be cleaned before any analysis is conducted. Zhou et al. (2007) applied data cleaning strategies to compile the Enron email dataset. We use their cleaned data for the subsequent analysis. The resulting data comprises 21, 635 messages sent between 156 employees with their covarites information. There are 6, 650 messages having more than one recipient across their 'To', 'CC' and 'BCC' fields, with a few messages having more than 50 recipients. For our analysis, we exclude messages with more than ten recipients, which is a subjectively chosen cut-off that avoids emails sent en masse to large groups. Each employee has three categorical variables: departments of these employees (Trading, Legal, Other), the genders (Male, Female) and seniorities (Senior, Junior). Employees are labelled from 1 to 156. The 3dimensional covariate vector z ij of edge (i, j) is formed by using a homophilic matching function between these 3 covariates of two employees i and j, i.e., if x ik and x jk are equal, then z ijk = 1; otherwise z ijk = 0. For our analysis, we removed the employees "32" and "37" with zero degrees, where the estimators of the corresponding node parameters do not exist. This leaves a connected network with 154 nodes. The minimum, 1/4 quantile, median, 3/4 quantile and maximum values of d are 1, 95, 220, 631 and 4637, respectively. It exhibits a strong degree heterogeneity. The estimators of α i with their estimated standard errors are given in Table 3. The estimates of degree parameters vary widely: from the minimum −4.36 to maximum 2.97. We then test three null hypotheses β 2 = β 3 , β 76 = β 77 and β 151 = β 154 , using the homogeneity test statistics ξ i,j = | β i − β j |/(1/ v i,i + 1/ v j,j ) 1/2 . The obtained p-values turn out to be 1.7 × 10 −24 , 1.8 × 10 −4 and 6.2 × 10 −23 , respectively, confirming the need to assign one parameter to each node to characterize the heterogeneity of degrees. The estimated covariate effects, their bias corrected estimates, their standard errors, and their p-values under the null of having no effects are reported in Table 4. From this table, we can see that the estimates and bias corrected estimates are almost the same, indicating that the bias effect is very small in the Poisson model and it corroborates the findings of simulations. The variables "department" and "seniority" are significant while "gender" is not significant. This indicates that the gender has no significant influence on the formation of organizational emails. The coefficient of variable "department" is positive, implying that a common value increases the probability of two employees in the same department to have more email connections. On the other hand, the coefficient of variable "seniority" is negative, indicating that two employees in the same seniority have less emails than those with unequal seniorities. This makes sense intuitively. Proof of Lemma 4. Recall that π ij = β i + β j + z ⊤ ij γ and
Node d i β i σ i Node d i β i σ i Node d i β i σ i Node d i β i σ i 1F i (β, γ) = j =i µ ij (β i + β j + z ⊤ ij γ) − d i , i = 1, . . . , n.
The Jacobian matrix F ′ γ (β) of F γ (β) can be calculated as follows. By finding the partial derivative of F i with respect to β, for i = j we have
∂F i (β, γ) ∂β j = µ ′ ij (π ij ), ∂F i (β, γ) ∂β i = j =i µ ′ ij (π ij ), ∂ 2 F i (β, γ) ∂β i ∂β j = µ ′′ ij (π ij ), ∂ 2 F i (β, γ) ∂β 2 i = j =i µ ′′ ij (π ij ).
When β ∈ B(β * , ǫ n1 ) and γ ∈ B(γ * , ǫ n2 ), by inequality (34b), we have
∂ 2 F i (β, γ) ∂β i ∂β j ≤ b n2 , i = j.
Therefore,
∂ 2 F i (β, γ) ∂β 2 i ≤ (n − 1)b n2 , ∂ 2 F i (β, γ) ∂β j ∂β i ≤ b n2 .(35)
Let
g ij (β) = ( ∂ 2 F i (β, γ) ∂β 1 ∂β j , . . . , ∂ 2 F i (β, γ) ∂β n ∂β j ) ⊤ .
In view of (35), we have
g ii (β) 1 ≤ 2(n − 1)b n2 ,
where x 1 = i |x i | for a general vector x ∈ R n . Note that when i = j and k = i, j,
∂ 2 F i (β, γ) ∂β k ∂β j = 0. Therefore, we have g ij (β) 1 ≤ 2b n2 , for j = i. Consequently, for vectors x, y, v ⊂ D, we have [F ′ γ (x)]v − [F ′ γ (y)]v ∞ ≤ max i { j [ ∂F i ∂β j (x, γ) − ∂F i ∂β j (y, γ)]v j } ≤ v ∞ max i n j=1 | ∂F i ∂β j (x, γ) − ∂F i ∂β j (y, γ)| = v ∞ max i n j=1 | 1 0 [g ij (tx + (1 − t)y)] ⊤ (x − y)dt| ≤ 4b n2 (n − 1) v ∞ x − y ∞ .
It completes the proof.
Proof of Lemma 5
To show this lemma, we need one preliminary result. We first introduce the concentration inequality. We say that a real-valued random variable X is sub-exponential with parameter
κ > 0 if E[|X| p ] 1/p ≤ κp for all p ≥ 1.
Note that if X is a κ-sub-exponential random variable with finite first moment, then the centered random variable X − E[X] is also sub-exponential with parameter 2κ. This follows from the triangle inequality applied to the p-norm, followed by Jensen's inequality for p ≥ 1:
E X − E[X] p ] 1/p ≤ [E |X| p ] 1/p + E[X] ≤ 2[E |X| p ] 1/p .
Sub-exponential random variables satisfy the following concentration inequality.
Lemma 10 (Vershynin (2012), Corollary 5.17). Let X 1 , . . . , X n be independent centered random variables, and suppose each X i is sub-exponential with parameter κ i . Let κ = max 1≤i≤n κ i . Then for every ǫ ≥ 0,
P 1 n n i=1 (X i − E(X i )) ≥ ǫ ≤ 2 exp −nγ · min ǫ 2 κ 2 , ǫ κ ,
where γ > 0 is an absolute constant.
Proof of Lemma 5. Recall that a ij − Ea ij , 1 ≤ i < j ≤ n, are independent and subexponential with respective parameters h ij and max i,j h ij ≤ h n . We set ǫ in Lemma 10 as
ǫ = h n 2 log(n − 1) γ(n − 1) 1/2 .
Assume n is sufficiently large such that ǫ/κ = 2 log(n − 1)/γ(n − 1) ≤ 1. By applying the concentration inequality in Theorem 10, we have for each i = 1, . . . , n,
P 1 n − 1 |d i − Ed i | ≥ h n 2 log(n − 1) γ(n − 1) 1/2 ≤ 2 exp −(n − 1)γ · 2 log n γ(n − 1) = 2 (n − 1) 2 .
By the union bound,
P d − Ed ∞ ≥ h n 2 γ (n − 1) log(n − 1) ≤ n i=1 P |d i − Ed i | ≥ h n 2 γ (n − 1) log(n − 1) ≤ 2n (n − 1) 2 .
Similarly, we have
P i<j z ij (a ij − Ea ij ) ∞ ≥ h n n log n ≤ 1 − 2p n .
Proof of Lemma 6
Proof of Lemma 6. Note that F ′ γ (β) ∈ L n (b n0 , b n1 ) when β ∈ B(β * , ǫ n1 ) and γ ∈ B(γ * , ǫ n2 ), and F γ ( β γ )=0. To prove this lemma, it is sufficient to show that the Kantovorich conditions for the function F γ (β) hold when D = B(β * , ǫ n1 ) and γ ∈ B(γ * , ǫ n2 ), where ǫ n1 is a positive number and ǫ n2 = o(b −1 n1 (log n/n) 1/2 ). The following calculations are based on the event E n :
E n = {d : max i |d i − Ed i | = O(h n (n log n) 1/2 )}.
In the Newton iterative step, we set the true parameter vector β * as the starting point
β (0) := β * .. Let V = (v ij ) = ∂F γ (β * )/∂β ⊤ and S = diag(1/v 11 , . . . , 1/v nn ). By Lemma 2, we have ℵ = V −1 ∞ = O((nb n0 ) −1 )
. Recall that F γ * (β * ) = Ed − d and γ ∈ B(γ * , (log n/n) 1/2 ) and Assumption 1 holds, Note that the dimension p of γ is a fixed constant. If ǫ n2 = o(b −1 n1 (log n) 1/2 n −1/2 ), by the mean value theorem, we have
F γ (β * ) ∞ ≤ d − Ed ∞ + max i | j =i [µ ij (β * , γ) − µ ij (β * , γ * )]| ≤ O(h n (n log n) 1/2 ) + max i j =i |µ ′ ij (β * ,γ)||z ⊤ ij (γ − γ * )| = O(h n (n log n) 1/2 ).
Repeatedly utilizing Lemma 2, we have
δ = [F ′ γ (β * )] −1 F γ (β * ) ∞ = [F ′ γ (β * )] −1 ∞ F γ (β * ) ∞ = O( h n b n0 log n n ) By Lemma 4, F γ (β) is Lipschitz continuous with Lipschitz coefficient λ = 4b n2 (n − 1). Therefore, if b n2 h n b 2 n0 = o( n log n ), then ρ = 2ℵλδ = O( 1 nb n0 ) × O(b n2 n) × O( h n b n0 log n n ) = O b n2 h n b 2 n0 log n n = o(1).
The above arguments verify the Kantovorich conditions. By Lemma 3, it yields that
β γ − β * ∞ = O h n b n0
log n n .
By Lemma 5, P (E n ) → 1 such that the above equation holds with probability at least 1 − O(n −1 ). It completes the proof.
Proof of Lemma 7
Proof of Lemma 7. Recall that
Q c (γ) = (Q c,1 (γ), . . . , Q c,p (γ)) ⊤ = j<i z ij (µ ij ( β γ , γ) − a ij ),
and Q ′ c (γ) is the Jacobian matrix of Q c (γ). When causing no confusion, we write Q c,k (γ) as Q c,k , k = 1, . . . , p. Note that
Q c,k = j<i z ijk (µ ij ( β γ , γ) − a ij ).
By finding the first order partial derivative of function Q c,k with respect to variable γ l , we have
∂Q c,k ∂γ l = j<i z ijk µ ′ ( π ij ) ∂ β γ,i ∂γ l + ∂ β γ,j ∂γ l + z ijl ,
where π ij = β γ,i + β γ,j + z ⊤ ij γ and β γ = ( β γ,1 , . . . , β γ,n ) ⊤ . Again, with the second order partial derivative, we have
∂ 2 Q c,k ∂γ ⊤ ∂γ l = j<i z ijk µ ′′ ( π ij ) ∂ β γ,i ∂γ ⊤ + ∂ β γ,j ∂γ ⊤ + z ij ∂ β γ,i ∂γ l + ∂ β γ,j ∂γ l + z ijl + z ijk µ ′ ( π ij ) ∂ 2 β γ,i ∂γ ⊤ ∂γ l + ∂ 2 β γ,j ∂γ ⊤ ∂γ l .
Recall that max i,j z ij ∞ = O(1) and when β ∈ B(β * , ǫ n1 ), γ ∈ B(γ * , ǫ n2 ), we have:
max i,j |µ ′ (π ij )| ≤ b n1 , max i,j |µ ′′ (π ij )| ≤ b n2 , max i,j |µ ′′′ (π ij )| ≤ b n3 .(36)
So, we have
∂ 2 Q c,k ∂γ ⊤ ∂γ l = O n 2 b n2 ( ∂ β γ ∂γ ⊤ ) 2 + b n1 max i ∂ 2 β γ,i ∂γ ⊤ ∂γ l .(37)
In view of (37), to derive the upper bound of
∂ 2 Q c,k ∂γ ⊤ ∂γ l , it is left to bound ∂ βγ ∂γ ⊤ and ∂ 2 β γ,i ∂γ ⊤ ∂γ l . Recall that F ( β γ , γ) = 0. With the derivative of function F ( β γ , γ) on variable γ, we have ∂F (β, γ) ∂β ⊤ β= βγ ,γ=γ ∂ β γ ∂γ ⊤ + ∂F (β, γ) ∂γ ⊤ β= βγ,γ=γ = 0.(38)
Thus, we have
∂ β γ ∂γ ⊤ = − ∂F (β, γ) ∂β ⊤ β= βγ ,γ=γ −1 ∂F (β, γ) ∂γ ⊤ β= βγ,γ=γ .(39)
To simplify notations, define
V = (v ij ) n×n := ∂F (β, γ) ∂β β= βγ,γ=γ , W = (w ij ) n×n := V −1 − S, F := F (β, γ),
where S = (s ij ) n×n and s ij = δ ij /v ii . Note that
∂F i ∂γ ⊤ β= βγ ,γ=γ = j =i z ij µ ′ ij ( β γ , γ).(40)
By inequality (36), we have
∂F ∂γ ⊤ β= βγ,γ=γ ≤ max i,k j =i |µ ′ ij ( β γ , γ)||z ijk | = O(b n1 n).(41)
By combing (39) and (41) and applying Lemma 2, we have
∂ β γ ∂γ ⊤ ∞ ≤ V ∞ ∂F ( β γ , γ) ∂γ ⊤ ∞ ≤ O( 1 nb n0 ) · O(b n1 n) = O( b n1 b n0 ).(42)
Next, we will evaluate ∂ 2 βγ ∂γ k ∂γ ⊤ . By (
∂ 2 β γ ∂γ k ∂γ ⊤ = −V −1 ∂ ∂γ k ∂F ∂β ⊤ β= βγ,γ=γ ∂ β γ ∂γ ⊤ − V −1 ∂ ∂γ k ∂F ∂γ ⊤ β= βγ,γ=γ := −I 1 − I 2 .(43)For i = j, we have ∂F ∂β ⊤ β= βγ ,γ=γ ij = µ ′ ij ( β γ , γ), ∂ ∂γ k ∂F ∂β ⊤ β= βγ,γ=γ ij = µ ′′ ij ( β γ , γ)(T ⊤ ij ∂ β γ ∂γ k + z ijk ).
Thus,
∂ ∂γ k ∂F ∂β ⊤ β= βγ,γ=γ ij ≤ b n2 2 ∂ β γ ∂γ ⊤ + max i,j z ij ∞ .(44)
Note that
∂F ∂β ⊤ β= βγ ,γ=γ ii = j =i µ ′ ij ( β γ , γ).
By (44)
≤ (n − 1)b n2 2 ∂ β γ ∂γ ⊤ + max i,j z ij ∞ .(45)
For all i = 1, . . . , n and j = 1, . . . , p, in view of (44) and (45), we have
∂ ∂γ k ∂F ∂β ⊤ β= βγ ,γ=γ ∂ β γ ∂γ ⊤ ij ≤ n ℓ=1 ∂ ∂γ k ∂F ∂β β= βγ,γ=γ iℓ ∂ β γ ∂γ ⊤ ℓj ≤ 2(n − 1)b n2 ∂ β γ ∂γ ⊤ 2 ∂ β γ ∂γ ⊤ + max i,j z ij ∞ = O(nb n2 ∂ β γ ∂γ ⊤ 2 ).
Thus,
I 1 = V −1 ∂ ∂γ k ∂F ∂γ β= βγ ,γ=γ ∂ β γ ∂γ ⊤ ≤ V −1 ∞ × max i p j=1 ∂ ∂γ k ∂F ∂β ⊤ β= βγ ,γ=γ ∂ β γ ∂γ ⊤ ij = O b n2 b n0 ∂ β γ ∂γ k 2 .(46)
Since
∂F i ∂γ ⊤ β= βγ ,γ=γ = j =i z ij µ ′ ij ( β γ , γ), we have ∂ ∂γ k ∂F i ∂γ ⊤ β= βγ ,γ=γ = j =i z ij µ ′′ ij ( β γ , γ)( ∂ β γ ∂γ k + z ijk ), such that ∂ ∂γ k ∂F ∂γ ⊤ β= βγ ,γ=γ ∞ ≤ (n − 1)b n2 (max i,j z ij ∞ )( ∂ β γ ∂γ k + max i,j z ij ∞ ).
Consequently, we have
I 2 ∞ = V −1 ∂ ∂γ k ∂F ∂γ ⊤ β= βγ,γ=γ ∞ ≤ V −1 ∞ ∂ ∂γ k ∂F ∂γ ⊤ β= βγ,γ=γ ∞ = O( 1 nb n0 + b 2 n1 nb 3 n0 ) × (n − 1)b n2 κ n ( ∂ β γ ∂γ k + κ n ) (47) = O b n2 b n0 × ∂ β γ ∂γ k ∞ .(48)
By combining (43), (46) and (48), it yields that
∂ 2 β γ ∂γ k ∂γ ⊤ = O b n2 b n0 ∂ β γ ∂γ k 2 .(49)
We will bound Q ′ β,ℓ SF , Q ′ β,ℓ W F and Q ′ β,ℓ V −1 R in turn as follows. Let z * = max i,j z ij ∞ . A direct calculation gives
Q ′ β,ℓ,i = n j=1,j =i z ijℓ µ ′ ij (π * ij ), such that |Q ′ β,ℓ,i | ≤ (n − 1)z * b n1 .(59)
Thus, by Lemmas 2 and 8, we have
|Q ′ β,ℓ V −1 R| ≤ i |Q ′ β,ℓ,i | V −1 R ∞ ≤ n(n − 1)b n1 O p ( b n2 h 2 n log n nb 3 n0 ) = O p ( nb n2 b n1 h 2 n log n b 3 n0 ).(60)
Then we bound Q ′ β,ℓ SF . A direct calculation gives that
Q ′ β,ℓ SF = n i=1 Q ′ β,ℓ,i v ii F i = n i=1 c i H i ,(61)
where
c i = Q ′ β,ℓ,i v ii , i = 1, . . . , n.
It is easy to show that max i=1,...,n
|c i | ≤ z * b n1 b n0 .
By expressing Q ′ β,ℓ SF as a sum of a ij s, we have
Q ′ β,ℓ SF = 2 1≤i<j≤n c i (µ ij − a ij ),
Note a ij (i < j) is independent and bounded by h n z * . By applying the concentration inequality for subexponential random variables to the above sum, we have
|Q ′ β,ℓ SF | = O p (h n n log n).(62)
Finally, we bound Q ′ β,ℓ W F . Let
σ 2 n = max i,j n 2 |(W ⊤ Cov(F )W ) ij |.
Therefore, by (59), we have
Var(Q ′ β,ℓ W F ) = [Q ′ β,ℓ ] ⊤ W ⊤ Cov(F )W Q ′ β,ℓ = i,j Q ′ β,ℓ,i (W ⊤ Cov(F )W ) ij Q ′ β,ℓ,j = O n 2 × n −2 σ 2 n × b 2 n1 n 2 = O(n 2 b 2 n1 σ 2 n ).
By Chebyshev's inequality, we have
P(|Q ′ β,ℓ W H| > nb 2 n1 σ n (log n) 1/2 ) ≤ O(n 2 σ 2 n b 2 n1 b −3 n0 ) n 2 b 2 n1 b −3 n0 σ 2 n log n → 0. It leads to Q ′ β,ℓ W H = O p (nb n1 σ n (log n) 1/2 ).(63)
By combining (58), (60) (62) , (63), it yields max ℓ=1,...,p
|Q ′ β,ℓ ( β * − β * )| = O p ( nb n2 b n1 h 2 n log n b 3 n0 ) + O p (h n n log n) + O p (nb n1 σ n (log n) 1/2 ) = O p nb n1 log n( h 2 n b n2 b 3 n0 + σ n ) .
In the case of V = Cov(F ), the equation (63) could be simplified. Denote W = V −1 − S. Then we have
Cov(W F ) = W ⊤ Cov(F )W = (V −1 − S)V (V −1 − S) = V −1 − S + SV S − S.
A direct calculation gives that
(SV S − S) ij = (1 − δ ij )v ij v ii v jj .
By Lemma 1, we have
|(W ⊤ Cov(F )W ) ij | = O b 2 n1 n 2 b 3 n0 .
Then, we have
Q ′ β,ℓ W F = O p nb 2 n1 b 3/2 n0
, which leads to the simplification:
max ℓ=1,...,p |Q ′ β,ℓ ( β * − β * )| = O p h 2 n b n1 b n2 n log n b 3 n0 .
It completes the proof.
Proofs for Theorem 2
Proof of Theorem 2. To simplify notations, write
µ ′ ij = µ ′ (β * i + β * j + z ⊤ ij γ * ) and V = ∂F (β * , γ * ) ∂β ⊤ , V γβ = ∂F (β * , γ * ) ∂γ ⊤ .
Let π * ij = β * i + β * j + z ⊤ ij γ * and π ij = β i + β j + z ⊤ ij γ. By a second order Taylor expansion, we have
µ( π ij ) − µ(π * ij ) = µ ′ ij ( β i − β i ) + µ ′ ij ( β j − β j ) + µ ′ ij z ⊤ ij ( γ − γ) + g ij ,(64)
where
g ij = 1 2 β i − β * i β j − β * j γ − γ * ⊤ µ ′′ ij ( π ij ) −µ ′′ ij ( π ij ) µ ′′ ij ( π ij )z ⊤ ij −µ ′′ ij ( π ij ) µ ′′ ij ( π ij ) −µ ′′ ij ( π ij )z ⊤ ij µ ′′ ij ( π ij )z ⊤ ij −µ ′′ ij ( π ij )z ⊤ ij µ ′′ ij ( π ij )z ij z ⊤ ij β i − β * i β j − β * j γ − γ * ,
and π ij lies between π * ij and π ij . By calculations, g ij can be simplified as
g ij = µ ′′ ( π ij )[( β i − β i ) 2 + ( β j − β j ) 2 + 2( β i − β i )( β j − β j )] +2µ ′′ ( π ij )z ⊤ ij ( γ − γ)( β i − β i + β j − β j ) + ( γ − γ) ⊤ µ ′′ ( π ij )z ij z ⊤ ij ( γ − γ)
Recall that z * := max i,j z ij ∞ = O(1). Note that |µ ′′ (π ij )| ≤ b n2 when β ∈ B(β * , ǫ n1 ) and γ ∈ B(γ * , ǫ n2 ). So we have
|g ij | ≤ 4b n2 β − β * 2 ∞ + 2b n2 β − β * ∞ γ − γ * 1 κ n + b n2 γ − γ * 2 1 κ 2 n ≤ 2b n2 [4 β − β * 2 ∞ + γ − γ * 2 1 z 2 * ].
Let g i = j =i g ij , g = (g 1 , . . . , g n ) ⊤ . If (4.11) in the main text holds and ). (65) Writing (64) into a matrix form, it yields
d − Ed = V ( β − β * ) + V γβ ( γ − γ * ) + g,
which is equivalent to
β − β * = V −1 (d − Ed) + V −1 V γβ ( γ − γ * ) + V −1 g.(66)
We bound the last two remainder terms in the above equation as follows. Let W = V −1 − S. Note that (Sg) i = g i /v ii and (n − 1)b n0 ≤ v ii ≤ (n − 1)b n1 . By Lemma 2 in the main text, we have
V −1 g ∞ ≤ V −1 ∞ g ∞ = O( 1 nb n0 × h 2 n b n2 log n b 2 n0
).
Note that the ith of V γβ is n j=1,j =1 µ ′ ij z ⊤ ij . So we have V γβ ( γ − γ * ) ∞ ≤ (n − 1)z * γ − γ * 1 = O p (κ n b n1 log n( b n2 h 2 n b 3 n0 + σ n )).
By Lemma 2 in the main text, we have
V −1 V γβ ( γ − γ * ) ∞ ≤ V −1 ∞ V γβ ( γ − γ * ) ∞ = O p κnb n1 log n nb n0 ( b n2 h 2 n b 3 n0 + σ n ) .(68)
Since max i |(W ΩW ⊤ ) ii | ≤ σ 2 n /n 2 , we have
P([W (d − Ed)] i > σ n log n/n) ≤ n 2 σ 2 n (log n) 2 |Var{[W (d − Ed)] i }| = 1 (log n) 2 .(69)
Consequently, by combining (66), (67), (68) and (69), we have
β i − β * i = [S(d − Ed)] i + O p ( κ n b n1 log n nb n0 ( b n2 h 2 n b 3 n0 + σ n )).
It completes the proof.
Proof of Theorem 3
Proof of Theorem 3. Assume that the conditions in Theorem 1 hold. A mean value expansion gives
Q c ( γ) − Q c (γ * ) = ∂Q c (γ) ∂γ ⊤ ( γ − γ * ),
whereγ lies between γ * and γ. By noting that Q c ( γ) = 0, we have
√ N( γ − γ * ) = 1 N ∂Q c (γ) ∂γ ⊤ −1 × Q c (γ * √ N ).
Note that the dimension of γ is fixed. By Theorem 1 and (4.10) in the main text, we have
1 N ∂Q c (γ) ∂γ ⊤ p →H := lim N →∞ 1 N H(β * , γ * ).
Write β * as β γ * for convenience. Therefore,
√ N ( γ − γ * ) =H −1 × (− Q( β * , γ * ) √ N ) + o p (1).(70)
By applying a third order Taylor expansion to Q( β * , γ * ), it yields − 1 √ N Q( β * , γ * ) = S 1 + S 2 + S 3 ,
where
S 1 = − 1 √ N Q(β * , γ * ) − 1 √ N ∂Q(β * ,γ * ) ∂β ⊤ ( β * − β * ), S 2 = − 1 2 √ N n k=1 ( β * k − β * k ) ∂ 2 Q(β * ,γ * ) ∂β k ∂β ⊤ × ( β * − β * ) , S 3 = − 1 6 √ N n k=1 n l=1 {( β * k − β * k )( β * l − β * l ) ∂ 3 Q(β * ,γ * ) ∂β k ∂β l ∂β ⊤ ( β * − β * )},
andβ * = tβ * + (1 − t) β * for some t ∈ (0, 1). Similar to the proof of Theorem 4 in Graham (2017), we will show that (1) S 2 is the bias term having a non-zero probability limit; (2) S 3 is an asymptotically negligible remainder term. We first evaluate the term S 3 . We calculate g ijk = ∂ 3 Q(β,γ) ∂β k ∂β i ∂β k according to the indices i, j, k as follows. Observe that g ijk = 0 when i, j, k are different numbers because µ ij only has two arguments β i and β j and its third partial derivative on three different β i , β j and β k is zero . So there are only two cases below in which g ijk = 0.
(1) Only two values among three indices i, j, k are equal. If k = i; i = j, g ijk = z ij ∂ 3 µ ij ∂π 3 ij ; for other cases, the results are similar.
(2) Three values are equal. g kkk = i =k z ki ∂ 3 µ ki ∂π 3 ki . Therefore, we have
S 3 = 1 6 √ N k,l,h ∂ 3 Q(β * , γ * ) ∂β k ∂β l ∂β h ( β * k − β * k )( β * l − β * l )( β * h − β * h ) = 1 6 √ N i<j ∂ 3 Q(β * , γ * ) ∂β 2 i ∂β j ( β * i − β * i ) 2 ( β * j − β * j ) + ∂ 3 Q(β * , γ * ) ∂β 2 j ∂β i ( β * j − β * j ) 2 ( β * i − β * i ) + i ∂ 3 Q(β * , γ * ) ∂β 3 i ( β * i − β * i ) 3 . So S 3 ∞ ≤ 4 3 √ N × max i,j | ∂ 3 µ ij (β * , γ * )) ∂π 3 ij | z ij ∞ × n(n − 1) 2 β * − β 3 ∞ .
By Lemma 6, we have
S 3 ∞ = O p ( b n3 h 3 n (log n) 3/2 n 1/2 b 3 n0
).
Similar to the calculation in the derivation of the asymptotic bias in Theorem 4 in Graham (2017), we have S 2 = B * + o p (1), where B * is defined at (4.12) in the main text.
Recall that V = ∂F (β * , γ * )/∂β ⊤ and V Qβ := ∂Q(β * , γ * )/∂β ⊤ . By noting that
d − Ed = 1≤i<j<n (a ij − Ea ij )T ij , we have −[Q(β * , γ * ) − V Qβ V −1 (d − Ed)] = 1≤i<j≤n (a ij − Ea ij )(z ij − V Qβ V −1 T ij ).
Similar to the calculation in the derivation of the asymptotic expression of S 1 in Graham (2017), we have
S 1 = 1 √ N j<i s ij (β * , γ * ) + o p (1),
Therefore, it shows that equation (71) is equal to
1 √ N j<i s ij ( β * , γ * ) = 1 √ N j<i s ij (β * , γ * ) + B * + o p (1),(72)with 1 √ N n i=1
j =i s * γ ij (β * , γ * ) equivalent to the first two terms in (71) and B * the probability limit of the third term in (71).
Simplifying expression of B * in (19)
In the case of V = ∂F (β * , γ * )/∂β = Var(d), B * can be simplified as follows. Let W = V −1 − S. A direct calculation gives that n k=1 ∂ 2 Q(β * , γ * ) ∂β k ∂β ⊤ Se k = n k=1 j =k z kj µ ′′ kj (π * ij ) j =k µ ′ kj (π * ij )
.
By Lemma 1, we have
n k=1 ∂ 2 Q ℓ (β * , γ * ) ∂β k ∂β ⊤ W e k = j =k z kjℓ µ ′′ (π * kj )(w kj + w kn ) = O b 2 n1 b n2 b 3 n0 n . So, if b 2 n1 b n2 b −3 n0 = o(n), then B * = 1 √ N n k=1 ∂ 2 Q(β * , γ * ) ∂β k ∂β ⊤ V −1 e k = − 1 √ N n k=1 j =k z kj µ ′′ kj (π * ij ) j =k µ ′ kj (π * ij ) ,(80)
Lemma 5 .
5With probability at least 1 − O(n −1 ), we have F (β * , γ * ) ∞ = O(h n n log n), Q(β * , γ * ) ∞ = O(h n n log n).
pages 210 - 268 .
210268Cambridge University Press. Yan, T., Jiang, B., Fienberg, S. E., and Leng, C. (2019). Statistical inference in a directed network model with covariates. Journal of the American Statistical Association, 114(526):857-868. Yan, T., Leng, C., and Zhu, J. (2016a). Asymptotics in directed exponential random graph models with an increasing bi-degree sequence. The Annals of Statistics, (44):31-57. Yan, T., Qin, H., and Wang, H. (2016b). Asymptotics in undirected random graph models parameterized by the strengths of vertices. Statistica Sinica, 26(1):273-293. Yan, T. and Xu, J. (2013). A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices. Biometrika, 100:519-524. Yan, T., Zhao, Y., and Qin, H. (2015). Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters. Journal of Multivariate Analysis, 133:61 -76. Yang, J., McAuley, J., and Leskovec, J. (2013). Community detection in networks with node attributes. In 2013 IEEE 13th international conference on data mining, pages 1151-1156. IEEE. Zhang, H. and Chen, S. X. (2021). Concentration inequalities for statistical inference. Communications in Mathematical Research, 37(1):1-85. Zhang, Y., Levina, E., and Zhu, J. (2016). Community detection in networks with node features. Electronic Journal of Statistics, 10(2):3153-3178. Zhang, Y., Wang, Q., Zhang, Y., Yan, T., and Luo, J. (2021). L-2 regularized maximum likelihood for β-model in large and sparse networks. arXiv preprint arXiv:2110.11856. Zhang, Y. and Xia, D. (2022). Edgeworth expansions for network moments. The Annals of Statistics, 50(2):726 -753. Zhou, Y., Goldberg, M., Magdon-Ismail, M., and Wallace, W. A. (2007). Strategies for cleaning organizational emails with an application to enron email dataset. In in 5th Conference of North American Association for Computatational Social Organization Science, Pittsburgh. North American Association for Computational Social and Organizational Science.
Figure 1 :
1The QQ plots of ξ i,j (n=100).
) .
)Example 2. (Unbounded discrete edges) This example generalizes the Poisson β-model in Section 3.4 of Yan et al. (2016b). Here, we model a ij ∼ Poisson λ ij , where λ ij = exp(β
Table 1 :
1The reported values are the coverage frequency (×100%) for β i − β j for a pair (i, j) / the length of the confidence interval(×10).n
(i, j)
L = 0
L = log(log n) L = (log n) 1/2 L = log n
100
(1, 2)
94.56/4.60
95.08/2.97
94.80/2.42
94.69/0.97
(50, 51) 94.72/4.60
94.93/2.04
94.89/1.43
94.83/0.31
(99, 100) 95.12/4.60
94.41/1.40
94.38/0.85
94.13/0.10
200
(1, 2)
95.20/3.24
94.79/2.01
94.76/1.63
95.09/0.52
(100, 101) 95.03/3.24
94.75/1.33
94.91/0.92
95.47/0.14
(199, 200) 94.58/3.24
95.05/0.88
94.63/0.52
93.90/0.04
are very small. All coverage frequencies are very close to the nominal level. The bias
under the case of the Poisson distribution is very small in our simulation design.
Table 2 :
2The reported values are the coverage frequency (×100%) for γ i for i / length
(×10) of confidence interval (γ * = (0.5, 1) ⊤ ).
n
γ
L = 0
L = log(log n) L = (log n) 1/2
L = log n
100 γ 1 95.13/0.52 95.25/0.22
94.92/0.15
95.04/0.02
γ bc,1 95.11/0.52 95.25/0.22
94.92/0.15
95.04/0.02
γ 2 94.98/3.08 95.28/1.31
95.00/0.88
95.06/0.15
γ bc,2 94.93/3.08 95.29/1.31
95.02/0.88
95.06/0.15
200 γ 1 94.87/0.26 95.49/0.10
95.07/0.07
94.92/0.007
γ bc,1 94.87/0.26 95.47/0.10
95.08/0.07
94.91/0.007
γ 2 95.31/1.52 95.12/0.59
94.97/0.39
94.49/0.041
γ bc,2 95.31/1.52 95.12/0.59
94.95/0.39
94.49/0.041
Table 3 :
3The estimates of β i and their standard errors in the Enron email dataset.
Table 4 :
4The estimators of γ i , the corresponding bias corrected estimators, the standard errors, and the p-values under the null γi = 0 (i = 1, 2, 3) for Enron email data.
Covariate
γ i
γ bc,i
σ i
p-value
Department 0.167
0.167 1.13 < 0.001
Gender
−0.006 −0.006 1.27
0.62
Seniority
−0.203 −0.203 1.09 < 0.001
9 Proof of Lemma 4
, we have∂
∂γ k
∂F
∂β ⊤
β= βγ,γ=γ ii
AcknowledgementsWe are grateful to Editor, Associate Editor and two anonymous referees for their insightful comments and suggestions. TY was partially supported by the National Natural Science Foundation of China (No. 11771171) and the Fundamental Research Funds for the Central Universities.Consequently, in view of (37), (42) and (49), we haveNote thatBy the mean value theorem, we havewhere t = αx + (1 − α)y for some α ∈ (0, 1). By combining inequalities(51),(52)andwhereThis completes the proof.13 Proof of Lemma 8. . , n. By applying a second order Taylor expansion to H( β γ * , γ * ), we havewhereβ * lies between β * and β * . We evaluate the last term in the above equation row by row. Its ℓth row isA directed calculation gives thatBy Lemma 6, we have that max ℓ=1,...,nNote that V ∈ L n (b n0 , b n1 ). By (56) and Lemma 1, we have).14 Proof of Lemma 9Proof of Lemma 9. For convenience, writeWhen evaluating functions f (β, γ) on (β, γ) at its true value (β * , γ * ), we suppress the argument (β * , γ * ). This is, writeIt completes the proof.17 Proof of(15)Recall that π ij = z ⊤ ij γ + β i + β j , µ ij (π ij ) = Ea ij and T ij is an n-dimensional vector with ith and jth elements 1 and other elements 0. By calculations, we haveNote thatTo simplify notations, letWhen emphasizing the arguments β and γ, we write A(β, γ * ) instead of A and so on. WhenRecall that z * = max i,j z ij ∞ and max i,j |µ ′ ij (β, γ * )| ≤ b n1 . To simplify notations, we suppress the subscript "max" in the matrix maximum norm · max in this section. It yieldsNow, we evaluate A(β, γ * ) − A(β * , γ * ). By the mean value theorem, we haveNext, we evaluate [BSD](β, γ * ) − [BSD](β * , γ * ). Note thatSince |µ ′ (π ij )| ≤ b n1 and |µ ′′ (π ij )| ≤ b n2 , we haveand for a vector v,By(75)and(76), we haveIt is easy to see thatBy the mean value theorem, we havewhere β lies between β and β * , andBy(77)and(78), we have).Consequently,By inequalities(73),(74)and(79)(1), then 1 n 2 H(β, γ * ) ij = 1 n 2 H(β * , γ * ) ij + o(1).
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"Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on R 1,d−1 *",
"Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on R 1,d−1 *"
] |
[
"Alexander A Reshetnyak \nInstitute of Strength Physics\nMaterials Science of SB RAS\nLaboratory of Computer-Aided Design of Materials\n634055TomskRussia\n"
] |
[
"Institute of Strength Physics\nMaterials Science of SB RAS\nLaboratory of Computer-Aided Design of Materials\n634055TomskRussia"
] |
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Gauge invariant Lagrangian descriptions of irreducible and reducible half-integer higher-spin mixed-symmetric massless and massive representations of the Poincare group with off-shell algebraic constraints are constructed within a metric-like formulation in a d-dimensional flat space-time on the basis of a suggested constrained BRST approach. A Lorentz-invariant resolution of the BRST complex within the constrained BRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely in terms of the initial triple gamma-traceless spin-tensor field Ψ (µ)n with gamma-traceless gauge parameter. The triplet and quartet formulations are derived. The minimal (un)constrained BRST-BV actions for above formulations are obtained, from proposed constrained BRST-BV approach to be by appropriate tools to construct interacting constrained Lagrangians.
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10.1134/s1063779618050349
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[
"https://arxiv.org/pdf/1803.05173v5.pdf"
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1803.05173
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fec0730fd9a9300c0fe0f1ac5291ecaea9d71560
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Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on R 1,d−1 *
Alexander A Reshetnyak
Institute of Strength Physics
Materials Science of SB RAS
Laboratory of Computer-Aided Design of Materials
634055TomskRussia
Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on R 1,d−1 *
Gauge invariant Lagrangian descriptions of irreducible and reducible half-integer higher-spin mixed-symmetric massless and massive representations of the Poincare group with off-shell algebraic constraints are constructed within a metric-like formulation in a d-dimensional flat space-time on the basis of a suggested constrained BRST approach. A Lorentz-invariant resolution of the BRST complex within the constrained BRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely in terms of the initial triple gamma-traceless spin-tensor field Ψ (µ)n with gamma-traceless gauge parameter. The triplet and quartet formulations are derived. The minimal (un)constrained BRST-BV actions for above formulations are obtained, from proposed constrained BRST-BV approach to be by appropriate tools to construct interacting constrained Lagrangians.
Introduction
Many modern issues of high-energy physics are related to higher-spin (HS) field theory, remaining by the part of the LHC experiment program. The tensionless limit of superstring theory [1] with help of BRST operator includes an infinite set of HS fields with integer and half-integer generalized spins and incorporates HS field theory into superstring theory and turns it into a method of studying the classical and quantum structure of the latter (for the present status of HS field theory, see the reviews [2], [3], [4]). Whereas (un)constrained BRST-BFV (see original papers to quantize constrained dynamical systems [5], [6]) and constrained BRST-BV approaches to construct respectively gauge-invariant Lagrangian formulations (LFs) and BV field-antifield actions for integer HS fields on constant curvature space-times with(out) off-shell holonomic constraints are known, see e.g. [7]. [8], [9], [10] and [11], [12], [13] constrained BRST-BFV and BRST-BV methods to be applied to the same aims for the half-integer HS fields have not been till developed. The paper suggests constrained BRST-BFV and BRST-BV approaches for construction of LFs and BRST-BV actions in the minimal sector of the field-antifield formalism for free (ir)reducible Poincare group representations with half-integer spins in a flat R 1,d−1 -space-time subject to an arbitrary Young tableaux (YT) with k rows, Y (s 1 , s 2 , ..., s k ), with spin s = (n 1 + 1 2 , ..., n k + 1 2 ) for n 1 ≥ n 2 ≥ ... ≥ n k [14], in a metric-like formalism (for the study of the LFs in the metric-like and the frame-like formalisms beyond the BRST methods, see e.g. [15] and [16], [17]). The latter constrained BRST-BV action presents a natural ground for the procedure of consistent construction of the interacting LFs for such HS fields. The paper is based on the research [18], [19] and organized as follows. In Section 2, we suggest the constrained BRST-BFV LFs for half-integer mixed-symmetric (MS) HS fields. In Section 3 a new formalism is applied for the case of totally-symmetric (TS), s = n + 1 2 , HS field in various representations. The construction of a minimal field-antifield actions on a base of natural extension of BRST-BFV approach up to a constrained BRST-BV method for half-integer HS fields is considered in Section 4.
The convention η µν = diag(+, −, ..., −) for the metric tensor, with the Lorentz indices µ, ν = 0, 1, ..., d − 1, the relations {γ µ , γ ν } = 2η µν for the Dirac matrices γ µ ,and the notation (A), [gh H , gh L , gh tot ](A) for the respective values of Grassmann parity, BFV, gh H , BV, gh L and total, gh tot = gh H +gh L , ghost numbers of a quantity A are used. The supercommutator [A, B} of quantities A, B with definite values of Grassmann parity is given by [
A , B} = AB − (−1) (A) (B) BA.
Constrained BRST-BFV Lagrangian formulations
There exists two equivalent ways of derivation constrained BRST-BFV approach for LFs for half-integer HS fields [18], first, from unconstrained BRST-BFV method, developed for arbitrary half-integer HS fields in Minkowski space R 1,d−1 [20] (see therein for the references on unconstrained BRST-BFV approach for half-integer HS fields), second, in self-consistent way. We consider here in details the second possibility. To do so remind, a massless halfinteger spin irreducible representation of the Poincare group in R 1,d−1 is described by a rank-k i=1 n i spin-tensor field Ψ (µ 1 )n 1 ,...,(µ k )n k ≡Ψ µ 1 1 ...µ 1 n 1 ,...,µ k 1 ...µ k n k A with generalized spin s (for suppressed Dirac index A), subject to a YT, Y (s 1 , ..., s k ) with k rows of length n 1 , ..., n k . The field (being symmetric with respect to permutations of each type of Lorentz indices µ i ) satisfies differential (Dirac) equation (1) and algebraic equations (γ-traceless and mixedantisymmetry ones) (2) :
ıγ µ ∂ µ Ψ (µ 1 )n 1 ,...,(µ k )n k = 0,(1)γ µ i l i Ψ (µ 1 )n 1 ,...,(µ k )n k = 0, Ψ (µ 1 )n 1 ,...,{(µ i )n i , ..., µ j 1 ... µ j l j }...µ j n j ,...(µ k )n k = 0,(2)(for 1 ≤ l i ≤ n i , 1 ≤ i ≤ k, i < j)
where the underlined figure bracket means that the indices inside do not take part symmetrization. Equivalently, the relations for general state (Dirac spinor) |Ψ from Fock space H generated by k pairs of bosonic (symmetric case) oscillators a i
µ i (x), a j+ ν j (x): [a i µ i , a j+ ν j ] = −η µ i ν j δ ij : t 0 |Ψ = t i |Ψ = t rs |Ψ = 0, g i 0 Ψ = (n i + d 2 )|Ψ (3) for t 0 , t i , t rs , g i 0 = −iγ µ ∂ µ ,γ µ a µ i , a + rµ a µ s , − 1 2 {a µ i , a + iµ } , r < s.(4)|Ψ = ∞ n 1 =0 n 1 n 2 =0 · · · n k−1 n k =0 ı i n i n 1 ! × ... × n k ! Ψ (µ 1 )n 1 ,...,(µ k )n k k i=1 n i l i =1 a +µ i l i i |0 ,(5)
with the generalized spin constraints imposed on |Ψ in terms of number particle operator g i 0 describe the irreducible massless of spin s = n + 1 2 Poincare group representation. In (4) we have used of d + 1 Grassmann-odd gamma-matrix-like objects {γ,γ µ } to be equivalent to the standard ones: γ µ , whose explicit realization differs in even, d = 2N , and odd, d = 2N + 1, N ∈ N dimensions (see for details [18]). :For even-valued dimension we have
{γ µ ,γ ν } = 2η µν , {γ µ ,γ} = 0,γ 2 = −1, so that γ µ =γ µγ , .(6)
whereas for odd-valued one, the second and third relations in (6) are changed on [γ µ ,γ] = 0, γ 2 = 1 with unchanged others. The set of primary constraints
{t 0 , t i , t ij , g i 0 }, (t 0 ) = (t i ) = 1, (t ij ) = (g i 0 ) = 0 will be closed with respect to the [ , }-multiplication if we add to them divergentless, l i = (1/2)[t 0 , t i }, traceless, l ij = (1/4)[t i , t j }, i ≤ j and D'alamber operators, l 0 = −t 2 0 : (l 0 , l i , l ij ) = ∂ µ ∂ µ , −ia iµ ∂ µ , 1 2 a µ i a jµ .(7)
The reality of the Lagrangian with consistent off-shell holonomic constraints requires a closedness for subset of differential constraints, o A , with respect to the appropriate hermitian conjugation defined by means of odd scalar product in H:
Φ |Ψ = d d x ∞ n 1 =0 n 1 n 2 =0 · · · n k−1 n k =0 ∞ p 1 =0 p 1 p 2 =0 · · · p l−1 p l =0 ı i n i (−ı) j p j n 1 ! × ... × n k !p 1 ! × ... × p l ! × 0| l j=1 p j m j =1 a ν j m j j Φ + (ν 1 )p 1 ,...,(ν l )p lγ 0 Ψ (µ 1 )n 1 ,...,(µ k )n k k i=1 n i l i =1 a +µ i l i i |0 = ∞ n 1 =0 n 1 n 2 =0 · · · n k−1 n k =0 k i=1 (−1) n i n i ! d d xΦ + (µ 1 )n 1 ,...,(µ k )n kγ 0 Ψ (µ 1 )n 1 ,...,(µ k )n k ,(8)
that means the set:
o A = {t 0 , l 0 , l i , l + i } (for l + i = −ia + iµ ∂ µ )
composes the first-class constraints subsystem. The holonomic constraints t i , t rs itself generate the superalgebra of total set of constraints {o a } = {t i , t rs , l lm } in H. The algebraically independent subset of {oā} ⊂ {o a } is given by:
{oā} = {t 1 , t 12 , t 23 , t 34 , . . . , t (k−1)k }.(9)
As the result the superalgebra A f c (Y (k), R 1,d−1 ) to be named as constrained half-integer HS symmetry algebra in Minkowski space with a YT having k rows with off-shell set of algebraic constraints {t i , t rs } and g i 0 appears by necessary objects to construct constrained LF for HS fields of spin s. The nilpotent constrained BRST operator for the system of
{o A } in the Hilbert space H c , H c = H ⊗ H o A
gh , BRST-extended independent algebraic constraints, constrained spin operator determined as
Q c (o A ) = q 0 t 0 + η 0 l 0 + η + i l i + l i+ η i + ı l η + l η l − q 2 0 P 0 ,(10)T i , T rs , σ i c (g) = t i + C A t B iA P B , t rs + C A t B rsA P B , g i 0 + C A g iB A P B + o(CP),(11)with gh H (Q c , T i , T rs , σ i c (g)) = (1, 0, 0, 0), [for (C A ; P B ) = (q 0 , η 0 , η + i , η i ; p 0 , P 0 , P j , P + j ) with [q 0 , p 0 ] = {η 0 , P 0 } = ı, {η i , P + j } = δ ij ]
should satisfy to the consistency conditions:
[Q c , T i } = 0, [Q c , T rs } = 0, [Q c , σ i c (g)} = 0,(12)
known [18] as the generating equations for superalgebra of the constrained BRST, Q c spin operators σ i c (g) and extended off-shell constraints T i , T rs . The exact solutions of (12) for unknown σ i c (g), T i , T rs exists in the form:
C A t B iA , t B rsA , g iB A P B = − ıη i p 0 − 2q 0 P i , −η + r P s − P + r η s , η + i P i − η i P + i .(13)
Presenting the general state vector
|χ c ∈ H c (for (gh H , gh L )|Ψ(a + i ) n b0 n f 0 ;(n) f i (n) pj = (0, 0) within the representation (η i , P i , p 0 , P 0 )|0 = 0): |χ c = n q n b0 0 η n f 0 0 i,j (η + i ) n f i (P + j ) n pj |Ψ(a + i ) n b0 n f 0 ;(n) f i (n) pj .(14)
and decomposing H c as H c = lim M →∞ ⊕ M l=−M H l c for gh H (|χ l c ) = −l, |χ l c ∈ H l c from the BRST condition, Q c |χ 0 c = 0, off-shell constraints, T i , T rs |χ c = 0 and σ i c (g)|χ c = R i |χ c for some R i ∈ R. we have the spectral problem analogous to one for unconstrained case [20]: that m i = n i (with n i from s = n + 1 2 ) and proper eigen-vectors |χ l c (n) k . The solution of the rest equations is written as the second-order equations of motion and sequence of the reducible gauge transformations (18) with off-shell constraints (19):
Q c |χ l c = δ|χ l−1 c , σ i c |χ l c = m i + d−2 2 |χ l c , ( , gh H ) (|χ l c ) = (l + 1, −l),(15)T i , T rs |χ l c = 0, l = 0, 1, ..., s c ,(16)Q c |χ 0 c (n) k = 0, δ|χ 0 c (n) k = Q c |χ 1 c (n) k , ... , δ|χ sc−1 c (n) k = Q c |χ sc c (n) k , δ|χ sc c (n) k = 0, (18) T i , T rs |χ l c (n) k = 0, l = 0, 1, ..., s c , for s c = k.(19)
The corresponding BRST-like constrained gauge-invariant action (as for integer HS field)
S (2) c|(n) k = dη 0(n) k χ 0 c |Q c |χ 0 c (n) k ,(20)
contains second order operator l 0 , but less terms in comparison with its unconstrained analog [20]. Repeating the procedure of the removing the dependence on l 0 , η 0 , q 0 from the BRST operator Q c (10) and from the whole set of the vectors |χ l c (n) k as it was done for unconstrained case by means of partial gauge-fixing we come to the: Statement: The first-order constrained gauge-invariant Lagrangian formulation for halfinteger HS field, Ψ (µ 1 )n 1 ,...,(µ k )n k (x) with generalized spin (s) k = (n) k +( 1 2 , ..., 1 2 ), is determined by the action,
S c|(n) k = (n) k χ 0 0|c | (n) k χ 1 0|c | t 0 ∆Q c ∆Q c t 0 η + i η i |χ 0 0|c (n) k |χ 1 0|c (n) k for ∆Q c = η + i l i + l + i η i ,(21)
invariant with respect to the sequence of the reducible gauge transformations (for s c − 1 = (k − 1)-being by the the stage of reducibility):
δ |χ (22) (for l = −1, 0, ..., k − 1 and |χ −1(m) 0|c = 0, m = 0, 1) with off-shell algebraically independent BRST-extended constraints imposed on the whole set of field and gauge parameters:
l(0) 0|c (n) k |χ l(1) 0|c (n) k = ∆Q c t 0 η + i η i t 0 ∆Q c |χ l+1(0) 0|c (n) k |χ l+1(1) 0|c (n) k , δ |χ k(0) 0|c (n) k |χ k(1) 0|c (n) k = 0T i |χ l(0) c (n) k + q 0 |χ l(1) c (n) k = 0, T rs |χ l(m) c (n) k = 0 l = 0, 1, ..., k; m = 0, 1.(23)
The first constraints in q 0 -independent form read
t i |χ l(0) 0|c − η i |χ l(1) 0|c = 0, t i |χ l(1) 0|c − 2P i |χ l(0) 0|c = 0, P i |χ l(1) 0|c = 0 .(24)
In the massive case for d = 2N , N ∈ N we should to add k pairs of additional even oscillators in differential constraints: (L i , L + i ) = (l i + mb i , l + i + mb + i ) and in decomposition for |χ l(1) 0|c (14) in accordance with [18]. For odd dimensions, d = 2N + 1 the Lagrangian formulation for massive HS fields may be extrapolated from one given in even dimension in terms of the ghost-independent or spin-tensor forms, which depends only on the standard Grassmann-even matrices γ µ , and does not depend onγ µ ,γ, due to the presence of the latter only as even degrees inside the Lagrangian, and due to the homogeneity of the gauge transformations w.r.t.γ (see [18] for details). In [18] it is shown the constrained Lagrangian formulation (21)-(23) is equivalent to unconstrained one for the same HS field and therefore equivalent to the dynamics to be determined by the initial irreps conditions (1), (2).
Fang-Fronsdal, triplet, quartet Lagrangians
We demonstrate the above general results on the example of TS (k = 1) spin-tensor field, Ψ (µ)n of spin n + 1 2 , which is subject to Dirac (1) and γ-traceless equation only from (2) and therefore is described by 2 Grassman-odd operators t 0 = −ıγ µ ∂ µ , t 1 =γ µ a µ (a µ ≡ a 1 µ ) and
g 1 0 ≡ g 0 = − 1 2 {a µ , a + µ } acting on the basic vector |Ψ = ∞ n=0 ı n n! Ψ (µ)n a +µ 1 . . . a +µn |0 .(25)
with respective proper eigen-values (0, 0, (n + d 2 )). The corresponding nilpotent constrained BRST operator for the differential first-class {t 0 , l 0 , l 1 , l + 1 }, off-shell independent BRST extended constraint T 1 and constrained spin operator σ c (g) in H c have the form:
Q c = q 0 t 0 + η 0 l 0 + η + 1 l 1 + l + 1 η 1 + ı η + 1 η 1 − q 2 0 P 0 ,(26)T 1 , σ c (g) = t 1 − ıη 1 p 0 − 2q 0 P 1 , g 0 + η + 1 P 1 − η 1 P + 1(27)
whose algebra satisfy to the relations (12). The first-order constrained irreducible gaugeinvariant LF for the field, Ψ (µ)n are given by the relations (21), (22) for l = −1, 0 with offshell algebraically independent BRST-extended constraint T 1 imposed on the fields, |χ m 0|c n , m = 0, 1 and gauge parameter |χ
1(0)
0|c n according to (23), (24) for |χ 1(1) 0|c n ≡ 0. The field vectors and gauge parameter being proper for σ c (g) have the decomposition in ghosts η + 1 , P + 1 :
|χ 0 0|c n = |Ψ n + η + 1 P + 1 |χ n−2 = |Ψ n +
ı n−2 (n − 2)! η + 1 P + 1 χ (µ) n−2 n−2 k=1 a + µ k |0 ,(28)
|χ 1 0|c n , |χ
1(0) 0|c n = P + 1 γ|χ 1 n−1 , |ξ n−1 = ı n−1 (n − 1)! P + 1 γχ (µ) n−1 1 , ξ (µ) n−1 n−1 k=1 a + µ k |0 ,(29)
The constraints (24) are resolved as the γ-traceless constraint for the gauge parameter ξ (µ) n−1 and triple γ-traceless one for Ψ (µ)n :
(t 1 ) 3 |Ψ n , t 1 |ξ n−1 = 0,γ|χ 1 n−1 = t 1 |Ψ n , |χ n−2 = − 1 2 (t 1 ) 2 |Ψ n(30)
and therefore, 3 i=1 γ µ i Ψ (µ)n = 0, γ µ ξ (µ) n−1 = 0. The action (21)) and the gauge transformations in terms of independent field vector |Ψ n in the ghost-free form look as
S c|(n) = n Ψ | t 0 − 1 4 (t + 1 ) 2 t 0 t 2 1 − t + 1 t 0 t 1 + l + 1 t 1 + t + 1 l 1 + 1 2 t + 1 l + 1 t 2 1 + 1 2 (t + 1 ) 2 l 1 t 1 |Ψ n , (31) δ|Ψ n = l + 1 |ξ n−1 .(32)
The gauge invariance for the action S c|(n) is easily checked with use of the Noether identity:
δS c|(n) ← − δ δ|ξ = n Ψ | t 0 − 1 4 (t + 1 ) 2 t 0 t 2 1 − t + 1 t 0 t 1 + l + 1 t 1 + t + 1 l 1 + 1 2 t + 1 l + 1 t 2 1 + 1 2 (t + 1 ) 2 l 1 t 1 l + 1 = 0,(33)
modulo the operators L(t + 1 , t 0 , l 1 , l + 1 )t 1 vanishing when acting on the γ-traceless vectors. The variational derivative of the functional δS c|(n) = Ψ |L(t 0 , t 1 , ...)|ξ + ξ |L + (t 0 , t 1 , ...)|Ψ (with the kernel L(t 0 , t 1 , ...) in (33)) with respect to the vector |ξ was introduced above.
In the spin-tensor form the action and the gauge transformations take the familiar form [21] with accuracy up to the common coefficient (n!) −1 :
S c|(n) (Ψ) = (−1) n d d xΨ (ν)n − ıγ µ ∂ µ Ψ (ν)n + 1 4 n(n − 1)η ν n−1 νn (ıγ µ ∂ µ )η µ n−1 µn Ψ (ν) n−2 µ n−1 µn (34) − nγ νn (ıγ µ ∂ µ )γ µn Ψ (ν) n−1 µn + n(ı∂ νn )γ µn Ψ (ν) n−1 µn + n(ı∂ µn )γ νn Ψ (ν) n−1 µn − 1 2 n(n − 1) γ ν n−1 (ı∂ νn )η µ n−1 µn Ψ (ν) n−2 µ n−1 µn + η ν n−1 νn γ µ n−1 (ı∂ µn )Ψ (ν) n−2 µ n−1 µn , = (−1) n d d x Ψp /Ψ − 1 4 n(n − 1)Ψ p /Ψ + nΨ p /Ψ − nΨ · pΨ − nΨ p · Ψ(35)+ 1 2 n(n − 1) Ψ · pΨ + Ψ p · Ψ , δΨ (µ)n = − n i=1 ∂ µ i ξ µ 1 ...µ i−1 µ i+1 ...µn ,(36)
where each term in (34) and (35) corresponds to the respective summand in (31), whereas for the last expression we have used Fang-Fronsdal notations [21] with identifications, p µ = −ı∂ µ , −ıγ µ ∂ µ = p / and p · Ψ = p µ 1 Ψ (µ)n .
The triplet formulation to describe Lagrangian dynamic of the field Ψ (µ)n with help of the triplet of spin-tensors Ψ (µ)n , χ (µ) n−1 1 , χ (µ) n−2 and gauge parameter ξ (µ) n−1 subject to the offshell 3 constraints on the field vectors, |Ψ n , |χ 1 n−1 , |χ n−2 (30) and γ-traceless constraint on |ξ n−1 in the ghost-independent form S c|(n) (Ψ, χ 1 , χ) = n Ψ |t 0 |Ψ n − n−2 χ|t 0 |χ n−2 + n−1 χ 1 |γt 0γ |χ 1 n−1 − n−1 χ 1 |γ l 1 |Ψ n − l + 1 |χ n + h.c. , (37) δ |Ψ n , |χ n−2 , |χ 1 n−1 = l + 1 , l 1 ,γt 0 |ξ n−1
coincides with one suggested in [22]. Without off-shell constraints the triplet formulation describes the free propagation of couple of massless particles with respective spins (n + 1 2 ), (n− 1 2 ),..., 1 2 . It was shown in [23] that this formulation maybe described within unconstrained quartet formulation with additional, to the triplet, compensator field vector |ς n−2 , whose gauge transformation is proportional to the constraint on |ξ n−1 : δ|ς n−2 =γt 1 |ξ n−1 and the whole off-shell constraints (30) are augmented by the terms proportional to |ς to provide theirs total gauge invariance with respect to (38) and above gauge transformations for |ς :
t 1 |Ψ −γ|χ 1 + l + 1γ |ς , |χ + 1 2 t 1γ |χ 1 + 1 2 t 0γ |ς , t 1 |χ + l 1γ |ς = 0, 0, 0 .(39)
Introducing the respective Lagrangian multipliers: fermionic n−1 λ 1 |, bosonic n−2 λ 2 |, fermionic n−3 λ 3 | with trivial gauge transformations, the equations (39) and theirs hermitian conjugated may be derived from the action functional S add|(n) (λ) = n−1 λ 1 | t 1 |Ψ n −γ|χ 1 n−1 + l + 1γ |ς n−2 + n−2 λ 2 | |χ n−2 (40) + 1 2 t 1γ |χ 1 n−1 + 1 2 t 0γ |ς n−2 + n−3 λ 3 | t 1 |χ n−2 + l 1γ |ς n−2 + h.c., so that, the gauge-invariant functional,
S (n) = S c|(n) (Ψ, χ 1 , χ) + S add|(n) (λ)(41)
determines the unconstrained LF for massless spin-tensor of spin (n + 1 2 ) in terms of quartet of spin-tensor fields Ψ (µ)n , χ (µ) n−1 1 , χ (µ) n−2 , ς (µ) n−2 with help of three Lagrangian multipliers λ (µ) n−i i , i = 1, 2, 3 with trivial, as it was shown in [23], dynamics. The constrained LF for the massive TS spin-tensor maybe explicitly obtained by means of procedure related to the dimensional reduction [18].
Constrained Minimal BRST-BV actions
Here we follow to the research [19], where the constrained BRST-BV approach to formulate minimal BV action for integer and half-integer HS fields on R 1.d−1 is suggested and, in part, to [24] for mixed-antisymmetric integer higher-spin fields. First of all, we weaken the vanishing of gh L on the component spin-tensors in the decomposition (14), when considering instead of field vector |χ c ∈ H c the generalized field-antifield vector |χ g|c ∈ H g|c = H g ⊗ H o A gh with Z-grading for H g|c = lim M →∞ ⊕ M l=−M H l g|c for gh tot (|χ l g|c c ) = −l, |χ l g|c ∈ H l g|c . For simplicity, we consider the case of TS HS fields. The total configuration space in the minimal sector contains (with off-shell constraints) in addition to the triplet Ψ (µ)n , χ 1|(µ) n−1 , χ (µ) n−2 the ghost spin-tensor field |C 1(0) c n introduced by the rule: ξ (µ) n−1 (x) = C (µ) n−1 (x)µ =⇒ |χ
1(0) 0|c n = |C 1(0) c n µ,(42)
with ( , gh tot , gh H , gh L ) C (µ) n−1 , |C 1(0) c n = (0, 1, 0, 1), (1, 0, −1, 1) ,
which due to the vanishing of the total ghost number and Grassmann parity may be combined with |χ 0 c n in generalized field vector :
|χ 0 gen|c n = |χ 0 c n + |C 1(0) c n , , gh tot |χ 0 gen|c = (1, 0).
with untouched |χ 1 gen|c n = |χ 1 0|c n , , gh tot |χ 1 gen|c = (0, −1). The corresponding antifield spin-tensors Ψ * (µ)n , χ equation for the deformed action with the interaction terms, thus producing a sequence of relations for these terms.
Notice, the metric-like LF (21)-(24) may be deformed to describe dynamic of both MS HS field with spin s = n + 1 2 on the AdS(d) space and, independently, dynamic of MS conformal HS field on R 1,d−1 which, in turn maybe used to study AdS/CFT correspondence problem.
some s c . Because of the representation (14) the physical state |Ψ (5) is contained in |χ 0 c = |χ c = |Ψ + |Ψ Ac for |Ψ Ac | C=0 = 0. The system (15), (16) is compatible, due to the closedness of the superalgebra {Q c , σ i c , T i , T rs }. Therefore, its resolution for the joint set of proper eigen-vectors permits one, first, to determine from the middle set and the spin and ghost numbers distributions for |χ l c :n i = p i + n f i + n pi , i = 1, . . . , k, |χ l c (n) k : n b0 + n f 0 + i n f i − n pi = −l.(17)
Acknowledgements The author is grateful to the organizers of the International Workshop SQS'17 for their hospitality. I also thank I.L. Buchbinder for the collaboration when solving the problem and to E.D.Skvortsov, D. Francia, A. Campoleoni for valuable correspondence. The work has been done in the framework of the Program of fundamental research of state academies of Sciences for 2013-2020., χ * (µ) n−2 , C * (µ) n−1 with ( , gh L )Ψ * = ( , gh L )χ * 1 = ( , gh L )χ * = (0, −1) and ( , gh L )C * = (1, −2)are combined into generalized antifield vectors as follows:|χ * 0 gen|c n = |χ * 0 c n + |C * 1(0) c n =γ |Ψ * (a + ) n + P + 1 η + 1 |χ * (a + ) n−2 +γη + 1 |C * n−1 ,(46) |χ * 1 gen|c n = |χ * 1 c n = η + 1 |χ * 1 (a + ) n−1 , , gh tot |χ * e gen|c = (1, e − 1), e = 0, 1for the ghost-andγ-independent antifield vectors |Ψ * (a + ) n , |χ * (a + ) n−2 , |χ * 1 (a + ) n−1 , |C * n−1 having the decompositions in powers of a + µ as for the respective field vectors (25), (28), (29) and for the ghost vector |C(a + ) n−1 instead of the gauge parameter. The generalized field and antifield vectors (44), (46), (47) can be uniquely written in terms of the generalized field-antifield vector: The minimal BV action for the massless spin-tensor field Ψ (µ)n in R 1,d−1 takes the formThe functional S c|(n) is invariant with respect to the Lagrangian BRST-transformationswhere the field, antifield vectors are subject to the off-shell BRST extended constraints according to(23), (24)The resolution of (52), (53) is reduced due to (30) to the formIn the ghost-independent form the expressions (50), (51) in terms of the triplets of field |Ψ n , |χ 1 n−1 , |χ n−2 and antifield |Ψ * n , |χ * 1 n−1 , |χ * n−2 vectors and singlets |C1 , χ ( * ) = S c|(n) Ψ, χ 1 , χ + n Ψ * |γl + 1 + n−2 χ * |γl 1 (56)and with independent field |Ψ n , ghost |C 1(0) c n−1 and antifield |Ψ * n , |C * 1(0) c n−1 vectors S c|(n) = S c|(n) |Ψ + n Ψ * |γ l + 1 +with the classical action given by(37)The minimal BV action and Lagrangian BRST-transformations for unconstrained quartet formulations for massless spin-tensor field are easily obtained from the respective BRST-BFV formulations. E.g. we get for the unconstrained minimal BV action S n :with account of (56), (41) respectively for S c|(n) Ψ ( * ) , χ1 , χ ( * ) , S add|(n) (λ) and antifield vector |ς * n−2 =γ|ς * (a + ) n−2 . The antifield vectors in (63) are considered without off-shell constraints (54), (55) as well as the antifield spin-tensors λ * (µ) n−i i , i = 1, 2, 3, determining the total field-antifield space, do not entered in S n .The obtained minimal BV actions for TS massless spin-tensor field Ψ (µ)n present the basic results of the Section.The actions serves to construct quantum actions under an appropriate choice of a gauge condition (e.g. for the TS field l 1 |Ψ n = 0), as well as to find an interacting theory, including both only the TS half-integer HS field Ψ (µ)n , with a vertex at least cubic in Ψ (µ)n and TS integer HS field Φ (µ)s ., as well as Ψ (µ)n interacting with an external electromagnetic field and some other HS fields. The consistency of deformation is to be controlled by the master
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|
[] |
[
"When Homomorphic Cryptosystem Meets Differential Privacy: Training Machine Learning Classifier with Privacy Protection",
"When Homomorphic Cryptosystem Meets Differential Privacy: Training Machine Learning Classifier with Privacy Protection"
] |
[
"Xiangyun Tang [email protected] \nBeijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n\n",
"Liehuang Zhu [email protected] \nBeijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n\n",
"Meng Shen [email protected] \nBeijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n\n",
"Xiaojiang Du \nBeijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n\n"
] |
[
"Beijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n",
"Beijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n",
"Beijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n",
"Beijing Institute of Technology\nBeijing Institute of Technology\nBeijing Institute of Technology\nTemple University\n"
] |
[] |
Machine learning (ML) classifiers are invaluable building blocks that have been used in many fields. High quality training dataset collected from multiple data providers is essential to train accurate classifiers. However, it raises concern about data privacy due to potential leakage of sensitive information in training dataset. Existing studies have proposed many solutions to privacy-preserving training of ML classifiers, but it remains a challenging task to strike a balance among accuracy, computational efficiency, and security.In this paper, we propose Heda, an efficient privacypreserving scheme for training ML classifiers. By combining homomorphic cryptosystem (HC) with differential privacy (DP), Heda obtains the tradeoffs between efficiency and accuracy, and enables flexible switch among different tradeoffs by parameter tuning. In order to make such combination efficient and feasible, we present novel designs based on both HC and DP: A library of building blocks based on partially HC are proposed to construct complex training algorithms without introducing a trusted thirdparty or computational relaxation; A set of theoretical methods are proposed to determine appropriate privacy budget and to reduce sensitivity. Security analysis demonstrates that our solution can construct complex ML training algorithm securely. Extensive experimental results show the effectiveness and efficiency of the proposed scheme.
| null |
[
"https://arxiv.org/pdf/1812.02292v1.pdf"
] | 54,447,686 |
1812.02292
|
0b09adc8805626c9a5092154087daede0f01cc8c
|
When Homomorphic Cryptosystem Meets Differential Privacy: Training Machine Learning Classifier with Privacy Protection
Xiangyun Tang [email protected]
Beijing Institute of Technology
Beijing Institute of Technology
Beijing Institute of Technology
Temple University
Liehuang Zhu [email protected]
Beijing Institute of Technology
Beijing Institute of Technology
Beijing Institute of Technology
Temple University
Meng Shen [email protected]
Beijing Institute of Technology
Beijing Institute of Technology
Beijing Institute of Technology
Temple University
Xiaojiang Du
Beijing Institute of Technology
Beijing Institute of Technology
Beijing Institute of Technology
Temple University
When Homomorphic Cryptosystem Meets Differential Privacy: Training Machine Learning Classifier with Privacy Protection
Machine learning (ML) classifiers are invaluable building blocks that have been used in many fields. High quality training dataset collected from multiple data providers is essential to train accurate classifiers. However, it raises concern about data privacy due to potential leakage of sensitive information in training dataset. Existing studies have proposed many solutions to privacy-preserving training of ML classifiers, but it remains a challenging task to strike a balance among accuracy, computational efficiency, and security.In this paper, we propose Heda, an efficient privacypreserving scheme for training ML classifiers. By combining homomorphic cryptosystem (HC) with differential privacy (DP), Heda obtains the tradeoffs between efficiency and accuracy, and enables flexible switch among different tradeoffs by parameter tuning. In order to make such combination efficient and feasible, we present novel designs based on both HC and DP: A library of building blocks based on partially HC are proposed to construct complex training algorithms without introducing a trusted thirdparty or computational relaxation; A set of theoretical methods are proposed to determine appropriate privacy budget and to reduce sensitivity. Security analysis demonstrates that our solution can construct complex ML training algorithm securely. Extensive experimental results show the effectiveness and efficiency of the proposed scheme.
I. INTRODUCTION
Machine learning (ML) classifiers are widely used in many fields, such as spam detection, image classification, and natural language processing. Many studies have modeled user data and obtained satisfactory classifiers that meet accuracy requirements [19], [36]. The accuracy of a classifier obtained from supervised learning is closely related to the quality of the training dataset, in addition to well-designed ML algorithms. An experimental study with a dataset of 300 million images at Google [29] demonstrates that the performance of classifiers increases as the order of magnitude of training data grows. However, training dataset is usually held by multiple data providers and may contain sensitive information, so it is important to protect data privacy in training of ML classifiers.
Consider the typical training process depicted in Figure 1. There are multiple data providers and a single data user. Upon receiving the request of dataset from the data user, each data provider applies privacy-preserving mechanisms (e.g., encryption or perturbation) to its own dataset. Then, the data user trains an ML classifier based on the dataset gathered from multiple data providers. During this process, each data provider Fig. 1. Application Scenario. Each non-shaded rectangle represents a type of role. Each shaded box indicates private data that should be accessible to only one party: a protected dataset to a data provider, and the model to the data user. Each solid arrow indicates an algorithm or a process. cannot know the classifier, while the data user cannot learn any sensitive information of the shared data.
More specifically, consider the following example of an advertisement recommendation task: In order to attract more consumers, an company wants to build a classifier to discern the most appropriate time for advertising. The training dataset used for constructing the classifier is extracted from the consumer purchase behavior data recorded by several online shopping sites. The consumer data is confidential because it contains sensitive information about consumers. Online shopping sites agree to share their data with companies, but refuse to reveal any privacy of the consumers. The company wants to construct a classifier based on the consumer data, but is unwilling to reveal the classifier to online shopping sites. Ideally, online shopping sites and the company run a privacy-preserving training algorithm, at the end of which the company learns the classifier parameters, and neither party learns anything else about the other party's input.
In general, supervised ML classifiers consist of two phases: the training phase and the classification phase. A series of secure schemes for the classification phase have been proposed [6], [10]. In this paper, we focus on the training phase, that is, privacy-preserving training of ML classifiers 1 .
Existing solutions to training ML classifier securely roughly depend on three types of techniques, namely secure multi-party computing (SMC), homomorphic cryptosystem (HC), and differential privacy (DP). SMC can construct many classifiers theoretically. But it relies on a trusted third-party for providing random number, and results in a large number of interactions and redundant computations for protecting data privacy [15], [23]. HC 2 allows the operation on ciphertext to be mapped to the corresponding plaintext. The secure training solutions based on HC [11], [17] may suffer from low efficiency. In addition, since partially HC only enables a single type of operation (e.g., addition or multiplication), HC-based solutions for training complex ML classifiers usually introduce a trusted third-party (e.g., the authorization server [10], [17]) or use an approximate equation that simplifies the complex iteration formula [2], [3]. DP can resist the attacker with the largest background knowledge [5], which ensures the security of the published data by adding noises. The computational efficiency of operations on perturbed data is significantly higher than those on ciphertext [5], [9]. Nevertheless, the quality of the published dataset is reduced due to the introduction of noises, and thereby the accuracy of the resulting classifiers is decreased inevitably.
As discussed above, HC is low efficient due to ciphertextbased computation, but can obtain a classifier with lossless accuracy. DP has high computational efficiency but leads to an inevitable loss of accuracy. Intuitively, we can take the strengths of HC and DP by adopting them simultaneously.
However, HC and DP are completely different systems: one for data encryption, and the other for data perturbation. It is a challenging task to combine them together. In particular, partially HC only supports one type of operation, which sets a barrier to the training of ML classifiers with complex operations such as power function, division, and square root. Furthermore, the noises added to sensitive data in DP determines the accuracy of classifiers and privacy of published data. The third challenge is how to archive high accuracy while ensuring privacy in DP.
In this paper, we propose Heda, an efficient privacypreserving scheme for training ML classifiers. By combining HC with DP, Heda obtains the tradeoffs between efficiency and accuracy and enables flexible switch among different tradeoffs by parameter tuning. Security analysis demonstrates that our building blocks can construct complex ML training algorithms. Extensive experimental results show the effectiveness and efficiency of the proposed scheme.
We address the above challenges by developing a set of key techniques.
We make an observation that different features 3 in a dataset usually contribute differently to the accuracy of classifiers [21], [31]. For the features with high contributions, we apply HC to these features such that the model parameters obtained from them are as accurate as possible. We apply DP to the rest features to improve the computational efficiency. The contribution of each feature in training ML classifiers can be evaluated using readily available techniques [21].
To address the second challenge, we employ two homomorphic encryption primitives: a multiplicative homomorphic encryption RSA and an additively homomorphic encryption Paillier. We carefully design a library of building blocks supporting for complex operations such as power function and dot product, which can handle ML classifiers with complex training operations. We take Logical Regression (LR) as an example to illustrate the power of our building blocks. The sigmoid function in the iterative formula of LR makes it difficult to construct a secure LR training algorithm based on HC. It is the first time that constructing a secure LR training algorithm by HC without an authorization server or any approximation. (Section VI)
In the face of the third challenge, we develop a formal method to determine the reasonable privacy budget, and we reduce the sensitivity by using insensitive microaggregation. We reduce the added noise and improve the usability of the noise dataset published by DP reasonably. (Section V)
To the best of our knowledge, it is the first study that achieves privacy-preserving training of ML classifiers by jointly applying HC and DP in an individual scheme. The rest of our paper is organized as follows. Section II describes related work, and Section III provides the background. Section IV describes the problem statement. Section V and Section VI present the special designs with DP and HC, respectively. Section VII describes the construction of Heda in detail. The security analysis is exhibited in Section VIII, and the evaluation results are provided in Section IX. Section X concludes this paper.
II. RELATED WORK
Since our work is related to secure ML classifiers algorithms which can be broadly divided into two categories: privacy-preserving classification and privacy-preserving training. We give the literature review of both subjects. Because Heda jointly applying HC and DP, and there are some studies about combining HC with DP but not about secure classifiers training, we present a discussion about these works. We give an analysis of our novelty at last.
A. Privacy-Preserving ML Classification
A series of techniques have been developed for privacypreserving ML Classification. Wang et al. [33] proposed an encrypted image classification algorithm based on multi-layer extreme learning machine that is able to directly classify encrypted images without decryption. They assumed the classifier had been trained, and the classifier not confidential. Grapel et al. [18] constructed several secure classification algorithms by HC, while the parameters of trained classifiers are not confidential for classifiers users. Zhu et al. [37] proposed a secure nonlinear kernel SVM classification algorithm, which is able to keep users' health information and healthcare provider's prediction model confidential.
Several works have designed general (non-application specific) privacy-preserving protocols and explored a set of common classifiers by HC [6], [10]. Usually, classification algorithms are simpler than training algorithms, building blocks that are able to build classification algorithms can be powerless for complex training algorithms.
B. Privacy-Preserving ML Classifier Training
Three techniques have been applied to privacy-preserving ML classifier training, they are SMC, HC, and DP. Constructing secure classifier training algorithms based on SMC relies on a large number of interactions and many redundant calculations for protect privacy, and it generally needs to introduce authoritative third parties to provide random number distribution services as well. In addition, SMC protocols for generic functions existing in practice rely on heavy cryptographic machinery. Applying them directly to model training algorithms would be inefficient [4], [23], [35].
HC is able to compute using only encrypted values. Employing HC, many secure algorithms have been developed for different specialized ML training algorithms such as Support Vector Machine (SVM) [17], [27], LR [11], [18], decision trees [30] and Naive Bayes [24]. However, partially HC only enables a single type of operation (e.g., addition or multiplication). In order to construct complex training algorithms, HC-based schemes usually need to rely on trusted third parties such as the Authorization Server [10], [17], or use an approximate equation to simplify the original complex iteration formula into a simple one [2], [3]. Gonzlez et al. [17] developed secure addition protocol and secure substractions protocol to construct the secure SVM training algorithm by employing Paillier, while some operations that are not supported by Paillier have to be implemented with the assistance of the Authorization Server in their scheme. Secure LR training algorithms existing implemented by HC are actually the linear regression [11], [18], because the sigmoid function contains power function and division operation, which makes LR training algorithms harder to be implemented by HC than other ML training algorithms. Several works solved the sigmoid function by an approximate equation 4 [2], [3].
Many secure ML classifier training algorithms have been explored in DP area such as decision tree [5], LR [9] and deep learning [1]. Blum et al. [5] proposed the first DP based decision tree training algorithm on the SuLQ platform. Abadi et al. [1] applied DP objective perturbation in a deep learning algorithm, where the noise was added to every step of the stochastic gradient descent. Due to the introduction of noise under DP mechanisms, the quality of the datasets were reduced, and the accuracy of these trained models was decreased inevitably. So the essential challenge for DP based frameworks is guaranteeing the accuracy by reducing the added noise, especially for the operation has high sensitivities [38]. According to the Laplace mechanism (cf. Definition 4), privacy budget and the sensitivity ∆f are two important factors affecting noise addition. In many papers, the value of is merely chosen arbitrarily or assumed to be given [1], [5]. Lee et al. [22] explored the selection rules of , but they have not given a way to determine the value of the privacy budget. Soria et al. [28] proposed a insensitive microaggregation-based DP mechanism, they found the amount of noise required to fulfill -DP can be reduced in insensitive microaggregation. Heda develops the insensitive microaggregation-based DP mechanism and decreases the amount of noise required to fulfill -DP again.
4 log( 1 1+exp(u) ) ≈ k j=0
a · u C. Homomorphic Cryptosystem Combine Differential Privacy Several works have studied combining HC with DP to solve a special security problem. Pathak et al. [26] proposed a scheme for composing a DP aggregate classifier using classifiers trained locally by separate mutually untrusting parties, where HC was used for composing the trained classifiers. Yilmaz et al. [34] proposed a scheme for optimal location selection utilizing HC as the building block and employing DP to formalize privacy in statistical databases. Aono et al. [3] constructed a secure LR training algorithm via HC and achieved DP to protect the model parameters. These works general constructed a secure algorithm via HC and used DP to protect the algorithm results. As we have discussed above, constructing a secure algorithm via HC is low efficient, and secure algorithm based on DP has inevitable loss in accuracy. We aim of constructing a secure classifier training algorithm jointly applying HC and DP in an individual scheme to obtain a tradeoff between efficiency and accuracy.
D. Novelty of Our Construction
Secure training algorithms based on HC have to handle datasets in ciphertext case, where the time consumption is considerable, while the accuracy is able to be guaranteed. Noise datasets published by DP mechanism are in plaintext case, it is efficient to train a model in plaintext case, while using the noise dataset may lead to a low accuracy. HC and DP have drawbacks as well as merits.
Heda takes the strengths of HC and DP to get a highefficiency and high-accuracy privacy-preserving ML classifier training algorithm. Heda is the first to combine these two techniques and construct a privacy-preserving ML classifier training algorithm in a multi-party setting, where feature evaluation techniques are employed to give the way of combination. By combining HC with DP, Heda obtains the tradeoffs between efficiency and accuracy, and enables flexible switch among different tradeoffs by parameter tuning. What's more, we develop a library of building blocks by HC that is able to construct complex training algorithms, and by using our building blocks this is the first time that solving the sigmoid function in secure LR training based on HC without any approximate equation. We develop the works of Lee et al. [22] and Soria et al. [28] giving a formula to determine the appropriate privacy budget and another lower sensitive solution.
III. PRILIMINARIES
A. Notation
A dataset D is an unordered set of n records with the size of |D|. x i ∈ R d , x i = (x i1 , x i2 , . . . , x id ) is the i-th record in dataset D, and y i is a class label correspond to x i . X = (x 1 , x 2 , . . . , x m ), Y = (y 1 , y 2 , . . . , y m ). β, ω, and b are the relevant parameters of the model trained by a ML algorithm. The subset A i corresponding to the i-th attribute in D. S is the scores assign to the features.
Cryptosystems define a plaintext space M, and a ciphertext space C. In Heda, we employ two public-key cryptosystems, Paillier and RSA. [[m]] and ||m|| are represented as the ciphertext of m under Paillier or RSA respectively. DP is generally achieved by a randomized algorithm M.
Ai
The subset of i-th attribute in D is the privacy budget in a DP mechanism. A query f maps dataset D to an abstract range f : D → R. The maximal difference in the results of query f is defined as the sensitivity ∆f . D is a neighboring dataset of D. Table I summarizes the notations used in the following sections.
B. Homomorphic Cryptosystem
Cryptosystems are composed of three algorithms: key generation (Gen) to generate the key, encryption (Enc) encrypting secret message and decryption (Dec) for decrypting ciphertext. Public-key cryptosystems employ a pair of keys (PK, SK), the public key (PK, the encryption key) and the private key (SK, the decryption key). Some cryptosystems are gifted with a property of homomorphic that makes cryptosystems perform a set of operations on encrypted data without knowledge of the decryption key. Formalized definition is given in Definition 1.
Definition 1: (homomorphic) [20]. A public-key encryption scheme (Gen, Enc, Dec) is homomorphic if for all n and all (PK, SK) output by Gen (1 n ), it is possible to define groups M, C (depending on PK only) such that:
(i) The message space is M, and all ciphertexts output by Enc pk are elements of C.
(ii) For any m 1 , m 2 ∈ M, any c 1 output by Enc pk (m 1 ), and any c 2 output by Enc pk (m 2 ), it holds that Dec sk (o (c 1 , c 2 )) = σ (m 1 , m 1 ).
In Heda, we employ two public-key cryptosystems, Paillier and RSA. Paillier possesses additively homomorphic property, and RSA possesses multiplicative. For more details about Paillier or RSA, we refer the reader to [20].
Paillier. The security of Paillier is based on an assumption related to the hardness of factoring. Assuming a pair of ciphertext (c 1 , c 2 ) is (m 1 , m 2 ) under the same Paillier encryption scheme where the public key is N , we have:
c 1 × c 1 = (1 + N ) m1+m2 r N modN 2 , where (m 1 + m 2 ) < N . The
additively homomorphic property in Paillier can be described as
[[m 1 + m 2 ]] = [[m 1 ]] × [[m 2 ]] modN 2 .
RSA. Based on the definition of a one-way trapdoor function, RSA gives the actual implementation of the first public key cryptosystem. RSA is a multiplicative HC, because that:
Enc RSA (m 1 )×Enc RSA (m 2 ) = ||(m 1 × m 2 ) e modN ||, where (m 1 × m 2 ) < N .
The multiplicative homomorphic property in RSA can be described as ||m 1 × m 2 || = ||m 1 || × ||m 2 || (modN ).
C. Differential Privacy
Definition 2: (Neighbor Dataset) [5]. The datasets D and D have the same attribute structure, and the symmetry difference between them is denoted as |D D |. We call D and D neighbour datasets if |D D | = 1.
Definition 3: ( -Differential Privacy) [5]. A randomized mechanism M gives -DP for every set of outputs R, and for any neighbor dataset of D and D , if M satisfies:
Pr [M (D) ∈ R] ≤ exp ( ) × Pr [M (D ) ∈ R].
A smaller represents a stronger privacy level [38]. While is equal to 0, for any neighbour dataset, the randomized mechanism M will output two identical results of the same probability distribution which cannot reflect any useful information. If is selected as a too large value in a DP mechanism, it does not mean that privacy is actually enforced by the mechanism. A composition theorem for named parallel composition (Theorem 1) is widely used. Lapace Mechanism (Definition 4) is the basic DP implementation mechanism and is suitable for the numerical data, which adds independent noise following the Laplace distribution to the true answer.
Definition 4: (Laplace mechanism) [14]. For a dataset D and a query function f : D → R with sensitive ∆f . Privacy mechanisms M (D) = f (D) + γ providers -DP, where γ ∼ lap ∆f represents the noise sampled from a Laplace distribution with a scaling of ∆f .
Definition 5: (Sensitivity) [5]. For a query f : D → R, and a pair of neighbor datasets (D, D ), the sensitivity of f is defined as: ∆f = max
D,D ||f (D) − f (D )|| 1 . Sensitivity ∆f is
only related to the type of query f . It considers the maximal difference between the query results.
IV. PROBLEM STATEMENT
We are devoted to addressing the problem on the secure training of ML classifier using private protected data gathered from different data providers. In this section, we introduce the overview of the system model and the roles involved in Heda. Then, we formally define the threat model and the security goal.
A. System Model
We target at the system application scenario which has been illustrated in Figure 1. There are n data providers P and a data user U in our model. Each P holds their own sensitive dataset D i and a pair of keys (PK, SK). P protects their sensitive data by applying privacy-preserving mechanisms (e.g., DP mechanism and HC). U holds his own keys (PK, SK). After obtaining the permission, U requests the sensitive data from P, and P returns the protected data. By running a sequence of secure interactive protocols with several P, U obtains the classifier parameters of being encrypted by U's keys.
As discussed in Section I and II, HC is able to construct accurate secure training algorithms, and DP mechanism providers high efficient secure training algorithms. However, it is low efficient that constructing a secure ML training algorithm by HC, and the model may poor in accuracy if the training data is under DP mechanism. We thereby desire to take the strengths of HC and DP, and feature evaluation techniques is used for providing a right combination method. We describe the overall idea of Heda as follows:
1) P scores all features by feature evaluation techniques and divides the dataset into two parts according to the scores (see Section VII-A).
2) P applies privacy-preserving mechanisms to the two parts respectively: the low scores part published by DP mechanism (see Section V); the high scores part encrypted by HC (see Section VI).
3) Upon receiving the query requests, P sends the protected data to U. 4) U trains a ML classifier under these two protected subdatasets (see Section VII-C).
B. Threat Model
U interacts with several P to obtain the protected data and performs training algorithms on the data. Each P trys to learn as much other P's sensitive data and U's trained classifier as possible by honestly executing pre-defined protocols. U follows the protocol honestly, but it tries to infer P's sensitive data as much as possible from the values he learns. As discussed above, we assume each participant is a passive (or honest-but-curious) adversary [16], that is, it does follow the protocols but tries to infer others' privacy as much as possible from the values they learn.
C. Security Goal
In Heda, we allow any two or more parties conspire to steal the privacy of other participants. We make the following assumptions: Each participate as a honest-but-curious adversary performs protocol honestly but may have interest in the private information of other domains. Any two or more participates may collude with each other. As passive adversaries, they do follow the protocol but try to infer other's privacy as much as possible from the values they learn.
The aim of Heda is achieving keeping privacy of each participant and computing model parameters securely when facing honest-but-curious adversaries or any collusion. To be specific, the privacy of U is model parameters, and each P is their sensitive data. We specify our security goals as follows:
1) When facing honest-but-curious adversaries, U and each P's privacy are confidential. 2) when facing any two or more parties collude with each other, U and each P's privacy are confidential.
V. ACCURACY AND PRIVACY DESIGN WITH DIFFERENTIAL PRIVACY DP ensures the security of the published data by adding noise. Insufficient noise leads to the security of the published data cannot be guaranteed, while excess noise causes the data unusable. Obviously, the key to using DP in the secure classifier training is to reduce the added noise while ensuring the security of the published data.
The two important parameters that determine the added noise are and ∆f (cf. Definition 4). A bigger or a smaller ∆f are able to reduce the added noise. However, if is selected as a too large value, although the system has been built upon DP framework, it dose not mean that privacy is actually enforced by the system. Therefore, must be combined with specific requirements to achieve the balance of security and usability of output results. On the other hand, ∆f is only determined by the type of query function (cf. Definition 5).
In this section, we develop a formula for reasonably determining the appropriate in DP mechanism, and we reduce the ∆f by using insensitive microaggregation.
A. Selection of Appropriate
In many papers, is chosen arbitrarily or assumed to be given, while decision on should be made carefully with considerations of the domain and the acceptable ranges of risk of disclosure. Lee et al. [22] explored the rule of , but they did not give a specific method for determining . We give a method for determining . It is worth noting that based on different criteria and backgrounds, can have different values, and we are trying to give a general one.
We follow some notations of Lee et al. [22]: If an adversary knows all the background knowledge, he tries to guess which one is the different values between D and D. Let W denotes the set of all possible combinations ω of D , ω ∈ W. For each possible ω, the adversary maintains a set of tuples α, µ . For a given query response, α and µ are the adversary's prior belief and posterior belief on D , i.e., ∀ω ∈ W, α (ω) = 1 m . For each possible ω, the adversary's posterior belief on ω is defined as
µ (ω) = P (D = ω|γ) = P (M(ω)=γ) ω∈W P (M(ω)=γ)
. Lee et al. [22] obtain the upper bound of through a series of derivations as Formula 1 (cf. Section V in [22])
≤ ∆f ∆v ln (m − 1) ρ 1 − ρ(1)
where ∆v = max 1≤i,j≤n,i =j |f (ω i ) − f (ω j )|, ρ is the probability that the adversary guessing success. Nevertheless, Lee et al. [22] did not give a method for setting ρ. We give a method for determining the upper bound of ρ (Proposition 1).
Proposition 1 (the upper bound of ρ for D ): Let A j is the subset of the j-th attribute in dataset D. Count max is Algorithm 1 Generating Appropriate Value of
Input : D = {(xi, yi)} m i=1
. Output : The appropriate on dataset D. 1: for j = 1 to d do 2: for i = 1 to m do 3: Computing ∆f and Countmax |Aj| in Aj.
4:
Obtaining j by Formula 1.
5: return = { 1, 2, . . . . . . , d }.
the occurrences number of the record which has the highest frequency in A j . Then Countmax |Aj | is the upper bound of ρ.
Proof 1 (Proof of Proposition 1): ρ is the probability that the adversary successfully guesses which instance is the different one between D and D. DP mechanism assumes that the adversary has a strong background knowledge, that is, he knows the value of each instance in D. x ij is the highest frequency instance in A j , so the adversary guesses x ij will get the highest probability of success. After DP mechanism, the adversary's probability of success should not be greater than the highest probability of random guessing and success, so the upper bound of ρ is Countmax
|Aj | .
The upper bound of i is obtained form each subset A i by Formula 1, then the dataset D provides max ( i )-DP according to Theorem 1. Algorithm 1 details the steps for generating the appropriate on dataset D.
B. Reducing ∆f by Insensitive Microaggregation
According to the Definition 4, the smaller the ∆f , the less noise is added, and thereby the more usable the data is. In this subsection, we detail the solution of reducing the ∆f in Heda. The amount of noise required to fulfill -DP can be greatly reduced if the query is run on a insensitive microaggregation version of all attributes instead of running it on the raw input data [28].
1) What is insensitive microaggregation:
Microaggregation is used to protect microdata releases and works by clustering groups of individuals and replacing them by the group centroid. DP makes no assumptions about the adversary's background knowledge. Microaggregation with DP can help increasing the utility of DP query outputs while making as few assumptions on the type of queries as microaggregation does [28]. However, if we modify one record in D, more than one clusters will differ from the original clusters generally. According to the Definition 3, we expect that we modify one record in D, each pair of corresponding clusters differs at most in single record. Microaggregation that satisfies this property is named insensitive microaggregation (IMA). Soria et al. [28] give a formal definition of IMA. Microaggregation is insensitive to input data if and only if the distance function Dist(x, y) is a fixed sequence of total order relations defined over the domain of D [28].
The sequence of total orders is determined by a sequence of reference points. The reference points are the two boundary points P and P , i.e. P = (p 1 , p 2 , . . . , p d ), p i = max (A i ), and P = (p 1 , p 2 , . . . , p d ), p i = min (A i ). The total order relations between two points in Heda is: Dist (x, y) = Algorithm 2 Generating an IMA Dataset Computing the boundary point P and P .
Input : D = {(xi, yi)} m i=1 ,
4:
Ci ← k nearest instances to P from D according to Dist (x, y), D := D\Ci.
5:
Ci+1 ←k nearest instances to P from D according to Dist (x, y), D := D\Ci+1. 6: i := i + 2 7: Ci ← remaining records. 8: Computing each centroid of Ci and use it to replace the records in each cluster. 9: return DIMA.
d i=0 (xi−yi) 2 (Pi−P i ) 2 .
Generating a IMA dataset is detailed in Algorithm 2.
2) Determining the sensitivity: As Definition 5, ∆f j = max (A j ) in dataset D, and the sensitivity in IMA is ∆fj k × m 2k , which is formalized in the Proposition 2. We detail Algorithm 3 constructing our DP mechanism. According to a distance function Dist() with an total order relation, IMA algorithm iteratively takes sets with cardinality k from the extreme points until less than 2k records are left. The less than 2k records are formed the last cluster C r that is the cluster at the center of the total order relation sequence. Every x i in D is ordered by Dist(). A pair of databases (D, D ) differing only in one instance x i means the larger database contains just one additional row [13]. The number of clusters on the left and the right of C r is equal, as shown in the Figure 2. If the different record in (D, D ) is x i on the left of C r and x i is located to C i . Then the changed clusters are the clusters from C i to C r , and the maximum change for each changed cluster is ∆fj k . Other clusters on the right side of C r will not be changed. The worst scenario is when x i is located to C 1 , there is the maximum number m 2k of changed clusters. The scenario on the left and the right sides of C r is symmetrical, so the number of changed clusters is at most m 2k . To make the sensitivity of D IM A smaller than the original dataset, we let ∆fj k × m 2k ≤ ∆f j , then we can get the best cluster size: k = m 2 . Soria et al. [13] thought that D and D differ in a "modified" record x i . The modification causes the whole sequence originally obtained by Dist() is changed from the position of x i in turn. So they considered the sensitive is ∆f j (D) = ∆fj k × m k . However, Dwork er al. [13] give that: "On pairs of Adjacent Dataset (D, D ) differing only in one row, meaning one is a subset of the other and the larger database contains just one additional row." Their sensitive method causes a greater sensitivity than ours, which reduces the usability of the dataset.
VI. PRIVACY DESIGN WITH HOMOMORPHIC CRYPTOSYSTEM
A homomorphic encryption algorithm can only support one type of operation (e.g., Addition or Multiplication). Existing HC-based secure training algorithms need to rely on trusted third parties such as the Authorization Server [10], [17], or use an approximate equation to simplify the original complex iteration formula [2], [3].
We elaborately design a library of building blocks by multiplicative homomorphic encryption RSA and additively homomorphic encryption Paillier, which is able to construct complex secure ML training algorithms needing no the Authorization Server or any approximate equation. In order to illustrate the power of our building blocks, we construct a secure LR training algorithm (see Section VII-B). It is the first solving the sigmoid function based on HC. In this section, we detail our building blocks. The security proofs for each building block are given in Section VIII.
ML training algorithms are computationally complex, so the building blocks need to support a range of choices including which party gets the input, which party gets the output, and whether the input or output are encrypted or not.
Protocol 2 Secure Subtraction Protocol
Input Alice: a = {a1, a2, . . . . . . , a d }, (PK Alice , SK Alice ).
Input Bob: PK Alice , [[b −1 ]] Alice = [[ b −1 1 , b −1 2 , . . . , b −1 d ]] Alice or b −1 = b −1 1 , b −1 2 , . . . , b −1 d . Output Bob: [[f (a, b)]] Alice = [[a − b]]Bob computes f (a, b) i Alice = b −1 i Alice × [[ai]] Alice . 4: return [[f (a, b)]] Alice to Bob.
Protocol 3 Secure Dot Product Protocol
Input Alice: a = {a1, a2, . . . . . . , a d }, (PK Alice , SK Alice ).
Input Bob: b = {b1, b2, . . . . . . , b d } and PK Alice . Output Bob: [[f (a, b)]] Alice = [[a × b]] Alice . 1: Alice sends [[a]] Alice to Bob. 2: Bob computes [[f (a, b)]] Alice = d i=1 [[ai]] b i Alice . 3: return [[f (a, b)]] Alice to Bob.
Protocol 4 Secure Multiplication Protocol
Input Alice: a, (PK Alice , SK Alice ). Input Bob: ||b|| Alice and PK Alice . Output Bob: ||f (a, b)|| Alice = ||a × b|| Alice . 1: Alice sends ||a|| Alice to Bob. 2: Bob computes ||f (a, b)|| Alice = ||a|| Alice × ||b|| Alice . 3: return ||f (a, b)|| Alice to Bob.
to obtain the secure addition protocol (Protocol 1) and secure subtraction protocol (Protocol 2).
2) Secure dot product and secure multiplication: Using Paillier's additive homomorphism property, we can construct a secure dot product protocol ( 3) Secure power function: In order to cope with more complex training algorithms, we design protocol 5 satisfying Condition 1 under RSA to obtain ||e ab || securely.
Condition 1 SK A , PK B , a SK B , PK A , b - [f (a, b)] A 1, 2, 3, 5 Condition 2 SK A , PK B , a SK B , PK A , [b] A - [f (a, b)] A 1, 2, 4 Condition 3 SK A , PK B SK B , PK A , ||f (a, b)|| A - [[f (a, b)]] A 6 Condition 4 SK A , PK B SK B , PK A , [[f (a, b)]] A - [[f (a, b
4) Secure changing the encryption cryptosystem:
There are multiple participants in Heda. Different participants have their own encryption scheme (i.e., the certain plaintext space and a pair of keys (PK,SK)). Homomorphic operations can only be operated in the same plaintext space. For completeness, we design two protocols converting ciphertext from one encryption scheme to another while maintaining the underlying plaintext.
VII. CONSTRUCTION OF HE D A
In this section, we introduce the overall framework of Heda. In Heda, U is able to learn a model without learning anything about the sensitive data of P, and in addition to U, others should learn nothing about the model.
Algorithm 4 Privacy-Preserving Training
Each P's Input: Di = {(xi, yi)} m i=1 , (PK P i −Paillier ,SK P i −Paillier ) and (PK P i −RSA ,SK P i −RSA ). U's Input: (PK U −Paillier , SK U −Paillier ). U's Output: β.
1: U initializes β. 2: According to S, each P divides all features in two part: the high scores part and the low scores part. 3: each P obtains the noise dataset from the low scores part subdataset by Algorithm 3 and sends it to U. 4: U trains a model by building blocks (Algorithm 5) combine with noised datasets. 5: return β to U.
proof will be given in Section VIII.
Heda is exhibited in Algorithm 4. We introduce the following details in this section: how to use feature evaluation technologies to divide a dataset into two parts (Algorithm 4
Step 2), how to construct a specific training algorithm using building blocks (Algorithm 4 Step 4), and how to combine DP mechanism with the building blocks (Algorithm 4 Step 4).
Proposition 4 (The Security of Heda): Algorithm 4 is secure in the honest-but-curious model.
A. Feature Partitioning
Obviously, It is best that each P conducts the feature evaluation locally. The locally computation does not require interaction with any other party which guarantee the privacy of each P. In addition, P can perform arbitrary computations on their sensitive data in plaintext case with high efficiency. After Several P who join in Heda locally implement feature evaluation, they communicate with each other negotiating the final scores S. Feature scores do not reveal the privacy of datasets, so it is feasible that several P share the scores of their datasets and negotiate the final feature scores.
According to the feature scores S, each P processes the dataset into an ordered dataset. Let an ordered dataset D = {A 1 , A 2 , ..A d } ordered by feature scores, i.e. S i is the scores of A i , S i > S j , 0 < i < j < d. Let the high scores part has ι features, then D = D 1 , D 2 ,
D 1 = {A 1 , A 2 , ..A ι }, D 2 = {A ι+1 , A ι+2 , ..A d }.
We assume that U spends t 1 in learning a classifier parameters on the low scores part (the noise dataset) and t 2 on the high scores part (the encrypted dataset), then the total time is T = t 1 + t 2 . Training a model in plaintext case usually takes time in milliseconds but usually takes thousands of seconds or longer in ciphertext case [10], [11]. There is a linear relationship between t 2 and ι, i.e. t 2 ≈ τ ι + b, where ι > 0, τ and b are two constants. The training time on noise dataset is much less than the time training a model on the encrypted dataset. Formula 2 shows the total time consumption.
T ≤ τ (ι + 1) + b (2)
Heda enables flexible switch among different tradeoffs between efficiency and accuracy by parameter tuning. With parameter ι, one is able to obtain the tradeoff between efficiency and accuracy. As the decreasing number of the ι, the total time consumption is consequent reduction. When the number of dimensions assigned to the high scores part is small, the accuracy is relatively low. According to the specific situations, ι is set appropriately.
As for the selection of feature evaluation techniques, many excellent feature evaluation techniques have been studied [8], [21]. When facing a dataset with different types, different backgrounds or different magnitude, different methods have their drawbacks as well as merits. We evaluate six widely used methods in our experiments. The methods we use are: Chi-square test, Kruskal-Wallis H (KW), Pearson correlation, Spearman correlation, Random forest and minimal Redundancy Maximal Relevance (mRMR). We are committed to finding a feature evaluation technique with the best robustness. After extensive experiments, we find that KW has the most stable effect when facing with different datasets (see Section IX-D).
B. Constructing Specific Training Algorithms using Building Blocks
There are rich variety of ML algorithms. Describing the implementation of building blocks towards each ML training algorithm naturally requires space beyond the page limit. We use LR 6 as an example to illustrate how to construct secure model training algorithms by our building blocks. LR training algorithm is not the most complicated one compared to other ML classifier training algorithms. However, the iterative process of LR involves sigmoid function (Sigmoid β T x i = e β T x i 1+e β T x i ) which makes it difficult to implement in ciphertext case. Most studies claimed they had constructed a secure LR training algorithm by HC which were the secure linear regression training algorithms actually, or they solved the sigmoid function by an approximate equation (cf. Section II-B). To best of our knowledge, it is the first 6 In order to maintain the continuity of our description, LR is also used as the example in the following Algorithm 5 privacy-preserving training of LR Each P's Input: Di = {(xi, yi)} m i=1 , (PK P i −Paillier ,SK P i −Paillier ) and (PK P i −RSA ,SK P i −RSA ). U's Input: (PK U −Paillier , SK U −Paillier ). U's Output: β.
1: U initializes a learning rate α, a fixed number of iterations Cycles and β. 2: while t in Cycles or minimum learning rate is not reached do 3:
Pi sends ( m i=1 XY P i ) to U.
4:
for i = 1 to n do 5: U sends e βX+r + e r P i to P.
Pi sends ( e X P i , m i=1 XY P i ) to U.U updates β by m i=1 X e βX 1+e βX − XY .
14: return β to U.
t+1 j = β t j − α m m i=1 x ij e β T xi 1 + e β T xi − y i(3)
Each building block is designed in a modular way, so carrying out secure LR training algorithm come down to invoking the right module. Suppose there are n data providers P, Algorithm 5 specifies our secure LR training algorithm. In all execution steps of Algorithm 5, when protocols are called, P is the role of Alice, and U is the role of Bob.
Proposition 5: Algorithm 5 is secure in the honest-butcurious model.
C. Combining Differential Privacy Mechanism with Our Building Blocks
In this sub-section we present our solution of combining DP mechanism with building blocks, that is, how to train a ML classifier on the mixed dataset that combine an encrypted dataset and a noise dataset.
The idea is simple but effective: We still let the high scores part has ι features. After requesting data from P, U obtains a dataset mixed with the encrypted dataset and the noise dataset, as depict in Figure 3. In the process of iteratively updating parameters, learning rate in the high scores part is computed by Algorithm 5, and learning rate in the low scores part is computed by Formula 3 just as in normal plaintext case.
Here, we present a simplified example to understand: Suppose we are computing w = X +Y .
X = {[x 1 ] , [x 2 ] , . . . , [x ι ] , x ι+1 , x ι+2 , . . . , x d }, Y = {[y 1 ], [ y 2 ], . . . , [ y ι ] , y ι+1 , y ι+2 , . . . , y d }. Then, w = {[x 1 + y 1 ], [x 2 + y 2 ], . . . , [x ι + y ι ] , x ι+1 + y ι+1 , x ι+2 + y ι+2 , . . . , x d + y d }.
Analogous, when the training algorithm updates parameters, the learning rate of the encrypted dataset is computed by Algorithm 5, and the learning rate of the noise dataset is computed as normal.
Encrypted Data
Noise Data
+ + …… 1(ι+1) 1(ι+2) ... 1 2(ι+1) 2(ι+2) ... 1 … … ... … (ι+1) (ι+2)
...
1 1 2 …… + + …… 1 [ 11 ] [ 12 ] ... [ 1ι ] 1(ι+1) 1(ι+2)
...
1 2 [ 21 ] [ 21 ] ... [ 2ι ] 2(ι+1) 2(ι+2) ... 2 … … … ... … … … ... … m [ 1 ] [ 1 ] ... [ ι ] (ι+1) (ι+2)
...
2 ……
[ 11 ] [x 12 ] ... [x 1ι ] [ 21 ] [x 21 ] ... [x 2ι ] … … ... … [ 1 ] [x m1 ] ... [x mι ]
VIII. SECURITY ANALYSIS
When facing the honest-but-curious adversaries, we follow a commonly used definition -secure two-party computation (cf. Appendix A) and a useful theorem -modular sequential composition (cf. Appendix B). We present our security proofs according to the ideas of these two definitions. For more details, we refer the reader to [16] for secure two-party computation and [7] for modular sequential composition.
Proof 3 (Proof of Proposition 3):
Security proof for Protocol 1, 2, 3 and 4: Protocol 1, 2 and 3 construct secure two-party computation using Paillier's additively homomorphic property. Protocol 4 constructs secure two-party computation using RSA's multiplicative homomorphic property. In these four protocols, Alice does not receive any message whose view only consists in its input, and Alice does not call to any other protocols. The security proofs of these four protocols can be summarized as follows.
The input of Alice is (a, PK A , SK A ), Bob is (b or [b] A , PK A ).
As Alice does not receive any message, Alice's view only consists in its input. Hence the simulator S π A (a, F (a, b)
) = view π A (a, PK A , SK A ). Bob's view is view π B (b or [b] A , PK A , [a] A , Output B )
. a is encrypted by PK A , and the confidentiality of [a] A is equivalent to the cryptosystem, thereby Bob cannot infer the value of [a] A straightforward. The simulator S π B does the following: (i) Generates l random coins, obtains
[c] A = [{c 1 , c 2 , . . . , c l }] A by P K A , where l is the length of a. (ii) Out- puts ([c] A , PK A , b or [b] A , [f (c, b)] A )
. By semantic security of the used cryptosystem:
([c] A , PK A , b or [b] A , [F (c, b)] A ) ≈ ([a] A , PK A , b or [b] A , [F (a, b)] A ).
Security proof for Protocol 5: The input of Alice is (a, PK A , SK A ), Bob is (b, PK A ).
As Alice does not receive any message in this hybrid model, Alice's view only consists in its input. Hence the
simulator S π A (a, F (a, b)) = view π A (a, PK A , SK A ). Bob's view is view π B (b, PK A , ||e a || A , F protocol−4 (||e a || A , ||b|| A ), Output B )
. e a is encrypted by PK A , and the confidentiality of ||e a || A is equivalent to the cryptosystem, So Bob cannot infer its value straightforward. The simulation S π B , on input (b, PK A ), dose the following: (i) Generates l random coins, then obtains c = {c 1 , c 2 , . . . , c l } and ||e c || A by PK A , where l is the length of a. (ii) Outputs c, PK A , SK A , ||e cb || A , F protocol−4 (||e c || A , ||b|| A )). As RSA is semantically secure, the distributions S π B = b, PK A , ||e c || A , ||e cb || A and view π B = b, PK A , ||e a || A , ||e ab || A are computationally indistinguishable.
As Protocol 4 is secure in the honest-but-curious model, we obtain the security of the Protocol 5 using Theorem 2.
Security proof for Protocol 6: The function is F : F ||e ab || A , PK P , SK P , PK R ,
SK R = φ, [[e ab ]] A .
Alice's view is view π A = [[e ab+r ]] A , PK P , SK P , PK R , SK R , output P rotocal−3 ,), and output P rotocal−3 = ||e ab+r || A . S π A runs as follows: (i) Generates l random coins, then obtains c = {c 1 , c 2 , . . . , c l } and [[e c ]] A by PK P , where l is the length of (ab + r). (ii) Outputs (||e c || A , [[e c ]] A , PK P , SK P , PK R , SK R ). (ab+r) and c are taken from the same distribution, independently from any other parameter. Paillier and RSA is seman- ]] A , r, PK P , SK P , PK R , SK R ). The distribution of c and ab are identical and RSA is semantically secure, so the real distribution r, ||e ab || A and the ideal distribution {r, ||e c || A } are statistically indistinguishable.
tically secure. So (||e c || A , [[e c ]] A , PK P , SK P , PK R , SK R ) ≈ ||e ab+r || A , [[e ab+r ]] A , PK P , SK P , PK R , SK R . Bob's view is view π B = ||e ab || A ,
As Protocol 3 is secure in the honest-but-curious model, we obtain the security of the Protocol 6 using Theorem 2. As Protocol 1 is secure in the honest-but-curious model, we obtain the security of the Protocol 7 using Theorem 2.
Proof 4 (Proof of Proposition 5): Since each P i behaves in the same way, we use P i substitute each P's behavior in this security proof.
U's view is view π
U = PK U , SK U , [[e X ]]= β, m i=1 X e βX 1+e βX − XY ,
that is, whether U can guess the sensitive information of each P i from Output U successfully.
Obviously a = m i=1 X e βX 1+e βX − XY is a no-solution equation for the unknown x and the known β. In addition to brute force cracking, there is no other better way to get the value of X. We assume a small dataset has 2-dimensional 100 instances, and the length of each dimension is 32 bits 7 . Then the probability that U guesses success is 1 2 32×d×m = 1 2 6400 . It is a negligible probability of success [20]. P i 's view is view π Pi = PK P i , SK P i , e βX+r + e r Pi , ||e X || Pi , [[e X ]] Pi . P i does not output any message. Hence the simulator S π Pi = view π Pi . P i runs as follows: (i) Generates l random coins and obtains c = {c 1 , c 2 , . . . , c l }, where l is the length of r. (ii) Uniformly picking
m = {m 1 , m 2 , . . . , m d }, where m ∈ M Pi . (iii) Output PK P i , SK P i , e mX+c + e c
Pi . The distribution of (c, m) and (r, β) are identical, so the real distribution (PK P i , SK P i , e βX+r + e r Pi and the ideal distribution (PK P i , SK P i , [[e mX+c + e c ]] Pi are statistically indistinguishable.
As those Protocols used in Algorithm 5 are secure in the honest-but-curious model, we obtain the security of Algorithm 5 using Theorem 2.
Proof 5 (Proof of Proposition 4):
Each P computes feature scores by feature evaluation technologies locally, obtains the noise dataset by Algorithm 3, and interacts with U to process the encrypted dataset by Algorithm 5. U trains a LR classifier with the encrypted datasets by Algorithm 5.
As those Protocols or Algorithms used in Algorithm 4 are secure in the honest-but-curious model, we obtain the security using Theorem 2.
IX. PERFORMANCE EVALUATION
We present the evaluation of Heda in this section. We answer the following questions in our evaluations toward Heda: (i) the performance of our DP components, (ii) the performance of our building blocks, and (iii) the performance overhead of Heda.
A. Preparations 1) Implementations: Our experiments were run using a desktop computer with configuration: single Intel i7 (i7-3770 64bit) processor for total 4 cores running at 3.40GHz and 8 GB RAM. We have implemented the building blocks, DP components, Heda, and mRMR in Java Development Kit 1.8. Feature selection algorithms including Chi-square test, Pearson correlation, Spearman correlation, Random forest and KW have been implemented in scikit-learn 8 .
The operations of Paillier and RSA are carried out in finite fields involving modular arithmetic operations on integers, while classifiers training generally use floating point numbers. In order to encrypted data taking real values, it is necessary to previously perform a format conversion into an integer representation. According to the international standard IEEE 754, binary floating point number D is expressed as D = (−1) s ×M ×2 E , where s is the sign bit, M is a significant number, and E is the exponent bit. We employ it to perform the format conversion toward our implementations. Overlength effective bits lead to inefficient algorithms, while underlength effective bits may low the accuracy. We retained two decimal places empirically.
Plaintext should be guaranteed in the plaintext space of the used cryptosystem. So we must consider the key length to avoid the possibility of overflow. After analyzing all the intermediate results, Paillier's key N is set to 2048-bit and RSA's key N is 2048-bit.
2) Datasets: Four datasets from the UCI ML repository [12] Table III. In order to avoid overfitting or contingent results, we show the average results of cross-validation of 10 runs. In each cross-validation, we randomly take 80% to train the model, and the remainder for testing.
B. Evaluation of Differential Privacy Components
In DP mechanism, we employ IMA to reduce the sensitivity ∆f and give a formula to determining the appropriate privacy budget . We recall that the best cluster size is k = m 2 , i.e. the best cluster size of the four datasets -BCWD, Adult, CAD and Car -are 16, 127, 18 and 29 respectively. Standard DP (cf. Definition 4) is the baseline for comparison, where the for each dataset is obtained by Algorithm 1. The scheme proposed by Soria et al. [28] is employed to serve as a control group named IMDAV, which is the state of the art for reducing the sensitivity in DP mechanisms based on IMA. Sum of Squared Errors (SSE) is used to evaluate the information loss of the dataset, and the percentage of Record Linkages (RL) is used to evaluate the risk of privacy leak.
For a given dataset D and an -DP dataset D with microaggregation of k size, SSE (Formula 4) is defined as the sum of squares of attribute distances between original records in D and their versions in the -DP dataset D .
SSE = Aj ∈D,A j ∈D xij ∈Aj ,x ij ∈A j x ij − x ij 2(4)
We implement our DP components (Algorithm 3), Standard DP and IMDAV on the four datasets with different cluster sizes, then record different SSE values. The results are depicted in Figure 4. It is clear that our DP components have low information loss compare to IMDAV. When k = m 2 , our DP components have lower SSE than Baseline.
Insufficient noise may cause privacy cannot be guaranteed. RL is the percentage of records of the original dataset D that can be correctly matched from the -DP dataset D . R is the set of original records that are at minimum distance from D, and RL is defined as: RL =
x i ∈D Pr x i n , where Pr x i = 0 if x i ∈ R 1 |R| if x i ∈ R .
We record the RL on the four datasets for different cluster sizes and show the results in Figure 5. We can see from the Figure 5 that our RL is roughly the same as IMDAV, though our SSE is much lower than it, and our RL is lower than the Baseline when k = m 2 . The lines in Figure 5(b) and 5(c) are not smooth, the phenomenon of which may be caused by the discrete attributes in Dataset Adult and Dataset CAD.
The smaller SSE means the less loss of information, and the smaller RL means the less risk of privacy leak. In conjunction with Figure 4 and Figure 5, comparing to the Baseline and IMDAV, our DP components in Heda have higher security and less information loss when the cluster size is given the best value k = m 2 .
C. Evaluation of Building Blocks
The way to use our building blocks constructing a secure ML training algorithm (cf. Section VII-B) and the security analysis of our building blocks (cf. Section VIII) have been given. In this subsection, we evaluate building blocks in terms of time consumption, accuracy and the number of interactions (round trips), taking Algorithm 5 as an example. A widely used criterion -accuracy ( #correctly classif ied instances #total instances ) is employed to evaluate the accuracy, and standard LR 9 is implemented as a control group. As the performance results in Table IV(a) and Table IV(b), training a model by Algorithm 5 not only has almost no loss of accuracy, but also has the acceptable time consumption. In experiments, we linearly simulate several P, and the time consumption of P in Table IV(b) is the accumulation of time spent by several P. In actual application scenarios, several P conduct algorithms in parallel, so that the time consumption of P and the total time consumption can be decreased sharply. We just show the raw running time to better illustrate the performance of building blocks. We believe our building blocks to be practical for sensitive applications.
When the datasets become very large, the trained models have almost no loss of accuracy as shown in Table IV(b). Results even show an increase in accuracy when evaluating on dataset Car. The increase is because local optimal values are chosen when initializing model parameters β, and it also shows secure training algorithms constructed by our building blocks would not affect the accuracy. It is worth mentioning that our building blocks construct a secure LR training algorithm without use the Authorization Server and any approximate equation and it is the first solving the sigmoid function in secure LR based on HC.
D. Evaluation of Heda
We want to find a more robust feature evaluation method for serving Heda, i.e., finding a method that is able to divide the dataset into the high scores part and the low scores part more accurately. We implement six widely used methods and evaluate each of them as following steps: (i) Computing the scores of each attribute by the feature evaluation method, (ii) According to the scores, obtaining ordered datasets (cf. Section VII-A), (iii) Obtaining a new sub-dataset by removing a feature with the lowest scores, then conducting an evaluation on this new sub-dataset, (iv) Repeating step (iii) until there is only one dimensional data left in the dataset. We suppose that step (iii) is able to get an ordered sub-dataset that gets the smallest decline in accuracy compared to the previous sub-dataset. We use the standard LR evaluating each sub-dataset obtained from step (iii) and record the results. Because of the layout restrictions, we only show the results of the dataset Adult and Car. Adult is the dataset with a large amount of data. Car represents the datasets that has only numerical attributes. Figure 6 visualizes the results, where accuracy is employed to measure the quality of datasets.
With respect to the six feature selection algorithms in Figure 6, we find that KW (Green lines) has a more stable performance no matter facing with a all numeric attribute dataset (Car as shown in Figure 6(b)) or a large amount of data dataset with numerical attributes and discrete attributes (Adult as shown in Figure 6(a)). More accurate scores are able to be obtained by KW, so we use KW as our feature evaluation algorithm in the following.
Recalling the steps of Heda, after obtaining the mixed dataset, it computes model parameters on the mixed dataset. Assuming the high scores part has ι features. We record and observe the changes in terms of accuracy and time consumption of Algorithm 4 when ι changes from 1 to d. Results are depicted in Figure 7, where "U Time in HC" and "U Time in DP" represent the training time spent by U on encrypted datasets and noise datasets respectively. After the dataset is published as the noise dataset by DP mechanism, several P are no longer involved in the training process, so the "P Time" is just the time consumption for several P participating in the encrypted dataset training.
1) Efficiency:
As can be seen from Figure 7, the total time consumption linearly increases as the ι grows, which is in line with our preset Formula 2. When ι = d, Heda has the lowest efficiency. But when ι changes to 1 (i.e., the number of the encrypted attributes are 1), we obtain the highest efficiency, and the total time consumption of Heda spent on each dataset can be reduced by more than 70% compare to when ι = d.
When ι becomes smaller, the efficiency of Heda becomes higher. The application needs to quickly train a model on sensitive datasets could set ι to one to obtain the model parameters as quickly as possible. Results show that Heda is able to train a model within one hour in the face with different scales encrypted datasets. We linearly simulate several P, and several P could conduct Algorithms in parallel in practical application, so the time consumption of P and the total time consumption can be decreased sharply again. The time consumption on noise datasets is less than 1 second, because noise datasets is in the plaintext case, the speed of computing in plaintext case is much faster than computing on the encrypted data.
2) Accuracy: We observe from Figure 7 that training speed become faster with smaller ι, but the accuracy is reduced, which indicates that the noise dataset does affect the dataset quality to a certain extent. Nevertheless, results show that even when the minimum value of ι is taken, the trained classifier accuracy does not be reduced much (not more than 8%). According to the requirements of specific applications, one can adjust ι appropriately to obtain a tradeoff between accuracy and time consumption.
Feature evaluation techniques help ML training algorithms remove redundant or noisy attributes, and adding appropriate noise is able to smooth out discrete attributes which has been quantized. So the accuracy is a slightly increased, when ι = 5 in the mixed dataset CAD, as depicted in Figure 7(c).
3) Some discussions: Heda obtains the tradeoffs between efficiency and accuracy by jointly applying HC and DP in an individual scheme, and enables flexible switch among different tradeoffs by parameter tuning. A small ι is able to achieve high efficiency and a slight loss in accuracy (In our experiments, the loss is not worse than 8%.). Different application scenarios pay different attention to efficiency and accuracy. According to the requirements of specific applications, developers can obtain a balance between accuracy and efficiency by adjusting ι appropriately.
X. CONCLUSION
In this paper, we proposed a novel efficient privacypreserving ML classifier training algorithm named Heda. By jointly applying HC and DP, Heda obtained the balance between efficiency and accuracy, enabled flexible switch among different tradeoffs by parameter tuning. In order to make Heda more efficient and accurate, we developed a library of building blocks base on HC, gave a formula for determining the appropriate privacy budget, and reduced the sensitivity of the query function by IMA in DP mechanism. We demonstrated the efficiency of Heda and our algorithms. In the future work, we plan to explore a generalized framework which enables constructing a wide range of privacy-preserving ML classifier training algorithms.
Theorem 1 :
1(Parallel Composition) [25]. Suppose we have a set of privacy mechanisms M = {M 1 , M 2 , . . . . . . , M m }. If each M i provides a i -DP guaranteed on a disjointed subset of the entire dataset, M will provide (max { 1 , 2 , . . . . . . , m })-DP.
Proposition 2 (
2∆f in IMA): f j (D) is a query function with -DP mechanism returning the noised values corresponding to the j-th attribute of D. After obtaining D IM A by Algorithm 2, the sensitivity of D IM A with cluster size k is∆f j (D) = ∆fj k × m 2k , where ∆f j = max (A j ).Proof 2 (Proof of Proposition 2): If M is an IMA algorithm, for every pair of datasets D and D differing in a single record, there is a bijection between the set of clusters {C 1 , . . . , C n } and C 1 , ..., C n such that each pair of corresponding clusters differs at most in a single record. So if the centroid is computed as the mean of the records in the same cluster, then the maximum change in any centroid is, at most, ∆fj k . The modification of single record may lead to multiple modifications of the centroid of clusters, and there are m k 5 different clusters in D.
Fig. 2 .
2Clusters in IMA Algorithm 3 IMA -DP Mechanism Input : D = {(xi, yi)} m i=1 ,k is the cluster size, m ≥ 2k. Output : An IMA -DP dataset DIMA− . 1: Generating the appropriate on dataset D by Algorithm 1. 2: Obtaining an IMA dataset DIMA from D by Algorithm 2. 3: Obtaining noise by using and ∆fj (cf. the definition of Laplace mechanism 4). 4: Adding x i = xi + noise to DIMA− . 5: return DIMA− .
Proposition 3 (
3The security of building blocks): Protocol 1 to 7 is secure in the honest-but-curious model.
a secure LR training algorithm by HC. Our HCbased secure LR training algorithm only needs 3 interactions (i.e. interactions between U and P) throughout each iteration process.LR is a binary classifier and try to learn a pair of parametersw and b, where w = (w 1 , w 2 , . . . , w d ) to satisfy f {x i } = w T x i + b and f {x i } ∼ = y i . LRuses Sigmoid Function to associate the true label y i with the prediction label f {x i }. Let β = (w, b), X = (x, 1), the iteration formula of LR is shown in Formula (3). The steps of LR training algorithm are as follows: (i) Initializing learning rate α, a fixed number of iterations and model parameters β. (ii) Updating β by Formula (3). (iii) If the maximum number of iterations or minimum learning rate is reached, output β; otherwise go to step 2.
Fig. 3 .
3A Mixed Dataset: the Encrypted Dataset Combine with the Noise Dataset.
||e ab+r || A , r, [[e ab+r ]] A , S π B runs as follows: (i) Generates l random coins, obtains c = {c 1 , c 2 , . . . , c l }, where l is the length of ab. (ii) obtains [e c ] A and [e c+r ] A by PK P . (ii) Outputs (||e c || A , ||e c+r || A , [[e c+r
Security proof for Protocol 7: The function is F : F ([[b]] A , PK A , SK A , PK B , SK B ) = (φ, [[b]] B ).Alice's view is view π A = ([[b + r]] A , PK A , SK A , PK B ). S π A runs as follows: (i) Generates l random coins and obtains[[c]] A = [[{c 1 , c 2 , . . . , c l }]] A by PK A , where l is the length of (m + r). (ii) Outputs ([[c]] B , [[c]] A , PK A , SK A , PK B ). (b + r)and c are taken from the same distribution, independently from any other parameter, and Paillier is semantically secure, so([[c]] B , [[c]] A , PK P , SK P , PK R , SK R ) ≈ ([[b + r]] B , [[b + r]] A , PK P , SK P , PK R , SK R ). Bob's view is view π B = ([[b]] A , [[b + r]] A , r, PK B , SK B ,PKA ), S π B runs as follows: (i) Generates l random coins and obtains [[c]] A = [[{c 1 , c 2 , . . . , c l }]] A by PK A , where l is the length of m. (ii) Outputs ([[c]] B , [[c + r]]] A , r, PK B , SK B , PK A ). The distribution of c and m are identical, and Paillier is semantically secure, so the real distribution {[[m]] B , [[m + r ]] A , r; [[m + r]] B } and the ideal distribution {[[c]] B , [[c + r]] A , r; [[c + r]] B } are statistically indistinguishable.
originated from several different application domains are used in our evaluation: (i) Breast Cancer Wisconsin (Diagnostic) Dataset (BCWD), (ii) Adult Dataset (Adult), (iii) Credit Approval Dataset (CAD), (iv) Car Evaluation Dataset (Car). The statistics are shown in
Fig. 5 .
5Differential Privacy Components Performance: Record Linkages.
Fig. 7 .
7Heda Performance. The bars represent the time consumption of Heda, and the line represents the accuracy.
TABLE I .
INOTATIONS
Notations Explanation
Notations Explanation
R
Set of real numbers
R d
d-dimension R
D
Dataset
|D|
The size of D
m
Size of dataset
D
Neighbour dataset
X
The record set in D
Y
The label set in D
xi
i-th Record in dataset
yi
Class label
o, σ
Functional operation
d
Dataset dimension
M
Plaintext space
β, b
Parameters of models
M
Mechanism
C
Ciphertext space
k
The cluster size
N
n-bit Primes
f
Query
Privacy budget
γ
Noise
∆f
Sensitivity
[[m]] Ciphertext under Pail-
lier
||m|| Ciphertext under RSA
ι
The number of en-
crypted features in D
S
The scores of features
[m]
The encryption of m
under a certain cryp-
tosystems
k is the cluster size, m ≥ 2k. Output : A IMA dataset DIMA that can perform DP. 1: Set i := 0 2: while |D| ≥ 2k do3:
Table IIshows the different conditions for building blocks. For all conditions, both parties Alice and Bob cannot obtain other useful information except for the legal information, the input and output of other parties are confidential.1) Secure addition and secure subtraction: Relying on Paillier's additive homomorphic property, it is straightforwardProtocol 1 Secure Addition Protocol Input Alice: a = {a1, a2, . . . . . . , a d }, (PK Alice , SK Alice ). Input Bob: PK Alice , [[b]] Alice = [[{b1, b2, . . . . . . , b d }]] Alice or b={b1, b2, . . . . . . , b d }. Output Bob: [[f (a, b)]] Alice = [[a + b]] Alice . 1: Alice sends [[a]] Alice to Bob. 2: for i to d do Bob computes f (a, b) i Alice = [[bi]] Alice × [[ai]] Alice . 4: return [[f (a, b)]] Alice to Bob.3:
Protocol 3) that satisfies Condition 1 easy (i.e., [[a × b]] = [[a]] b ). However, when Bob only has ciphertext [[b]] Alice who is unable to perform [[a]]Alice . Paillier fails to construct a secure dot product protocol that satisfies Condition 2. Because the length of Paillier's ciphertext is 1024 bits or longer usually (We will discuss the key length setting in detail in Section IX.), so the computational complexity of[[a]] [[b]] is awfully large. Therefor, when faced with Condition 2, we use RSA's multiplicative homomorphism property to construct secure multiplication protocol (Protocol 4).b
TABLE II .
IITHE CONDITIONS OF BUILDING BLOCKSConditions
Input
Output
Protocols
Alice
Bob
Alice
Bob
Input Alice: a = {a1, a2, . . . . . . , a d }, (PK Alice , SK Alice ).Input Bob: b = {b1, b2, . . . . . . , b d } and PK Alice .Input Alice: (PK Alice−Paillier , SK Alice−Paillier ) and (PK Alice−RSA , SK Alice−RSA ).)]] B
7
Protocol 5 Secure Power Function Protocol
Output Bob: e ab
Alice .
1: Alice sends e a
Alice = {e a 1 , e a 2 , . . . e a d } Alice to Bob.
2: Bob Initializes W .
3: In Bob:
4: for i to d do
5:
Letting wi = e x i
Alice .
6:
for t to bi − 1 do
7:
wi = wi × e a i
Alice by protocol 4.
8:
W = W × wi by protocol 4.
9: return W Alice = e ab
Alice to Bob.
Protocol 6 Converting Ciphertext: e ab
Alice to e ab
Alice
Input Bob: e ab
Alice .
Output Bob: e ab
Alice .
1: Bob uniformly picks r and computes e ab+r
Alice by Protocol
5.
2: Bob sends e ab+r
Alice to Alice.
3: Alice decrypts e ab+r
Alice , obtains and sends e ab+r
Alice
to Bob.
4: Bob computes e ab
Alice =
e ab+r
Alice × (e r ) −1 by
protocol 3.
5: return e ab
Alice to Bob.
Alice to [[b]] Bob satisfying Condition 4.The first (Protocol 6) switches e ab
Alice to
e ab
Alice satisfying Condition 3, and the other (Protocol 7)
switches [[b]]
The security Protocol 7 Converting Ciphertext: [[b]] Alice to [[b]] Bob Input Alice: (PK Alice , SK Alice ) and PK Bob . Input Bob: [[b]] Alice . Output Bob: [[b]] Bob . 1: Bob uniformly picks r and computes [[b + r]] Alice by Protocol 1. 2: Bob sends [[b + r]] Alice to Alice. 3: Alice decrypts [[b + r]] Alice , obtains and sends [[b + r]] Bob to Bob. 4: return [[b]] Bob to Bob.
Pi , ||e X || Pi ,[[e Xβ ]] Pi , [[
X
e βX+r +e r ]]
Pi , [[
m
i=1
X e βX
1+e βX ]]
Pi
, Output U . As
intermediate results are encrypted by the public key of each P i ,
and Paillier and RSA is semantically secure, thus the sensitive
information of each P i is computationally indistinguishable
in intermediate results. What we need to discuss is the
confidentiality of Output U
TABLE III .
IIISTATISTICS OF DATASETS
Datasets
Instances
number
Attributes
number
Discrete
attributes
Numerical
attributes
BCWD
699
9
0
9
Adult
32561
14
8
6
CAD
690
15
9
6
Car
1728
6
0
6
Table IV(a) gives the running time of each building block (Protocol 1 to Protocol 7) with encrypted four datasets on Algorithm 5, where Protocol 6 named Exchange 1, and Protocol 7 named Exchange 2. Table IV(b) gives the total time consumption, accuracy and communication overhead.
TABLE IV .
IVBUILDING BLOCKS PERFORMANCE (a) The running time of each building block Datasets Add Subtraction Dot Product Multiplication Power Function Exchange 1 Exchange 2 The total time, accuracy and communication overhead Datasets Standard LR Secure LR Total Time U Time P Time InteractionsBCWD
1462ms
268ms
372ms
2400ms
2452ms
61510ms
18ms
Adult
17026ms
5710ms
2759ms
47012ms
52291ms
1112344ms
17ms
CAD
8643ms
546ms
1114ms
18882ms
20906ms
411493ms
16ms
Car
2752ms
374ms
890ms
5800ms
6515ms
295934ms
22ms
(b) BCWD
96.595%
95.422%
2428s
32s
2396s
189
Adult
82.590%
81.578%
41145s
459s
40685s
1500
CAD
85.607%
84.463%
17346s
252s
17094s
1173
Car
72.32%
72.68%
10458s
122s
10335s
1122
(a) Adult
(b) Car
Fig. 6. Feature Evaluation Performance
In this paper, ML classifers and ML models are used interchangeably. arXiv:1812.02292v1 [cs.CR] 6 Dec 2018
In this paper, we only consider partially HC due to the computational inefficiency of fully HC.3 Without loss of generality, when facing the same training task, we assume that all the dataset has been locally preprocessed and represented with the same feature vectors[17],[32].
denotes a ceiling functions.
Typically, single-precision floating-point occupies 4 bytes (32-bit) memory space.
http://scikit-learn.org
A LR model that are trained and tested non-privately using scikit-learn (http://scikit-learn.org).
≈ denotes computational indistinguishability against probabilistic polynomial time adversaries with negligible advantage in the security parameter λ.
APPENDIXA. Secure two-party computationFor all our two-party protocols, to ensure security, we have to show that whatever Alice (Bob) can compute from its interactions with Bob (Alice) can be computed from its input and output, which leads to a commonly used definition secure two-party computation (e.g.,[6],[11],[16]).We follow the notations of Bost et al.[6]: Let F = (F A , F B ) be a (probabilistic) polynomial function and π a protocol computing F ; Alice and Bob want to compute F (a, b) where a is Alice's input and b is Bob's input, using π and with the security parameter λ; The view of party Alice during the execution of π is the tuple view π Alice (λ, a, b) = (λ; a; m 1 , m 2 , ..., m n ) where m 1 , m 2 , ..., m n are the messages received by Alice. We define the view of Bob similarly. Let output π Alice (a, b) and output π Bob (a, b) denote Alice's and Bob's outputs respectively. The global output as output π (a, b) = (output π Alice (a, b), output π Bob (a, b)). Definition 6: (Secure Two-Party Computation)[6],[16]. A protocol π privately computes f with statistical security if for all possible inputs (a, b) and simulators S Alice and S Bob hold the following properties10:B. Modular sequential compositionFor our protocols and algorithms are constructed in a modular way, we use the following useful sequential modular composition (Theorem 2)[7]: The Parties run a protocol π and use calls to an ideal functionality F in π (e.g. A and B compute F privately by sending their inputs to a trusted third party and receiving the results); If we can show that π respects privacy in the honest-but-curious model and if we have a protocol ρ that privately computes F in the same model, then we can replace the ideal calls for F by the execution of ρ in π; the new protocol, denoted π ρ is then secure in the honest-but-curious model[6],[7]. We assume an incorruptible trusted party T for evaluating functionalities (F 1 , F 2 , . . . , F n ) − hybrid model.
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All ideals calls of π to the trusted party for F i is replaced by a real execution of ρ i. T Zhu, G Li, W Zhou, P S Yu, ; . , F n ) − hybrid model. 29Parties not communicate until receiving T's output (i.e. sequential composition). Modular Sequential Composition. restated asT. Zhu, G. Li, W. Zhou, and P. S. Yu, "Differentially private data publishing and analysis: A survey," IEEE Transactions on Knowledge and Data Engineering, vol. 29, no. 8, pp. 1619-1638, Aug 2017. Parties not communicate until receiving T's output (i.e. se- quential composition). Let π be a two-party protocol in the (F 1 , F 2 , . . . , F n ) − hybrid model, and ρ 1 , ρ 2 , . . . , ρ n be real protocols (i.e. protocols in the semi-honest model) computing F 1 , F 2 , . . . , F n . All ideals calls of π to the trusted party for F i is replaced by a real execution of ρ i . Theorem 2: (Modular Sequential Composition) restated as
F n in the presence of semihonest adversaries. Let G be a two-party probabilistic polynomial time functionality and π a protocol that securely computes G in the (F 1 , F 2 , . . . , F n ) − hybrid model in the presence of semi-honest adversaries. . Let F 1 , F 2, F n be two-party probabilistic polynomial time functionalities and ρ 1. ρ n protocols that compute respectively F 1. Then π ρ1,ρ2,...,ρn securely computes G in the presence of semi-honest adversariesLet F 1 , F 2 , . . . , F n be two-party probabilistic poly- nomial time functionalities and ρ 1 , ρ 2 , . . . , ρ n protocols that compute respectively F 1 , F 2 , . . . , F n in the presence of semi- honest adversaries. Let G be a two-party probabilistic polyno- mial time functionality and π a protocol that securely computes G in the (F 1 , F 2 , . . . , F n ) − hybrid model in the presence of semi-honest adversaries. Then π ρ1,ρ2,...,ρn securely computes G in the presence of semi-honest adversaries.
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[] |
[
"Simple Codes and Sparse Recovery with Fast Decoding",
"Simple Codes and Sparse Recovery with Fast Decoding"
] |
[
"Mahdi Cheraghchi ",
"João Ribeiro "
] |
[] |
[] |
Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. In this work, we study a very simple construction of binary linear codes that correct a given number of errors K. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the code, and nearly linear in the number of errors K. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in non-adaptive group testing, and construct simple explicit measurement schemes with O(K 2 log 2 N ) tests and O(K 3 log 2 N ) recovery time for identifying up to K defectives in a population of size N .
|
10.1109/isit.2019.8849702
|
[
"https://export.arxiv.org/pdf/1901.02852v3.pdf"
] | 57,759,331 |
1901.02852
|
1884426c1db199390123568c01157e845d0ec4a2
|
Simple Codes and Sparse Recovery with Fast Decoding
10 Dec 2022
Mahdi Cheraghchi
João Ribeiro
Simple Codes and Sparse Recovery with Fast Decoding
10 Dec 2022
Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. In this work, we study a very simple construction of binary linear codes that correct a given number of errors K. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the code, and nearly linear in the number of errors K. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in non-adaptive group testing, and construct simple explicit measurement schemes with O(K 2 log 2 N ) tests and O(K 3 log 2 N ) recovery time for identifying up to K defectives in a population of size N .
Introduction
The problem of constructing low-redundancy codes with practical decoding algorithms that handle a prescribed number of adversarial errors has been extensively studied in coding theory. We distinguish between two standard decoding settings for linear codes: Syndrome decoding, where one has access to the syndrome of the corrupted codeword, and full decoding, where one has access to the corrupted codeword itself. In both cases, the goal is to return the error pattern.
The syndrome holds extra pre-computed information about the corrupted codeword. As a result, we expect to be able to perform syndrome decoding much faster than full decoding. In fact, while full decoding has complexity at least linear in the block length of the code, syndrome decoding can be potentially accomplished in time sublinear in the block length. Syndrome decoding is important for various reasons: In many cases, the most efficient way we have of performing full decoding for a given linear code is to first compute the syndrome from the corrupted codeword, and then run a syndrome decoding algorithm. Furthermore, as we shall see, syndrome decoding is connected to other widely studied recovery problems.
Two examples of families of high-rate linear codes with good decoding guarantees are the classical BCH codes, which are widely used in practice, and expander codes, introduced by Sipser and Spielman [1]. The properties of BCH codes derive from the theory of polynomials over large finite fields. On the other hand, the guarantees behind expander codes follow from the combinatorial properties of the underlying expander graphs. Due to their combinatorial nature, expander codes have simple decoders, while decoding BCH codes requires algorithms which perform arithmetic over large fields. However, while BCH codes support sublinear syndrome decoding [2,3], no such sublinear syndrome decoders are known for expander codes.
Syndrome decoding of linear codes can be interpreted as sparse recovery over a finite field. In exact for-all sparse recovery, we aim to recover all sparse vectors x from W x, where W is a measurement matrix (which may be sampled with high probability from some distribution). The goal is to minimize recovery time and number of measurements (i.e., rows of W ) with respect to the sparsity and length of the vectors. We say a for-all sparse recovery scheme is approximate if it allows recovery (within some error) of the best sparse approximation of arbitrary vectors. There has been significant interest in developing combinatorial algorithms for sparse recovery. Unlike their geometric counterparts, such procedures have the advantage of supporting sublinear time recovery. Furthermore, while sparse recovery is normally studied over the reals, such algorithms can usually be modified to perform sparse recovery over any field.
In this work, we study a simple combinatorial construction of high-rate binary linear codes that support nearly optimal syndrome decoding. While we present our decoding algorithms over F 2 , our syndrome decoder can be adapted to work over any field, improving upon previous combinatorial algorithms for exact for-all sparse recovery with the same number of measurements. This adaptation to arbitrary fields is discussed in more detail at the end of Section 4.1.1. We remark that the parity-check matrix of our code remains 0-1-valued even when we work over fields other than GF (2).
A different sparse recovery problem that has been extensively studied is that of Non-Adaptive Group Testing (NAGT). In this setting, our goal is to identify all defectives within a given population. To this end, we may perform tests by pooling items of our choice and asking whether there is a defective item in the pool. In NAGT, tests are fixed a priori, and so cannot depend on outcomes of previous tests. A main problem in this area consists in finding NAGT schemes supporting sublinear recovery time with few tests in the zero-error regime, where it is required that the scheme always succeeds in recovering the set of defectives. In the second part of this work, we present a zero-error NAGT scheme with few tests and a competitive sublinear time recovery algorithm.
Related Work
Sublinear time decoding of BCH codes was studied by Dodis, Reyzin, and Smith [2,3], who gave a poly(K log N ) time syndrome decoder, where N is the block length and K is the maximum number of errors. More precisely, according to [3, Proof of Lemma 1] their decoder asymptotically requires Θ(K log 2 K · log log K + K 2 log N ) multiplications over the extension field of order N + 1. In turn, we can perform multiplications over this field in time Θ(log N · log log N ) under a believable assumption [4]. Combining both results yields asymptotic syndrome decoding complexity
Θ((K log 2 K · log log K + K 2 log N ) · log N · log log N ).
In contrast, the best syndrome decoders for expander codes run in time O(DN ), where D is the left degree of the expander [5]. Full decoding of expander codes takes time O(N ) in the regime where K = Θ(N ) [1], but it takes time O(N log N ) when K is small and we want the rate of the expander code to be large [5].
The work on combinatorial for-all sparse recovery algorithms was initiated by Cormode and Muthukrishnan [6], and several others soon followed [7,8,9] with improved recovery time and number of measurements. More recently, sublinear time combinatorial algorithms for approximate for-all sparse recovery with optimal number of measurements [10] or very efficient recovery [11] were given (both with strong approximation guarantees). We compare the results obtained by these works in the context of for-all sparse recovery with the result we obtain in this work in Section 1.2.
The first zero-error NAGT schemes supporting sublinear recovery time with few tests were given independently by Cheraghchi [12] and Indyk, Ngo, and Rudra [13]. In [13], the authors present an NAGT scheme which requires T = O(K 2 log N ) tests (which is order-optimal) and supports recovery in time poly(K) · T log 2 T + O(T 2 ), where N is the population size and K is the maximum number of defectives.
However, their scheme is only explicit when K = O log N log log N . Cheraghchi [12] gives explicit schemes which require a small number of tests and handle a constant fraction of test errors. However, his schemes may output some false positives. While this can be remedied, the resulting recovery time will be worse. Later, Ngo, Porat, and Rudra [14] also obtained explicit sublinear time NAGT schemes with near-optimal number of tests that are robust against test errors. In particular, they obtain schemes requiring T = O(K 2 log N ) tests with recovery time poly(T ). Subsequently to the publication of a shortened version of the present work [15], Cheraghchi and Nakos [16] constructed explicit NAGT schemes requiring T = O(K 2 log N ) tests and recovery time nearly linear in T .
Contributions and Techniques
Our binary linear codes are a bitmasked version of expander codes. Roughly speaking, bitmasking a binary M ×N matrix W consists in replacing each entry in W by the log N -bit binary expansion of its corresponding column index, multiplied by that entry of W . This gives rise to a new bitmasked matrix of dimensions M log N × N . The bitmasking technique has already been used in [6,7,8,9,11] in several different ways to obtain sublinear time recovery algorithms for approximate sparse recovery. We provide a detailed explanation of this technique and its useful properties in Section 3.
The parity-check matrices of our codes are obtained by following the same ideas as [9,11]: We take the parity-check matrix of an expander code, bitmask it, and stack the two matrices. Note that these codes have a blowup of log N , where N is the block length, in the redundancy when compared to expander codes. However, we show that these codes support randomized and deterministic syndrome decoding in time O(K log K ·log N ) under a random expander, where K is the number of errors. We remark that this is within an O(log K) factor of the optimal recovery time for small K. In particular, this also leads to randomized and deterministic full decoders running in time O(log K · N log N ).
Our syndrome decoders can be made to work over any field with almost no modification. As a result, we obtain a recovery algorithm for exact for-all sparse recovery over any field with nearly optimal recovery time O(K log K · log N ) from O(K log 2 N ) measurements. This improves upon the recovery time of previous schemes using the same number of measurements in the exact for-all sparse recovery setting [9,11]. Although sublinear time recovery is possible with fewer measurements (the optimal number of measurements is O(K log(N/K))), the dependency on N in that case is generally worse than what we obtain, which is optimal. A detailed comparison between our work and previous results in the for-all sparse recovery setting can be found in Table 1.
Our randomized decoders have several advantages over their deterministic counterparts that make them more practical. First, the hidden constants in the runtime are smaller. Second, the runtime is independent of the degree of the underlying expander. As a result, we can instantiate our codes under explicit expanders with sub-optimal degree without affecting the decoding complexity. Third, the failure probability of the algorithm has a negligible effect on its runtime for large block lengths. Therefore, it can be set to an arbitrarily small constant of choice with limited effect on the runtime. In particular, our randomized full decoder is faster than the expander codes decoder even under a random expander if K is small.
Recovery time
Approximate? (Y/N)
[6] K 2 log 2 N N [7] K log 2 K · log 2 N Y [8] K 2 · polylog(N ) Y [9]
K log K · log 2 N N [11] K log K · log 2 N Y This work (see Remark 1) K log K · log N N < K log 2 N measurements [10] poly(K log N ) Y BCH codes [3,4] K 2 log 2 N · log log N N Table 1: Summary of best known previous results and the result obtained in this work on sublinear recovery in the for-all sparse recovery setting. Here, K denotes the sparsity and N the vector length. We omit the O(·) notation in recovery times for simplicity, and in the third column we distinguish between schemes that work for approximate sparse recovery (i.e., arbitrary vectors), versus those that work only in the exact for-all setting. Since it is not relevant for us, we do not distinguish between the approximation guarantees obtained in each work for approximate sparse recovery. All works except the last two rows require at least O(K log 2 N ) measurements, which is the number of measurements our scheme requires. While sublinear decoding is possible in those cases, the dependency on N is generally worse than what we obtain. For a more detailed description of such schemes, see [10, Table 1] and [7, Table 1]. As mentioned before, our measurement matrices remain 0-1-valued even when we work over fields other than GF(2).
Remark 1.
We assume that reading an integer from memory takes time O(1). If instead we assume that reading an L-bit integer takes time O(L), then we incur an extra log K factor in our deterministic recovery time, for a total runtime of O(K log 2 K · log N ).
In the second part of our work, we present a simple, explicit zero-error NAGT scheme with recovery time O(K 3 log 2 N ) from O(K 2 log 2 N ) tests, where N is the population size and K is the maximum number of defectives. Such a scheme is obtained by bitmasking a disjunct matrix. At a high-level, the recovery algorithm works as follows: First, we use the bitmask to obtain a small superset of the set of defectives. Then, we simply apply the naive recovery algorithm for general disjunct matrices to this superset to remove all false positives. The recovery time of this scheme is better than of those presented in Section 1.1, albeit we are an O(log N ) factor away from the optimal number of tests. Moreover, unlike our scheme, those schemes make use of algebraic list-decodable codes and hence require sophisticated recovery algorithms with large constants. Finally, we note that the bitmasking technique has been used in a different way to obtain efficient NAGT schemes which recover a large fraction of defectives, or even all defectives, with high probability [17].
Organization
In Section 2, we introduce several concepts and results in coding and group testing that will be relevant for our work in later sections. Then, in Section 3 we present the bitmasking technique and its original application in sparse recovery. We present our code construction and the decoding algorithms in Section 4. Finally, our results on non-adaptive group testing can be found in Section 5.
Preliminaries
Notation
We denote the set {0, . . . , N − 1} by [N ]. Given a vector x, we denote its support {i :
x i = 0} by supp(x).
We say a vector is K-sparse if |supp(x)| ≤ K. We index vectors and matrix rows/columns starting at 0. Sets are denoted by calligraphic letters like S and X . In general, we denote the base-2 logarithm by log. Given a graph G and a set of vertices S, we denote its neighborhood in G by Γ(S). For a matrix W , we denote its i-th row by W i· and its j-th column by W ·j .
Unbalanced Bipartite Expanders
In this section, we introduce unbalanced bipartite expander graphs. We will need such graphs to define our code in Section 4.
Definition 2 (Bipartite Expander). A bipartite graph G = (L, R, E) is said to be a (D, K, ǫ)-bipartite expander if every vertex u ∈ L has degree D (i.e., G is left D-regular) and for every S ⊆ L satisfying
|S| ≤ K we have |Γ(S)| ≥ (1 − ǫ)D|S|.
Such a graph is said to be layered if we can partition R into disjoint subsets R 1 , . . . , R D with |R i | = |R|/D for all i such that every u ∈ L has degree 1 in the induced subgraph G i = (L, R i , E). We call such i ∈ [D] seeds and denote the neighborhood of S in G i by Γ i (S).
The graph G can be defined by the function C :
L × [D] → R which maps (u, i) ∈ L × [D]
to the neighbor of u in R i . For convenience, we also define C i = C(·, i), which defines the subgraph G i .
Informally, we say that a bipartite expander graph is unbalanced if |R| is much smaller than |L|. The next lemma follows immediately from Markov's inequality.
Lemma 3. Fix some c > 1 and a set S such that |S| ≤ K, and let G be a (D, K, ǫ)-layered bipartite expander. Then, for at least
a (1 − 1/c)-fraction of seeds i ∈ [D], it holds that |Γ i (S)| ≥ (1 − cǫ)|S|.
We will also need the following lemma that bounds the number of right vertices with a single neighbor in a given subset of left vertices. Proof. Suppose that there are fewer than (1 − 2δ)|S| vertices in Γ i (S) with only one neighbor in S. Since G i has left-degree 1, this means that there are more than 2δ|S| vertices in S which are adjacent to right vertices with degree at least 2. Therefore, we can upper bound |Γ i (S)| as
|Γ i (S)| < (1 − 2δ)|S| + 1 2 · 2δ|S| = (1 − δ)|cS|,
which contradicts our assumption.
The next theorem states we can sample nearly-optimal layered unbalanced bipartite expanders with high probability.
Coding Theory
In this section, we define some basic concepts from coding theory that we use throughout our paper. We point the reader to [19] for a much more complete treatment of the topic.
Given an alphabet Σ, a code over Σ of length N is simply a subset of Σ N . We will be focusing on the case where codes are binary, which corresponds to the case Σ = {0, 1}. For a code C ⊆ Σ N , we call N the block length of C. If |Σ| = q, the rate of C is defined as
log q |C| N .
We will study families of codes indexed by the block length N . We do not make this dependency explicit, but it is always clear from context. If Σ is a field, we say C is a linear code if c 1 + c 2 ∈ C whenever c 1 , c 2 ∈ C. In other words, C is a linear code if it is a vector space over Σ. To each linear code C we can associate a unique parity-check matrix H such that C = ker H. Given such a parity-check matrix H and a vector x ∈ Σ N , we call Hx the syndrome of x. Clearly, we have x ∈ C if and only if its syndrome is zero.
Group Testing
As mentioned in Section 1, in group testing we are faced with a set of N items, some of which are defective. Our goal is to correctly identify all defective items in the set. To this end, we are allowed to test subsets, or pools, of items. The result of such a test is 1 if there is a defective item in the pool, and 0 otherwise. Ideally, we would like to use as few tests as possible, and have efficient algorithms for recovering the set of defective items from the test results.
In Non-Adaptive Group Testing (NAGT), all tests are fixed a priori, and so cannot depend on the outcome of previous tests. While one can recover the defective items with fewer tests using adaptive strategies, practical constraints preclude their use and make non-adaptive group testing relevant for most applications.
It is useful to picture the set of N items as a binary vector x ∈ {0, 1} N , where the 1's stand for defective items. Then, the T tests to be performed can be represented by a T × N test matrix W satisfying
W ij = 1, if item j is in test i 0, else.
The outcome of the T tests, which we denote by W ⊙ x, corresponds to the bit-wise union of all columns corresponding to defective items. In other words, if S denotes the set of defective items, we have
W ⊙ x = j∈S W ·j ,
where the bit-wise union of two N -bit vectors, x ∨ y, is an N -bit vector satisfying
(x ∨ y) i = 1, if x i = 1 or y i = 1 0, else.
We say an NAGT scheme is zero-error if we can always correctly recover the set of defectives. Zero-error NAGT schemes are equivalent to disjunct matrices.
Definition 6 (Disjunct matrix). A matrix W is said to be d-disjunct if the bit-wise union of any up to d columns of W does not contain any other column of W .
The term contains used in Definition 6 is to be interpreted as follows: A vector x is contained in a vector y if y i = 1 whenever x i = 1, for all i. If we know there are at most K defective items, taking the rows of a K-disjunct T × N matrix as the tests to be performed leads to a NAGT algorithm with T tests and a simple O(T N ) recovery algorithm: An item is not defective if and only if it is part of some test with a negative outcome, so one can just check whether each item participates in a negative test.
As a result, much effort has been directed at obtaining better randomized and explicit 1 constructions of K-disjunct matrices, with as few tests as possible with respect to number of defectives K and population size N . The current best explicit construction of a K-disjunct matrix due to Porat and Rothschild [20] requires O(K 2 log N ) rows (i.e., tests), while a probabilistic argument shows that it is possible to sample K-disjunct matrices with O(K 2 log(N/K)) rows with high probability. We remark that both these results are optimal up to a log K factor [21].
Theorem 7 ([21, 20]). There exist explicit constructions of K-disjunct matrices with T = O(K 2 log N ) rows.
Moreover, it is possible to sample a K-disjunct matrix with T = O(K 2 log(N/K)) rows with high probability. Both these results are optimal up to a log K factor, and the matrix columns are O(K log N )-sparse in the two constructions.
Bitmasked Matrices and Exact Sparse Recovery
In this section, we describe the bitmasking technique, along with its application in sparse recovery presented by Berinde et al. [9]. As already mentioned, later on Cheraghchi and Indyk [11] modified this algorithm to handle approximate sparse recovery with stronger approximation guarantees, and gave a randomized version of this algorithm that allows for faster recovery.
Fix a matrix W with dimensions M × N . Consider another matrix B of dimensions log N × N such that the j-th column of B contains the binary expansion of j with the least significant bits on top. We call B a bit-test matrix. Given W and B, we define a new bitmasked matrix W ⊗ B with dimensions M log N × N by setting the i-th row of W ⊗ B for i = q log N + t as the coordinate-wise product of the rows W q· and B t· for q ∈ [M ] and t ∈ [log N ]. This means that we have
(W ⊗ B) i,j = W q,j · B t,j = W q,j · bin t (j),
for i = q log N + t and j ∈ [N ], where bin t (j) denotes the t-th least significant bit of j.
Let W be the adjacency matrix of a (D, K, ǫ)-bipartite expander graph G = (L = [N ], R = [M ], E). We proceed to give a high level description of the sparse recovery algorithm from [9]. Suppose we are given access to (W ⊗ B)x for some unknown K-sparse vector x of length N . Recall that our goal is to recover x from this product very efficiently. A fundamental property of the bitmasked matrix W ⊗ B is the following: Suppose that for some q ∈ [M ] we have
supp(x) ∩ supp(W q· ) = {u}(1)
for some u ∈ [N ]. We claim that we can recover the binary expansion of u directly from the log N products
(W ⊗ B) q log N · x, (W ⊗ B) q log N +1 · x, . . . , (W ⊗ B) q log N +log N −1 · x.
1 By an explicit construction, we mean one in which we can construct the matrix in time polynomial in N .
In fact, if (1) holds, then for i = q log N + t it is the case that
(W ⊗ B) i · x = N j=1 W q,j · bin t (j) · x j = bin t (u), since W q,j · x j = 0 only if j = u.
In words, if u is the unique neighbor of q in supp(x), then the entries of (W ⊗ B)x indexed by q log N, . . . , q log N + log N − 1 spell out the binary expansion of u. By the discussion above, if the edges exiting supp(x) in G all had different neighbors in the right vertex set, we would be able to recover x by reading off the binary expansion of the elements of supp(x) from the entries of (W ⊗ B)x. However, if there are edges (u, v) and (u ′ , v) for u, u ′ ∈ supp(x) in G, it is not guaranteed that we will recover u and u ′ as elements of supp(x). While such collisions are unavoidable, and thus we cannot be certain we recover x exactly, the expander properties of G ensure that the number of collisions is always a small fraction of the total number of edges. This means we will make few mistakes when reconstructing supp(x). As a result, setting ǫ to be a small enough constant and using a simple voting procedure (which we refrain from discussing as it is not relevant to us), we can recover a sparse vector y that approximates x in the sense that
x − y 0 ≤ x 0 2 .
We can then repeat the procedure on input (W ⊗ B)(x − y) to progressively refine our approximation of x.
As a result, we recover a K-sparse vector x exactly in at most log K iterations.
Code Construction and Decoding
In this section, we define our code that is able to tolerate a prescribed number of errors, and analyze efficient syndrome decoding and full decoding algorithms. Let N be the desired block-length of the code, K an upper bound on the number of adversarial errors introduced, and ǫ ∈ (0, 1) a constant to be determined later. We fix an adjacency matrix W of a (D, K, ǫ)- It follows that H has dimensions (D · M (1 + log N )) × N , and so C has rate at least
1 − D · M (1 + log N ) N = 1 − O K · log 2 N ǫ 2 N .
In particular, if K and ǫ are constants, then C has rate at least 1 − O log 2 N N . Note that instead of storing the whole parity-check matrix H in memory, one can just store the function table of C, which requires space N D log M , along with a binary lookup table of dimensions log N × N containing the log N -bit binary expansions of 0, . . . , N − 1.
Syndrome Decoding
In this section, we study algorithms for syndrome decoding of C. Fix some codeword c ∈ C, and suppose c is corrupted by some pattern of at most K (adversarially chosen) errors. Let x denote the resulting corrupted codeword. We have x = c + e, for a K-sparse error vector e. The goal of syndrome decoding is to recover e from the syndrome Hx as efficiently as possible. Our decoder is inspired by the techniques from [9] presented in Section 3, and also the sparse recovery algorithms presented in [11].
For the sake of exposition, we consider only the case of decoding over GF(2) -the adaptation to arbitrary fields is simple and is discussed at the end of Section 4.1.1.
A Deterministic Algorithm
In this section, we present and analyze our deterministic decoder which on input Hx recovers the error vector e in sublinear time. Before we proceed, we fix some notation: For s ∈ [D], let G s denote the subgraph of G induced by the function C s (recall Definition 2), and let W s be its adjacency matrix. Informally, our decoder receives
Hx = W x (W ⊗ B)x
as input, and works as follows:
1. Estimate the size of supp(e). This can be done by computing W s · x 0 for all seeds s ∈ [D] and taking the maximum;
2. Using information from W s · x and (W s ⊗ B)x for the good seed fixed in Step 2, recover a K-sparse vector y which approximates the error pattern e following the ideas from Section 3;
3. If needed, repeat these three steps with x − y in place of x.
A detailed description of the deterministic decoder can be found in Algorithm 1. We will proceed to show the procedure detailed in Algorithm 1 returns the correct error pattern e, provided ǫ is a small enough constant. if (W s · x) q = 0 then 9:
Let u ∈ [N ] be the integer with binary expansion
(W s ⊗ B) q log N · x, (W s ⊗ B) q log N +1 · x, . . . , (W s ⊗ B) q log N +log N −1 · x,
10:
Set y u = 1.
11:
Output y Output y + z
We begin by showing that procedure Estimate in Algorithm 1 returns a good approximation of the size of supp(e) along with a good seed. where the parameter ǫ comes from the underlying expander graph.
Proof. Recall that we defined L s = W s · x 0 and L = max s∈[D] W s · x 0 . First, observe that W s · x = W s · e for all s. Then, the upper bound L ≤ |supp(e)| follows from the fact that W s · e 0 ≤ |supp(e)| for all seeds s and all vectors e, since each column of W s has Hamming weight 1 by the fact that the underlying graph G is layered.
It remains to lower bound L. Since e is K-sparse, we know that
|Γ(supp(e))| ≥ (1 − ǫ)D|supp(e)|.
By an averaging argument, it follows there is at least one seed s ∈ [D] such that We now show that, provided a set X ⊆ L = [N ] has good expansion properties in G s , then procedure Approximate in Algorithm 1 returns a good approximation of X . Proof. First, it is immediate that the procedure returns an |X |-sparse vector y. This is because |Γ s (X )| ≤ |X | as G s is 1-left regular, and the procedure adds at most one position to y per element of Γ s (X ). In order to show the remainder of the lemma statement, observe that we can bound x − y 0 as
|Γ s (supp(e))| ≥ (1 − ǫ)|supp(e)|.x − y 0 ≤ A + B,
where A is the number of elements of X that the procedure does not add to y, and B is the number of elements outside X that the procedure adds to y. First, we bound A. Let Γ ′ s (X ) denote the set of neighbors of X in G s that are adjacent to a single element of X . Then, the lower bound in (2) and Lemma 4 ensure that
|Γ ′ s (X )| ≥ (1 − 4ǫ)|X |.
Note that, for each right vertex q ∈ Γ ′ s (X ), the bits
(W ⊗ B) q log N · x, (W ⊗ B) q log N +1 · x, . . . , (W ⊗ B) q log N +log N −1 · x
are the binary expansion of u for a distinct u ∈ X , and u is added to supp(y). As a result, we conclude that
A ≤ 4ǫ|X |.
It remains to bound B. Note that the procedure may only potentially add some u ∈ X if the corresponding right vertex q is adjacent to at least three elements of X . This is because right vertices q that are adjacent to exactly two elements of X satisfy (W s · x) q = 0, and so are easily identified by the procedure and skipped. Then, the lower bound in (2) ensures that there are at most 1 2 · 2ǫ|X | = ǫ|X | such right vertices. As a result, we conclude that
B ≤ ǫ|X |,
and hence x − y 0 ≤ 5ǫ|X |.
We combine the lemmas above to obtain the desired result. iterations.
Proof. Under the conditions in the corollary statement, combining Lemmas 8 and 9 guarantee that in the first iteration of Decode we obtain a K-sparse vector y 1 such that e − y 1 0 ≤ 5ǫ e 0 ≤ 5ǫK.
Recursively applying this result shows that after ℓ iterations we have a vector y = y 1 + y 2 + · · · + y ℓ with sparsity at most
ℓ i=1 (5ǫ) i−1 K ≤ 2K satisfying e − y 0 ≤ (5ǫ) ℓ K. Setting ℓ = 1 + log K log( 1 5ǫ )
ensures that e − y 0 < 1, and hence e = y.
To conclude this section, we give a detailed analysis of the runtime of the deterministic syndrome decoder. We have the following result. Proof. We begin by noting that, for a K-sparse vector y and seed s ∈ [D], we can compute W s · y and (W s ⊗ B)y in time O(K) and O(K log N ), respectively, with query access to the function table of C and the lookup table of binary expansions. We now look at the costs incurred by the different procedures. We will consider an arbitrary iteration of the algorithm. In this case, the input vector is of the form x + y, where x is the corrupted codeword and y is a 2K-sparse vector. iterations.
Remark 12.
In the proof of Theorem 11, we assume that reading an integer from memory (e.g., from the support of a sparse vector y or the function table of C) takes time O(1). If instead we assume that reading an L-bit integer from memory takes time O(L), then we obtain runtime
O log K log 1 5ǫ (K log M + M )(D + log N ) .
instead.
We conclude this section by briefly describing how to adapt Algorithm 1 to perform sparse recovery over any field. There are several ways to do this. One possibility is to replace the check in Line 8 of Algorithm 1 by the following: (W s · x) q = 0 and all non-zero entries of
(W s ⊗ B) q log N · x, . . . , (W s ⊗ B) q log N +log N −1 · x equal (W s · x) q .
As a result, right vertices in Γ s (supp(e)) with exactly two neighbors in supp(e) are skipped by the algorithm. Observe that this additional condition is trivially satisfied over GF(2) if (W s · x) q = 0. Then, in Line 17 one should set y u = (W s · x) q instead.
We remark that the condition in Line 8 could be simplified in general to only checking whether (W s ·x) q = 0, at the expense of obtaining worse constants in the lemmas from this section. In particular, we would have to make ǫ smaller. Since we care about the practicality of our algorithms, we made an effort to have ǫ be as large as possible.
A Randomized Algorithm
In this section, we analyze a randomized version of Algorithm 1 that is considerably faster. The main idea behind this version is that in procedure Estimate we can obtain a good estimate of |supp(e)| with high probability by relaxing ǫ slightly and sampling W s · x 0 for a few i.i.d. random seeds only.
In Algorithm 2, we present the randomized decoder, which uses an extra slackness parameter δ. We will show that the procedure detailed in Algorithm 2 returns the correct error pattern e with probability at least 1 − η, provided ǫ is a small enough constant depending on δ.
We begin by showing that procedure Estimate in Algorithm 2 returns a good approximation of the size of supp(e) with high probability. Lemma 13. Procedure Estimate(r, W x) in Algorithm 2 returns a seed s ∈ [D] satisfying
(1 − 2(1 + δ)ǫ)|supp(e)| ≤ |Γ s (supp(e))| ≤ |supp(e)|(3)
with probability at least 1 − (1 + δ) −r .
Proof. Similarly to the proof of Lemma 8, the desired result will follow if we show that with probability at least 1 − (1 + δ) −r over the choice of the seeds s 1 , . . . , s r it holds that
L s ≥ (1 − 2(1 + δ)ǫ)|supp(e)|(4)
for some seed s ∈ {s 1 , . . . , s r }. We note that (4) holds if |Γ s (supp(e))| ≥ (1 − (1 + δ)ǫ)|supp(e)|. The probability that a random seed fails to satisfy this condition is at most 1 1+δ by Lemma 3. Therefore, the probability that none of the seeds s 1 , . . . , s r satisfy the condition is at most (1 + δ) −r , as desired.
After we obtain a good seed via the Estimate procedure, invoking Lemma 9 with ǫ(1 + δ) in place of ǫ ensures that we can obtain a good sparse approximation of supp(e) with high probability. We thus obtain the following corollary. For each s i , compute L i = W si · x 0
4:
Output (max i∈{1,...,r} L i , arg max i∈{1,...,r} L i ) if (W s · x) q = 0 then 9:
5: procedure Approximate(W s · x, (W s ⊗ B)x) ⊲
Let u ∈ [N ] be the integer with binary expansion Proof. The proof is analogous to that of Theorem 11, except that in the procedure Estimate in Algorithm 2 we only test r seeds. This means procedure Estimate now takes time O(r(K + M )).
(W s ⊗ B) q log N · x, (W s ⊗ B) q log N +1 · x, . . . , (W s ⊗ B) q log N +log N −1 · x,
We note that the runtime of the randomized decoder in Theorem 15 is independent of the degree D of the expander. This has two advantages: First, it means the hidden constants in the runtime are considerably smaller than in the deterministic case from Theorem 11, even assuming we use a near-optimal expander with degree D = O( log N ǫ ). Second, it means that replacing the near-optimal non-explicit expander graph by an explicit construction with sub-optimal parameters will affect the runtime of the randomized decoder only marginally. Furthermore, the failure probability η only affects lower order terms of the runtime complexity. Therefore, we can (for example) set η to be any arbitrarily small constant with only negligible effect in the runtime for large block lengths. Finally, we observe that computing the 0-norm of vectors can be sped up with a randomized algorithm. One can simply sample several small subsets of positions and estimate the true 0-norm with small error and high probability by averaging the 0-norm over all subsets. As mentioned before, we will consider instantiations of our code with an explicit expander in Section 4.3.
Full Decoding
In this section, we study the decoding complexity of our code in the setting where we only have access to the corrupted codeword x = c + e. This mean that if we want to perform syndrome decoding, we must compute the parts of the syndrome that we want to use from x.
Recall that we have access to the function table of C, as well as a lookup table of log N -bit binary expansions. As a result, we can compute products of the form W s · x and (W s ⊗ B)x in time O(N ) and O(N log N ), respectively. This is because all columns of W s (resp. W s ⊗ B) have 1 nonzero entry (resp. at most log N nonzero entries), and the nonzero entry (resp. entries) of the j-th column are completely determined by C s (j) = C(s, j) and the j-th column of the lookup table of binary expansions.
We now analyze the runtimes of both the deterministic and randomized decoders from Algorithms 1 and 2 in this alternative setting. We have the following results.
O(K log K · N log N ).
Proof. The proof is analogous to that of Theorem 11, except we now must take into account the time taken to compute products of the form W s · x and (W s ⊗ B)x:
• The procedure Estimate in Algorithm 1 requires computing D products of the form W s (x + y), which in total take time O(D(N + K + M )), along with computing the 0-norm of all resulting vectors. Since The desired statements now follow by noting that there are at most 1 + log K log ( 1 5ǫ ) iterations.
Theorem 18.
On input x = c + e with c ∈ C and e a K-sparse error vector, procedure Decode from Algorithm 2 returns e with probability at least 1 − η in expected time
O log K log 1 5ǫ(1+δ) (N + K + M )(r + log N ) , where r = 1 + log(1/η) + log log K − log log 1 5ǫ(1+δ) log(1 + δ) .
Proof. The proof is analogous to that of Theorem 17, except that in the procedure Estimate in Algorithm 2 we only need to test r seeds. This means procedure Estimate now takes time O(q(N + K + M )).
There are important properties that are not explicit in the proof of Theorem 18. Observe that only one computation takes O(N log N ) per iteration of the Algorithm 2; Namely, the computation of (W s ⊗ B)x for a fixed seed s. Consequently, the hidden constant in the computation time is small. Moreover, as already discussed for the randomized syndrome decoder, the runtime is independent of the degree of the expander, and the effect of the failure probability on the runtime is negligible for large block lengths. This means that the decoder described in Algorithm 2 is also faster in the full decoding setting than the O(N D) expander codes decoder adapted from [5], whose running time depends on the degree of the expander graph, as long as the number of iterations is not too large. This is the case if the number of errors K allowed is a small constant (e.g., K ≤ 5) and we set ǫ to be not too large. Furthermore, observe that, unlike our decoder, the runtime of the expander codes decoder is affected by a sub-optimal choice of unbalanced bipartite expanders. Finally, if we want a faster decoder for an arbitrary but fixed error threshold K, we can also set ǫ to be small enough so that the maximum number of iterations is sufficiently small for our needs. In this case, the rate of our code becomes smaller since we must make ǫ smaller.
Instantiation with Explicit Expanders
In this section, we analyze how instantianting our construction with an explicit layered unbalanced expander with sub-optimal parameters affects the properties of our codes.
More precisely, we consider instantiating our code with the GUV expander introduced by Guruswami, Umans, and Vadhan [22], and an explicit highly unbalanced expander constructed by Ta-Shma, Umans, and Zuckerman [23]. For simplicity, in this section we will assume that all parameters not depending on N (such as K and ǫ) are constants.
Fix constants α, ǫ, K > 0. Then, the GUV graph is a (D, K, ǫ)-layered bipartite expander with degree
D = O log N · log K ǫ 1+1/α
, and, for each layer, a right vertex set of size
M = D 2 · K 1+α .
Observe that, although the GUV expander is unbalanced, the size of its right vertex set grows with the degree. Ta-Shma, Umans, and Vadhan [23] provided explicit constructions of highly unbalanced layered expanders. In particular, they give a construction of a (D, K, ǫ)-layered bipartite expander with degree
D = 2 O(log log N ) 3 ,
and, for each layer, a right vertex set of size
M = K O(1/ǫ) .
Syndrome decoding Deterministic
Randomized Graphs from [22] O(log N ) 3+3/α O(log N ) 3+2/α Graphs from [23] 2 O(log log N ) 3 O(log N ) Full decoding Deterministic Randomized Graphs from [22] Table 3: Complexity of deterministic and randomized full decoding for different explicit graphs when the number of errors K and expander error ǫ are constants.
O(N log 1+1/α N ) O(N log N ) Graphs from [23] O(N 2 O(log log N ) 3 ) O(N log N )
Plugging the parameters of both graphs presented in this section into the runtimes in Theorems 11, 15, 17, and 18 and treating K, ǫ, and α as constants immediately yields explicit high-rate codes with syndrome and full decoding complexity displayed in Table 2 (for syndrome decoding), and in Table 3 (for full decoding).
Observe that for both graphs there is a substantial decrease in complexity for randomized decoding versus deterministic decoding for the same setting. This is due to the fact that the decoding complexity of our randomized decoding algorithms is independent of the degree of the underlying expander, which we have already discussed before, and that the degree of explicit constructions is sub-optimal. Using the highly unbalanced explicit graphs from [23], the decoding complexity of our randomized algorithms essentially matches that of the case where we use a random expander with near-optimal parameters (we are ignoring the contribution of K, which we assume to be small).
We conclude by noting that the full decoding complexity of expander codes under the explicit graphs from this section matches the second column of Table 3. In comparison, our randomized algorithm performs better under both graphs.
Group Testing
In this section, we show how we can easily obtain a scheme for non-adaptive group testing with few tests and sublinear time recovery. More precisely, we will prove the following: Theorem 19. Given N and K, there is an explicit test matrix W of dimensions T × N , where T = O(K 2 log 2 N ), such that it is possible to recover a K-sparse vector x from W ⊙ x in time O(K 3 log 2 N ).
We begin by describing the test matrix W . Let W ′ be an explicit K-disjunct matrix of dimensions M ×N with M = O(K 2 log N ). Such explicit constructions exist as per Theorem 7. Then, our test matrix W is defined as
W = W ′ W ′ ⊗ B ,
where B is the log N × N bit-test matrix from Section 3. It follows immediately that W has dimensions T × N with T = M log N = O(K 2 log 2 N ). It remains to describe and analyze the recovery algorithm that determines x from
W ⊙ x = W ′ ⊙ x (W ′ ⊗ B) ⊙ x = y (1) y (2)
, whenever x is K-sparse. At a high-level, the algorithm works as follows:
1. For q ∈ [M ], let s q be the integer in [N ] with binary expansion y (2) q log N , y (2) q log N +1 , . . . , y (2) q log N +log N −1 . Recover the (multi) set S = {s 0 , s 1 , . . . , s M−1 }. The disjunctness property of the underlying matrix W ′ ensures that supp(x) ⊆ S; 2. Similarly to the original recovery algorithm for disjunct matrices, run through all s ∈ S and check whether W ′ s is contained y (1) . Again, the fact that W ′ is disjunct ensures that this holds if and only if s ∈ supp(x).
A rigorous description of this recovery procedure can be found in Algorithm 3. We now show that procedure Recover(W ⊙ x) indeed outputs supp(x), provided that x is K-sparse. First, we prove that supp(x) ⊆ S.
Lemma 20. Suppose that x is K-sparse. Then, if S = SuperSet(W ⊙ x), we have supp(x) ⊆ S.
Proof. It suffices to show that if supp(x) = {s 0 , . . . , s t−1 } for t ≤ K, then there is q ∈ [M ] such that W ′ q,s0 = 1 and W ′ q,sj = 0 for all 1 ≤ j ≤ t − 1. If this is true, then y (2) q log N , y (2) q log N +1 , . . . , y (2) q log N +log N −1 would be the binary expansion of s 0 . The desired property follows since W ′ is K-disjunct. In fact, if this was not the case, then W ′ ·s0 would be contained in t−1 i=1 W ′ ·si , and hence W ′ would not be K-disjunct. The next lemma follows immediately from the fact that W ′ is K-disjunct. We conclude this section by analyzing the runtime of procedure Recover(W ⊙ x). We have the following result.
Theorem 23. On input a K-sparse vector x, procedure Recover(W ⊙x) returns supp(x) in time O(K 3 log 2 N ).
Proof. We analyze the runtime of procedures SuperSet and Remove separately:
Lemma 4 .
4Let G = (L, R, E) be a layered bipartite graph. If S ⊆ L satisfies |Γ i (S)| ≥ (1 − δ)|S|, then the number of right vertices in Γ i (S) with only one neighbor in S (with respect to the subgraph G i ) is at least (1 − 2δ)|S|.
Theorem 5 ([ 18 ,
518Lemma 4.2]). Given any N , K, and ǫ, we can sample a layered (D, K, ǫ)-bipartite expander graph G = (L = [N ], R = [M D], E) with high probability for D = O log N ǫ and M = O K ǫ . The graph in Theorem 5 can be obtained by sampling a random function C : [N ] × [D] → [M ], and choosing (x; (s, y)) as an edge in G if C(x, s) = y.
layered unbalanced bipartite expander graph G = (L, R, E) with L = [N ] and R = [D · M ], where D = O log N ǫ and M = O K ǫ . Such an expander G can be obtained with high probability by sampling a random function C : [N ]×[D] → [M ] as detailed in Section 2.2. We define our code C ⊆ {0, 1} N by setting its parity-check matrix H as H = W W ⊗ B .
Output (max s L s , arg max s L s )5: procedure Approximate(W s · x, (W s ⊗ B)x)⊲ Computes a good approximation of supp(e)
Set y = Approximate(W s · x, (W s ⊗ B)
Lemma 8 .
8Procedure Estimate(W x) in Algorithm 1 returns a seed s ∈ [D] satisfying (1 − 2ǫ)|supp(e)| ≤ |Γ s (supp(e))| ≤ |supp(e)|,
a result, by Lemma 4 the number of vertices in Γ s (supp(e)) adjacent to a single vertex in supp(e) is at least (1 − 2ǫ)|supp(e)|, and thus |Γ s (supp(e))| ≥ W s · x 0 ≥ (1 − 2ǫ)|supp(e)|.
Lemma 9 .
9Fix a vector x ∈ {0, 1} N and denote its support by X . Suppose that|Γ s (X )| ≥ (1 − 2ǫ)|X | (2)for a given seed s ∈ [D]. Then, procedure Approximate(W s · x, (W s ⊗ B)x) returns an |X |-sparse vector y ∈ {0, 1} N such that x − y 0 ≤ 5ǫ x 0 .
Corollary 10 .
10Suppose that ǫ < 1/10. Then, on input Hx for x = c + e, Decode(Hx) in Algorithm 1 recovers the error vector e after at most 1 + log K
Theorem 11 .
11On input Hx for x = c + e with c ∈ C and e a K-sparse error vector, procedure Decode(Hx) in Algorithm 1 returns e in time O log K log 1 5ǫ (K + M )(D + log N ) . In particular, if ǫ is constant, M = O(K/ǫ), and D = O(log N/ǫ), procedure Decode takes time O(K log K · log N ).
•
The procedure Estimate in Algorithm 1 requires computing D products of the form W s (x + y), which in total take time O(D(K + M )), along with computing the 0-norm of all resulting vectors. SinceW s (x + y) has length M , doing this for all seeds takes time O(DM ). In total, the Estimate procedure takes time O(D(K + M )); • The procedure Approximate in Algorithm 1 requires the computation of W s (x+y) and (W s ⊗B)(x+y) for a fixed seed s, which take time O(K + M ) and O((K + M ) log N ), respectively. The remaining steps can be implemented in time O(M log N ) for a total time of O((K + M ) log N );Note that the time required to compute the sum of sparse vectors in Line 19 is absorbed into the O((K + M )(D + log N )) complexity of previous procedures. The desired statements now follow by noting that there are at most 1 + log K
Set y = Approximate(W s · x, (W s ⊗ B)x)19: Set z = Decode(H(x − y), η, δ, ǫ)20: Output y + z Corollary 14. Suppose that ǫ(1 + δ) < 1/10. Then, on input Hx for x = c + e, procedure Decode in Algorithm 2 returns the error vector e with probability at least 1 − η in at most 1 + The statement follows by repeating the proof of Corollary 10 but replacing Lemma 8 with Lemma 13 and by invoking Lemma 9 with ǫ(1 + δ) in place of ǫ. Since each iteration succeeds with probability at least 1 − (1 + δ) −r and there are at most (1 + δ) −r ≤ η by the choice of r in Algorithm 2.We conclude this section by analyzing the runtime of the randomized decoder.Theorem 15. On input Hx for x = c + e with c ∈ C and e a K-sparse error vector, procedure Decode Algorithm 2 returns e with probability at least 1 − η in time 1/η) + log log K − log log 1 5ǫ(1+δ) log(1 + δ).
Remark 16 .
16As in the proof of Theorem 11, we assume in this section that reading an integer from memory takes time O(1). If instead we assume that reading an L-bit integer from memory takes time O(L), · (K(log M + log D) + M ) + (K + M ) log N ) instead. The preceding arguments still stand, as even for explicit expanders it holds that log M and log D are negligible compared to log N .
Theorem 17 .
17On input x = c + e with c ∈ C and e a K-sparse error vector, procedure Decode from Algorithm 1 returns e in timeO log K log 1 5ǫ (N + K + M )(D + log N ) . In particular, if ǫ is constant, M = O(K/ǫ), and D = O(log N/ǫ), procedure Decode takes time
W s (x + y) has length M , doing this for all seeds takes time O(DM ). In total, the Estimate procedure takes time O(D(N + K + M )); • The procedure Approximate in Algorithm 1 requires the computation of W s (x+y) and (W s ⊗B)(x+y) for a fixed seed s, which take time O(N + K + M ) and O((N + K + M ) log N ), respectively. The remaining steps can be implemented in time O(M log N ) for a total time of O((N + K + M ) log N ).
Lemma 21 .
21If x is K-sparse and supp(x) ⊆ S, then Remove(W ⊙ x, S) returns supp(x).Combining Lemmas 20 and 21 with Algorithm 3 leads to the following result.
Corollary 22 .
22If x is K-sparse, then Recover(W ⊙ x) returns supp(x).
•
In procedure SuperSet(W ⊙ x), for each q ∈ [M ] we need O(log N ) time to compute s q and add it to S. It follows that SuperSet(W ⊙ x) takes time O(M log N ) = O(K 2 log 2 N ); • In procedure Remove(W ⊙ x, S), it takes time O(|supp(W ′ ·j )|) to decide whether W ′ ·j is contained in W ′ ⊙ x = y (1) . Therefore, in total the procedure takes time O(|S| · max j∈S |supp(W ′ ·j )|). Noting that |S| ≤ M and that |supp(W ′ ·j )| = O(K log N ) for all j implies that the procedure takes time O(M · K log N ) = O(K 3 log 2 N ).
Table 2 :
2Complexity of deterministic and randomized syndrome decoding for different explicit graphs when the number of errors K and expander error ǫ are constants.
AcknowledgmentsThe authors thank Shashanka Ubaru and Thach V. Bui for discussions on the role of the bitmasking technique in sparse recovery. M. Cheraghchi's research was partially supported by the National Science Foundation under Grants No. CCF-2006455 and CCF-2107345. J. Ribeiro's research was partially supported by the NSF grants CCF-1814603 and CCF-2107347, the NSF award 1916939, DARPA SIEVE program, a gift from Ripple, a DoE NETL award, a JP Morgan Faculty Fellowship, a PNC center for financial services innovation award, and a Cylab seed funding award.Let s q be the integer with binary expansion y(2)q log N , y(2)q log N +1 , . . . , y(2)q log N +log N −15:Add s q to S if W ′ ·s is not contained in y (1) then 10:Remove s from S 11:Output S 12: procedure Recover(W ⊙ x) ⊲ Main recovery procedure13:Set S = SuperSet(W ⊙ x)14:Output Remove(W ⊙ x, S)
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"Deep Diffusion Models for Seismic Processing",
"Deep Diffusion Models for Seismic Processing"
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"Ricard Durall \nFraunhofer ITWM\n\n\nFraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n\n",
"Ammar Ghanim \nFraunhofer ITWM\n\n\nFraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n\n",
"Mario Fernandez \nFraunhofer ITWM\n\n\nFraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n\n",
"Norman Ettrich \nFraunhofer ITWM\n\n\nFraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n\n",
"Janis Keuper \nFraunhofer ITWM\n\n\nFraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n\n\nIMLA\nOffenburg University\n\n"
] |
[
"Fraunhofer ITWM\n",
"Fraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n",
"Fraunhofer ITWM\n",
"Fraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n",
"Fraunhofer ITWM\n",
"Fraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n",
"Fraunhofer ITWM\n",
"Fraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n",
"Fraunhofer ITWM\n",
"Fraunhofer Center Machine Learning 3É\ncole Normale Supérieure\n",
"IMLA\nOffenburg University\n"
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[] |
Seismic data processing involves techniques to deal with undesired effects that occur during acquisition and pre-processing. These effects mainly comprise coherent artefacts such as multiples, non-coherent signals such as electrical noise, and loss of signal information at the receivers that leads to incomplete traces. In the past years, there has been a remarkable increase of machine-learning-based solutions that have addressed the aforementioned issues. In particular, deep-learning practitioners have usually relied on heavily fine-tuned, customized discriminative algorithms. Although, these methods can provide solid results, they seem to lack semantic understanding of the provided data. Motivated by this limitation, in this work, we employ a generative solution, as it can explicitly model complex data distributions and hence, yield to a better decision-making process. In particular, we introduce diffusion models for three seismic applications: demultiple, denoising and interpolation. To that end, we run experiments on synthetic and on real data, and we compare the diffusion performance with standardized algorithms. We believe that our pioneer study not only demonstrates the capability of diffusion models, but also opens the door to future research to integrate generative models in seismic workflows.
|
10.1016/j.cageo.2023.105377
|
[
"https://arxiv.org/pdf/2207.10451v1.pdf"
] | 250,916,931 |
2207.10451
|
551bb52f6cb72f404e6ad92cb3b5d6209e8fc0c6
|
Deep Diffusion Models for Seismic Processing
Ricard Durall
Fraunhofer ITWM
Fraunhofer Center Machine Learning 3É
cole Normale Supérieure
Ammar Ghanim
Fraunhofer ITWM
Fraunhofer Center Machine Learning 3É
cole Normale Supérieure
Mario Fernandez
Fraunhofer ITWM
Fraunhofer Center Machine Learning 3É
cole Normale Supérieure
Norman Ettrich
Fraunhofer ITWM
Fraunhofer Center Machine Learning 3É
cole Normale Supérieure
Janis Keuper
Fraunhofer ITWM
Fraunhofer Center Machine Learning 3É
cole Normale Supérieure
IMLA
Offenburg University
Deep Diffusion Models for Seismic Processing
Seismic data processing involves techniques to deal with undesired effects that occur during acquisition and pre-processing. These effects mainly comprise coherent artefacts such as multiples, non-coherent signals such as electrical noise, and loss of signal information at the receivers that leads to incomplete traces. In the past years, there has been a remarkable increase of machine-learning-based solutions that have addressed the aforementioned issues. In particular, deep-learning practitioners have usually relied on heavily fine-tuned, customized discriminative algorithms. Although, these methods can provide solid results, they seem to lack semantic understanding of the provided data. Motivated by this limitation, in this work, we employ a generative solution, as it can explicitly model complex data distributions and hence, yield to a better decision-making process. In particular, we introduce diffusion models for three seismic applications: demultiple, denoising and interpolation. To that end, we run experiments on synthetic and on real data, and we compare the diffusion performance with standardized algorithms. We believe that our pioneer study not only demonstrates the capability of diffusion models, but also opens the door to future research to integrate generative models in seismic workflows.
Introduction
Deep generative learning has become an important research area in the machine learning community, being more relevant in many applications. Namely, they are widely used for image synthesis and various image-processing tasks such as editing, interpolation, colourization, denoising, and super-resolution. Recently, diffusion probabilistic models [1,2] have emerged as a novel, powerful class of generative learning methods. In a short period of time, these models have achieved surprisingly high performance [3,4,5,6], and have even surpassed state-of-the-art algorithms like generative adversarial networks [7] (GANs) and variational autoencoders [8] (VAEs).
At the same time, the geophysics community has been actively adopting deep-learning techniques to boost and automate numerous seismic interpretation tasks including fault picking [9,10], salt delineation [11,12], well-to-seismic tie [13,14], horizon tracking [15,16], multiple removal [17,18], etc. Nonetheless, to the best of our knowledge, there has not been yet any work exploring the application of diffusion models to seismic data and thus, studying their potential advantages to already established deep-learning approaches in this domain. Driven by this motivation, in this work, we study the applicability of diffusion models for seismic processing. Seismic imaging is essential to discover and characterize economically worthwhile geological reservoirs, such as hydrocarbons accumulations, and to manage the extraction of the resources stored in them. Unfortunately, recorded seismic signals at the surface are inevitably contaminated by coherent and incoherent noise of various nature. The process of removing the noise, while retaining the primary signal, is called seismic processing. In this paper, we focus on three relevant, well-known seismic processing tasks: demultiple, denoising and interpolation. Demultiple and denoising are both removing unwanted signals from the seismic section; the first gets rid of coherent noise caused by reverberations of waves between strong reflectors, whereas the latter removes incoherent noise of miscellaneous causes. The goal of interpolation is to fill-in gaps in the image caused by limitations during acquisition. Although at the first glance the nature of these problems might look different or unrelated, it is possible to formulate a common framework, in which they can be solved. This is feasible, due to the fact that the diffusion models, like most of generative models, learn the density distribution of the input data. In other words, unlike discriminative approaches which draw boundaries in the data space, the generative approaches model how data is placed throughout the space [19]. As a result, they are powerful algorithms that can be independently applied to a large diversity of problems.
Background
Generative models for modelling estimate the marginal distribution, denoted as p(x), over observable variables x, e.g., images. In the literature, we can find different formulations that tackle this problem such as autoregressive generative models, latent variable models, flow-based models, and energybased models.
Latent Variable Models
The main idea of this type of models is to utilize latent variables z to formulate the joint distribution p(x, z), which describes the marginal distribution as a function of learnable parameters θ (likelihood).
Mathematically, it can be written as:
z ∼ p θ (z) x ∼ p θ (x|z) p θ (x) = z p θ (x, z) = z p θ (x|z)p θ (z).(1)
Unfortunately, for most of the problems we do not have access to the true distribution p(x) and hence, we need to fit our model to some empirically observed subset. One solution is to use Monte Carlo sampling to approximate the integral over z to try to estimate the model parameters θ. Nonetheless, this approach does not scale to high dimensions of z and consequently, we will suffer from issues associated with the curse of dimensionality. Another solution is to use variational inference, e.g., VAE [8]. In particular, the lower bound of the log-likelihood function, called the Evidence Lower BOund (ELBO). The ELBO provides a joint optimization objective, which simultaneously updates the variational posterior q φ (z|x) and likelihood model p θ (x|z). The objective is written as:
log p(x) ≥ E z∼q φ (z|x) [log p θ (x|z)] − KL[q φ (z|x)||p(z)],(2)
where KL stands for the Kullback-Leibler divergence.
Hierarchical Latent Variable Models
Once defined a single stochastic layer, it is straightforward to derive hierarchical extensions. For example, let us consider a latent variable model with two latent variables z 1 and z 2 . We can define the joint distribution p(x, z 1 , z 2 ) and marginalizing out the latent variables:
p θ (x) = z 1 z 2 p θ (x, z 1 , z 2 ) = z 1 z 2 p θ (x|z 1 )p θ (z 1 |z 2 )p θ (z 2 ).(3)
Similar to the single latent model, we can derive the variational approximation (ELBO) to the true posterior as:
log p(x) ≥ E z 1 ∼q φ (z 1 |z 2 ) [log p θ (x|z 1 )] − KL[q φ (z 1 |x)||p θ (z 1 |x)] − KL[q φ (z 2 |z 1 )||p(z 2 )].(4)
Diffusion Models
Diffusion models belong to the latent variable family as well. In fact, we can think of them as a specific realization of a hierarchical latent variable model, where the inference model 1 does not have learnable parameters. Instead, it is constructed so that the final latent distribution q(x T ) converges to a standard Gaussian (where T is the number of latent variables). The objective function of diffusion models is written as:
log p(x) ≥ E x 1:T ∼q(x 1:T |x 0 ) [KL(q(x T |x 0 )||p θ (x T )) + T t=2 KL(q(x t−1 |x t , x 0 )||p θ (x t−1 |x t )) −log p θ (x 0 |x 1 )].(5)
Under certain assumptions, this objective can be further simplified, leading to the following approxima- 1 Remember that the inference model relates a set of observable variables to a set of latent variables, e.g., q(z|x). tion:
log p(x) T t=2 KL(q(x t−1 |x t , x 0 )||p θ (x t−1 |x t )) = T t=2 || − θ ( √ᾱ t x 0 + √ 1 −ᾱ t , t)|| 2 .(6)
Note that we drop the expectation for clarity. The exact derivation can be found in [2].
Methodology
In this section, we provide a brief overview of diffusion models formulation. Note that we do not aim at covering the entire derivations. For a more indepth, detailed mathematical description, we refer the reader to [2].
Background
On a high level, diffusion models consist of two parts: forward diffusion and parametrized reverse. The forward diffusion part can be described as a process, where Gaussian noise is gradually applied to the input image x 0 until the image becomes entirely unrecognizable from a normal distribution x T ∼ N (0, I) (T is the number of transformation steps). That is to say, at each step of this process, the noise is incrementally added to the data,
x 0 + − → x 1 + − → ... + − → x T .
This procedure together with the Markov assumption 2 leads to a simple parameterization forward process expressed as:
q(x 1:T |x 0 ) = T t=1 q(x t |x t−1 ) = T t=1 N (x t ; 1 − β t x t−1 , β t I),(7)
where the variable β defines a fixed variance schedule, chosen such that q(x T |x 0 ) ≈ N (0, I).
The second part, the parametrized reverse process, represents the data synthesis. Thus, it undoes the forward diffusion process and performs iterative denoising. To that end, the reverse process is trained to generate data by converting random noise into realistic data. Formally, this generative process is defined as a stochastic process, which iteratively removes noise from the input images using deep neural networks. Starting with the pure Gaussian noise p(x T ) = N (x T , 0, I), the model learns the joint distribution p θ (x 0:T ) as:
p θ (x 0:T ) = p(x T ) T t=1 p θ (x t−1 |x t ) = p(x T ) T t=1 N (x t−1 ; µ θ (x t , t), Σ θ (x t , t)),(8)
where the time-dependent parameters of the Gaussian transformations θ are learned. Note in particular that the Markov formulation asserts that a given reverse diffusion transformation distribution depends only on the previous timestep.
Training
A diffusion model is trained by finding the reverse Markov transitions that maximize the likelihood of the training data. In practice, this process consists of optimizing the variational lower bound on the log likelihood. Hereunder the simplified expression derived by [2]: Figure 3: In each reverse step t, the model θ is fed with the semi-denoised multiple-free image x t and the multiple-infested input. As an output, the network generates the image x t−1 , which should have less noise and no multiples.
log p(x) T t=2 || − θ ( √ᾱ t x 0 + √ 1 −ᾱ t , t)|| 2 ,(9)where α t = 1 − β t andᾱ t = T i=1 α i .(10)
Note, ultimately, the deep neural network learns to predict the noise component at any given timestep.
Experiments
In this section, we validate the flexibility of diffusion models for different seismic tasks. In particular, we analyse three case studies: demultiple, denoising and interpolation. To do that, we present an end-to-end deep-learning approach that can deal (separately) with demultiple, denoising and interpolation scenarios. Furthermore, we benchmark the results with alternative paradigms that are currently employed in both academia and industry domains. The implementation details are as following: In all our experiment, we train the diffusion model for 200,000 iterations with a batch size of 32; we set β to follow a linear schedule, and we use a depth of 2000 timesteps for both the forward process (see Equation 7) and the reverse denoising process (see Equation 8).
Architecture
Image diffusion models commonly employ a timeconditional U-net [20], parametrized as θ (•, t), as a neural backbone. This architecture was initially introduced in [2], where the main motivation for this topology choice was the requirement for the model to have identical input and output dimensionality. The architecture consists of a stack of residual layers and downsampling convolutions, followed by a stack of residual layers with upsampling convolutions; skip connections connect the layers with the same spatial size. Furthermore, it uses a global attention layer with a single head to add a projection of the timestep embedding into each residual block.
Demultiple
Primary seismic reflections are events which have reflected only once, and they are employed to describe the subsurface interfaces. Multiples, on the contrary, are events which appear when the signal has not taken a direct path from the source to the receiver after reflecting on a subsurface boundary. The presence of multiples in a recorded dataset can trigger erroneous interpretations, since they do not only interfere with the analysis in the post-stack domain, e.g., stratigraphic interpretation, but also with the analysis in the pre-stack domain, e.g., amplitude variation with offset inversion. Thereby, the demultiple process plays a crucial role in any seismic processing workflow.
In this first experiment, we follow the approach from [21,18], and generate synthetic pairs of multiple-infested and multiple-free gathers. This data setup allows us to train the model in a supervised manner and therefore, we can frame the demultiple problem as an image-to-image transformation task, where the network learns to remove the multiples without removing primary energy. As in [18], the training dataset is designed to include a rich amount of features present in real datasets, to maximize transferability to real case uses. To that end, we employ as a baseline a conditional diffusion models proposed by [22]. More specifically, we condition our model by concatenating the semi-denoised multiple-free image x t with the multiple-infested input (see Figure 3). Ideally, the network should return an improved semi-denoised multiple-free gather x t−1 that after T reverse steps should converge into a noise-and multiple-free gather x 0 .
Once the model is trained, it is crucial to assess the inference capabilities of the network when working on real data, i.e., generalizability. Nonetheless, this is not a granted property in deep-learning models due to the distribution gap between different datasets, e.g., the gap between synthetic and real datasets [23]. In our experiments, we test the Figure 4: This figure displays two cropped gathers that contain multiples (input), and the results after applying the demultiple algorithms. Moreover, we plot the difference between the input and the output to check the content that has been removed. Note that we apply a scaling factor of 3 in the differences to stress the changes. diffusion approach on the dataset from the Volve field made available under Equinor Open Data Licence. Furthermore, we compare the outcomes with two other multiple-attenuation methodologies: one based on Radon-transform [24] and one based on deep learning [18]. Figure 4 shows an example of such a comparison, where we can observe how the diffusion solution offers competitive results, despite minimal hyperparameter tuning involved. For additional results, see Figure 9 in the Appendix.
Denoising
Incoherent noise can be caused by superposition of numerous unwanted signals from various sources such as ocean waves, wind and electrical instrument noise among others. Removing such incoherent noise can improve the overall signal-to-noise ratio and, consequently, increase the certainty of interpretation. Traditional approaches can be subdivided into two main categories: the prediction filtering methods and domain transform methods. The first type assumes linearity and predictability of the signal, and constructs a predictive filter to suppress the noise [25,26]. These methods have been widely adopted by the industry due to their efficiency, although they tend to under-suppress noise and occasionally suffer from signal leakage [27]. The second type of methods uses mathematical transformations, e.g., Fourier transform [28], wavelet transform [29], curvelet transform [30,31], to steer the seismic data into domains, where seismic signals and noise can be Finally, the third and four rows display the difference between the results and the original and the input data, respectively. easier separated and then leverage the sparse characteristics of seismic data. This approach, however, often requires a time-consuming transform coefficient tuning. To cope with this drawback, a new trend based on deep-learning algorithms has emerged, resulting in optimized solutions that remove incoherent noise from seismic data as well as speed up the inference time [32,33].
Similar to the demultiple scenario, we create pairs of images to train our diffusion model. Nonetheless, this time, the objective is to eliminate undesired uncorrelated noise, while preserving the inherent characteristics of the data. To that end, the pairs of training data consist of a real image and their noisy version. To create the noisy images, we synthetically add Gaussian noise to the original real images with a variability of the 50% of their energy. For this second case of study, we train on 1994 BP [34] dataset, from which we extract random patches (from different shot gathers) that neither overlap among each other, nor have more than 40% of their content equal to 0. In this fashion, we try to guarantee certain level of variety in the training data.
For the testing set, we apply the same conditions as for training. Additionally, we employ a second dataset (Model94 [34]) to evaluate the generalization capacity of our system. As for comparison, we use a spectral filtering technique based on the Fourier transform, namely a complex Wiener prediction filter called FX-Decon [25,26], which is dedicated for signal extraction and non-coherent noise suppression in the frequency domain. To assess the results, we use structural similarity index (SSIM) and signalto-noise ratio (SNR) as quantitative metrics. Figure 5 displays them for each configuration, i.e., different datasets and methods, and we can observe how the diffusion model provides the best scores when we test on data coming from the same dataset as the one used for training. However, as expected, it has a drastic drop when we test on a new dataset, e.g., Model94. This phenomenon is mainly caused by the distribution gap between different datasets. On the other hand, FX-Decon achieves similar performance on both datasets (no drop), as this method does not involve any learning, i.e., data fitting. Finally, Figure 6 illustrates a denoising example for both algorithms. The difference between the outputs and the original data (third row in Figure 6) allows us to see that diffusion model removes some coherent signal, while FX-Decon does not. Ideally, this should be corrected, but we leave this improvement for future work. Nevertheless, overall, the diffusion approach leads to less noisy outputs, as can be noticed in the output image. For additional results, see Figure 10 in the Appendix.
Interpolation
Seismic data processing algorithms greatly benefit from regularly sampled and reliable data. However, it is rarely the case where the acquired data is presented flawless, i.e., complete shot gathers without missing traces. Frequently, the reason for that are acquisition constraints such as geophones issues, topography, and economical limitations. As a consequence, interpolation techniques are a fundamental key for most seismic processing systems.
In this last case of study, we evaluate the capacity of our diffusion model to interpolate missing traces.
To that end, we follow the evaluation methodology introduced by [35], namely, we consider the scenario with irregular missing traces and with a level of decimation set to 50% (see Figure 7). Regarding the data for this experiment, we repeat the setup presented in the denoising section, using 1994 BP dataset for training and testing, and Model94 for testing on a new dataset. Finally, to have a baseline to compare with, we implement the so-called "standard" topology from [35], which is essentially a U-net-like network. Figure 8 shows the qualitative evaluation of the diffusion approach and of the U-net baseline. Although results from the latter are superior, the improvement could be considered marginal given the small metric differences. Furthermore, both algorithms seem to struggle when inferring on unseen datasets. On the other hand, besides the quantitative results, the potential that diffusion models might bring is objectively higher than discriminative models, as the former are generative models and therefore, can capture more advanced data properties. For additional results, see Figure 11 in the Appendix.
Discussion
In this work, we propose a generative framework based on diffusion models to address several seismic tasks. In particular, our case studies include demultiple, denoising and interpolation. To solve them, we define the problem as an image-to-image transformation, where we have an input image that requires certain modifications so that, the output result belongs to the target domain. For example, in the demultiple scenario, given a multiple-infested gather (input domain), our diffusion approach has to identify the multiples and cancel them out, leading to a multiple-free output gather (target domain).
The results of our experimental evaluations are fairly encouraging, as they show competitive performance, when comparing with standardized, customized algorithms. As we pointed out before, diffusion models for seismic data is an unexplored field to date and hence, the ultimate goal of this project is not to outperform these current algorithms in their respective areas, but to provide a solid analysis of the applicability and flexibility of this novel framework. Therefore, the main success of our implementation can be regarded as proof of concept that can be used to adopt generative models, namely diffusion models, in the geoscience community. We believe that our work can help to lay the foundation for future research that can benefit both academia and industry.
Acknowledgement
Figure 1 :
1Scheme of the different latent variable models. (Top) Single latent variable model. (Center) Hierarchical latent variable model. (Bottom) Diffusion model.
Figure 2 :
2Denoising diffusion process. While the Markov chain of the forward diffusion gradually adds noise to the input (dash arrows), the reverse process removes it stepwise (solid arrows).
Figure 5 :
5Mean and standard deviation of SSIM and SNR metrics calculated on 500 random denoised images. Results from diffusion and FX-Decon scenarios.
Figure 6 :
6This figure shows an example of denoising. The first row contains the original image and the input image (original with noise). The second row presents the diffusion and the FX-Decon results.
Figure 7 :
7This figure shows an example of interpolation. From left to right: the original image, the mask, the input image (original with mask), the diffusion result and its difference with respect to the original image.
Figure 8 :
8Mean and standard deviation of SSIM and SNR metrics calculated on 500 random interpolated images. Results from diffusion and U-net scenarios.
Figure 9 :
9This figure displays demultiple results at different intermediate steps for the reverse process. Note that the first two rows show synthetic data examples, while the last two from the Volve dataset.
Figure 10 :
10This figure displays denoising results at different intermediate steps for the reverse process. Note that the examples belong to the Model94 dataset.
Figure 11 :
11This figure displays interpolation results at different intermediate steps for the reverse process. Note that the examples belong to the Model94 dataset.
Markov assumption is used to describe a model that holds the memoryless property of a stochastic process.
This work was developed in the Fraunhofer Cluster of Excellence Cognitive Internet Technologies. The authors would like to acknowledge the members of the Fraunhofer ITWM DLSeis consortium (http://dlseis.org) for their financial support. We are also grateful to Equinor and Volve Licence partners for releasing Volve seismic field data under an Equinor Open Data Licence.AppendixMore ResultsWe provide additional results, where we can visualize the evolution of the reverse process for all the aforementioned case studies (seeFigure 9,Figure 10andFigure 11). Note that the subindexes of the x indicate the output of an intermediate step during the inference process, being x 1999 random noise and x 0 the final output of the model.
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"FAVIQ: FAct Verification from Information-seeking Questions",
"FAVIQ: FAct Verification from Information-seeking Questions"
] |
[
"Jungsoo Park [email protected] \nKorea University\n\n",
"Sewon Min \nUniversity of Washington\n\n",
"Jaewoo Kang [email protected] \nKorea University\n\n",
"Luke Zettlemoyer \nUniversity of Washington\n\n",
"Hannaneh Hajishirzi [email protected] \nUniversity of Washington\n\n\nAllen Institute of AI\n\n"
] |
[
"Korea University\n",
"University of Washington\n",
"Korea University\n",
"University of Washington\n",
"University of Washington\n",
"Allen Institute of AI\n"
] |
[
"Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics"
] |
Despite significant interest in developing general purpose fact checking models, it is challenging to construct a large-scale fact verification dataset with realistic real-world claims. Existing claims are either authored by crowdworkers, thereby introducing subtle biases that are difficult to control for, or manually verified by professional fact checkers, causing them to be expensive and limited in scale. In this paper, we construct a large-scale challenging fact verification dataset called FAVIQ, consisting of 188k claims derived from an existing corpus of ambiguous information-seeking questions. The ambiguities in the questions enable automatically constructing true and false claims that reflect user confusions (e.g., the year of the movie being filmed vs. being released). Claims in FAVIQ are verified to be natural, contain little lexical bias, and require a complete understanding of the evidence for verification. Our experiments show that the stateof-the-art models are far from solving our new task. Moreover, training on our data helps in professional fact-checking, outperforming models trained on the widely used dataset FEVER or in-domain data by up to 17% absolute. Altogether, our data will serve as a challenging benchmark for natural language understanding and support future progress in professional fact checking.
|
10.18653/v1/2022.acl-long.354
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[
"https://www.aclanthology.org/2022.acl-long.354.pdf"
] | 235,731,930 |
2107.02153
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db3ae3d766495361153c004b3526ad2cd0d2baf2
|
FAVIQ: FAct Verification from Information-seeking Questions
Long PapersCopyright Long PapersMay 22-27, 2022
Jungsoo Park [email protected]
Korea University
Sewon Min
University of Washington
Jaewoo Kang [email protected]
Korea University
Luke Zettlemoyer
University of Washington
Hannaneh Hajishirzi [email protected]
University of Washington
Allen Institute of AI
FAVIQ: FAct Verification from Information-seeking Questions
Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics
the 60th Annual Meeting of the Association for Computational LinguisticsLong Papers1May 22-27, 2022
Despite significant interest in developing general purpose fact checking models, it is challenging to construct a large-scale fact verification dataset with realistic real-world claims. Existing claims are either authored by crowdworkers, thereby introducing subtle biases that are difficult to control for, or manually verified by professional fact checkers, causing them to be expensive and limited in scale. In this paper, we construct a large-scale challenging fact verification dataset called FAVIQ, consisting of 188k claims derived from an existing corpus of ambiguous information-seeking questions. The ambiguities in the questions enable automatically constructing true and false claims that reflect user confusions (e.g., the year of the movie being filmed vs. being released). Claims in FAVIQ are verified to be natural, contain little lexical bias, and require a complete understanding of the evidence for verification. Our experiments show that the stateof-the-art models are far from solving our new task. Moreover, training on our data helps in professional fact-checking, outperforming models trained on the widely used dataset FEVER or in-domain data by up to 17% absolute. Altogether, our data will serve as a challenging benchmark for natural language understanding and support future progress in professional fact checking.
Introduction
Fact verification, the task of verifying the factuality of the natural language claim, is an important NLP application (Cohen et al., 2011) and has also been used to evaluate the amount of external knowledge a model has learned (Petroni et al., 2021). However, it is challenging to construct fact verification data with claims that contain realistic and implicit misinformation. Crowdsourced claims from prior work such as FEVER (Thorne et al., 2018a) are Figure 1: An example of a refute claim on FAVIQ, constructed using ambiguity in the information-seeking question, e.g., through a crossover of the year of the film being released and being filmed.
written with minimal edits to reference sentences, leading to strong lexical biases such as the overuse of explicit negation and unrealistic misinformation that is less likely to occur in real life (Schuster et al., 2019). On the other hand, data constructed by professional fact-checkers are expensive and are typically small-scale (Hanselowski et al., 2019).
In this paper, we show it is possible to use information-seeking questions (Kwiatkowski et al., 2019) and their ambiguities to construct a large-scale, challenging, and realistic fact verification dataset. Information-seeking questions are inherently ambiguous because users do not know the answers to the questions they are posing. For example, in Figure 1, the question is ambiguous because the filming of the movie and the release of the movie can both be seen as the creation time.
We introduce a new dataset FAVIQ-FAct Verification derived from Information-seeking Questions, which uses such ambiguities to generate challenging fact verification problems. For instance, the claim in Figure 1 requires the model to identify that the movie released in 2001 is in fact filmed in 2000 and to return refute. Like this, claims generated through the crossover of the disambiguation of information-seeking questions are likely to contain misinformation that real users are easily confused with. We automatically generate such claims by composing valid and invalid question-answer pairs and transforming them into textual claims using a neural model. The data is further augmented by claims from regular questionanswer annotations.
In total, FAVIQ consists of 188k claims. We manually verified a subset of claims to ensure that they are as natural as human-written claims. Our analysis shows that the claims have significantly lower lexical bias than existing crowdsourced claims; claims involve diverse types of distinct entities, events, or properties that are semantically close, being more realistic and harder to verify without a complete understanding of the evidence text.
Our experiments show that a model with no background knowledge performs only slightly better than random guessing, and the state-of-the-art model achieves an accuracy of 65%, leaving significant room for improvement. Furthermore, training on FAVIQ improves the accuracy of verification of claims written by professional fact checkers, outperforming models trained on the target data only or pretrained on FEVER by up to 17% absolute. Together, our experiments demonstrate that FAVIQ is a challenging benchmark as well as a useful resource for professional fact checking.
To summarize, our contributions are three-fold:
1. We introduce FAVIQ, a fact verification dataset consisting of 188k claims. By leveraging information-seeking questions and their natural ambiguities, our claims require the identification of entities, events, or properties that are semantically close but distinct, making the fact verification problem very challenging and realistic.
2. Our experiments show that the state-of-the-art fact verification models are far from solving FAVIQ, indicating significant room for improvement.
3. Training on FAVIQ significantly improves the verification of claims written by professional fact checkers, indicating that FAVIQ can support progress in professional fact checking.
Related Work
Fact verification Fact verification is crucial for real-world applications (Cohen et al., 2011) and as a benchmark to evaluate the knowledge in a model (Petroni et al., 2021). One line of work has studied professional fact checking, dealing with claims collected by professional fact checkers in specific domains (Vlachos and Riedel, 2014;Ferreira and Vlachos, 2016;Augenstein et al., 2019;Hanselowski et al., 2019). While such data contains realistic claims that have occurred in the real world, it is expensive to construct as it requires labor from professional fact checkers. Moreover, it is less suitable as a benchmark due to lack of a standard evidence corpus such as Wikipedia 2 and ambiguities in labels. 3 Other fact verification datasets are collected through crowdsourcing (e.g., FEVER (Thorne et al., 2018a) and its variants (Thorne et al., 2018b;Thorne and Vlachos, 2019)) by altering a word or negating the reference text to intentionally make true or false claims. This process leads to largescale datasets but with strong artifacts and unrealistic claims (Schuster et al., 2019;Thorne and Vlachos, 2019;Eisenschlos et al., 2021). Consequently, a trivial claim-only baseline with no evidence achieves near 80% (Petroni et al. (2021), verified in Section 4.1). While more recent work proposes new crowdsourcing methods that alleviate artifacts (Schuster et al., 2021;Eisenschlos et al., 2021), their claims are still written given particular evidence text, being vulnerable to subtle lexical biases that can be hard to explicitly measure.
We construct a fact verification dataset from highly ambiguous information-seeking questions. Our claims have significantly less lexical bias than other crowdsourced ones (Figure 3), contain realistic misinformation that people are likely to be confused about (Table 4), and are challenging to current state-of-the-art models (Section 4.1). Moreover, training a model on our data improves professional fact checking (Section 4.2).
QA to Verification Task Prior work has also used QA data to create entailment or fact verification benchmarks. Most make use of synthetic or annotated questions (Demszky et al., 2018;Jiang et al., 2020;Pan et al., 2021;Chen et al., 2021) 4 f Figure 2: Overview of the data creation process. The data consits of two sets (A and R). For A, we use the disambiguated question-answer pairs and generate support and refute claims from matching pairs (filmed-2000, released-2001) and crossover pairs (filmed-2001, released-2000), respectively. For R, we use the reference answer (Deckard Shaw) and the incorrect prediction from DPR (Dominic Toretto) to generate support and refute claims, respectively. f is a T5 model that transforms question-answer pairs to claims (Section 3.1.3).
while we use questions posed by real users to reflect confusions that naturally occur while seeking information. Thorne et al. (2021) use informationseeking questions, by converting yes/no questions to support/refute claims, but at a small scale and with unambiguous questions. Instead, our work uses large-scale information-seeking questions (with no restriction in answers) to claims. We are also unique in using highly ambiguous QA pairs to obtain claims that are more challenging to verify and have significantly fewer lexical cues (quantitative comparisons in Section 3.3).
Data
Data Construction
We construct FAVIQ-FAct Verification derived from Information-seeking Questions, where the model is given a natural language claim and predicts support or refute with respect to the English Wikipedia. The key idea to construct the data is to gather a set of valid and invalid questionanswer pairs (Section 3.1.2) from annotations of information-seeking questions and their ambiguities (Section 3.1.1), and then convert each questionanswer pair (q, a) to a claim (Section 3.1.3). Figure 2 presents an overview of this process.
Data Sources
We use QA data from Natural Questions (NQ, Kwiatkowski et al. (2019)) and AmbigQA . NQ is a large-scale dataset consisting of the English information-seeking questions mined from Google search queries. AmbigQA provides disambiguated question-answer pairs for NQ questions, thereby highlighting the ambiguity that is inherent in information-seeking questions. Given an ambiguous question, it provides a set of multiple distinct answers, each paired with a new disambiguated question that uniquely has that answer.
Composing Valid and Invalid QA Pairs
FAVIQ is constructed from ambiguous questions and their disambiguation (A set) and is further augmented by using unambiguous question-answer pairs (R set).
From ambiguous questions (A set)
We use the data consisting of a set of (q, {q 1 , a 1 }, {q 2 , a 2 }), where q is an information seeking question that has a 1 , a 2 as multiple distinct answers. 5 q 1 and q 2 are disambiguated questions for the answers a 1 and a 2 , i.e., q 1 has a 1 as a valid answer and a 2 as an invalid answer. We use (q 1 , a 1 ) and (q 2 , a 2 ) as valid question-answer pairs, and (q 1 , a 2 ) and (q 2 , a 1 ) as invalid question-answer pairs. This data is particularly well suited to fact checking because individual examples require identification of entities, events, or properties that are semantically close but distinct: the fact that a user asked an ambiguous question q without realizing the difference between (q 1 , a 1 ) and (q 2 , a 2 ) indicates that the distinction is non-trivial and is hard to notice without sufficient background knowledge about the topic of the question.
From regular questions (R set) We use the QA data consisting of a set of (q, a): an informationseeking question q and its answer a. We then obtain an invalid answer to q, denoted as a neg , from an off-the-shelf QA model for which we use the model from Karpukhin et al. (2020) with heuristics to obtain hard negatives but not the false negative; details provided in Appendix A. We use (q, a) and (q, a neg ) as a valid and an invalid question-answer pair, respectively. We can think of (q, a neg ) as a hard negative pair chosen adversarially from the QA model. 6 This data can be obtained on a much larger scale than the A set because annotating a single valid answer is easier than annotating disambiguations.
Transforming QA pairs to Claims
We transform question-answer pairs to claims by training a neural model which maps (q, a) to a claim that is support if and only if a is a valid answer to q, otherwise refute. We first manually convert 250 valid and invalid question-answer pairs obtained through Section 3.1.2 to claims. We then train a T5-3B model (Raffel et al., 2020), using 150 claims for training and 100 claims for validation. The model is additionally pretrained on data from Demszky et al. (2018), see Appendix A.
Obtaining silver evidence passages
We obtain silver evidence passages for FAVIQ by (1) taking the question that was the source of the claim during the data creation (either a user question from NQ or a disambiguated question from AmbigQA), (2) using it as a query for TF-IDF over the English Wikipedia, and (3) taking the top passage that contains the answer. Based on our manual verification on 100 random samples, the precision of the silver evidence passages is 70%. We provide silver evidence passages primarily for supporting training of the model, and do not explicitly evaluate passage prediction; more discussion in Appendix A. Future work may use human annotations on top of our silver evidence passages in order to further improve the quality, or evaluate passage prediction. 188k 12.0 9 10.5 13.5 Table 2: Statistics of a variety of fact verification datasets. Avg and Q1-3 are the average and quantiles of the length of the claims based on whitespace tokenization on the validation data; for FAVIQ, we report the macro-average of the A set and the R set. dataname is as large as FEVER and has a distribution of claim lengths that is much closer to that of professional fact checking datasets (SNOPES and SCIFACT).
Data Validation
In order to evaluate the quality of claims and labels, three native English speakers were given 300 random samples from FAVIQ, and were asked to: (1) verify whether the claim is as natural as a human-written claim, with three possible ratings (perfect, minor issues but comprehensible, incomprehensible), and (2) predict the label of the claim (support or refute). Validators were allowed to use search engines, and were encouraged to use the English Wikipedia as a primary source. Validators found 80.7% of the A set and 89.3% of the R set to be natural, and 0% to be incomprehensible. The rest have minor grammatical errors or typos, e.g., missing "the". In most cases the errors actually come from the original NQ questions which were human-authored, indicating that these grammatical errors and typos occur in real life. Lastly, validators achieved an accuracy of 95.0% (92.7% of A and 97.3% of R) when evaluated against gold labels in the data-this indicates high-quality of the data and high human performance. This accuracy level is slightly higher than that of FEVER (91.2%).
Data Analysis
Data statistics for FAVIQ are listed in Table 1. It has 188k claims in total, with balanced support and refute labels. We present quantitative and qualitative analyses showing that claims on FAVIQ contain much less lexical bias than other crowdsourced datasets and include misinformation that is realistic and harder to identify. S and R denotes support and refute, respectively. BOOLQ-FV indicates data from Thorne et al. (2021) that uses BOOLQ. LMI scores of FAVIQ are significantly lower than those of FEVER and FM2, indicating significantly less lexical overlap. Table 2 compares statistics of a variety of fact verification datasets: SNOPES (Hanselowski et al., 2019), SCI-FACT (Wadden et al., 2020), FEVER (Thorne et al., 2018a), FM2 (Eisenschlos et al., 2021), BOOLQ-FV (Thorne et al., 2021) and FAVIQ.
Comparison of size and claim length
FAVIQ is as large-scale as FEVER, while its distributions of claim length is much closer to claims authored by professional fact checkers (SNOPES and SCIFACT). FM2 is smaller scale, due to difficulty in scaling multi-player games used for data construction, and has claims that are slightly longer than professional claims, likely because they are intentionally written to be difficult. BOOLQ-FV is smaller, likely due to relative difficulties in collecting naturally-occurring yes/no questions.
Lexical cues in claims We further analyze lexical cues in the claims on FEVER, FM2, BOOLQ-FV and FAVIQ by measuring local mutual information (LMI; Schuster et al. (2019); Eisenschlos et al. (2021)). LMI measures whether the given bigram correlates with a particular label. More specifically, LMI is defined as:
LM I(w, c) = P (w, c) log P (w, c) P (w) · P (c) ,
where w is a bigram, c is a label, and P (·) are estimated by counting (Schuster et al., 2019).
Dataset Top Bigrams by LMI
FEVER-S is a, a film, of the, is an, in the, in a FEVER-R is only, only a, incapable of, is not, was only, is incapable A set-S on the, was the, the date, date of, in episode, is what A set-R of the, the country, at the, the episode, started in, placed at R set-S out on, on october, on june, released on, be 18, on august R set-R out in, on september, was 2015, of the, is the, released in Table 3: Top bigrams with the highest LMI for FEVER and FAVIQ. S and R denotes support and refute respectively. Highlighted bigrams indicate negative expressions, e.g., "only", "incapable" or "not".
The distributions of the LMI scores for the top-100 bigrams are shown in Figure 3. The LMI scores of FAVIQ are significantly lower than those of FEVER, FM2, and BOOLQ-FV, indicating that FAVIQ contains significantly less lexical bias.
Tables 3 shows the top six bigrams with the highest LMI scores for FEVER and FAVIQ. As highlighted, all of the top bigrams in refute claims of FEVER contain negative expressions, e.g., "is only", "incapable of", "did not". In contrast, the top bigrams from FAVIQ do not include obvious negations and mostly overlap across different labels, strongly suggesting the task has fewer lexical cues. Although there are still top bigrams from FAVIQ causing bias (e.g., related to time, such as 'on October'), their LMI values are significantly lower compared those from other datasets.
Qualitative analysis of the refute claims We also analyzed 30 randomly sampled refute claims from FAVIQ and FEVER respectively. We categorized the cause of misinformation as detailed in Appendix B, and show three most common categories for each dataset as a summary in Table 4.
On FAVIQ, 60% of the claims involve entities, events or properties that are semantically close, but still distinct. For example, they are specified with conjunctions (e.g., "was foreign minister" and "signed the treaty of versailles from germany"), or share key attributes (e.g., films with the same title). This means that relying on lexical overlap or partially understanding the evidence text would lead to incorrect predictions; one must read the full evidence text to realize that the claim is false. Furthermore, 16.7% involve events, e.g., from filing for bankruptcy for the first time to completely ceasing operations (Table 4). This requires full understanding of the underlying event and tracking of state changes (Das et al., 2019;Amini et al., 2020).
The same analysis on FEVER confirms the findings from Schuster et al. (2021); many of claims contain explicit negations (30%) and antonyms (13%), with misinformation that is less likely to occur in the real world (20%). 7
Experiments
We first evaluate state-of-the-art fact verification models on FAVIQ in order to establish baseline performance levels (Section 4.1). We then conduct experiments on professional fact-checking datasets to measure the improvements from training on FAVIQ (Section 4.2).
Baseline Experiments on FAVIQ
Models
We experiment with two settings: a zero-shot setup where models are trained on FEVER, and a standard setup where models are trained on FAVIQ. For FEVER, we use the KILT (Petroni et al., 2021) version following prior work; we randomly split the official validation set into equally sized validation and test sets, as the official test set is hidden.
All models are based on BART (Lewis et al., 2020), a pretrained sequence-to-sequence model which we train to generate either support or refute. We describe three different variants which differ in their input, along with their accuracy on FEVER by our own experiments.
Claim only BART takes a claim as the only input. Although this is a trivial baseline, it achieves an accuracy of 79% on FEVER.
TF-IDF + BART takes a concatenation of a claim and k passages retrieved by TF-IDF from Chen et al. (2017). It achieves 87% on FEVER. We choose TF-IDF over other sparse retrieval methods like BM25 (Robertson and Zaragoza, 2009) because Petroni et al. (2021) report that TF-IDF outperforms BM25 on FEVER.
DPR + BART takes a concatenation of a claim and k passages retrieved by DPR (Karpukhin et al., 2020), a dual encoder based model. It is the stateof-the-art on FEVER based on Petroni et al. (2021) and Maillard et al. (2021), achieving an accuracy of 90%.
Implementation details
We use the English Wikipedia from 08/01/2019 following KILT (Petroni et al., 2021). We take the plain text and lists provided by KILT and create a collection of passages where each passage has up to 100 tokens. This results in 26M passages. We set the number of input passages k to 3, following previous work (Petroni et al., 2021;Maillard et al., 2021). Baselines on FAVIQ are jointly trained on the A set and the R set.
Training DPR requires a positive and a negative passage-a passage that supports and does not support the verdict, respectively. We use the silver evidence passage associated with FAVIQ as a positive, and the top TF-IDF passage that is not the silver evidence passages as a negative. More training details are in Appendix C. Experiments are reproducible from https://github.com/ faviq/faviq/tree/main/codes. Table 5 reports results on FAVIQ. The overall accuracy of the baselines is low, despite their high performance on FEVER. The zero-shot performance is barely better than random guessing, indicating that the model trained on FEVER is not able to generalize to our more challenging data. When the baselines are trained on FAVIQ, the best model achieves an accuracy of 65% on the A set, indicating that existing state-of-the-art models do not solve our benchmark. 8
Results
Impact of retrieval
The performance of the claim only baseline that does not use retrieval is almost random on FAVIQ, while achieving nearly 80% accuracy on FEVER. This result suggests significantly less bias in the claims, and the relative importance of using background knowledge to solve the task. When retrieval is used, DPR outperforms TF-IDF, consistent with the finding from Petroni et al. (2021).
A set vs. R set The performance of the models on the R set is consistently higher than that on the A set by a large margin, implying that claims based on ambiguity arisen from real users are more challenging to verify than claims generated from regular question-answer pairs. This indicates clearer contrast to prior work that converts regular QA data to declarative sentences (Demszky et al., 2018;Pan et al., 2021).
Error Analysis We randomly sample 50 error cases from DPR + BART on the A set of FAVIQ and categorize them, as shown in Table 6.
• Retrieval error is the most frequent type of errors. DPR typically retrieves a passage with the correct topic (e.g., about "Lie to Me") but that is missing more specific information (e.g., the end date). We think the claim having less lexical overlap with the evidence text leads to low recall@k of the retrieval model (k = 3). 8 We additionally show and discuss the model trained on FAVIQ and tested on FEVER in Appendix D. They achieve non-trivial performance (67%) although being worse than FEVER-trained models that exploit bias in the data.
• 28% of error cases involve events. In particular, 14% involve procedural events, and 6% involve distinct events that share similar properties but differ in location or time frame.
• In 18% of error cases, retrieved evidence is valid but not notably explicit, which is naturally the case for the claims occurring in real life. FAVIQ has this property likely because it is derived from questions that are gathered independently from the evidence text, unlike prior work (Thorne et al., 2018a;Schuster et al., 2021;Eisenschlos et al., 2021) with claims written given the evidence text.
• 16% of the failure cases require multi-hop inference over the evidence. Claims in this category usually involve procedural events or compositions (e.g. "is Seth Curry's brother" and "played for Davidson in college"). This indicates that we can construct a substantial portion of claims requiring multi-hop inference without having to make data that artificially encourages such reasoning (Yang et al., 2018;Jiang et al., 2020).
• Finally, 10% of the errors were made due to a subtle mismatch in properties, e.g., in the example in Figure 6, the model makes a decision based on "required minimum number" rather than "exact number" of a particular brand.
Professional Fact Checking Experiments
We use two professional fact-checking datasets.
SNOPES (Hanselowski et al., 2019) consists of 6,422 claims, authored and labeled by professional fact-checkers, gathered from the Snopes website. 9 We use the official data split.
SCIFACT (Wadden et al., 2020) consists of 1,109 claims based on scientific papers, annotated by domain experts. As the official test set is hidden, we use the official validation set as the test set, and separate the subset of the training data as the validation set to be an equal size as the test set.
For both datasets, we merge not enough info (NEI) to refute, following prior work that converts the 3-way classification to the 2-way classification Sathe et al., 2020;Petroni et al., 2021).
Models
As in Section 4, all models are based on BART which is given a concatenation of the claim and Table 6: Error analysis on 50 samples of the A set of FAVIQ validation data. C and E indicate the claim and retrieved evidence passages from DPR, respectively. Gold and blue indicate gold label and prediction by the model, respectively. The total exceeds 100% as one example may fall into multiple categories.
the evidence text and is trained to generate either support or refute. For SNOPES, the evidence text is given in the original data. For SCIFACT, the evidence text is retrieved by TF-IDF over the corpus of abstracts from scientific papers, provided in the original data. We use TF-IDF over DPR because we found DPR works poorly when the training data is very small.
We consider two settings. In the first setting, we assume the target training data is unavailable and compare the model trained on FEVER and FAVIQ in a zero-shot setup. In the second setting, we allow training on the target data and compare the model trained on the target data only and the model with the transfer learning-pretrained on either FEVER or FAVIQ and finetuned on the target data.
To explore models pretrained on NEI labels, we add a baseline that is trained on a union of the KILT version of FEVER and NEI data from the original FEVER from Thorne et al. (2018a). For FAVIQ, we also conduct an ablation that includes the R set only or the A set only.
Implementation details When using TF-IDF for SCIFACT, we use a sentence as a retrieval unit, and retrieve the top 10 sentences, which average length approximates that of 3 passages from Wikipedia. When using the model trained on either FEVER or FAVIQ, we use DPR + BART by default, which gives the best result in Section 4.1.
As an exception, we use TF-IDF + BART on SCI-FACT for a more direct comparison with the model trained on the target data only that uses TF-IDF.
When the models trained on FEVER or FAVIQ are used for professional fact checking, we find models are poorly calibrated, likely due to a domain shift, as also observed by Kamath et al. (2020) and Desai and Durrett (2020). We therefore use a simplified version of Platt scaling, a post-hoc calibration method (Platt et al., 1999;Guo et al., 2017;Zhao et al., 2021). Given normalized probabilities of support and refute, denoted as p s and p r , modified probabilities p s and p r are obtained via:
p s p r = Softmax p s + γ p r ,
where −1 < γ < 1 is a hyperparameter tuned on the validation set. Table 7 reports accuracy on professional factchecking datasets, SNOPES and SCIFACT.
Results
Impact of transfer learning We find that transfer learning is effective-pretraining on large, crowdsourced datasets (either FEVER or FAVIQ) and finetuning on the target datasets always helps. Improvements are especially significant on SCI-FACT, likely because its data size is smaller. Using the target data is still important-models finetuned on the target data outperform zero-shot models by up to 20%. This indicates that crowdsourced data cannot completely replace professional fact checking data, but transfer learning from crowdsourced data leads to significantly better professional fact checking performance. Table 7: Accuracy on the test set of professional factchecking datasets. Training on FAVIQ significantly improves the accuracy on SNOPES and SCIFACT, both in the zero-shot setting and in the transfer learning setting.
FAVIQ vs. FEVER Models that are trained on FAVIQ consistently outperform models trained on FEVER, both with and without the target data, by up to 4.8% absolute. This demonstrates that FAVIQ is a more effective resource than FEVER for professional fact-checking. The model on FEVER is more competitive when NEI data is included, by up to 3% absolute. While the models on FAVIQ outperform models on FEVER even without NEI data, future work can possibly create NEI data in FAVIQ for further improvement.
Impact of the A set in FAVIQ The performance of the models that use FAVIQ substantially degrades when the A set is excluded. Moreover, models trained on the A set (without R set) perform moderately well despite its small scale, e.g., on SNOPES, achieving the second best performance following the model trained on the full FAVIQ. This demonstrates the importance of the A set created based on ambiguity in questions.
SNOPES benefits more from the A set than the R set, while SCIFACT benefits more from the R set than the A set. This is likely because SCIFACT is much smaller-scale (1k claims) and thus benefits more from the larger data like the R set. This suggests that having both the R set and the A set is important for performance.
Conclusion & Future Work
We introduced FAVIQ, a new fact verification dataset derived from ambiguous information-seeking questions. We incorporate facts that real users were unaware of when posing the question, leading to false claims that are more realistic and challenging to identify without fully understanding the context. Our extensive analysis shows that our data contains significantly less lexical bias than previous fact checking datasets, and include refute claims that are challenging and realistic. Our experiments showed that the state-of-the-art models are far from solving FAVIQ, and models trained on FAVIQ lead to improvements in professional fact checking. Altogether, we believe FAVIQ will serve as a challenging benchmark as well as support future progress in professional fact-checking.
We suggest future work to improve the FAVIQ model with respect to our analysis of the model prediction in Section 4.1.2, such as improving retrieval, modeling multi-hop inference, and better distinctions between entities, events and properties. Moreover, future work may investigate using other aspects of information-seeking questions that reflect facts that users are unaware of or easily confused with. For example, one can incorporate false presuppositions in questions that arise when users have limited background knowledge (Kim et al., 2021). As another example, one can explore generating NEI claims by leveraging unanswerable information-seeking questions. Furthermore, FAVIQ can potentially be a challenging benchmark for the claim correction, a task recently studied by Thorne and Vlachos (2021) that requires a model to correct the refute claims.
A Details in Data Construction
Details of obtaining a neg We obtain an invalid answer to the question, denoted as a neg , using an off-the-shelf QA model, for which we use DPR followed by a span extractor (Karpukhin et al., 2020).
The most naive way to obtain a neg is to take the highest scored prediction that is not equal to a. We however found such prediction is likely to be a valid answer to q, either because it is semantically the same as a, or because the ambiguity in the question leads to multiple distinct valid answers. We therefore use two heuristics that we find greatly reduce such false negatives. First, instead of taking the top incorrect prediction, we obtain the top k predictions p 1 ...p k from the model and randomly sample one from {p 1 ...p k } \ {a}. We use k = 50.
Although this is not a fundamental solution to remove false negatives, it significantly alleviates the problem, drastically dropping the portion of false negatives from 14% to 2% based on our manual verification on 50 random samples. Second, we train a neural model that is given a pair of the text and classifies whether they are semantically equivalent or not. This model is based on T5-large, trained and validated respectively on 150 and 100 pairs of (a, p i ) (i = 1...k) which we manually label. We then exclude the predictions in {p 1 ...p k } which are classified as semantically equivalent to a by the classifier.
QA-to-claim converter We use a pretrained sequence-to-sequence model trained on a small number of our own annotations. We first manually write 250 claims given valid or invalid questionanswer pairs. We then train a T5-3B model (Raffel et al., 2020), using 150 claims for training and 100 claims for validation. Each question-answer pair is fed into T5 with special tokens question: and answer:, respectively.
When training, we evaluate on the validation data every epoch and stop training when the validation accuracy does not increase for ten epochs. The accuracy is measured by the exact match score of the generated and the reference text after normalization, which we found to correlate well with the quality of the generated claims. The final model we train achieves 83% on the validation data. At inference time, we filter claims that do not contain the answer string, which may happen when the question is overly specific.
Why don't we evaluate evidence prediction? Unlike FEVER (Thorne et al., 2018a), which includes evidence prediction as part of the task, our paper does not report the evidence prediction performance and mainly reports the classification accuracy. There are three reasons for this change:
1. As claims on FAVIQ were written independent from any reference text, gold evidence text must be gathered through a separate process, which greatly increases the cost. This is different from other annotated fact checking datasets where a crowdworker wrote a claim based on the reference text and therefore the same reference text can be considered as gold evidence.
2. Finding gold evidence text is an inherently incomplete process; no human can get close to, or even measure the upperbound. Therefore, even after exhaustive human annotation, evaluation against annotated evidence leads to significant amount of false negatives. For example, when manually evaluating the top negatives of TF-IDF on 50 random samples from FEVER, 42% are false negatives.
3. Including evidence prediction as part of evaluation significantly restricts the approach models can take. For instance, one may choose not to use the text corpus provided in the dataset (e.g., Wikipedia), and decide to use other sources such as structured data (e.g. knowledge bases) or implicit knowledge stored in large neural models.
Nonetheless, as described in Section 3.1.4, we still provide the silver evidence passages which is useful to train a model, e.g., DPR, and supports future work to evaluate the evidence prediction accuracy.
B Analysis of refute Claims
We randomly sample 30 refute claims from FAVIQ and FEVER, respectively, and categorize the cause of the misinformation, as shown in Table 8. See Section 3.3 for discussion.
C Details of Experiments
DPR training for FEVER As FEVER provides the annotated evidence passage, we use it as a positive training example. We obtain a negative by querying the claim to TF-IDF and taking the passage that is not the positive passage and has the second highest score. We initially considered using the negative with the highest score, but found that many of them (37%) are false negatives based on our manual evaluation of 30 random samples. This is likely due to incomprehensive evidence annotation as discussed in Appendix A. We find using the negative with the second highest instead decreases the portion of false negatives from 37% to 13%.
Other details Our implementations are based on PyTorch 10 (Paszke et al., 2019) and Huggingface Transformers 11 (Wolf et al., 2020). When training a BART-based model, we map support and refute labels to the words 'true' and 'false' respectively so that each label is mapped to a single token. This choice was made against mapping to 'support' and 'refute'because the BART tokenizer maps 'refute' into two tokens, making it difficult to compare probabilities of support and refute.
By default, we use a batch size of 32, a maximum sequence length of 1024, and 500 warmup steps 10 https://pytorch.org/ 11 https://github.com/huggingface/ transformers using eight 32GB GPUs. For SCIFACT, we use a batch size of 8 and no warmup steps using four 32G GPUs. We tune the learning rate in between {7e-6, 8e-6, 9e-6, 1e-5} on the validation data. Table 9 reports the model performance when trained on FAVIQ and tested on FEVER. The best-performing model achieves non-trivial performance (67%). However, their overall performance is not as good as model performance when trained on FEVER, likely because the models do not exploit the bias in the FEVER dataset. Nonetheless, we underweight the test performance on FEVER due to known bias in the data.
D Additional Experiments
Figure 3 :
3Plot of LMI scores of top 100 predictive bigrams for FEVER, FM2, BOOLQ-FV and FAVIQ (macro-averaged over the A set and the R set).
(2019); Eisenschlos et al. Conjunctions (33.3%) C: Johannes bell was the foreign minister that signed the treaty of versailles from germany. / E: Johannes bell served as Minister of Colonial Affairs ... He was one of the two German representatives who signed the Treaty of Versailles. Shared attributes (26.7%) C: Judi bowker played andromeda in the 2012 remake of the 1981 film clash of the titans called wrath of the titans. E: Judi bowker ... Clash of the Titans (1981). Procedural event (16.7%) C: Mccrory's originally filed for bankruptcy on february 2002. / E: McCrory Stores ... by 1992 it filed for bankruptcy. ... In February 2002 the company ceased operation. Negation (30.0%) C: Southpaw hasn't been released yet. E: Southpaw is an American sports drama film released on July 24, 2015. Cannot find potential cause (20.0%) C: Mutiny on the Bounty is Dutch. E: Mutiny on the Bounty is a 1962 American historical drama film. Antonym (13.3%) C: Athletics lost the world series in 1989. E: The 1989 World Series ... with the Athletics sweeping the Giants.
Athletics lost the world series in 1989. E: The 1989 World Series was ... with the Athletics sweeping the Giants in four games. Johannes bell was the foreign minister that signed the treaty of versailles from germany. / E: Johannes bell served as Minister of Colonial Affairs ... He was one of the two German representatives who signed the Treaty of Versailles. Judi bowker played andromeda in the 2012 remake of the 1981 film clash of the titans called wrath of the titans. / E: Judi bowker ... Clash of the Titans (Mccrory's originally filed for bankruptcy on february 2002. / E: McCrory Stores ... by 1992 it filed for bankruptcy. ... In February 2002 the company ceased operation. Tyler, the Creator is the name of the song at the end of who dat boy. E: "Who Dat Boy" is a song by American rapper Tyler, the Creator. Mutiny on the Bounty is Dutch. (from FEVER) E: Mutiny on the Bounty is a 1962 American Technicolor epic historical drama film. Annotation error 10.0 20.0 C: Pasek and paul were the individuals that wrote the lyrics to the greatest showman.
-DPR followed by a span extraction model. We choose a negTotal Support Refute
Train
A
17,008
8,504
8,504
R 140,977
70,131 70,846
Dev
A
4,260
2,130
2,130
R
15,566
7,739
7,827
Test
A
4,688
2,344
2,344
R
5,877
2,922
2,955
Table 1 :
1FAVIQ statistics. A includes claims derived
from ambiguous questions, while R includes claims
from regular question-answer pairs.
Table 4 :
4Three most common categories based on 30 refute claims randomly sampled from the validation set, for FAVIQ (top) and FEVER (bottom) respectively. Full statistics and examples in Appendix B. C and E indicate the claim and evidence text, respectively. Refute claims in FAVIQ are more challenging, not containing explicit negations or antonyms.
Table 5 :
5Fact verification accuracy on FAVIQ. DPR + BART achieves the best accuracy; however, there is overall significant room for improvement.
Table 8 :
8Categorization of 30 refute claims on FAVIQ and FEVER, randomly sampled from the validation set. C and E indicate the claim and evidence text, respectively. Examples are from FAVIQ unless otherwise specified.Model
Dev
Test
Clain only BART
47.2
48.3
TF-IDF + BART
67.8
66.6
DPR + BART
67.2
66.5
Table 9 :
9Fact verification accuracy on FEVER of different models when trained on FAVIQ.
For this reason, prior work on professional fact checking assumes gold evidence document.3 Most claims fall into the mixture label, rather than support or refute.4 Annotated questions are simulated by crowdworkers given the evidence text and the answer, having largely different distributions from information-seeking questions(Lee et al., 2019;Gardner et al., 2019).
If q has more than two distinct answers, we sample two. This is to construct a reasonable number of claims per q.
It is possible that the R set contains bias derived from the use of DPR. We thus consider the R set as a source for data augmentation, while A provides the main data.
For instance, consider the claim "Mutiny on the Bounty is Dutch" inTable 4. There is no Dutch producer, director, writer, actors, or actress in the film-we were not able to find a potential reason that one would believe that the film is Dutch.
https://www.snopes.com
AcknowledgementsWe thank Dave Wadden, James Thorn and Jinhyuk Lee for discussion and feedback on the paper. We thank James Lee, Skyler Hallinan and Sourojit Ghosh for their help in data validation. This work was supported by NSF IIS-2044660, ONR N00014-18-1-2826, an Allen Distinguished Investigator Award, a Sloan Fellowship, National Research Foundation of Korea (NRF-2020R1A2C3010638) and the Ministry of Science and ICT, Korea, under the ICT Creative Consilience program (IITP-2022-2020-0-01819).
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Active learning promises to alleviate the massive data needs of supervised machine learning: it has successfully improved sample efficiency by an order of magnitude on traditional tasks like topic classification and object recognition. However, we uncover a striking contrast to this promise: across 5 models and 4 datasets on the task of visual question answering, a wide variety of active learning approaches fail to outperform random selection. To understand this discrepancy, we profile 8 active learning methods on a per-example basis, and identify the problem as collective outliers -groups of examples that active learning methods prefer to acquire but models fail to learn (e.g., questions that ask about text in images or require external knowledge). Through systematic ablation experiments and qualitative visualizations, we verify that collective outliers are a general phenomenon responsible for degrading pool-based active learning. Notably, we show that active learning sample efficiency increases significantly as the number of collective outliers in the active learning pool decreases. We conclude with a discussion and prescriptive recommendations for mitigating the effects of these outliers in future work.
|
10.18653/v1/2021.acl-long.564
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Mind Your Outliers! Investigating the Negative Impact of Outliers on Active Learning for Visual Question Answering
August 1-6, 2021
Siddharth Karamcheti Ranjay
Department of Computer Science
Stanford University
Krishna Li
Department of Computer Science
Stanford University
Fei-Fei Christopher [email protected]
Department of Computer Science
Stanford University
D Manning [email protected]
Department of Computer Science
Stanford University
Mind Your Outliers! Investigating the Negative Impact of Outliers on Active Learning for Visual Question Answering
Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing
the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language ProcessingAugust 1-6, 20217265
Active learning promises to alleviate the massive data needs of supervised machine learning: it has successfully improved sample efficiency by an order of magnitude on traditional tasks like topic classification and object recognition. However, we uncover a striking contrast to this promise: across 5 models and 4 datasets on the task of visual question answering, a wide variety of active learning approaches fail to outperform random selection. To understand this discrepancy, we profile 8 active learning methods on a per-example basis, and identify the problem as collective outliers -groups of examples that active learning methods prefer to acquire but models fail to learn (e.g., questions that ask about text in images or require external knowledge). Through systematic ablation experiments and qualitative visualizations, we verify that collective outliers are a general phenomenon responsible for degrading pool-based active learning. Notably, we show that active learning sample efficiency increases significantly as the number of collective outliers in the active learning pool decreases. We conclude with a discussion and prescriptive recommendations for mitigating the effects of these outliers in future work.
Introduction
Today, language-equipped vision systems such as VizWiz, TapTapSee, BeMyEyes, and CamFind are actively being deployed across a broad spectrum of users. 1 As underlying methods improve, these systems will be expected to operate over diverse visual environments and understand myriad language inputs (Bigham et al., 2010;Tellex et al., 2011;Mei et al., 2016;Anderson et al., 2018b;Park et al., 2019). Visual Question Answering (VQA), the task of answering questions about Figure 1: We systematically evaluate active learning on VQA datasets and isolate their inability to perform better than random sampling due to the presence of collective outliers. Active learning methods prefer to acquire these outliers, which are hard and often impossible for models to learn. We show that Dataset Maps, like the one shown here, can heuristically identify these collective outliers as examples assigned low model confidence and prediction variability during training. visual inputs, is a popular benchmark used to evaluate progress towards such open-ended systems (Agrawal et al., 2015;Krishna et al., 2017;Gordon et al., 2018;Hudson and Manning, 2019). Unfortunately, today's VQA models are data hungry: Their performance scales monotonically with more train-ing data (Lu et al., 2016;Lin and Parikh, 2017), motivating the need for data acquisition mechanisms such as active learning, which maximize performance while minimizing expensive data labeling.
While active learning is often key to effective data acquisition when such labeled data is difficult to obtain (Lewis and Catlett, 1994;Tong and Koller, 2001;Culotta and McCallum, 2005;Settles, 2009), we find that 8 modern active learning methods (Gal et al., 2017;Siddhant and Lipton, 2018;Lowell et al., 2019) show little to no improvement in sample efficiency across 5 models on 4 VQA datasets -indeed, in some cases performing worse than randomly selecting data to label. This finding is in stark contrast to the successful application of active learning methods on a variety of traditional tasks, such as topic classification (Siddhant and Lipton, 2018;Lowell et al., 2019), object recognition (Deng et al., 2018), digit classification (Gal et al., 2017), and named entity recognition (Shen et al., 2017). Our negative results hold even when accounting for common active learning ailments: cold starts, correlated sampling, and uncalibrated uncertainty. We mitigate the cold start challenge of needing a representative initial dataset by varying the size of the seed set in our experiments. We account for sampling correlated data within a given batch by including Core-Set selection (Sener and Savarese, 2018) in the set of active learning methods we evaluate. Finally, we use deep Bayesian active learning to calibrate model uncertainty to high-dimensional data (Houlsby et al., 2011;Gal and Ghahramani, 2016;Gal et al., 2017).
After concluding that negative results are consistent across all experimental conditions, we investigate active learning's ineffectiveness on VQA as a data problem and identify the existence of collective outliers (Han and Kamber, 2000) as the source of the problem. Leveraging recent advances in model interpretability, we build Dataset Maps (Swayamdipta et al., 2020), which distinguish between collective outliers and useful data that improve validation set performance (see Figure 1). While global outliers deviate from the rest of the data and are often a consequence of labeling error, collective outliers cluster together; they may not individually be identifiable as outliers but collectively deviate from other examples in the dataset. For instance, VQA-2 (Goyal et al., 2017) is riddled with collections of hard questions that require external knowledge to answer (e.g., "What is the symbol on the hood often associated with?") or that ask the model to read text in the images (e.g., "What is the word on the wall?"). Similarly, GQA (Hudson and Manning, 2019) asks underspecified questions (e.g., "what is the person wearing?" which can have multiple correct answers). Collective outliers are not specific to VQA, but can similarly be found in many open-ended tasks, including visual navigation (Anderson et al., 2018b) (e.g., "Go to the grandfather clock" requires identifying rare grandfather clocks), and open-domain question answering (Kwiatkowski et al., 2019), amongst others.
Using Dataset Maps, we profile active learning methods and show that they prefer acquiring collective outliers that models are unable to learn, explaining their poor improvements in sample efficiency relative to random sampling. Building on this, we use these maps to perform ablations where we identify and remove outliers iteratively from the active learning pool, observing correlated improvements in sample efficiency. This allows us to conclude that collective outliers are, indeed, responsible for the ineffectiveness of active learning for VQA. We end with prescriptive suggestions for future work in building active learning methods robust to these types of outliers.
Related Work
Our work tests the utility of multiple recent active learning methods on the open-ended understanding task of VQA. We draw on the dataset analysis literature to identify collective outliers as the bottleneck hindering active learning methods in this setting.
Active Learning. Active learning strategies have been successfully applied to image recognition (Joshi et al., 2009;Sener and Savarese, 2018), information extraction (Scheffer et al., 2001;Finn and Kushmerick, 2003;Jones et al., 2003;Culotta and McCallum, 2005), named entity recognition (Hachey et al., 2005;Shen et al., 2017), semantic parsing (Dong et al., 2018), and text categorization (Lewis and Gale, 1994;Hoi et al., 2006). However, these same methods struggle to outperform a random baseline when applied to the task of VQA (Lin and Parikh, 2017;Jedoui et al., 2019). To study this discrepancy, we systematically apply 8 diverse active learning methods to VQA, including methods that use model uncertainty (Abramson and Freund, 2004;Collins et al., 2008;Joshi et al., 2009), Bayesian uncertainty (Gal andGhahramani, 2016;Kendall andGal, 2017), disagreement (Houlsby et al., 2011;Gal et al., 2017), and Core-Set selection (Sener and Savarese, 2018).
Visual Question Answering. Progress on VQA has been heralded as a marker for progress on general open-ended understanding tasks, resulting in several benchmarks (Agrawal et al., 2015;Malinowski et al., 2015;Ren et al., 2015a;Goyal et al., 2017;Krishna et al., 2017;Suhr et al., 2019;Hudson and Manning, 2019) and models (Zhou et al., 2015;Fukui et al., 2016;Lu et al., 2016;Zhu et al., 2016;Wu et al., 2016;Anderson et al., 2018a;Tan and Bansal, 2019;Chen et al., 2020). To ensure that our negative results are not dataset or model-specific, we sample 4 datasets and 5 representative models, each utilizing unique visual and linguistic features and employing different inductive biases.
Interpreting and Analyzing Datasets. Given the prevalence of large datasets in modern machine learning, it is critical to assess dataset properties to remove redundancies (Gururangan et al., 2018;Li and Vasconcelos, 2019) or biases (Torralba and Efros, 2011;Khosla et al., 2012;Bolukbasi et al., 2016), both of which negatively impact sample efficiency. Prior work has used training dynamics to find examples which are frequently forgotten (Krymolowski, 2002;Toneva et al., 2019) versus those that are easy to learn (Bras et al., 2020). This work suggests using two model-specific measures confidence and prediction variance -as indicators of a training example's "learnability" (Chang et al., 2017;Swayamdipta et al., 2020). Dataset Maps (Swayamdipta et al., 2020), a recently introduced framework uses these two measures to profile datasets to find learnable examples. Unlike prior datasets analyzed by Dataset Maps that have a small number of global outliers as hard examples, we discover that VQA datasets contain copious amounts of collective outliers, which are difficult or even impossible for models to learn.
Active Learning Experimental Setup
We adopt the standard pool-based active learning setup from prior work (Lewis and Gale, 1994;Settles, 2009;Gal et al., 2017;Lin and Parikh, 2017), consisting of a model M, initial seed set of labeled examples (x i , y i ) ∈ D seed used to initialize M, an unlabeled pool of data D pool , and an acquisition function A(x, M). We run active learning over a series of acquisition iterations T where at each iteration we acquire a batch of B new examples per:x ∈ D pool to label per x = arg max x∈D pool A(x, M).
Acquiring an example often refers to using an oracle or human expert to annotate a new example with a correct label. We follow prior work to simulate an oracle using existing datasets, forming D seed from a fixed percentage of the full dataset, and using the remainder as D pool (Gal et al., 2017;Lin and Parikh, 2017;Siddhant and Lipton, 2018). We re-train M after each acquisition iteration.
Prior work has noted the impact of seed set size on active learning performance (Lin and Parikh, 2017;Misra et al., 2018;Jedoui et al., 2019). We run multiple active learning evaluations with varying seed set sizes (ranging from 5% to 50% of the full pool size). We keep the size of each acquisition batch B to a constant 10% of the overall pool size.
Models
Visual Question Answering (VQA) requires reasoning over two modalities: images and text. Most models use feature "backbones" (e.g., features from object recognition models pretrained on Ima-geNet, and pretrained word vectors for text). We evaluate with a representative sample of existing VQA models, including the following: 2
LogReg is a logistic regression model that uses either ResNet-101 or Faster R-CNN image features with mean-pooled GloVe question embeddings (Pennington et al., 2014). Although these models 7268 are not as performant as the subsequent models, logistic regression has been effective on VQA (Suhr et al., 2019), and is pervasive in the active learning literature (Schein and Ungar, 2007;Yang and Loog, 2018;Mussmann and Liang, 2018).
LSTM-CNN is a standard model introduced with VQA-1 (Agrawal et al., 2015). We use more performant ResNet-101 features instead of the original VGGNet features as our visual backbone.
BUTD (Bottom-Up Top-Down Attention) uses object-based features in tandem with attention over objects (Anderson et al., 2018a). BUTD won the 2017 VQA Challenge (Teney et al., 2018), and has been a consistent baseline for recent work in VQA.
LXMERT is a large multi-modal transformer model that uses BUTD's object features and contextualized BERT (Devlin et al., 2019) language features (Tan and Bansal, 2019). LXMERT is pretrained on a corpus of aligned image-and-textual data spanning MS COCO, Visual Genome, VQA-2, NLVR-2, and GQA (Lin et al., 2014;Krishna et al., 2017;Goyal et al., 2017;Suhr et al., 2019;Hudson and Manning, 2019), initializing a cross-modal representation space conducive to fine-tuning. 3
Acquisition Functions
Several active learning methods have been developed to account for different aspects of the machine learning training pipeline: while some acquire examples with high aleotoric uncertainty (Settles, 2009) (having to do with the natural uncertainty in the data) or epistemic uncertainty (Gal et al., 2017) (having to do with the uncertainty in the modeling/learning process), others attempt to acquire examples that reflect the distribution of data in the pool (Sener and Savarese, 2018). We sample a diverse set of these methods:
Random Sampling serves as our baseline passive approach for acquiring examples.
Least Confidence acquires examples with lowest model prediction probability (Settles, 2009).
Entropy acquires examples with the highest entropy in the model's output (Settles, 2009).
MC-Dropout Entropy (Monte-Carlo Dropout with Entropy acquisition) acquires examples with
high entropy in the model's output averaged over multiple passes through a neural network with different dropout masks (Gal and Ghahramani, 2016). This process is a consequence of a theoretical casting of dropout as approximate Bayesian inference in deep Gaussian processes.
BALD (Bayesian Active Learning by Disagreement) builds upon Monte-Carlo Dropout by proposing a decision theoretic objective; it acquires examples that maximise the decrease in expected posterior entropy (Houlsby et al., 2011;Gal et al., 2017;Siddhant and Lipton, 2018) -capturing "disagreement" across different dropout masks.
Core-Set
Selection samples examples that capture the diversity of the data pool (Sener and Savarese, 2018;Coleman et al., 2020). It acquires examples to minimize the distance between an example in the unlabeled pool to its closest labeled example. Since Core-Set selection operates over a representation space (and not an output distribution, like prior strategies) and VQA models operate over two modalities, we employ three Core-Set variants: Core-Set (Language) and Core-Set (Vision) operate over their respective representation spaces while Core-Set (Fused) operates over the "fused" vision and language representation space.
Experimental Results
We evaluate the 8 active learning strategies across the 5 models described in the previous section. Figures 2-5 show a representative sample of active learning results across datasets. Due to space constraints, we only visualize 4 active learning strategies -Least-Confidence, BALD, CoreSet-Fused, and the Random Baseline -using 3 models (LSTM-CNN, BUTD, LXMERT). 4 Results and trends are consistent across the different acquisition functions, models and seed set sizes (see the appendix for results with other models, acquisition functions, and seed set sizes). We now go on to provide descriptions of the datasets we evaluate against, and the corresponding results. : Results for varied active learning methods on VQA-Sports, a simplified VQA dataset. Strategies perform on par with or worse than the random baseline, when using 10% of the full dataset as the seed set. : Results for the full VQA-2 dataset, also using 10% of the full dataset as a seed set. Similar to the plot above, all active learning methods perform similar to a random baseline. : Results on VQA-2 using 50% of the dataset as a seed set. While methods are relatively better when using a larger seed set-confirming results from (Lin and Parikh, 2017)-no methods outperform random. : Results on GQA using 10% of the dataset for the seed set. Even with different question structures, the above trends hold, with strategies performing worse than or equivalent to random. Acquisitions by Difficulty BALD Figure 6: We visualize the difference in acquisition preferences between random and active learning acquisitions (least confidence and BALD) across multiple iterations. Active learning methods prefer to sample impossible examples which models are unable to learn, hurting sample efficiency relative to the random baseline.
5 0 0 1 K 1 . 5 K 2 K 2 . 5 K 3 K 3 . 5 K 4 K 4 . 5 K 5 K Number of4 0 K 8 0 K 1 2 0 K 1 6 0 K 2 0 0 K 2 4 0 K 2 8 0 K 3 2 0 K 3 6 0 K 4 0 0 K Number of Training Examples 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Validation Accuracy LSTM-CNN -VQA-2 Random Baseline Least-Confidence BALD Core-Set (Fused) 4 0 K 8 0 K 1 2 0 K 1 6 0 K 2 0 0 K 2 4 0 K 2 8 0 K 3 2 0 K 3 6 0 K 4 0 0 K Number of7270 4 0 K 8 0 K 1 2 0 K 1 6 0 K 2 0 0 K 2 4 0 K 2 8 0 K 3 2 0 K 3 6 0
Simplified VQA Datasets
One complexity of VQA is the size of the output space and the number of examples present (Agrawal et al., 2015;Goyal et al., 2017); VQA-2 has 400k training examples, and in excess of 3k possible answers (see Table 1). However, prior work in active learning focuses on smaller datasets like the 10-class MNIST dataset (Gal et al., 2017), binary classification (Siddhant and Lipton, 2018), or small-cardinality (≤ 20 classes) text categorization (Lowell et al., 2019). To ensure our results and conclusions are not due to the size of the output space, we build two meaningful, but narrow-domain VQA datasets from subsets of VQA-2. These simplified datasets reduce the complexity of the underlying learning problem and provide a fair comparison to existing active learning literature.
VQA-Sports.
We generate VQA-Sports by compiling a list of 20 popular sports (e.g., soccer, football, tennis, etc.) in VQA-2, and restricting the set of questions to those with answers in this list. We picked the sports categories by ranking the GloVe vector similarity between the word "sports" to answers in VQA-2, and selected the 20 most commonly occurring answers.
VQA-Food. We generate the VQA-Food dataset similarly, compiling a list of the 20 commonly occurring food categories by GloVe vector similarity to the word "food."
Results. Figure 2 presents results for VQA-Sports, with an initial seed set restricted to 10% of the total pool (500 examples). The appendix reports similar results on VQA-Food. For LSTM-CNN, Least-Confidence appears to be slightly more sample efficient, while all other strategies perform on par with or worse than random. For BUTD, all methods are on par with random; for LXMERT, they perform worse than random. Generally on VQA-Sports, active learning performance varies, but fails to outperform random acquisition.
VQA-2
VQA-2 is the canonical dataset for evaluating VQA models (Goyal et al., 2017). In keeping with prior work (Anderson et al., 2018a; Tan and Bansal, 2019), we filter the training set to only include answers that appear at least 9 times, resulting in 3130 unique answers. Unlike traditional VQA-2 evaluation, which treats the task as a multi-label binary classification problem, we follow prior active learning work on VQA (Lin and Parikh, 2017), which formulates it as a multi-class classification problem, enabling the use of acquisition functions such as uncertainty sampling and BALD. the right of?". We use the standard GQA training set of 943k questions, 900k of which we use for the active learning pool.
Results
Results. Figure 5 shows results on GQA using a seed set of 10% of the full pool (90k examples). Despite its notable differences in question structure to VQA-2, active learning still performs on par with or slightly worse than random.
Analysis via Dataset Maps
The previous section shows that active learning fails to improve over random acquisition on VQA across models and datasets. A simple question remains -why? One hypothesis is that sample inefficiency stems from the data itself: there is only a 2% gain in validation accuracy when training on half versus the whole dataset. Working from this, we characterize the underlying datasets using Dataset Maps (Swayamdipta et al., 2020) and discover that active learning methods prefer sampling "hard-tolearn" examples, leading to poor performance.
Mapping
Collective Outliers
This leaves two questions: 1) can we characterize these "hard" examples, and 2) are these examples responsible for the ineffectiveness of active learning on VQA? We first identify hard-to-learn examples as collective outliers and explain why active learning methods prefer to acquire them. Next, we perform ablation experiments, removing these outliers from the active learning pool iteratively, and demonstrate a corresponding boost in sample efficiency relative to random acquisition. (Figure 7), in VQA-2, we identify clusters of hard-to-learn examples that require optical character recognition (OCR) for reasoning about text (e.g., "What is the first word on the black car?"); another cluster requires external knowledge to answer ("What is the symbol on the hood often associated with?"). In GQA, we identify different clusters of collective outliers; one cluster stems from innate underspecification (e.g., "what is on the shelf?" with multiple objects present on the shelf); another cluster requires multiple reasoning hops difficult for current models (e.g., "What is the vehicle that is driving down the road the box is on the side of?"). We sample 100 random "hard-to-learn" examples from both VQA-2 and GQA and find that 100% of the examples belong to one of the two aforementioned collectives. Since hard-to-learn examples constitute 25-30% of the data pool, active learning methods cannot avoid them. Uncertainty-based methods (e.g., Least-Confidence, Entropy, Monte-Carlo Dropout) identify them as valid acquisition targets because models lack the capacity to correctly answer these examples, assigning low confidence and high uncertainty. Disagreementbased methods (e.g., BALD) are similar; model confidence is generally low but high variance (lower middle/lower right of the Dataset Maps). Finally, diversity methods (e.g., Core-Set selection) identify these examples as different enough from the existing pool to warrant acquisition, but fail to learn meaningful representations, fueling a vicious cycle wherein they continue to pick these examples.
Ablating Outliers. To verify that collective outliers are responsible for the degradation of active learning performance, we re-run our experiments using active learning pools with varying numbers of outliers removed. To remove these outliers, we sort and remove all examples in the data pool using the product of their model confidence and prediction variability (x and y-axis values of the Dataset Maps). We systematically remove examples with a low product value and observe how active learning performance changes (see Figure 8).
We observe a 2-3x improvement in sample efficiency when removing 50% of the entire data pool, consisting mainly of collective outliers (Figure 8c). This improvement decreases if we only remove 25% of the full pool (Figure 8b), and further degrades if we remove only 10% (Figure 8a). This ablation demonstrates that active learning methods are more sample efficient than the random baseline when collective outliers are absent from the unlabelled pool.
Discussion and Future Work
This paper asks a simple question -why does the modern neural active learning toolkit fail when applied to complex, open ended tasks? While we focus on VQA, collective outliers are abundant in tasks such as natural language inference (Bowman et al., 2015;Williams et al., 2018) and opendomain question answering (Kwiatkowski et al., 2019), amongst others. More insidious is their nature; collective outliers can take multiple forms, requiring external domain knowledge or "commonsense" reasoning, containing underspecification, or requiring capabilities beyond the scope of a given model (e.g., requiring OCR ability). While we perform ablations in this work removing collective outliers, demonstrating that active learning fails as collective outliers take up larger portions of the dataset, this is only an analytical tool; these outliers are, and will continue to be, pervasive in open-ended datasets -and as such, we will need to develop better tools for learning (and performing active learning) in their presence.
Selective Classification. One potential direction for future work is to develop systems that abstain when they encounter collective outliers. Historical artificial intelligence systems, such as SHRDLU (Winograd, 1972) and QUALM (Lehnert, 1977), were designed to flag input sequences that they were not designed to parse. Ideas from those methods can and should be resurrected using modern techniques; for example, recent work suggests that a simple classifier can be trained to identify out-ofdomain data inputs, provided a seed out-of-domain dataset (Kamath et al., 2020). Active learning methods can be augmented with a similar classifier, which re-calibrates active learning uncertainty scores with this classifier's predictions. Other work learns to identify novel utterances by learning to intelligently set thresholds in representation space (Karamcheti et al., 2020), a powerful idea especially if combined with other representation-centric active learning methods like Core-Set Sampling (Sener and Savarese, 2018).
Active Learning with Global
Reasoning. Another direction for future work to explore is to leverage Dataset Maps to perform more global, holistic reasoning over datasets, to intelligently identify promising examples -in a sense, baking part of the analysis done in this work directly into the active learning algorithms. A possible instantiation of this idea would be in training a discriminator to differentiate between "learnable" examples (upper half of each Dataset Map) from the "unlearnable", collective outliers with low confidence and low variability. Between each active learning acquisition iteration, one can generate an updated Dataset Map, thereby reflecting what models are learning as they obtain new labeled examples.
Machine learning systems deployed in realworld settings will inevitably encounter open-world datasets, ones that contain a mixture of learnable and unlearnable inputs. Our work provides a framework to study when models encounter such inputs. Overall, we hope that our experiments serve as a catalyst for future work on evaluating active learning methods with inputs drawn from open-world datasets.
Reproducibility
All code for data preprocessing, model implementation, and active learning algorithms is made available at https://github.com/siddk/vqa-outliers. Additionally, this repository also contains the full set of results and dataset maps as well.
The authors are fully committed to maintaining this repository, in terms of both functionality and ease of use, and will actively monitor both email and Github Issues should there be problems.
A Overview
Due to the broad scope of our experiments and analysis, we were unable to fit all our results in the main body of the paper. Furthermore, given the limited length provided by the appendix, we provide only salient implementation details and other representative results here; however, we make all code, models, data, results, active learning implementations available at this link: https:
//github.com/siddk/vqa-outliers.
Generally, any combination of {active learning strategy × model × seed set size × analysis/acquisition plot} is present in this paper, and is available in the public code repository.
B Implementation Details
B.1 Models & Training
Where applicable, we implement our models based on publicly available PyTorch implementations. For the LSTM-CNN model, we base our implementation off of this repository: https://github.com/ Shivanshu-Gupta/Visual-Question-Answering, while for the Bottom-Up Top-Down Attention Model, we use this repository: https://github.com/ hengyuan-hu/bottom-up-attention-vqa, keeping default hyperparameters the same.
Logistic Regression. When implementing Logistic Regression, we base our PyTorch implementation on the broadly used Scikit-Learn (https: //scikit-learn.org) implementation, using the default parameters (including L2 weight decay). We optimize our models via stochastic gradient descent.
LXMERT. As mentioned in Section 3, the default LXMERT checkpoint and fine-tuning code made publicly available in Tan and Bansal (2019) (associated code repository: https://github.com/ airsplay/lxmert) is pretrained on data from VQA-2 and GQA, leaking information that could substantially affect our active learning results. To mitigate this, we contacted the authors, who kindly provided us with a checkpoint of the model without VQA pretraining.
However, in addition to this model obtaining different results from those reported in the original work, the provided pretrained checkpoint behaves slightly differently during fine-tuning, requiring different hyperparameters than provided in the original repository. We perform a coarse grid search over hyperparameters, using the LXMERT implementation provided by HuggingFace Transformers (Wolf et al., 2019), and find that using an AdamW optimizer rather than the BERT-Adam Optimizer used in the original work without any special learning rate scheduling results in the best fine-tuning performance.
B.2 Acquisition Functions
We use standard implementations of the 8 active learning strategies described, borrowing from prior implementations (Mussmann and Liang, 2018) and existing code repositories (https://github.com/ google/active-learning). We provide additional details below.
Monte-Carlo Dropout. For our implementations of the deep Bayesian active learning methods (Monte-Carlo Dropout w/ Entropy, BALD), we follow Gal and Ghahramani (2016) and estimate a Dropout distribution via test-time dropout, running multiple forward passes through our neural networks, with different, randomly sampled Dropout masks. We use a value of k = 10 forward passes to form our Dropout distribution.
Amortized Core-Set Selection. In the original Core-Set selection active learning work introduced by Sener and Savarese (2018), it is shown that Core-Set selection for active learning can be reduced to a version of the k-centers problem, which can be solved approximately (2-OPT) with a greedy algorithm. However, running this algorithm on highdimensional representations, across large pools can be prohibitive; Core-Set selection is batch-aware, requiring recomputing distances from each "clustercenter" (points in the set of acquired examples) to all points in the active learning pool after each acquisition in a batch. While we can run this out completely for smaller datasets (and indeed, this is what we do for our small datasets VQA-Sports and VQA-Food), a single acquisition iteration for a large dataset for the full VQA-2 dataset takes approximately 20 GPU-hours on the resources we have available, or up to 9 days for a single Core-Set selection run. For GQA, performing exact Core-Set selection takes at least twice as long.
To still capture the spirit of Core-Set diversitybased selection in our evaluation, we instead introduce an amortized implementation of Core-Set selection, which is comprised of two steps. We first downsample the high-dimensional representations (of either the fused language and text, or either unimodal representations) via Principal Component Analysis (PCA) to make the distance computation faster by an order of magnitude. Then, rather than updating distances from examples in our acquired set to points in our pool after each acquisitionx, we delay updates, instead only refreshing the distance computation every 2000 acquisitions (roughly 5% of an acquisition batch for VQA-2). This allows us to report results for Core-Set selection with the three different proposed representations (Fused, Language-Only, Vision-Only) for VQA-2; unfortunately, for GQA and LXMERT (due to the high cost of training), even running this amortized version of Core-Set selection is prohibitive, so we report a subset of results, and omit the rest.
C Active Learning Results
We include further results from our study of active learning applied to VQA, including results on VQA-Food (not included in the main body), active learning results for the two logistic regression models -Log-Reg (ResNet-101) and Log-Reg (Faster R-CNN), as well as with the 4 acquisition strategies not included in the main body of the paper -Entropy, Monte-Carlo Dropout w/ Entropy, Core-Set (Language), and Core-Set (Vision). Figure 9 shows results on VQA-Food with the LSTM-CNN, BUTD, and LXMERT models, with a seed set comprised of 10% of the total pool. The results are mostly similar to those reported in the paper; strategies track or underperform random sampling, with the exception of Least-Confidence for the LSTM-CNN model -however, this is the sole exception, and the LSTM-CNN has the highest training variance of all the models we try. Figure 10 shows active learning results for the Lo-gReg (ResNet-101) model on VQA-Sports (seed set = 10%), and VQA-2 (seed set = 10%, 50%). Results are similar to those reported in the paper, with active learning failing to outperform random acqusition. Figure 11 presents the same set of experiments as the prior section, except with the LogReg (Faster R-CNN) model. While the object-based Faster R-CNN representation enables much higher performance than the ResNet-101 representation, active learning results are consistent with those reported in the paper. Figure 12 presents results for the four other active learning strategies we implement -Entropy, Monte Carlo Dropout w/ Entropy, Core-Set (Language), and Core-Set (Vision) -for the BUTD model. Results are across VQA-Sports (seed set = 10%), and VQA-2 (seed set = 10%, 50%) -despite the unique features of each strategy, the trends remain consistent with those in the paper. Figure 10: Active learning results using the Logistic Regression (ResNet-101) model on VQA-Sports (10% seed set), and VQA-2 (10% and 50% seed set). Most strategies either track or underperform random acquisition. 5 0 0 1 K 1 . 5 K 2 K 2 . 5 K 3 K 3 . 5 K 4 K 4 . 5 K 5 K Number of Training Examples Figure 11: Active learning results using the Logistic Regression (Faster R-CNN) model on VQA-Sports (10% seed set), and VQA-2 (10% and 50% seed set). While the Faster R-CNN representation leads to better validation accuracies, active learning performance remains consistent. 5 0 0 1 K 1 . 5 K 2 K 2 . 5 K 3 K 3 . 5 K 4 K 4 . 5 K 5 K Number of Training Examples Figure 12: Results with the BUTD on VQA-Sports, VQA-2 and GQA using the alternative 4 acquisition strategies not included in the main body of the paper. Unsurprisingly, results are consistent with those reported in the paper.
C.1 VQA-Food
C.2 Logistic Regression (ResNet-101)
C.3 Logistic Regression (Faster R-CNN)
C.4 Other Acquisition Strategies
D Dataset Maps & Acquisitions
For image features we use grid-based features from ResNet-101 (He et al., 2016), or object-based features from Faster R-CNN (Ren et al., 2015b) finetuned on Visual Genome (Anderson et al., 2018a).
Figure 2
2Figure 2: Results for varied active learning methods on VQA-Sports, a simplified VQA dataset. Strategies perform on par with or worse than the random baseline, when using 10% of the full dataset as the seed set.
Figure 3
3Figure 3: Results for the full VQA-2 dataset, also using 10% of the full dataset as a seed set. Similar to the plot above, all active learning methods perform similar to a random baseline.
Figure 4
4Figure 4: Results on VQA-2 using 50% of the dataset as a seed set. While methods are relatively better when using a larger seed set-confirming results from (Lin and Parikh, 2017)-no methods outperform random.
Figure 5
5Figure 5: Results on GQA using 10% of the dataset for the seed set. Even with different question structures, the above trends hold, with strategies performing worse than or equivalent to random.
Figure 7 :
7Example groups of collective outliers in the VQA-2 and GQA datasets.
VQA Datasets. A Dataset Map (Swayamdipta et al., 2020) is a model-specific graph for profiling the learnability of individual training examples. Dataset Maps present holistic pictures of classification datasets relative to the training dynamics of a given model; as a model trains for multiple epochs and sees the same examples repeatedly, the mapping process logs statistics about the confidence assigned to individual predictions. Maps then visualize these statistics against two axes: the y-axis plots the average model confidence assigned to the correct answer over training epochs, while the x-axis plots the spread, or variability, of these values. This introduces a 2D representation of a dataset (viewed through its relationship with individual model) where examples are placed on the map by coarse statistics describing their "learnability". We show the Dataset Map for BUTD trained on VQA-2 inFigure 1. For our work, we build this map post-hoc, training on the entire pool as a means for analyzing what active learning is doing -treating it as a diagnostic tool for identifying the root cause why active learning seems to fail for VQA.In an ideal setting, the majority of examples in the training set should lie in the upper half of the graph -i.e., the mean confidence assigned to the correct answer should be relatively high. Examples towards the upper-left side represent the "easy-tolearn" examples, as the variability in the confidence assigned by the model over time is fairly low.A curious feature of VQA-2 and other VQA datasets is the presence of the 25-30% of examples in the bottom-left of the map (shown in red inFigure 1) -examples that have low confidence and variability. In other words, models are unable to learn a large proportion of training examples. While prior work attributes examples in this quadrant to "labeling errors" (Swayamdipta et al., 2020), labeling errors in VQA are sparse, and cannot account for the density of such examples in these maps. Interpreting Acquisitions. We profile the acquisitions made by each active learning method, contextualizing the acquired examples via their placement on the associated Dataset Map. We segregate training examples into four buckets using the map's y-axis: easy (≥ 0.75), medium (≥ 0.50), hard (≥ 0.25), and impossible (≥ 0.00). Ideally, active learning should be robust to "hard-to-learn" examples, focusing instead on learnable, high uncertainty examples towards the upper-right portion of the Dataset Map. Instead, we find that active learning methods acquire a large proportion of impossible examples early on and concentrate on the easier examples only after the impossible examples dwindle (see Figure 6). In contrast, the random baseline acquires examples proportional to each bucket's density in the underlying map; acquiring easier examples earlier and performing on par with or better than all others.
Figure 8 :
8Using Dataset Maps, we remove hard-to-learn examples, which we identify as collective outliers. With the outliers removed, active learning methods demonstrate up to 2-3x sample efficiency versus random sampling.
Figure 9 :
9Results for the representative active learning methods on VQA-Food, a simplified VQA dataset similar to VQA-Food, across LSTM-CNN, BUTD, and LXMERT.
Figure 13 :
13Dataset Maps for the Bottom-Up Top-Down Attention model on VQA-Sports, VQA-Food, and GQA respectively. Note that VQA-Sports and VQA-Food have fewer "hard-to-learn" examples.
.Figures 3 and 4show results on VQA-2 with different seed set sizes -10% (40k examples) and50% (200k examples). Active learning performs relatively better with larger seed sets but still underperforms random. Surprisingly, when initialized with 50% of the pool as the seed set, the gain in validation accuracy after acquiring the entire pool of examples (400k examples total) is only 2%. This is an indication that the lack of sample efficiency might be a result of the underlying data, a problem we explore in the next section.GQA was introduced as a means for evaluating compositional reasoning(Hudson and Manning, 2019). Unlike VQA's natural human-written questions, GQA contains synthetic questions of the form "what is inside the bottle the glasses are to4.3 GQA
Underspecification:
What is on the shelf?
Multi-hop reasoning:
What is the vehicle that is
driving down the road the box
is on the side of?
GQA
VQA-2
External knowledge:
What does the symbol on the
blanket mean?
OCR:
What is the first word on the
black car?
To provide further context around active learning acquisitions across datasets, Figures 13-16 present Dataset Maps and acquisitions for the BUTD Model across VQA-Sports, VQA-Food, and GQA respectively. Interesting to note is that while VQA-Sports and VQA-Food are generally easier, with fewer "hard-to-learn" examples, active learning still has a bias for picking those examples. For GQA, our earlier analysis is confirmed; active learning is picking the collective outliers populating the bottom half of the Dataset Map.LSTM-CNN -VQA-Food4 0 0
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Applications can be found at https://vizwiz.org/, https://taptapsee.com/, https://www.bemyeyes.com/, and https://camfindapp.com/
Key implementation details can be found in the appendix. In the interest of full reproducibility and further work in active learning and VQA, we release our code and results here: https://github.com/siddk/vqa-outliers.
Results for LXMERT in Tan and Bansal (2019) are reported after pretraining on training and validation examples from the VQA datasets we use. While this is fair if the goal is optimizing for test performance, this exposure to training and validation examples leaks important information; to remedy this, we obtained a model checkpoint from the LXMERT authors trained without VQA data. This is also why our LXMERT results are lower than the numbers reported in the original paper -however, the general boost provided by crossmodal pretraining holds.
For LXMERT, running Core-Set selection is prohibitive, so we omit these results; please see Appendix B for more details.
AcknowledgementsWe thank Kaylee Burns, Eric Mitchell, Stephen Mussman, Dorsa Sadigh, and our anonymous ACL reviewers for their useful feedback on earlier versions of this paper. We are also grateful to Hao Tan for providing us with the LXMERT checkpoint trained without access to VQA datasets, as well as for general LXMERT fine-tuning pointers.Siddharth Karamcheti is graciously supported by the Open Philanthropy Project AI Fellowship. Christopher D. Manning is a CIFAR Fellow.Acquisitions by Difficulty BALDFigure 16: Acquisitions with the BUTD Model on the full GQA dataset. Given that the map for GQA is similar to the map for VQA-2, it is not surprising that the active learning acquisitions follow a similar trend, preferring to select "hard-to-learn" examples.
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In this work, we propose a frequency bin-wise method to estimate the single-channel speech presence probability (SPP) with multiple deep neural networks (DNNs) in the short-time Fourier transform domain. Since all frequency bins are typically considered simultaneously as input features for conventional DNN-based SPP estimators, high model complexity is inevitable. To reduce the model complexity and the requirements on the training data, we take a single frequency bin and some of its neighboring frequency bins into account to train separate gate recurrent units. In addition, the noisy speech and the a posteriori probability SPP representation are used to train our model. The experiments were performed on the Deep Noise Suppression challenge dataset. The experimental results show that the speech detection accuracy can be improved when we employ the frequency bin-wise model. Finally, we also demonstrate that our proposed method outperforms most of the state-of-the-art SPP estimation methods in terms of speech detection accuracy and model complexity.Index Terms-frequency bin-wise, speech presence probability, a posteriori probability, gated recurrent units
|
10.1109/icassp49357.2023.10096321
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2302.12048
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2a9e62488214b8faa671caeb22c94ebe2894bbc4
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FREQUENCY BIN-WISE SINGLE CHANNEL SPEECH PRESENCE PROBABILITY ESTIMATION USING MULTIPLE DNNS
23 Feb 2023
Shuai Tao [email protected]
Audio Analysis Lab, CREATE
Aalborg University
AalborgDenmark
JesperHimavanth Reddy
Audio Analysis Lab, CREATE
Aalborg University
AalborgDenmark
MadsRindom Jensen
Audio Analysis Lab, CREATE
Aalborg University
AalborgDenmark
Graesbøll Christensen
Audio Analysis Lab, CREATE
Aalborg University
AalborgDenmark
FREQUENCY BIN-WISE SINGLE CHANNEL SPEECH PRESENCE PROBABILITY ESTIMATION USING MULTIPLE DNNS
23 Feb 2023
In this work, we propose a frequency bin-wise method to estimate the single-channel speech presence probability (SPP) with multiple deep neural networks (DNNs) in the short-time Fourier transform domain. Since all frequency bins are typically considered simultaneously as input features for conventional DNN-based SPP estimators, high model complexity is inevitable. To reduce the model complexity and the requirements on the training data, we take a single frequency bin and some of its neighboring frequency bins into account to train separate gate recurrent units. In addition, the noisy speech and the a posteriori probability SPP representation are used to train our model. The experiments were performed on the Deep Noise Suppression challenge dataset. The experimental results show that the speech detection accuracy can be improved when we employ the frequency bin-wise model. Finally, we also demonstrate that our proposed method outperforms most of the state-of-the-art SPP estimation methods in terms of speech detection accuracy and model complexity.Index Terms-frequency bin-wise, speech presence probability, a posteriori probability, gated recurrent units
INTRODUCTION
Noise estimation is one of the key components to realize singlechannel and multi-channel speech enhancement, most of which rely on the speech presence probability (SPP) to update the noise statistics [1][2][3]. Available noise power spectral density (PSD) estimators also make use of the SPP to decide when to update the noise PSD [4][5][6]. Compared to voice activity detectors (VAD), SPP is a soft-decision approach that depends on the correlation of inter-bands and inter-frames [7]. Accurate SPP estimation can greatly improve the effectiveness of speech enhancement [8,9].
In the short time-frequency transform (STFT) domain, some conventional statistical signal processing methods commonly assume that the spectral coefficients of speech and noise are independent and follow the complex Gaussian distribution [10,11]. Therefore, the SPP can be derived from the a posteriori probability of the time-frequency (T-F) bins of the noisy speech. According to this assumption, [4] applied the minima values of a smoothed periodogram to estimate the SPP which enables the SPP estimation to be more robust under the effect of non-stationary noise. In [5], to achieve a highly accurate SPP estimate with low latency and computational complexity, an optimal fixed a priori SNR was used to guarantee the a posteriori SPP to be close to zero when speech is absent. In addition, [7] takes the correlation of inter-band and inter-frame into account when designing a general SPP estimator.
Recently, deep neural networks (DNNs) have been proven to be effective at processing non-stationary noise, and many novel DNN-based approaches have been proposed to estimate SPP accurately, which have been applied to speech enhancement and speech recognition successfully [12][13][14]. In these methods, recurrent neural networks (RNNs) [15] are commonly used to acquire information from neighboring frames since the frames contain temporal information which can improve the accuracy of SPP estimation. In [14], a bidirectional long short-term memory (BLSTM) was trained by the input features of multi-time frames with all frequency bins to estimate the SPP. In [12], considering the ideal ratio mask (IRM) [16] ranges from 0 to 1 at each T-F bin, they selected different DNN models, such as LSTM, BLSTM, gate recurrent units (GRUs), and bidirectional GRU (BGRU) to estimate the IRM and approximate the SPP. However, the problem that arises here is that as the complexity of the model goes up and more training data is applied to the model, more powerful hardware is required to train the models.
Inspired by conventional SPP estimation methods, our model estimates the SPP based on the correlation of several neighboring T-F bins in contrast to the typical DNN-based SPP estimation approach where all frequency bins are regarded as the input features. This allows us to use DNNs on a one-to-one basis with frequency bins therefore vastly reducing the number of parameters in the model and the amount of computations taking place. In this work, we thus propose a frequency bin-wise SPP estimation model in the STFT domain that relies on using multiple DNNs to estimate the SPP. For our proposed model architecture, the GRU module is used to extract time and frequency information from each frequency bin and several of its neighbors. Additionally, since IRM-based SPP estimation methods may misclassify the T-F bins dominated by non-speech and noise [12,17,18], we choose the a posteriori probability to represent the SPP in the STFT domain.
The work is organized as follows. In Section 2, the problem of frequency bin-wise single channel SPP estimation is formulated. In Section 3, the SPP estimation model with multiple DNNs is designed. In Section 4 and Section 5, the experimental procedures and results are provided, respectively. Finally, Section 6 presents the conclusion. The work can be found on GitHub 1 .
FREQUENCY BIN-WISE SPP ESTIMATION
Signal Modeling
For the single channel speech signal x(n), we assume that it is corrupted by the additive noise d(n). That is, in the STFT domain, we can obtain the noisy speech y(n) representation as follows:
Y (k, l) = X(k, l) + D(k, l),(1)
where k ∈ {0, ..., K − 1} denotes the frequency bin index and K is the number of frequency bins, l ∈ {0, ..., L − 1} denotes the time frame index and L is the number of time frames. With the assumption of a zero-mean complex Gaussian distribution and independence for X and D, we have
φY (k, l) = E[|Y (k, l)| 2 ] = φX (k, l) + φD(k, l),(2)
where E[·] is the statistical expectation operator, φX (k, l) = E[|X(k, l)| 2 ] and φD(k, l) = E[|D(k, l)| 2 ]. The PSD of the clean and the noisy speech can be represented by φX (k, l) and φD(k, l), respectively. In the STFT domain, there exists a correlation between the neighboring T-F bins [7]. Therefore, the SPP estimate can be improved using the correlation. The first step in creating our input signal vector is to obtain a vector corresponding to each individual frequency bin,
ϕY (k) = [φY (k, 0), ..., φY (k, l), ...φY (k, L − 1)] T .(3)
Each frequency bin vector contains L consecutive time frames, which contain relevant contextual information for the estimation of the SPP. Since RNNs are effective at processing temporal information [19,20], we employ RNNs in this work to extract time correlations from the neighboring time frames.
To improve the SPP estimation accuracy, we take a few neighboring frequency bin vectors into consideration to extract frequency correlations from the input signal matrix. Therefore, the input signal matrix ΦY (k) can be obtained as
ΦY (k) = [ϕY (k − I), ..., ϕY (k), ..., ϕY (k + I)] T ,(4)
where I is the number of neighboring frequency bin vectors. Now, the time correlation and frequency correlation of neighboring time-frequency bins can be extracted according to the input signal matrix ΦY (k). In this work, the SPP is represented by the a posteriori probability [5], and the DNN is used to estimate the SPP from the noisy observation.
Since the typical DNN-based approach takes all the frequency bins into account to estimate the SPP, the model complexity may be increased. In this section, we, therefore, design multiple specific DNNs to estimate the frequency bin-wise SPP. Additionally, since the a posteriori probability is derived by the correlation of neighboring T-F bins, the a posteriori probability SPP representation of the clean speech and the noisy speech PSD are used as the training data pairs to train our model.
SPP Estimation Model and Loss Function
To extract the time and frequency correlation of the consecutive T-F bins in the input signal matrix ΦY (k) from the observed noisy PSD φY (k, l), we set K specific DNNs as the regression module. As mentioned in (4), the coefficient of the k'th input signal matrix can be used to train the k'th DNN for the SPP estimate in the k'th frequency bin.
First, to train the DNN model, we choose the log-power periodogram as the input feature [21,22]. Therefore, the input features of each individual DNN are obtained from the log input signal matrix ΦY (k). It can be expressed as
Φ ′ Y (k) = log(ΦY (k)),(5)where Φ ′ Y (k) isSPPY (k) = F θ k (Φ ′ Y (k)),(6)where spp Y (k) = [ SPPY (k, 0), ..., SPPY (k, l), ..., SPPY (k, L − 1)] T is the SPP estimate of the k'th input features, F θ k is the k'th DNN with the parameter θ.
To update the DNN parameters, the loss between the target and the estimated SPP is calculated by meansquared error (MSE), i.e.,
LMSE = 1 L L−1 l=0 (SPPY (k) − SPP Y (k)) 2 ,(7)
where SPPY (k) = [SPPY (k, 0), ..., SPPY (k, l), ..., SPPY (k, L − 1)] T is the target function. In this work, the a posteriori probability is regarded as the SPP representation, therefore SPPY (k, l) can be represented by
SPPY (k, l) = 1 + p(H0) p(H1) (1 + ξH 1 ) e − |Y | 2 φ D ξ H 1 1+ξ H 1 −1(8)
where p(H0) and p(H1) denote a priori speech absence and presence probability, ξH 1 is the a priori SNR during speech presence [5].
Model Architecture
In this work, since a GRU can outperform an LSTM both in terms of convergence in CPU time, and in terms of parameter updates and generalization [23], we choose GRUs to design the SPP estimation model. The model training strategy is shown in Fig. 1 and the DNN model is trained by the input features of the logarithmic power spectral T-F bins. The training strategy of the typical DNN-based SPP estimation model in Fig. 1(a) shows that a GRU module is trained using K frequency bins (all frequency bins) and L consecutive time frames. The typical DNN-based model input size is K and, in this work, the size of the hidden layer is the same as the size of the input layer. The proposed training strategy of the frequency bin-wise SPP estimation model is shown in Fig. 1(b). When I neighboring frequency bins are introduced to estimate the SPP of a single frequency bin, the input size is 2I + 1, and one hidden layer is set. The output of each hidden layer state is regarded as the value of the SPP estimate at the current time. Finally, to restrict the output range of the DNN to [0, 1], the output layer is the activation function Sof tplus with a fixed parameter β.
EXPERIMENTAL SETTINGS
In this work, the sub-band DNS dataset is used to train our designed model. During testing, 200 noisy utterances (1.1 hours) and 1800 noisy utterances (1 hour) were collected from the DNS dataset [24], and the TIMIT dataset [25], respectively. Each clean utterance is corrupted by a random noise utterance selected from the noise dataset, each noisy utterance SNR ranging from -5dB to 25 dB. The noise data includes 150 different types of noise taken from Audioset [26] Freesound [27] and Demand datasets [28].
The receiver operating characteristic (ROC) [29] curve is used to evaluate the SPP estimation method performance and the falsealarm probability Pfa = 0.05 given in [7] is used to calculate the speech detection probability, Pd. Additionally, we apply the area under curve (AUC) metric which is derived from ROC and ranges between [0, 1] to represent overall performance. We also adopt the adaptive threshold set to -60 dB below the maximum instantaneous power across all TF bins shown in [7] to distinguish the speech and non-speech bins across all T-F bins of clean speech.
The sampling rate of all utterances is 16 kHz. Hann window is applied to STFT analysis and the length of the time window for STFT is 16 ms and the hop length is 8 ms. We use the mean and standard derivation to normalize the dataset. During training, the Adam optimizer [30] is utilized to optimize the neural network parameters. The learning rate is set to 0.001. Weight decay is set to 0.00001 to prevent overfitting. The parameter will be updated at the 50th and 100th epochs for the implemented DNN models. Pytorch is used to implement the frequency bin-wise SPP estimation model and the reference DNN-based model.
RESULTS AND DISCUSSION
In this section, to prove the effectiveness of our method, a comparison is shown between a typical DNN-based model and our proposed method using ROC curves. Moreover, some numerical results are provided to evaluate the accuracy of the SPP estimators and the model complexity, respectively.
Examination of ROC Curves
To investigate the performance of the proposed method, 200 training utterances (1.1 hours) are used to train our proposed frequency binwise model. In addition, 200 utterances (1.1 hours), 1000 utterances (5.5 hours), and 3000 utterances (16.6 hours) are used to train the typical DNN-based model, respectively. To investigate the effect of using neighboring frequency bins for the proposed method, we set I = 0 (no neighboring frequency bins), I = 1 (with 1 neighboring frequency bin), and I = 2 (with two neighboring frequency bins) to train the frequency bin-wise model. Fig. 2 shows an example of SPP estimation results. A noisy utterance of length 20 seconds and input SNR of 11 dB taken from the DNS dataset, is used for testing by the typical DNN-based SPP estimation model and the frequency bin-wise model. From Fig. 2, we can observe that the typical DNN-based method and the proposed frequency bin-wise method are able to estimate the SPP with similar accuracy. In addition, we also investigate the impact of the training data volume on SPP estimation accuracy for the typical DNN-based SPP estimation model. From Fig. 3, we can find that when we increase training data from 1.1 hours to 5.5 hours and then to 16.6 hours for the typical DNN-based model, there is a gradual increase in AUC but still falls short of our proposed method in terms Pd.
Numerical Results
To evaluate the performance of the proposed method, the speech detection probability and the AUC are calculated from the ROC curves to represent the speech detection accuracy and the effectiveness of the SPP estimation method, respectively. In addition, we also investigate the effect of model complexity on SPP estimation accuracy. Inspired by [31] and [32], we compare our method with the state-ofthe-art self-attention model and, in this work, 3 self-attention heads and 2 encoder layers are used to estimate the SPP. The self-attention model is trained in a typical way where all the frequency bins are treated as input features. During training, the frequency bin-wise SPP estimation model and the self-attention-based SPP estimation model are trained with 1.1 hours of training data pairs. The typical DNN-based model is trained with 1.1 and 16.6 hours of training data pairs, respectively. All training data pairs come from the DNS
Methods
Pd (Pfa = 0.05) AUC IMCRA [4] 0.1183 0.6504 Unbiased [5] 0.3460 0.7348 General [7] 0.1132 0.6229 Self-Attention [31] In Table 1, we show how the proposed model compares to other conventional methods and a few DNN-based methods using Pd and AUC as metrics. The results in Table 1 are obtained from testing using the TIMIT dataset (1 hour).
With 1.1 hours of training data, we can observe that the frequency bin-wise model AUC (0.7986) is lower than the typical DNN-based model and the self-attention-based model, it is still higher than IM-CRA [4] (0.6504), Unbiased MMSE [5] (0.7348) and General SPP estimator [7] (0.6229). Especially, when we set I = 1 and I = 2, the sub-frequency bin-based model achieved higher AUCs of 0.8011 and 0.7988, respectively. For the speech detection accuracy, all the frequency bin-wise models achieved higher speech detection accuracy than other methods and when we take one neighboring frequency bin (I = 1) into account the speech detection probability can reach 0.5038.
According to the results, we can confirm that an increase in model complexity can improve the performance of DNN-based applications, and in this work, the SPP estimation accuracy can also be improved, which is consistent with the experimental results shown in [33]. The reason is that the complex model can extract more global information than the simple model to estimate the SPP from all frequency bins. Additionally, a remarkable improvement in speech detection accuracy appears when we employ our proposed method to estimate the SPP, especially when we set I = 1, the model performance and Pd are improved. The reason for the improved performance could be that the DNNs can extract specific contextual information for each frequency bin which is not possible when I = 0 due to the lack of inclusion of its neighbors.
Finally, by comparing the AUC of different SPP estimation methods, we can observe that all DNN-based models can achieve higher performance of SPP estimation than the conventional methods. For DNN-based SPP estimation models, although all the presented models demonstrate similar performance, the speech detection accuracy is different. Therefore, it can be observed that more details can be detected by the bin-wise model leading to better detection accuracy.
Computational Complexity
To evaluate the complexity of the proposed model relative to its counterparts, we use the number of parameters and floating point operations (FLOPs) as the metrics. For our proposed frequency bin-wise model, the total parameters and FLOPs of all the models are used to represent computational complexity. We use the ptflops 2 python 2 https://pypi.org/project/ptflops/ 3024 3920 library to calculate the total parameters and FLOPs for our method and the reference DNN-based methods. Table 2 shows that our proposed method has fewer parameters and FLOPs than the other methods. The reason is that although we use multiple DNNs to estimate the SPP, each DNN has less input size than the typical DNN-based model. Furthermore, although we introduced the neighboring frequency bins to estimate the SPP in 4.2, from Table 2, we can also observe that the increase in computational complexity is minimal even with the inclusion of additional neighboring frequency bins. From the above experimental results, we can confirm that although increasing the training data and using complex models can contribute to the improvement of the performance of the typical DNN-based SPP model, high computational complexity is inevitable. However, it can be observed that the proposed frequency bin-wise model not only shows an improvement in Pd while maintaining similar performance in terms of the AUC but also reduces the computational complexity while using the same amount of training data.
CONCLUSION
In this work, we proposed an effective frequency bin-wise SPP estimation method that shows good performance with a limited amount of training data while also maintaining low model complexity. Experimental results show that in addition to reducing the model complexity, the frequency bin-wise model also shows better performance even in comparison with the typical DNN-based model that is trained with increasing amounts of training data. The experimental observations involving the inclusion of neighboring frequency bins show that there is an increase in speech detection accuracy as well as the AUC (compared to its counterpart that does not include any neighboring frequency bins) due to being exposed to local contextual information. Since multiple DNNs are employed to estimate the SPP in the STFT domain, the frequency bin-wise model's computational complexity is much lower than its DNN-based counterparts.
Fig. 1 .
1Typical DNN-based model training strategy vs our proposed method. (a) Typical DNN-based SPP estimation model (with all frequency bins), and (b) Proposed frequency bin-wise SPP estimation model, a frequency bin along with 2I neighboring frequency bins are treated as the input features.
Fig. 2 . 05 .Fig. 3 .
2053ROC curves comparison of the typical DNN-based model and the frequency bin-wise model. Both models are trained with the same amount of training data (1.1 hours). The vertical dotted line indicates the false-alarm probability Pfa = 0.Input ROC curves comparison of the typical DNN-based model and the frequency bin-wise model with an increase in training data for the typical DNN-based model. The vertical dotted line indicates the false-alarm probability Pfa = 0.05. Input SNR = 11 dB.
the input feature for the k'th DNN. Also, during training, we haveL
DNN
Power Spectrogram
SPP
.
.
.
.
.
.
.
.
.
.
.
.
K
(a) The typical DNN-based SPP estimation model training strategy
DNN
Power Spectrogram
SPP
DNNk
DNN0
DNNK-1
.
.
.
.
.
.
k I
+
k I
-
2 1
I +
.
.
.
.
.
.
k
(b) Frequency bin-wise SPP estimation model training strategy
Table 1 .
1Speech detection probability Pd and AUC for different SPP estimation methods. Input SNR ∈ [-5 dB, 25 dB].
Table 2 .
2Parameters and FLOPs comparison of the DNN-based SPP estimation model.Methods
Para
FLOPs (Mac)
Self-Attention [31]
867.12K
85.6M
Typical DNN-based 100.62K
13.1M
Proposed (I = 0)
1548
2451
Proposed (I = 1)
2292
3188
Proposed (I = 2)
https://github.com/Shuaitaoaau/SPP
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This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1-10, 2014) that almost surely lim δ→0 U (δ)N (t, δ) = t, where N (t, δ) is the minimal number of boxes of size at most δ needed to cover a subordinator's range up to time t, and U (δ) is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for N (t, δ), complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N (t, δ), but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than δ. This new process can be manipulated with remarkable ease in comparison with N (t, δ), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process. result in this paper is a central limit theorem for N (t, δ), complementing the almost sure convergence result lim δ→0 U (δ)N (δ, t) = t, almost surely, where U (δ) denotes the renewal function of the subordinator, see [24, Theorem 1.1].Prior to the results in[24], most works on box-counting dimension focused only on finding the value of lim δ→0 log(N (t, δ))/ log(1/δ), which defines the box-counting dimension. However, working with N (t, δ) itself allows precise understanding of its fluctuations around its mean, inaccessible at the log scale.We will introduce an alternative "box-counting scheme"to N (t, δ), which allows us to understand the dimension of the range in terms of the Lévy measure, complementing results formulated in terms of the renewal function.The fractal dimensional study of sets such as the range or graph of Lévy processes, and especially subordinators, has a very rich history. There are many works which study the box-counting, Hausdorff, and packing dimensions of sets related to Lévy processes[4,6,9,[11][12][13][16][17][18][19][20][24][25][26].A Lévy process is a stochastic process in R d which has stationary, independent increments, and starts at the origin. A subordinator X := (X t ) t≥0 is a non-decreasing real-valued Lévy process. The Laplace exponent of a subordinator X is defined by the relation e − (λ) = E[e −λX 1 ] for λ ≥ 0. By the Lévy Khintchine formula [1, p. 72], can always be expressed as (λ) = dλ +
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10.1007/s10959-018-0813-5
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30fc48ecb9ad7f8df72877ff9683509e1727d83c
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Fractaldimensional properties of subordinators
2018
Open Access
A Barker
Fractaldimensional properties of subordinators
Journal of Theoretical Probability
2018Article Published Version Creative Commons: Attribution 4.0 (CCBY) Received: 22 June 2017 / Revised: 19 January 20189840 doi: https://doi.org/10.1007/s1095901808135 Available at http://centaur.reading.ac.uk/77131/ It is advisable to refer to the publisher's version if you intend to cite from the work. See Guidance on citing . To link to this article DOI: http://dx. Publisher: Springer CentAUR Central Archive at the University of Reading Reading's research outputs online J Theor ProbabLévy processes · Subordinators · Fractal dimension · Box-counting dimension Mathematics Subject Classification (2010) Primary 60G51 · 28A80; Secondary 60G75 · 60F05 · 60F15
This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1-10, 2014) that almost surely lim δ→0 U (δ)N (t, δ) = t, where N (t, δ) is the minimal number of boxes of size at most δ needed to cover a subordinator's range up to time t, and U (δ) is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for N (t, δ), complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N (t, δ), but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than δ. This new process can be manipulated with remarkable ease in comparison with N (t, δ), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process. result in this paper is a central limit theorem for N (t, δ), complementing the almost sure convergence result lim δ→0 U (δ)N (δ, t) = t, almost surely, where U (δ) denotes the renewal function of the subordinator, see [24, Theorem 1.1].Prior to the results in[24], most works on box-counting dimension focused only on finding the value of lim δ→0 log(N (t, δ))/ log(1/δ), which defines the box-counting dimension. However, working with N (t, δ) itself allows precise understanding of its fluctuations around its mean, inaccessible at the log scale.We will introduce an alternative "box-counting scheme"to N (t, δ), which allows us to understand the dimension of the range in terms of the Lévy measure, complementing results formulated in terms of the renewal function.The fractal dimensional study of sets such as the range or graph of Lévy processes, and especially subordinators, has a very rich history. There are many works which study the box-counting, Hausdorff, and packing dimensions of sets related to Lévy processes[4,6,9,[11][12][13][16][17][18][19][20][24][25][26].A Lévy process is a stochastic process in R d which has stationary, independent increments, and starts at the origin. A subordinator X := (X t ) t≥0 is a non-decreasing real-valued Lévy process. The Laplace exponent of a subordinator X is defined by the relation e − (λ) = E[e −λX 1 ] for λ ≥ 0. By the Lévy Khintchine formula [1, p. 72], can always be expressed as (λ) = dλ +
Introduction and Background
We shall mostly study the minimal number, N (t, δ), of intervals of length at most δ needed to cover the range {X s : 0 ≤ s ≤ t} of a subordinator (X s ) s≥0 . The main
∞ 0 (1 − e −λx ) (dx),(1)
where d is the linear drift, and is the Lévy measure, which determines the size and intensity of the jumps (discontinuities) of X , and moreover satisfies the condition If the Lévy measure is infinite, then infinitesimally small jumps occur at an infinite rate, almost surely. We will not study processes with finite Lévy measure, as they have only finitely many jumps, and hence no fractal structure.
The box-counting dimension of a set in R d is lim δ→0 log(N (δ))/ log(1/δ), where N (δ) is the minimal number of d-dimensional boxes of side length δ needed to cover the set. The limsup and liminf, respectively, define the upper and lower box-counting dimensions. For further background reading, we refer to [1,2] for subordinators, [7,21,23] for Lévy processes, and [9,26] for fractals.
The paper is structured as follows: Sect. 2 outlines the statements of all of the main results; Sect. 3 contains the proof of the CLT result for N (t, δ) and the lemmas required for this proof; Sect. 4 contains the proofs of all of the main results on the new process L(t, δ); Sect. 5 extends this work to the graph of a subordinator, and considers the special case of a subordinator with regularly varying Laplace exponent.
A Central Limit Theorem for N(t, δ)
Expanding upon Bertoin's result [2,Theorem 5.1], the following almost sure limiting behaviour of N (t, δ) was determined by Savov [ (Savov [24,Theorem 1]) If a subordinator has infinite Lévy measure or a nonzero drift, then for all t > 0, lim δ→0+ U (δ)N (t, δ) = t almost surely.
We will complement and refine this work with a CLT on N (t, δ). When the subordinator has no drift, we require a mild condition on the Lévy measure:
lim inf δ→0 I (2δ) I (δ) > 1,(2)
where I (u) := u 0 (x)dx, and (x) := ((x, ∞)). (2) (2) is far less restrictive than regular variation (or even O-regular variation) of the Laplace exponent, and appears naturally in the context of the law of the iterated logarithm (see e.g. [1, p. 87]).
Remark 2.2 Condition
Theorem 2.3
For every driftless subordinator with Lévy measure satisfying (2), for any t > 0, N(t, δ) satisfies the following central limit theorem:
N (t, δ) − ta(δ) t 1 2 b(δ) d → N (0, 1),(3)as δ → 0, where a(δ) := U (δ) −1 , and b(δ) := U (δ) − 3 2 Var(T δ ) 1 2 .
An Alternative Box-Counting Scheme, L(t, δ)
Definition 2.4
The process of δ-shortened jumps,X δ := (X δ t ) t≥0 , is obtained by shortening all jumps of X of size larger than δ to instead have size δ. That is,X δ is the subordinator with Laplace exponent˜ δ (u) = du + δ 0 (1 − e −ux )˜ δ (dx) and Lévy measure˜ δ (dx) = (dx)1 {x<δ} + (δ) δ (dx), where δ denotes a unit point mass at δ, and is the Lévy measure of X .
Definition 2.5 For δ, t > 0, L(t, δ) is defined by L(t, δ) := 1 δX δ t .
We will see in Theorem 2.7 that L(t, δ) can replace N (t, δ) in the definition lim δ→0 log(N (t, δ))/ log(1/δ) of the box-counting dimension of the range of X . Now let us formally state the main results on L(t, δ).
Remark 2.6
The log scale at which box-counting dimension is defined allows flexibility among functions to be taken in place of the optimal count. In particular, there is freedom between functions related by f g asymptotically, where the notation means that there exist positive constants A, B such that A f (x) ≤ g(x) ≤ B f (x) for all x. For more details, we refer to [9, p. 42].
Theorem 2.7
For all δ, t > 0, for every subordinator, N (t, δ) L(t, δ). In particular, by Remark 2.6, L(t, δ) can be used to define the box-counting dimension of the range, i.e. lim δ→0 log(N (t, δ))/ log(1/δ) = lim δ→0 log(L(t, δ))/ log(1/δ).
L(t, δ) μ(δ) = t,(4)L(t, δ) − tμ(δ) t 1 2 v(δ) d → N (0, 1) (5) as δ → 0, where μ(δ) = 1 δ (d + I (δ)), and v(δ) := 1 δ ∞ 0 (x ∧ δ) 2 (dx) 1 2 .
Remark 2.11 Applying Remark 2.4, the Lévy Khintchine formula (1), and the fact that for any integrable function f ,
δ 0 f (x)˜ δ (dx) = ∞ 0 f (x ∧ δ) (dx)
, it follows that for all δ, t > 0, the mean and variance of L(t, δ) are given by
E[L(t, δ)] = tμ(δ), Var(L(t, δ)) = tv(δ).
Computing the moments of L(t, δ) is remarkably simple in comparison with the moments of N (t, δ), which are not well known. This is a key benefit of using L(t, δ) to study the box-counting dimension of the range of a subordinator.
Lemma 3.1 For every subordinator with infinite Lévy measure, a sufficient condition for the convergence in distribution
(3), with σ 2 δ := Var(T δ ), is lim δ→0 U (δ) 7 3 σ 2 δ = 0.(6)
The proof of Lemma 3.1 relies upon the Berry-Esseen theorem, a very useful result for proving central limit theorem results as it provides the speed of convergence, which is stated here in Lemma 3.
P Y 1 − E[Y ] + · · · + Y n − E[Y ] Var(Y ) 1 2 √ n ≥ x − P(Z ≥ x) ≤ cE[|Y − E[Y ]| 3 ] Var(Y ) 3 2 √ n .(7)
For brevity, we will only provide calculations for t = 1. The proofs for different values of t are essentially the same. Recall the definitions a(δ) := U (δ) −1 , σ 2 δ := Var(T δ ), and b(δ) := U (δ) − 3 2 σ δ . We shall aim to prove that for all x ∈ R,
lim δ→0 P N (1, δ) − a(δ) b(δ) ≤ x − P(Z ≤ x) = 0.
For each δ > 0, (7) provides an upper bound, and then under condition (2), we can prove that this bound converges to zero as δ → 0.
Proof of Lemma 3.1 Let T (k) δ denote the kth time at which N (t, δ) increases, and let T δ,k , k ∈ N, denote iid copies of T (1) δ . By the strong Markov property, T (k) δ and k i=1 T δ,i have the same distribution. Then, with n := a(δ) + xb(δ) , where · denotes the ceiling function,
P N (1, δ) − a(δ) b(δ) ≤ x = P(N (1, δ) ≤ a(δ) + xb(δ)),(8)
and since N (1, δ) only takes integer values, using the fact that T (n) δ has the same distribution as the sum of n iid copies of T (1) δ , it follows that
(8) = P (N (1, δ) ≤ n) = P(T (n) δ ≥ 1) = P n i=1 T δ,i ≥ 1 = P n i=1 T δ,i − U (δ) ≥ 1 − nU (δ) = P ⎛ ⎝ n i=1 T δ,i − U (δ) nσ 2 δ ≥ 1 − nU (δ) nσ 2 δ ⎞ ⎠ .(9)
It follows from Lemma 3.3 that σ 2 δ ≤ E[T 2 δ ] ≤ cU (δ) 2 , which then implies that b(δ) = o(a(δ)) as δ → 0. Then, as δ → 0, the asymptotic behaviour of n is
n = a(δ) + xb(δ) ∼ a(δ) + xb(δ) = a(δ) + o(a(δ)) ∼ a(δ) = U (δ) −1 . (10)
It follows, with x depending on x and δ, that as δ → 0,
−x : = 1 − nU (δ) nσ 2 δ = 1 − a(δ) + xb(δ) U (δ) ( a(δ) + xb(δ) ) 1 2 σ δ ∼ 1 − (a(δ) + xb(δ))U (δ) (a(δ) + xb(δ)) 1 2 σ δ (11) = 1 − 1 − xb(δ)U (δ) (a(δ) + xb(δ)) 1 2 σ δ ∼ −xb(δ)U (δ) U (δ) − 1 2 σ δ = −xb(δ)U (δ) 3 2 σ δ = −x.(12)
Now, by the triangle inequality and symmetry of the normal distribution, combining (9) and (12), it follows that as δ → 0, for any x ∈ R,
P N (1, δ) − a(δ) b(δ) ≤ x − P(Z ≤ x) ≤ P(Z ≥ −x ) − P(Z ≥ −x) + P ⎛ ⎝ 1 nσ 2 δ n i=1 (T δ,i − U (δ)) ≥ −x ⎞ ⎠ − P( Z ≥ −x ) = P ⎛ ⎝ 1 nσ 2 δ n i=1 (T δ,i − U (δ)) ≥ −x ⎞ ⎠ − P( Z ≥ −x ) + o(1).(13)
Recall that we wish to show that (13) converges to zero. By the Berry-Esseen theorem and (10), it follows that as δ → 0,
(13) ≤ C E[|T δ − U (δ)| 3 ] σ 3 δ n 1 2 + o(1) ∼ C U (δ) 1 2 E[|T δ − U (δ)| 3 ] σ 3 δ .
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J Theor Probab
Applying the triangle inequality, then Lemma 3.3 with m = 2 and m = 3 to
E[|T δ − U (δ)| 3 ], it follows that (13) ≤ 8C U (δ) 1 2 U (δ) 3 σ 3 δ = 8C U (δ) 7 3 σ 2 δ 2 3 .
Therefore if the condition (6) as in the statement of Lemma 3.1 holds, then the desired convergence in distribution (3) follows, as required.
Lemma 3.3 For every subordinator with infinite Lévy measure, for all
m ≥ 1, lim sup δ→0+ E[T m δ ] U (δ) m < ∞.
Proof of Lemma 3.3 First, by the moments and tails lemma (see [15, p. 26]),
E[T m δ ] U (δ) m = E T δ U (δ) m = ∞ 0 my m−1 P T δ U (δ) > y dy.
By the definition of T δ , it follows that X u ≥ δ if and only if T δ ≤ u, and then
E[T m δ ] U (δ) m = ∞ 0 my m−1 P(X yU (δ) ≤ δ)dy = ∞ 0 my m−1 P e − 1 δ X yU (δ) ≥ e −E[T m δ ] U (δ) m ≤ ∞ 0 my m−1 e 1−yU (δ) (1/δ) dy ≤ ∞ 0 my m−1 e 1−cy dy,
which is finite and independent of δ. Therefore the lim sup is finite, as required.
Proof of Theorem 2.3
Theorem 2.3 is proven by a contradiction, using Lemma 3.4 to show that the sufficient condition in Lemma 3.6 holds. (2) implies that for each η ∈ (0, 1), there exists a sufficiently large integer n such that
Lemma 3.4 Recall the definition I
(δ) := δ 0 (x)dx. The conditionlim inf δ→0 I (δ) I (2 −n δ) > 1 η .(14)
Proof of Lemma 3.4 The integral condition (2) imposes that for some B > 1,
lim inf δ→0 I (δ) I (δ/2) = lim inf δ→0 δ 0 (y)(dy) δ/2 0 (y)dy = B.(15)
Then, by effectively replacing 1/2 with 2 −n (so 1/2 is replaced by a smaller constant), we can replace B with B n , which can be made arbitrarily large by choice of n. This follows by splitting up the fraction,
lim inf δ→0+ I (δ) I (2 −n δ) = lim inf δ→0+ I (δ) I (2 −1 δ) I (2 −1 δ) I (2 −2 δ) · · · I (2 −(n−1) δ) I (2 −n δ) ≥ lim inf δ→0+ I (δ) I (2 −1 δ) lim inf δ→0+ I (2 −1 δ) I (2 −2 δ) · · · lim inf δ→0+ I (2 −(n−1) δ) I (2 −n δ) = B n > 1 η ,
where we simply take n sufficiently large that B n > 1/η.
Using Lemma 3.4 for a contradiction is the step in the proof of Theorem 2.3 which requires the condition (2). In order to prove Theorem 2.3, we require the notation introduced in Definition 3.5. We refer to [14, p93] for more details.
Definition 3.5 Recalling from Remark 2.4 that the processX δ has Laplace exponent
δ (u) = du + δ 0 (1 − e −ux ) (dx) + (1 − e −uδ ) (δ), we define: (i) g(u) := d du˜ δ (u) = d + δ 0 xe −ux˜ δ (dx), (ii) R(u) :=˜ δ (u) − ug(u) = δ 0 (1 − e −ux (1 + ux))˜ δ (dx), (iii) λ δ denotes the unique solution to g(λ δ ) = x δ , for d < x δ < d + δ 0 x˜ δ (dx).
One can ignore the drift d in Definition 3.5, since d = 0 throughout Sect. 3. The proof of Theorem 2.3 now requires the following lemma:
Lemma 3.6 For α > 0, t = (1 + α)U (δ), and g(λ δ ) = x δ = δ/t, if lim sup δ→0 δλ δ < ∞,
then the desired convergence in distribution (3), as in Theorem 2.3, holds.
Proof of Theorem 2.3 Assume for a contradiction that there exists a sequence (δ m ) m≥1 converging to zero, such that lim m→∞ λ δ m δ m = ∞. That is to say, assume that the sufficient condition in Lemma 3.6 doesn't hold. For brevity, we omit the dependence of δ m on m. Hence for all fixed η, n > 0, η ≥ e −λ δ 2 −n δ for all small enough δ > 0. By Fubini's theorem,
I (δ) = δ 0 (x)dx = δ 0 x˜ δ (dx), so ηI (δ) + I (2 −n δ) ≥ e −λ δ 2 −n δ I (δ) + I (2 −n δ) ≥ e −λ δ 2 −n δ δ 0 x˜ δ (dx) + 2 −n δ 0 x (dx) = e −λ δ 2 −n δ δ (δ) + e −λ δ 2 −n δ δ 0 x (dx) + 2 −n δ 0 x (dx).(16)
Removing part of the first integral and noting 1 ≥ e −λ δ x for all x > 0,
(16) ≥ e −λ δ 2 −n δ δ (δ) + δ 2 −n δ e −λ δ 2 −n δ x (dx) + 2 −n δ 0 e −λ δ x x (dx). Now, e −λ δ 2 −n δ ≥ e −λ δ x for x ≥ 2 −n δ. So for g(λ δ ) = x δ = δ (1+α)U (δ) ,(16) ≥ e −λ δ 2 −n δ δ (δ) + δ 2 −n δ e −λ δ x x (dx) + 2 −n δ 0 e −λ δ x x (dx) = e −λ δ 2 −n δ δ (δ) + δ 0 e −λ δ x x (dx) ≥ g(λ δ ) = δ (1 + α)U (δ) ≥ I (δ) (1 + α)K ,
where the last two inequalities, respectively, follow by Definitions 2.4, 3.5 (i) with d = 0, and the relation U (δ) −1 I (δ)/δ, see [1, p74]. So for a constant K > 0, for all sufficiently small δ > 0, we have shown ηI (δ) + I (2 −n δ) ≥ I (δ) (1+α)K . Taking η > 0 small enough that 1 (1+α)K ≥ 2η, it follows that I (2 −n δ) ≥ ηI (δ), and hence I (δ)/I (2 −n δ) ≤ 1/η. But in Lemma 3.4 we showed that for each fixed η > 0, there is sufficiently large n such that lim inf δ→0 I (δ)/I (2 −n δ) > 1/η, which is a contradiction, so the sufficient condition as in Lemma 3.6 must hold. Y (x) < ∞. This is proven using Remark 3.10, the inequality P(Y t < a) ≥ 1 − Cth(a) for all Lévy processes (see [22, p954] for details), and the asymptotic expansion of U (δ) as in [8,Theorem 4]. 3.9, 3.12, 3.6 Lemmas 3.9, 3.12, and 3.6 give sufficient conditions for Theorem 2.3 to hold. The proofs for these lemmas are facilitated by Lemma 3.11, which was proven in 1987 by Jain and Pruitt [14, p94]. Recall thatX δ denotes the process with δ-shortened jumps, as defined in Definition 2.4.
Proofs of Lemmas
Lemma 3.9 The convergence in distribution (3) as in Theorem 2.3 holds if for some
α ∈ (0, 1], lim inf δ→0 [P(X δ (1+α)U (δ) ≤ δ) + P(X δ (1−α)U (δ) ≥ δ)] > 0.
Proof of Lemma 3.9 For all α > 0, recalling that E[T δ ] = U (δ),
σ 2 δ = Var(T δ ) ≥ Var(T δ ; |T δ − U (δ)| ≥ αU (δ)) ≥ α 2 U (δ) 2 [P(T δ ≥ (1 + α)U (δ)) + P(T δ ≤ (1 − α)U (δ))].
For the desired convergence in distribution (3) to hold, it is sufficient by Lemma 3.1 to show that lim δ→0 U (δ) 7 3 /σ 2 δ = 0. Now,
U (δ) 7 3 σ 2 δ ≤ U (δ) 1 3 α 2 [P(T δ ≥ (1 + α)U (δ)) + P(T δ ≤ (1 − α)U (δ))] .
Note that T δ ≥ t if and only ifX δ t ≤ δ since jumps of size larger than δ do not occur in either case, and so
X t =X δ t when T δ ≥ t. It follows that (3) holds if lim inf δ→0 [P(X δ (1+α)U (δ) ≤ δ) + P(X δ (1−α)U (δ) ≥ δ)] > 0.
Remark 3.10 The condition in Lemma 3.9 is not optimal. If for ε ∈ (0, 1/6),
lim δ→0 U (δ) 2ε− 1 3 [P(X δ U (δ)+U (δ) 1+ε ≤ δ) + P(X δ U (δ)−U (δ) 1+ε ≥ δ)]
= ∞, then the convergence in distribution (3) follows too. This stronger condition does not lead to any more generality than the condition (2) for driftless subordinators. Proof of Lemma 3.12 Applying the inequality (17) from Lemma 3.11,
d = g(∞) < x δ < g(0) = d + δ 0 x˜ δ (dx), P(X δ t ≤ t x δ ) ≥ 1 − (1 + ε)c ε 2 t R(λ δ ) e −(1+2ε)t R(λ δ ) .(17)P(X δ (1+α)U (δ) ≤ δ) ≥ 1 − (1 + ε)c ε 2 t R(λ δ ) e −(1+2ε)t R(λ δ ) .(18)
Now, letting lim sup δ→0 t R(λ δ ) < ∞, we will consider two separate cases: (i) If lim inf δ→0 t R(λ δ ) = β > 0, then by choice of ε > 0 such that 1+ε ε 2 = β 2c , the lower bound in (18) is larger than a positive constant as δ → 0. (ii) If lim inf δ→0 t R(λ δ ) = 0, then imposing ε = 2c/(t R(λ δ )), the lower bound in (18) is again larger than a positive constant as δ → 0. The desired convergence in distribution (3) then follows in each case by Lemma 3.9.
Proof of Lemma 3.6 Noting that 1 − e −y (1 + y) ≤ y for all y > 0,
t R(λ δ ) = (1 + α)U (δ) δ 0 1 − e −λ δ x (1 + λ δ x) ˜ δ (dx) ≤ (1 + α)U (δ) δ 0 λ δ x˜ δ (dx) = (1 + α)U (δ) δ 0 x (dx) + δ (δ) λ δ . (19)
Then by the relation U (δ)I (δ) ≤ Cδ for a constant C (see [2,Prop 1.4]),
(19) = (1 + α)U (δ)I (δ)λ δ ≤ Cδλ δ .
So we can conclude that if lim sup δ→0 δλ δ < ∞, then the desired convergence in distribution (3) follows by Lemma 3.12.
Proofs of Results on L(t, δ)
Firstly, we prove Theorem 2.7, which confirms that L(t, δ) can replace N (t, δ) in the definition of the box-counting dimension of the range. This is done by showing that L(t, δ) N (t, δ), which is known to be sufficient by Remark 2.6.
Proof of Theorem 2.7
The jumps of the original subordinator X and the process with shortened jumpsX δ are all the same size, other than jumps bigger than size δ. The optimal number of intervals to cover the range, N (X, t, δ), always increases by 1 at each jump bigger than size δ, regardless of its size, so it follows that N (X, t, δ) = N (X δ , t, δ), with the obvious notation.
Instead of counting the number N (X, t, δ) of boxes needed to cover the range of X , consider those needed for the range of the subordinator X (0,δ) with Lévy measure (dx)1 {x<δ} (so all jumps of size larger than δ are removed), and adding Y δ t , which counts the number of jumps larger than size δ of X . Then one can easily verify that 2N (X, t, δ). Consider M (X (0,δ) , t, δ), the number of intervals in a lattice of side length δ to intersect with the range of X (0,δ) . It is easy to show N (t, δ) M(t, δ) (see [9, p 42]).
N (X, t, δ) ≤ N (X (0,δ) , t, δ) + Y δ t ≤Also, M(X (0,δ) , t, δ) = 1 δ X (0,δ) t
, since X (0,δ) has no jumps of size larger than δ.
Now, 1 δ X (0,δ) t 1 δ X (0,δ) t
for small enough δ, and hence
L(X, t, δ) = 1 δX δ t = 1 δ X (0,δ) t + Y δ t M(X (0,δ) , t, δ) + Y δ t N (X (0,δ) , t, δ) + Y δ t N (X, t, δ).
By Remark 2.6, lim δ→0 log(L(t,δ)) log(1/δ) = lim δ→0 log (N (t,δ)) log(1/δ) , and hence L(t, δ) can be used to define the box-counting dimension of the range of any subordinator.
Next we will prove the CLT result for L(t, δ), working with t = 1 for brevity. The proof is essentially the same for other values of t > 0. We will show convergence of the Laplace transform of 1 v(δ) (L(1, δ) − μ(δ)) to that of the standard normal distribution. Recall that Z ∼ N (0, 1) has Laplace transform E[e −λZ ] = e λ 2 /2 . (1), δL(t, δ) =X δ t is a subordinator with Laplace exponent˜ δ , and it follows that for any λ ≥ 0,
Proof of Theorem 2.10 By Remark 2.4 and
lim δ→0 E exp −λ L(1, δ) − μ(δ) v(δ) = e λ 2 2 ⇐⇒ lim δ→0 λμ(δ) v(δ) −˜ δ λ δv(δ) = λ 2 2 .
Recalling the definition μ(δ) = 1 δ (d + I (δ)), where I (δ) := δ 0 x˜ δ (dx), and writing δ in the Lévy Khintchine representation as in (1), it follows that
λμ(δ) v(δ) −˜ δ λ δv(δ) = λ(d + I (δ)) δv(δ) − dλ δv(δ) − δ 0 1 − e − λx δv(δ) ˜ δ (dx) = λI (δ) δv(δ) − δ 0 1 − e − λx δv(δ) ˜ δ (dx) = δ 0 λx δv(δ)˜ δ (dx) − δ 0 1 − e − λx δv(δ) ˜ δ (dx).(20)
Then applying the fact that y 2 2 − y 3 6 ≤ y − 1 + e −y ≤ y 2 2 for all y > 0,
δ 0 λ 2 x 2 2δ 2 v(δ) 2 − λ 3 x 3 6δ 3 v(δ) 3 ˜ δ (dx) ≤ (20) ≤ δ 0 λ 2 x 2 2δ 2 v(δ) 2˜ δ (dx).
By the definition of v(δ), it follows that v(δ) 2 = 1
δ 2 δ 0 x 2˜ δ (dx), and so δ 0 λ 2 x 2 2δ 2 v(δ) 2˜ δ (dx) = λ 2 2 .
It is then sufficient, in order to show that (20) converges to λ 2 2 , to prove that
lim δ→0 ∞ 0 x 3 δ 3 v(δ) 3˜ δ (dx) = 0.(21)
Again by the definition of v(δ), for (21) to hold we require both
lim δ→0 δ 0 x 3 (dx) δ 0 x 2 (dx) + δ 2 (δ) 3 2 = 0,(22)lim δ→0 δ 3 (δ) δ 0 x 2 (dx) + δ 2 (δ) 3 2 = 0.(23)
Squaring the expression in (22), since x ≤ δ within each integral, it follows that
δ 0 x 3 (dx) 2 δ 0 x 2 (dx) + δ 2 (δ) 3 ≤ δ 2 δ 0 x 2 (dx) 2 δ 0 x 2 (dx) + δ 2 (δ) 3 .
By the binomial expansion, (a + b) 3 ≥ 3a 2 b for a, b > 0, and then as δ → 0,
(22) ≤ δ 2 δ 0 x 2 (dx) 2 3 δ 0 x 2 (dx) 2 δ 2 (δ) = 1 3 (δ) → 0,
since the Lévy measure is infinite. For (23), simply observe that as δ → 0,
δ 3 (δ) δ 0 x 2 (dx) + δ 2 (δ) 3 2 ≤ δ 3 (δ) δ 2 (δ) 3 2 = 1 (δ) 1 2 → 0.
Next we will prove the almost sure convergence result for L(t, δ). If there is a drift and the Lévy measure is finite, then the result is trivial. So we need only consider cases with infinite Lévy measure, and begin with the zero drift case. Using a Borel-Cantelli argument (see [15, p. 32] for details), we shall prove that almost surely lim inf δ→0 L(t, δ)/μ(δ) = lim sup δ→0 L(t, δ)/μ(δ) = t.
First, we will prove the almost sure convergence to t along a subsequence δ n converging to zero. Then, by monotonicity of μ(δ) and L(t, δ), we will deduce that for all δ between δ n and δ n+1 , L(t, δ)/μ(δ) also tends to t as δ n → 0.
Proof of Theorem 2.8 For all ε > 0, by Chebyshev's inequality and Remark 2.11,
n P L(t, δ n ) tμ(δ n ) − 1 > ε ≤ 1 ε 2 n Var (L(t, δ n )) t 2 μ(δ n ) 2 = 1 ε 2 n t δ 2 n δ n 0 x 2 (dx) + δ 2 n (δ n ) t 2 δ 2 n δ n 0 x (dx) + δ n (δ n ) 2 = 1 tε 2 n δ n 0 x 2 (dx) + δ 2 n (δ n ) δ n 0 x (dx) + δ n (δ n ) 2 ≤ 1 tε 2 n δ n δ n 0 x (dx) + δ n (δ n ) δ n 0 x (dx) + δ n (δ n ) 2 = 1 tε 2 n 1 μ(δ n ) .(24)
Recall that μ(δ)
= ∞ 0 1 δ (x ∧ δ) (dx), so since 1 δ (x ∧ δ)
is non-decreasing as δ decreases, it follows that μ(δ) is non-decreasing as δ decreases. Now, μ is continuous, and lim δ→0 μ(δ) = ∞, so it follows that for any fixed r ∈ (0, 1) there is a decreasing sequence δ n such that μ(δ n ) = r −n for each n. Then (24) is finite, so by the Borel-Cantelli lemma, lim n→∞ L(t, δ n )/μ(δ n ) = t almost surely.
When there is no drift, L(t, δ) is given by changing the original subordinator's jump sizes from y to 1 δ (y ∧ δ). By monotonicity of this map, it follows that for a fixed sample path of the original subordinator, each individual jump of the process L(t, δ n+1 ) is at least as big as the corresponding jump of the process L(t, δ n ). So L(t, δ) is non-decreasing as δ decreases, and so for all δ n+1 ≤ δ ≤ δ n ,
L(t, δ n ) tμ(δ n ) μ(δ n ) μ(δ n+1 ) ≤ L(t, δ) tμ(δ) ≤ L(t, δ n+1 ) tμ(δ n ) = L(t, δ n+1 ) tμ(δ n+1 ) μ(δ n+1 ) μ(δ n ) .
Then by our choice of the subsequence δ n , it follows that for all δ n+1 ≤ δ ≤ δ n ,
r L(t, δ n ) tμ(δ n ) ≤ L(t, δ) tμ(δ) ≤ 1 r L(t, δ n+1 ) tμ(δ n+1 ) ,(25)
and since lim n→∞ L(t, δ n )/μ(δ n ) = t, it follows that
rt ≤ lim inf δ→0 L(t, δ) μ(δ) ≤ lim sup δ→0 L(t, δ) μ(δ) ≤ t r .
Taking limits as r → 1, it follows that lim δ→0 L(t, δ)/μ(δ) = t almost surely. For a process with a positive drift d > 0 and infinite Lévy measure, denote the scaling term obtained by removing the drift asμ(δ) := μ(δ) − d/δ. Then the above Borel-Cantelli argument forμ yields the almost sure limit along a subsequenceδ n as in (24). Then since the functions μ(δ) and L(t, δ) are again monotone in δ when there is a drift, the argument applies as in (25). N (t, δ), the almost sure behaviour in Theorem 2.1 is formulated in terms of the renewal function, and in order to write this in terms of the characteristics, the expression is more complicated than for L(t, δ). For details, see [24,Corollary 1] and [8,Prop 1], the latter of which is very powerful for understanding the asymptotics of U (δ) for subordinators with a positive drift, significantly improving upon results in [5].
Extensions and Special Cases
Extensions: Box-Counting Dimension of the Graph
The graph of a subordinator X up to time t is the set {(s, X s ) : 0 ≤ s ≤ t}. The boxcounting dimensions of the range and graph are closely related. This is evident when we consider the mesh box-counting schemes M G (t, δ), M R (t, δ), denoting graph and range, respectively. The mesh box-counting scheme counts the number of boxes in a lattice of side length δ to intersect with a set. Remark 5.2 It follows that the graph of every subordinator X has the same boxcounting dimension as the range of X t := t + X t , the original process plus a unit drift.
T 1 + δ] × [X T 1 , X T 1 + δ]
, and so on. Denote the number of these boxes by N G (t, δ), and write N R (t, δ) as the optimal number of boxes needed to cover the range.
If d ≥ 1, then we have T 1 = T (δ,∞) because X δ ≥ dδ. It follows that each time N G (t, δ) increases by 1, so does N R (t, δ), and vice versa, so N G (t, δ) = N R (t, δ), and the box-counting dimension of the range and graph are equal when d ≥ 1.
For d ∈ (0, 1), a similar argument applies with a covering of δ d × δ rectangles rather than δ × δ squares. Starting with [0, δ d ] × [0, δ], at time T 1 , add a new box
[T 1 , T 1 + δ d ] × [X T 1 , X T 1 + δ]
, and so on. The number of these boxes is again N R (δ, t), since X δ d ≥ δ. By Remark 2.6 , this covering of rectangles can still be used to define the box-counting dimension of the range, since for k := 1 d , with N G (t, δ) and N G (t, δ) as the number of squares and of rectangles, respectively, N G (t, δ) ≤ N G (t, δ) ≤ k N G (t, δ/k).
Remark 5.4
The box-counting dimension of the graph of every subordinator is 1 almost surely, since subordinators have bounded variation (BV) almost surely. The same is true for the graph of all BV functions/processes, including in particular every Lévy process without a Gaussian component, whose Lévy measure satisfies (1 ∧ |x|) (dx) < ∞. By Proposition 5.3, the box-counting dimension of the range of every subordinator with drift d > 0 is 1 almost surely.
Special Cases: Regular Variation of the Laplace Exponent
Corollary 5.5 is analogous to [24,Corollary 2], with L(t, δ) in place of N (t, δ). This allows very fine comparisons, not visible at the log scale, to be made between subordinators whose Laplace exponents are regularly varying with the same index.
Corollary 5.5
Consider a subordinator whose Laplace exponent is regularly varying at infinity, such that (λ) ∼ λ α F(λ) for α ∈ (0, 1), where F(·) is a slowly varying function. Then almost surely as δ → 0, for all t > 0,
L(t, δ) ∼ tδ −α F 1 δ (2 − α) .
Proof of Corollary 5.5 Note that d = 0, i.e. there is no drift, when the Laplace exponent is regularly varying of index α ∈ (0, 1). By Theorem 2.8, as δ → 0,
L(t, δ) ∼ tμ(δ) = t I (δ) δ = t δ δ 0 (x)dx.
Since is regularly varying at 0, as x → 0, (x) ∼ ( 1 x )/ (1 − α) (see [1, p. 75]). Then by Karamata's theorem (see [3,Prop. 1.5.8]), almost surely as δ → 0,
L(t, δ) ∼ tδ −α F 1 δ (2 − α) .
Corollary 5.6 strengthens the result of Theorem 2.7 when the Laplace exponent is regularly varying. The result cannot be strengthened in general, as the relationship between μ(δ) and U (δ) −1 is " " rather than "∼" (see [2,Prop. 1.4]).
Corollary 5.6
For a subordinator with Laplace exponent regularly varying at infinity with index α ∈ (0, 1), for all t > 0, almost surely as δ → 0, N (t, δ) ∼ (2 − α) (1 + α)L(t, δ) Corollary 5.6 follows immediately from Corollary 5.5 and [24, Corollary 2], which says that when the Laplace exponent is regularly varying at infinity, such that (λ) ∼ λ α F(λ) for α ∈ (0, 1), where F(·) is a slowly varying function, for all t > 0, almost surely as δ → 0,
N (t, δ) ∼ (1 + α)tδ −α F 1 δ .
Remark 5.7 For α ∈ (0, 1), (2 − α) (1 + α) takes values between π/4 and 1. So L(t, δ) and N (t, δ) are closely related when the Laplace exponent is regularly varying, but as δ → 0, L(t, δ) grows to infinity slightly faster than N (t, δ).
∞0
(1 ∧ x) (dx) < ∞. The renewal function is the expected first passage time above δ, U (δ) := E[T δ ], where T δ := ∞ 0 1 {X t ≤δ} dt.
where α > 0 is fixed and chosen sufficiently large that x δ < δ 0 x˜ δ (dx) for all δ (this is possible by the relation U (δ) −1 I (δ)/δ, see [2, Prop 1.4]),
Remark 3. 7
7For a driftless subordinator, Theorem 2.3 holds under the same condition (2) applied to the function H (y) := y 0 x (dx) rather than the integrated tail function I . The integrated tail I (y) = H (y) + y (y) depends on the large jumps of X since (x) = ((x, ∞)), but H does not depend on the large jumps, so these conditions are substantially different. With only minor changes, the argument as in the proof of Theorem 2.3 works with H in place of I . Under condition (2) for H in place of I , one can prove that Lemma 3.4 holds with H in place of I . Then we assume for a contradiction that there exists a sequence (δ m ) m≥1 converging to zero, such that lim m→∞ λ δ m δ m = ∞. But then as in the proof of Theorem 2.3, one can deduce that ηH (δ)+ H (2 −n δ) ≥ 1 (1+α)K H (δ), which contradicts the analogous Lemma 3.4 result with H in place of I .
Remark 3. 8
8Theorem 2.3 can also be proven for subordinators with a drift d > 0, under a stronger regularity condition. For Y t := X t −dt, define Y as the Laplace exponent of Y . The convergence in distribution (3) holds whenever lim sup x→0 x −5/6
Lemma 3. 11 (
11Jain, Pruitt [14, Lemma 5.2]) There exists c > 0 such that for every ε > 0, t ≥ 0 and x δ > 0 satisfying
Lemma 3. 12
12For α > 0, t = (1 + α)U (δ), and g(λ δ ) = x δ = δ/t, if lim sup δ→0 t R(λ δ ) < ∞,then the desired convergence in distribution(3), as in Theorem 2.3, holds.
Remark 4. 1
1Theorem 2.8 is formulated in terms of the characteristics of the subordinator (i.e. the drift and Lévy measure). For
Remark 5. 1
1For every subordinator with infinite Lévy measure or a positive drift, M G (t, δ) = t/δ + M R (t, δ), where · denotes the floor function. Indeed, M R (t, δ) increases by 1 if and only if M G (t, δ) increases by 1 and the new box for the graph lies directly above the previous box. For each integer n, M G (t, δ) also increases at time nδ, the new box directly to the right of the previous box.
Proposition 5. 3
3For every subordinator with drift d > 0, the box-counting dimensions of the range and graph agree almost surely. Proof of Proposition 5.3 Letting T (δ,∞) denote the first passage time of the subordinator above δ, consider an optimal covering of the graph with squares of side length δ as follows: Starting with [0, δ] × [0, δ], at time T 1 := min(T (δ,∞) , δ), add a new box [T 1 ,
24, Theorem 1.1].Theorem 2.1
has many equivalent formulations, see [1, Ex. III.7], and [3, Sect. 2.1]. We emphasise that
Theorem 2.8 For every subordinator with infinite Lévy measure, for all t > 0,lim
δ→0
for any subordinator. Theorems 2.1, 2.7 and 2.8 allow us to understand this relationship in terms of geometric properties of subordinators. Theorem 2.10 For every subordinator with infinite Lévy measure, for all t > 0,almost surely, where μ(δ) := 1
δ (d + I (δ)), and I (δ) =
δ
0
(y)dy.
Remark 2.9 It can be deduced from [2, Prop 1.4] that U (δ) −1
1
δ (d + I (δ)),
1 dy.Now, applying Markov's inequality, the definition E[e −λX t ] = e −t (λ) , and the fact
that U (δ) (1/δ) ≥ c for some constant c (see [2, Prop 1.4]),
Proof of Theorem 2.33.1 A Sufficient Condition for Theorem 2.3We will first work towards proving the following sufficient condition:
Acknowledgements Many thanks to Mladen Savov for guiding the author towards this interesting topic, and for numerous helpful discussions related to this work. Further thanks to Ron Doney for his feedback on an early draft of this paper, and thanks to an anonymous referee for their valuable comments on this work. Funding was provided by Engineering and Physical Sciences Research Council.Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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[] |
[
"Many-body non-equilibrium effects in all-electric electron spin resonance",
"Many-body non-equilibrium effects in all-electric electron spin resonance"
] |
[
"Jose Reina-Gálvez \nCenter for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain\n",
"Christoph Wolf \nCenter for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain\n",
"Nicolás Lorente \nCenter for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain\n"
] |
[
"Center for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain",
"Center for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain",
"Center for Quantum Nanoscience\nInstitute for Basic Science (IBS)\nDonostia International Physics Center (DIPC)\nCentro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)\nEwha Womans University\n03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain"
] |
[] |
Motivated by recent developments in measurements of electron spin resonances of individual atoms and molecules with the scanning tunneling microscope (ESR-STM), we study electron transport through an impurity under periodic driving as a function of the transport parameters in a model junction. The model consists of a single-orbital quantum impurity connected to two electrodes via time-dependent hopping terms. The hopping terms are treated at the lowest order in perturbation theory to recover a Lindblad-like quantum master equation with electron transport. As in the experiment, the ESR-STM signal is given by the variation of the long-time DC current with the driving frequency. The density-matrix coherences play an important role in the evaluation of the ESR-STM signal. Electron correlation is included in our impurity mode. The charging energy U has significant influence on the spin dynamics depending on the sign and magnitude of the applied DC bias. Our model allows direct insight into the origin of the ESR signal from the many-body dynamics of the impurity. arXiv:2303.09944v1 [cond-mat.mes-hall]
| null |
[
"https://export.arxiv.org/pdf/2303.09944v1.pdf"
] | 257,623,034 |
2303.09944
|
bef866dc3d4a20938d0a9a41aa8eba72332b85b3
|
Many-body non-equilibrium effects in all-electric electron spin resonance
Jose Reina-Gálvez
Center for Quantum Nanoscience
Institute for Basic Science (IBS)
Donostia International Physics Center (DIPC)
Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)
Ewha Womans University
03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain
Christoph Wolf
Center for Quantum Nanoscience
Institute for Basic Science (IBS)
Donostia International Physics Center (DIPC)
Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)
Ewha Womans University
03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain
Nicolás Lorente
Center for Quantum Nanoscience
Institute for Basic Science (IBS)
Donostia International Physics Center (DIPC)
Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)
Ewha Womans University
03760, 03760, 20018, 20018Seoul, Seoul, Donostia-San Sebastián, Donostia-San SebastiánKorea, Korea, Spain, Spain
Many-body non-equilibrium effects in all-electric electron spin resonance
(Dated: March 20, 2023)
Motivated by recent developments in measurements of electron spin resonances of individual atoms and molecules with the scanning tunneling microscope (ESR-STM), we study electron transport through an impurity under periodic driving as a function of the transport parameters in a model junction. The model consists of a single-orbital quantum impurity connected to two electrodes via time-dependent hopping terms. The hopping terms are treated at the lowest order in perturbation theory to recover a Lindblad-like quantum master equation with electron transport. As in the experiment, the ESR-STM signal is given by the variation of the long-time DC current with the driving frequency. The density-matrix coherences play an important role in the evaluation of the ESR-STM signal. Electron correlation is included in our impurity mode. The charging energy U has significant influence on the spin dynamics depending on the sign and magnitude of the applied DC bias. Our model allows direct insight into the origin of the ESR signal from the many-body dynamics of the impurity. arXiv:2303.09944v1 [cond-mat.mes-hall]
I. INTRODUCTION
The use of time-dependent techniques in the scanning tunneling microscope (STM) at GHz frequencies ushered in the acquisition of electron-spin resonances (ESR) with the STM [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These developments grant access to new phenomena thanks to the unprecedented high-energy resolution of ESR combined with the subatomic precision of the STM. Examples are the analysis of elusive atomic configurations on surfaces by measuring the actual magnetic moment of f-electron atoms [2], or the determination of isotopes of single adsorbates [6].
The ESR-STM technique measures the direct current (DC) through a localized spin impurity, such as single atoms or molecules, in the STM junction as the tipsample bias is periodically modulated. At a certain modulation frequency, the DC current experiences a variation that can be detected. The ESR spectra are values of the junction DC current as a function of driving frequency, typically in the GHz (µeV) range. To drive the localized electron spin, a suitable alternating electric field is fed either directly to the tip [1][2][3][4][5][6][7][8][9][10][11] or to the entire sample via an antenna [1,4]. The mechanism that couples the electric field to a local magnetic moment is not clear and substantial effort has been devoted to try to understand under what circumstances ESR is produced (for a recent review please refer to Ref. [17]). Clarifying the origin of ESR in the STM is not only conceptually, but also practically important, because the full development of the ESR-STM technique requires a high degree of control to * [email protected] † [email protected] ‡ [email protected] acquire meaningful signals.
In the present work, we address the effect of the transport parameters in the ESR signal. Understanding how transport affects the signal yields key information on the way the ESR is produced. In previous publications [18,19], we have shown that a time-dependent modulation of the tunneling matrix elements between electrodes and impurity is sufficient to produce a sizable ESR signal. It is well-known that electric fields efficiently modulate these transfer matrix elements [20,21], and in turn, this modulation drives the spin [18,19]. Moreover, the suggested adiabatic motion of the impurity in the timedependent electric field [1,22] would only increase the tunneling modulation. Our model is based on a transport description of the electron current in the presence of driving via the modulation of the tunneling matrix elements. We treat the spin degrees of freedom via a reduced density matrix, which allows us to develop a quantum master equation for the transport process under driving and with quantum spins [19].
The article is organized as follows. In Sec. II, we summarize the model and the theoretical approach. In the present article, we put special emphasis on clarifying the different equations and on how to treat the extended basis set to include electronic correlations under a finite charging energy U . We present the results of simulations with a set of parameters compatible with experimental ESR-STM setups in Sec. III. The calculations explore the behavior of the continuous wave (CW) ESR-STM signal (change in DC current as the driving frequency is changed) as a function of the DC bias. The results clearly show the role of the involved states, the importance of having changing populations and coherences, as well as their influence in the DC current that is ultimately the experimental observable. The results corroborate the im- The model used in our Hamiltonian representing the system in (a). A single orbital is connected via timedependent hopping elements, TL(t) and TR(t), to the left and right free-electron electrodes. Under an external magnetic field, the singly occupied spin up (↑) and down (↓) levels are split by the Zeeman energy with Larmor or resonance frequency f0. The orbital contains electron-electron correlation by the introduction of a charging energy (or intra-orbital Coulomb repulsion) U .
portance of coherent charge fluctuations to have a measurable signal in ESR-STM. Figure 1 shows a representative model of the type of system considered in this work. A central region that can be solved exactly is coupled via some hopping matrix elements to electron reservoirs. These hoppings need to be small compared to the typical energies of the central region in order to obtain a quantum master equation (QME) as will be shown in the following. We solve this QME in the long-time limit using Floquet's theorem that treats linear differential equations under a periodic drive [23]. Finally in this section, we derive the equation for the time-dependent electronic current and for its DC component in this long-time limit.
II. THEORETICAL APPROACH
A. The model Hamiltonian
The model for the full quantum system consists of a quantum impurity (a magnetic adsorbate, such as a single atom or molecule) tunnel-coupled to two electron reser-voirs, see Fig. 1. The full system is described by
H(t) = H elec + H I + H T (t),(1)
where the first term describes the two electrodes, the second term is the impurity Hamiltonian and the third term is the tunneling Hamiltonian, which is the only timedependent one. The electrodes are assumed to be described by one-electron states,
H elec = αkσ ε αk c † αkσ c αkσ ,(2)
α identifies the electrode (α = L,R), while σ =↑, ↓ is the electron spin projection along the quantization axis and k is its momentum. Each electrode is characterized by a chemical potential µ α such that the total DC bias is eV DC = µ L −µ R . Following Ref. [19], the quantum impurity consists of a single orbital with intra-orbital correlation represented by the charging energy U . The impurity is subjected to an external magnetic field such that its Hamiltonian is given by
H I = σ εd † σ d σ + Un d↑nd↓ + gµ B B ·ŝ,(3)
where ε is the orbital energy of the impurity, U is the corresponding Coulomb repulsion, andn dσ = d † σ d σ is the occupation operator of the orbital. Its spin operator, s has componentsŝ j = σ,σ d † σσ j σσ d σ /2, whereσ j (j = x, y, z) are the Pauli matrices. The last term of Eq.
(3) is the Zeeman contribution to the Hamiltonian.
The coupling between the impurity and the two reservoirs is described by the tunneling Hamiltonian
H T (t) = αkσ T α (t)c † αkσ d σ + T * α (t)d † σ c αkσ .(4)
The periodic drive is introduced by a time-dependent hopping, T α (t), parameterized as:
T α (t) = T 0 α [1 + A α cos(ωt)] ,(5)
following Refs. [18,19]. This approximation captures the effect of the driving electric field on the electron transfer probability because of the changing tunneling barrier. Figure 2 (a) shows a simple scheme for the modulation of the transmission of the wave function across one of the barriers, under varying external electric field. Although not needed, the presence of piezoelectric effects [24] would enhance the tunneling modulation in the time-dependent electric field. Tunneling modulation is very efficient in driving the spin. In Ref. [18], we showed that the tunneling modulation directly enters the Rabi flip-flop rate in an effective two-level system where electrons hop in and out the impurity. Indeed, the tunneling modulation implies an effective change of state of the impurity due to charge transfer from the electrodes.
To formulate the problem in terms of the reduced density matrix, we consider all possible configurations for To simplify, we assume that the eigenstate basis |l is, in first approximation, the same as the |p -basis. In order to be able to compare the energies of states with different number of electrons, we assume that the missing electrons are at the chemical potential of the electrodes at zero bias. The zero-electron state is at zero energy.
zero, one and two electrons in the impurity. Figure 2 (b) shows an energy diagram with the four possible eigenstates of the simplest spin-1/2 system. These configurations are |p with p =↑, ↓, 2, ∅. The first two account for one electron states while the third and forth are labelling the spin singlets with two and zero electrons, respectively. The impurity Hamiltonian in this basis is given by:
H I = p ε p |p p| + gµ B 2 {(B x + iB y )| ↓ ↑ | + (B x − iB y )| ↑ ↓ |}(6)
where ε p takes the values ε σ = ε + gµ B B z σ for σ =↑ or ↓, ε 2 = 2ε + U and ε ∅ = 0. The tunneling Hamiltonian in this |p basis set is:
H T (t) = αkσ T α (t)c † αkσ |∅ σ| + T α (t)c † αkσ |σ 2| + h.c.
(7) whereσ indicates the opposite to the σ spin projection.
Since the impurity Hamiltonian does not depend on time, we can use the eigenbasis of the impurity to describe the reduced density matrix. This eigenstate basis is given by
H I |l = E l |l .(8)
From now on, Latin characters (l, j, u, v, . . . ) refer to eigenstates that are combined electronic and spin configurations of the impurity. Accordingly, we write H T (t) in terms of the Hubbard operators |l j| obtained from these impurity many-body eigenstates [25],
H T (t) = αkσlj T α (t)c † αkσ λ ljσ |l j| + h.c. ,
that explicitly contains the matrix element that reflects the change of the many-body configurations of the impurity j of N + 1 electrons to l of N electrons:
λ ljσ = l|d σ |j = l|∅ σ|j + l|σ 2|j .(9)
B. The quantum master equation
We derive the QME by treating the coupling between the impurity and the reservoirs to the lowest order in perturbation theory in H T like in Refs. [19,[26][27][28][29][30][31][32]. This approximation amounts to the Born-Markov approximation [33,34]). The reduced density matrix in the impurity eigenstate basis set is
ρ lj (t) = Tr [ρ T (t)|l j|] ,(10)
with the trace taken over all the degrees of freedom of the total system andρ T (t) the time-dependent density matrix of also the total system [19,32].
The QME for ρ lj (t) is ρ lj (t) − i∆ lj ρ lj (t) = vu Γ vl,ju (t) + Γ * uj,lv (t) ρ vu (t) − Γ jv,vu (t)ρ lu (t) − Γ * lv,vu (t)ρ uj (t) ,(11)
where we have denoted ∆ l,j = E l − E j . All indices (l, j, v, u) refer only to many-body eigenstates of the impurity Hamiltonian, H I . The above QME, Eq. (11), is physically meaningful in the limit of weak coupling between impurity and electrodes. Here, weak means that the induced broadening of the impurity levels is smaller than the typical separation between levels, ∆ l,j . In this way, we make sure that the dynamics induced by the electrode is a small perturbation of the intrinsic impurity dynamics. In this limit, the different approaches to obtain a linear equation in the reduced density matrix yield the same QME [35].
The rates Γ(t) can be written as the sum of two contributions per electrode α:
Γ vl,ju (t) = α Γ − vl,ju,α (t) + Γ + vl,ju,α (t) .(12)
These rates can be expressed as
Γ − vl,ju,α (t) = i 2π σ λ vlσ λ * ujσ (1 + A α cos(ωt)) γ ασ × ∞ −∞ d f α ( ) 1 − ∆ ju + i /τ c + e iωt A α /2 − ∆ ju ω + i /τ c + + e −iωt A α /2 − ∆ ju − ω + i /τ c(13)
and
Γ + vl,ju,α (t) = − i 2π σ λ * lvσ λ juσ (1 + A α cos(ωt)) γ ασ × ∞ −∞ d (1 − f α ( )) 1 + ∆ ju − i /τ c + e iωt A α /2 + ∆ ju + ω − i /τ c + e −iωt A α /2 + ∆ ju − ω − i /τ c .(14)
The Fermi occupation function is given by f α ( ) = 1/ e βα( −µα) + 1 where β α is the inverse temperature times the Boltzmann constant for electrode α. Additionally, γ ασ is the level broadening due to the hopping, T 0 α , to electrode α for spin σ:
γ ασ = 2πD ασ |T 0 α | 2 ,(15)
that depends on the spin-dependent density of states, given by
D ασ = D α (1/2 + σP α ).(16)
Here, σ = ±1/2 and P α is a real number between -1 and 1 giving the spin polarization of the electrode, and D α is the density of states of the electrode at the Fermi energy. The above rate expressions neglect the timedependence of the electrodes [19]. This approximation is valid when the AC amplitude is much smaller than the DC component of the bias, otherwise Eq. (13) should include further Bessel functions to take into account the time-dependence of the electrode's Green's function [20,21].
The physical interpretation of the rates is straightforward. The rates are proportional to γ ασ , Eq. (15), that is the usual broadening induced by the hoppingmatrix elements and the density of states of the electrodes. Whether the process involves electrons or holes is contained in the appearance of the Fermi occupation factors. The expressions given in Eqs. (13) and (14) contain the λ matrix elements that take in the right weights of each impurity state. Finally, the factors including the magnitude of the hopping modulation A α take into account whether the electron-transfer process involves the absorption or emission of a photon from the microwave field.
Finally, a finite 1/τ c improves the convergence of the Green's function. Its inclusion leads to a small renormalization (or Lamb shift) of the spectrum.
C. The long-time limit
The rate is periodic in time at a fixed drive of frequency ω/2π and can be expanded in terms of Fourier components allowing us to express all equations in Floquet components. We introduce the Floquet index n as the Fourier index of the rate [23]: Γ vl,ju,α (t) = n e −inωt Γ vl,ju,α;n (ω).
From Eqs. (11) and (17) The current flowing out of electrode α is defined as
I α = −e d Nα
dt . This translates into the usual Meir-Wingreen formula [36], where now the matrix elements of all quantities appear in terms of many-body eigenstates, l, j, u (see Ref. [19]):
I α (t) = 2e lju Re ρ lu (t) Γ − lj,ju,α (t) − Γ + lj,ju,α (t) .(19)
Using I L = −I R , we symmetrize the current by making I = (I L + I L )/2 = (I L − I R )/2 and the above expression can be rewritten as
I(t) = − 2e lju Re ρ lu (t) Γ − lj,ju,R (t) + Γ + lj,ju,L (t)− Γ − lj,ju,L (t) − Γ + lj,ju,R (t) .(20)
This expression differs from previous approaches because it now contains the contribution of the coherences of the density matrix not only the populations [37][38][39]. We will show that under certain conditions the coherences are crucial for the correct calculation of the ESR signal. Since CW ESR-STM experiments measure the DC current in the long-time limit, we express it in Floquet components as:
I(ω) = −
Finally, let us emphasize that the full description based on a QME is possible when keeping to the lowest order in the hopping terms. This order is sufficient when the impurity level lies within the two Fermi levels. However, outside this bias window, higher-order terms may become comparable to or larger than the lower-order term. These higher-order terms contain sums over intermediate states opening the possibility to co-tunneling processes and to Kondo scattering (see for example Ref. [18] and [40,41]). In the present approach these processes are absent.
III. RESULTS
Our model consists of a S = 1/2 impurity that is weakly connected to two electrodes under a finite DC bias and a CW drive. Our aim is to explore the behavior of the ESR signal as the DC voltage is varied for a set of parameters intended to mimic conditions found in ESR-STM experiments.
A. Model parameters
The model parameters are chosen under the proviso of obtaining a strong ESR signal of a S = 1/2 system weakly connected to two electrodes under electrical driving. To achieve this, we need:
1. an imbalance in the transport-electron spin in order to make the main rates different from zero. This is achieved by having different spin-polarization of the electrodes.
2. a predominant long-time average population of one electron in the impurity, otherwise the system does not behave like a S = 1/2.
3. an electronic level, ε, within the DC-bias range.
4. to flip the transport spin using a magnetic field transversal to the electron spin polarization.
5. a modulation of the tunneling hopping with the spin-polarized electrode by the oscillating electric field.
6. low temperature. We take 1 K for both electrodes.
In our calculations, we achieve the above conditions with the following parameters: 1. The left electrode has a polarization of P L = 0.45 in Eq. (16). Increasing the polarization up to 100% will increase the ESR signal amplitude. 2. To stabilize the charge state, we apply different couplings with γ R = 20 × γ L = 5 µeV. This coupling asymmetry is often found in experiments, where the impurity couples more strongly to the substrate than the STM tip. The DC bias drop is eV DC = µ L − µ R . We use the model of a double-barrier tunnel unction [42] and assume an asymmetric DC bias drop where µ L = (1 − η)eV DC and µ R = −ηeV DC with the factor η = γ L /(γ L + γ R ) = 1/21. This means that the bias drop takes places mostly on the left electrode. 3. The energy of our model is set by ε = −10 meV. In addition, the electronic states are assumed to have an intrinsic width of /τ c = 10 µeV. In order to explore the interplay of the many-body states in ESR processes, we take a fixed charging energy close to the electronic level energy, of U = 3|ε|/2 = 15 meV. 4. In order to flip the spin, defined along the z-axis of the spin polarization, P L , we apply a B-field component along the x-axis perpendicular to the z-axis component. The magnetic field is taken as B = (0.6, 0, 0.1) T, which gives a Larmor frequency of approximately 17 GHz. The largest ESR signal takes place for a magnetic field completely aligned with the x-axis in good agreement with experiments [43]. 5. The modulation of the tunneling matrix element is A L = 50%, Eq. (5) and applied only to the left electrode, which is the polarized one. Since the right electrode is not spin polarized, A R does not contribute to the resonance, but only to the background current.
B. Non-zero rates: the opening of transport channels with applied bias A transport channel opens when the corresponding rates, Eq. (12) are different from zero. Inspection of Eq. (13) shows that this occurs when two conditions are met: The first one is energy conservation, largely controlled by the Fermi factors. The energy conservation implies that the change of state has to be compensated by the applied bias. Under our present conditions, the bias drop takes place largely at the left electrode, then ∆ v,l = E v − E l has to be larger than µ L = eV L = (1 − η)eV DC . This is due to the appearance of a term f (∆ v,l ) in Eq. (13) when 1/τ c → 0 + . The second condition is that the sequential transport process leads to a change in the charge state of the impurity such that λ vlσ = 0 when v and l differ in one electron of spin σ. Then, the difference in energy ∆ v,l in the rates Eq. (12) always addresses states differing by one electron.
C. DC-bias dependence of the ESR signal
The DC-bias will determine when the transport channels of the system opens. But the occurrence of ESR further depends on the possibility of a spin-flip process. For this, the transport channel must be compatible with spin-flip processes.
First, we study the dependence of the magnitude and sign of the ESR signal ∆I as function of the magnitude and sign of the applied DC bias. Figure 3 (a) and (b) show two representative spectra taken at opposite signs of the DC bias. The difference between both spectra is more than a change of sign. To better understand this behavior, Fig. 3 (c) shows the ESR peak intensity as function of V DC . Take, for example, a positive bias where we obtain a large negative value of the ESR signal. This correlates with a large contribution of the coherence-term ρ ↑↓ between spin up and down (Fig. 3 (d)). We emphasize that this occurs in the long-time limit under substantial decoherence of the system as long as the drive sustains the coherences. The connection between ESR signal and coherences of the density matrix can be understood by studying the behavior of the electronic current, Eq. (19).
When the applied bias is positive (µ L − µ R > 0), spinpolarized electrons flow from the left electrode into the impurity. A negative ion is formed if µ L > ∆ 2,↓ = E 2 − FIG. 3. a) and b) ESR signal ∆I(f ) = I(f )−IBG as function of relative frequency δ = f − f0 for two different signs of the DC bias. in a) the DC bias is negative and in b) the DC bias is positive which inverts the ESR amplitude. For this system the Larmor frequency is f0 = 17.025 GHz which is the natural resonance frequency of the Hamiltonian plus the re-normalization imposed by the Lamb shift. c) ESR signal and d) real part of the coherence ρ ↑↓ between spin up and down as a function of DC bias when on resonance (δ = 0). The transport channels are closed for VDC U +ε (neglecting the Zeeman energy) and VDC ≥ ε. In this work we took U = 3|ε|/2, so the ESR signal is zero between VDC /|ε| 0.5 and VDC /|ε| ≥ −1. The behavior of the ESR signal reflects the behavior of the coherences except for a sign. E ↓ ≈ U + (we have neglected the Zeeman energy), that corresponds to a transition from a singly-occupied level (with spin down, u =↓) to a doubly charged level (v = 2). At the same time, we need that µ R < ∆ 2,↓ , as is the case at positive bias. Similarly, the formation of the positively-charge ion is energetically possible. However, there is an important asymmetry due to the very different couplings between impurity and electrodes (γ L γ R ) as well as in the bias drop. As a consequence the formation of the negative ion is favored over the positive one for this present case.
Then, we can simplify the expression for the electron current, Eq.
where, for instance, Γ − ↓2,2↓,L;0 is the electron rate for a process that involves a non-spin-flip transition (spin-up state) through the doubly-occupied one by exchanging an electron with the left electrode, Floquet index n = 0. At the same time, ρ ↓ = ρ ↓↓,0 while ρ ↓↑ = ρ ↓↑,1 and ρ ↑↓ = ρ ↑↓,−1 where −1, 0, 1 are Floquet indices.
At a large-enough bias, all channels are open giving a background current, I BG :
I BG = 2e Re ρ ↓ (ω)Γ − ↓2,2↓,L;0 + ρ ↑ (ω)Γ − ↑2,2↑,L;0 ,
which recovers the usual expression for the current for very asymmetrical couplings [36]. The background current shows a small frequency dependence as it is largely given by the rates with Floquet index n = 0. Indeed, there is no coherence in the density matrix when the driving frequency is different from the Larmor frequency (off resonance) and I(ω) = I BG .
Only on resonance, is the coherence ρ ↓,↑ (ω) different from zero. Then, there is a clear frequency-dependent contribution to the current at the Larmor frequency that originates in the coherences of the density matrix. Accordingly, the coherences contribution to the DC current depends on the Floquet indices n = ±1.
Increasing the value of the charging energy, U , moves the doubly-occupied state energy (E 2 = 2 + U ). For U → +∞, it becomes impossible to open the channel connecting the single-electron states with the doublyoccupied one. As a consequence, the ESR signal completely disappears for positive bias.
At negative bias, µ L < ∆ ↓,∅ = −10 meV marks the threshold for having a current, where v = ∅ corresponds to the positively charged impurity. As in the discussion above, we have neglected the Zeeman energy. The ESR signal also follows the behavior of −ρ ↑↓ as above, Fig. 3.
The intermediate state mediating the transport process at negative bias is the one corresponding to the positive ion, v = ∅. Then Eq. (19) can be simplified by taking the positive ion contribution:
I(ω) = − 2e Re ρ ↓ (ω)Γ + ↓∅,∅↓,L;0 + ρ ↑ (ω)Γ + ↑∅,∅↑,L;0 + ρ ↓,↑ (ω)Γ + ↓∅,∅↑,L;−1 + ρ ↑,↓ (ω)Γ + ↑∅,∅↓,L;1 ,(23)
where again, the ESR signal originates in the coherences of the density matrix. Contrary to the positive-bias case, the limit U → ∞ does not alter the results since the doubly-occupied level is not involved. The presence of a finite charging energy then leads to breaking the electron-hole symmetry. At U → ∞ the electron-hole asymmetry becomes the largest, with no ESR signal for positive bias and a large signal for negative bias at the bias threshold marked by the impurity level. For DC bias below the threshold (at 5 mV here) the DC current drops dramatically as the channel closes and the line shape as a function of frequency becomes increasingly asymmetric. Moreover, the width of the resonance also increases with the DC bias, leading to smaller T2 times as the decoherence is enhanced. The more asymmetric Fano profiles are found near the transportchannel thresholds.
D. ESR-STM linewidths
for positive DC bias. At threshold, V DC ≈ U +ε ≈ 5 mV, a strongly asymmetric Fano profile is obtained. This behavior can be traced back to the interference between the on-resonance scattering with the background. As the bias is further reduced, the transmission channel is increasingly closed, leading to a smaller background current and a smaller signal. In this regime, the ESR signal also depends on the change of the populations, in stark contrast to the open channel case, where the ESR signal is basically determined by the coherences.
This closed-channel region is of practical importance because here the system exhibits an enhanced coherence time. The present treatment of this regime is valid as long as higher-order transport processes such as cotunneling are not dominating.
IV. SUMMARY AND CONCLUSIONS
This work explores the ESR signal in the DC current through a quantum impurity connected to two electron reservoirs under bias. The model is intended to reproduce the conditions of ESR-STM, where the applied bias contains a DC component and a AC component usually in the GHz frequency range. We extend previous work [19] to include finite intra-atomic correlation, and we show its impact on the DC-bias dependence of the ESR signal. Our theory is based on a Linblad-like QME that was obtained by keeping the modulation of the tunneling matrix element to lowest order. This limits the transport regime to the sequential or on-resonance one. This situation seems to be similar to transition-metal impurities, molecules, or alkali metal dimers that have s-electrons close to the Fermi energy of the substrate [44][45][46][47]. We only treat spin-1/2 systems in the present study but the extension to larger spin systems can be achieved with relative ease.
The addition of the impurity charging energy, U , breaks the electron-hole symmetry of the system. This has wide ranging implications for the transport when taking into account the opening and closing of different transport channels as the applied DC bias varies. As a consequence, spin-1/2 systems such as the ones of Refs. [44][45][46][47] should exhibit a bias-sign dependence of the ESR signal in the experiment.
Our work highlights the importance of properly including the complete reduced density matrix in the calculation of the ESR signal. In the open-channel case, we found that the ESR signal is proportional to the coherences or off-diagonal elements of the density-matrix. However, in the closed-channel region, the diagonal elements or populations play a significant role. Therefore, the ESR-induced change in the DC current can be indicative of coherences or population changes of the system depending on the transport regime.
The present theory is based on a charge-fluctuation description where the impurity charge is changing during the electron transport process, and the fluctuations induce the spin-flip processes that in turn lead to the ESR-signal as long as driving and polarization are maintained. Our results emphasize the need to correctly treat the coherent charge fluctuation and include the coherence in the description of the full transport processes, not only for the evaluation of the impurity's population but also in the equation of the electron current.
ACKNOWLEDGMENTS
We are pleased to thank our collaborators for important discussions. A non-exhaustive list of the many contributors to our discussions is: L. Arrachea Further financial support from projects RTI2018-097895-B-C44 and PID2021-127917NB-I00 funded by MCIN/AEI/10.13039/501100011033 is gratefully acknowledged. Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.
FIG. 1 .
1(a) Scheme of the electron transport geometry for an impurity (here a molecule) in a junction under an external drive given by the applied bias. In an ESR-STM setup, the left (L) and right (R) electrodes represent STM tip and substrate. (b)
FIG. 2 .
2(a) Barrier modulation represented here by two barriers at two different external electric fields. The transmitted wavefunction (given by the log of the density in thick orange and cyano curves) is much larger for one of the applied bias, illustrating the effect of the modulation of the hopping that provides the electric-field coupling to the impurity spin. (b) Energy scheme of the quantum impurity. The four possible states are |p =↑, ↓, 2, ∅.
vl,ju;n (ω) + Γ * uj,lv;−n (ω)]ρ vu;n−n −Γ * lv,vu;−n (ω)ρ uj;n−n − Γ jv,vu;n (ω)ρ lu;n−n }.(18) D. Expressions for the electronic current
Re ρ lu;−n (ω) × Γ − lj,ju,R;n + Γ + lj,ju,L;n − Γ − lj,ju,L;n − Γ + lj,ju,R;n .
(19), by neglecting the involvement of the positive ion, and only considering the negative ion as the intermediate step in the electron transfer between electrodes through the impurity:I(ω) = 2e Re ρ ↓ (ω)Γ − ↓2,2↓,L;0 + ρ ↑ (ω)Γ − ↑2,2↑,L;0 + ρ ↓,↑ (ω)Γ − ↓2,2↑,L;−1 + ρ ↑,↓ (ω)Γ − ↑2,2↓,L;1 ,
Figure 4
4shows four characteristic CW ESR-STM signals as a function of the frequency of the drive, f = ω/2π, FIG. 4. DC current as a function of the driving frequencyδ = f − f0, for the four different positive voltages a) VDC = 0, b) VDC = 2.5 mV, c) VDC = 5 mV and d) VDC = 7.5 mV. The background current was not removed. The current changes in a small interval about the resonance frequency.
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[] |
[] |
[
"Ying Cui [email protected]. \nDepartment of Industrial and Systems Engineering\nUniversity of Minnesota\nMinneapolisU.S.A. 55455\n",
"Junyi Liu [email protected]. \nDepartment of Industrial Engineering\nTsinghua University\n100084BeijingChina\n",
"Jong-Shi Pang [email protected]. \nThe Daniel J. Epstein Department of Industrial and Systems Engineering\nUniversity of Southern California\n90089Los AngelesU.S.A\n"
] |
[
"Department of Industrial and Systems Engineering\nUniversity of Minnesota\nMinneapolisU.S.A. 55455",
"Department of Industrial Engineering\nTsinghua University\n100084BeijingChina",
"The Daniel J. Epstein Department of Industrial and Systems Engineering\nUniversity of Southern California\n90089Los AngelesU.S.A"
] |
[] |
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of "stationary solutions" one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed "pseudo B-(for Bouligand) stationary solution", for a broad class of discontinuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example.
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"https://export.arxiv.org/pdf/2305.14798v1.pdf"
] | 258,865,667 |
2305.14798
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1d1679be9341527670ba6bd45d7011bb523def6e
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24 May 2023
Ying Cui [email protected].
Department of Industrial and Systems Engineering
University of Minnesota
MinneapolisU.S.A. 55455
Junyi Liu [email protected].
Department of Industrial Engineering
Tsinghua University
100084BeijingChina
Jong-Shi Pang [email protected].
The Daniel J. Epstein Department of Industrial and Systems Engineering
University of Southern California
90089Los AngelesU.S.A
24 May 2023Original: January 30, 2022; Revised: July 1, 2022The Minimization of Piecewise Functions: Pseudo Stationarity This paper is dedicated to Professor Roger J.B. Wets on the occasion of his 85th birthday, and for his pioneering research on the subject of our work.
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of "stationary solutions" one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed "pseudo B-(for Bouligand) stationary solution", for a broad class of discontinuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example.
Introduction
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. Like all nonconvex problems, the computational task of a minimizer, local or global, is prohibitively challenging, if not impossible. Thus a realistic goal is to try to compute a solution that is a promising candidate for a minimizer, in particular, one that satisfies some necessary conditions of a local minimizer, i.e., a stationary solution of some sort. There is a very long tradition of investigation of such conditions, starting from the classical one of a zero objective gradient for an unconstrained differentiable optimization problem to the use of some advanced subdifferentials from variational analysis [47] to obtain a set-inclusion problem. With the functions involved being discontinuous, an insightful understanding, constructive characterization, and profitable employment of the latter subdifferentials all are not easy tasks, especially when there are constraints that are embedded in the objective function taking on infinite value.
As an alternative, the idea of approximating the discontinuous functions by smooth, say continuously differentiable, functions appears promising. One of the first smoothing methods for optimization of discontinuous functions was proposed in [26]. In general, such a method involves smoothing a locally integrable function by mollifier approximations using integration. A sequence of the resulting smoothed problems is then solved and their stationary solutions are used to define a stationarity concept for the original problem. Studied comprehensively in the seminal paper [18] under the framework of minimizing an extended-valued function, this approach leads to the definition of "mollifier subgradients" that serve as the target of a computational resolution for a discontinuous optimization problem. See [34,35] for the applications of mollifier induced smoothing functions, called "averaged function" in [18,Definition 3.1], to nonlinear and semi-infinite programming, respectively. The paper [7] gives a survey of smoothing methods for nonconvex, nonsmooth optimization problems with the underlying functions being continuous. It is important to point out while the theory in [7,18] is quite general, the applied problems surveyed in [7] are all of the kind of a univariate nonsmooth function composite with a smooth function; for these composite functions, smoothing is applied to the univariate component while the composition is maintained. Thus it is fair to say that to date, the practical aspect of smoothing is restricted to this class of composite functions where the convolutional operation requires only the evaluation of integrals of scalar functions of one real variable; the approach becomes most effective when the resulting averaged functions are explicitly available for computational purposes with the convolution operation staying in the background for general analysis. A related paper [17] discusses smoothing by introducing artificial random variables that also allow the use of stochastic gradient methods for solving the approximated problems.
Supported by diverse source problems, this paper studies a broad class of discontinuous optimization problems with objective and constraints defined by piecewise functions modeled as the products of nonconvex nonsmooth functions and the indicator functions of the positive/nonnegative real axis, called Heaviside functions. Invented by Oliver Heaviside (1850Heaviside ( -1925 in his pioneering work on differential equations for the study of electromagnetic waves [29], the "open" Heaviside function is the indicator of the open interval ( 0, ∞ ). For our purpose, we include the indicator of the closed interval [ 0, ∞ ) also as a Heaviside function. With the advance of nonconvex nonsmooth optimization as documented in [12], our goal in approximating these problems is to least disrupt the nonsmoothness and nonconvexity of the given functions and do so only when needed, for instance in designing practical computational methods. Thus, unlike the previous work [3,[16][17][18]26] that smooth the discontinuous Heaviside function, the approximating problems developed in the present paper remain nonsmooth and nonconvex but are solvable by various surrogation methods; see [12,Chapter 7]. In addition to this departure from the smooth approximations, we adopt the approach to define an approximation-independent and subdifferential-free stationarity condition, called pseudo B-(for Bouligand) stationarity, that is necessarily satisfied by a local minimizer of the given discontinuous optimization problem. The definition is motivated by a recent study [27] of the ℓ 0 -optimization problem that lies at the center of sparse optimization in statistical estimation [28]. The well-known ℓ 0 -function is defined by: | t | 0 1 if t = 0 0 otherwise. We describe two constructive approaches for computing/approximating such a pseudo B-stationary solution. Omitting the details that are left for a follow-up algorithmic study, we emphasize that these approaches can be computationally implemented in practice by difference-of-convex programming based algorithms [37,40,42,43,45] when the involved functions are of this kind, and more generally, by surrogation methods [12,Chapter 7] for broad classes of nonsmooth nonconvex optimization problems.
The rest of the paper is organized as follows. The next section begins with the formal definition of the problem to be studied followed by some preliminary remarks. Section 3 presents a host of discontinuous piecewise functions arising from various optimization contexts that are unified by our central problem. A main result in Section 4 identifies a principal sign condition that plays a central role throughout the paper; this result connects our problem with two related problems in the literature in terms of their global minima. As a remedy to the computational intractability of these global minima, Section 5 defines a pseudo B-stationarity concept that is amenable to computation. Two constructive approaches for computing such a stationary solution is presented in Sections 6 and 8, respectively. In particular, the approximation approach described in the last section can be traced back to a pioneering paper by Roger Wets and his collaborators that we expand in Section 7 and to whom we dedicate our work.
Problem Definition and Preliminary Discussion
Consider the following piecewise optimization problem:
minimize x∈X Φ(x) c(x) + K k=1 ϕ k (x) 1 ( 0,∞ ) (g k (x)) subject to L ℓ=1 φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) ≤ b, called the functional constraint,(1)ψ ′ (x; v) lim τ ↓0 ψ(x + τ v) − ψ(x) τ
exists for all v ∈ R n . We make several immediate remarks about the formulation (1); foremost is to note that the functional constraint adds considerable challenges to this problem, without which the analysis simplifies somewhat. Other remarks are as follows:
• While the open Heaviside function 1 ( 0, ∞) (•) is lower semicontinuous on the real line, a product such as ϕ k (•) 1 ( 0,∞ ) (g k (•)) may not be lower semicontinuous atx ∈ g −1 k (0) unless ϕ k is nonnegative there. For the problem (1), this sign condition on the pairs of functions {ϕ k , g k } K k=1 and {φ ℓ , h ℓ } L ℓ=1 will persist throughout the paper. This restriction is responsible for the closedness of the feasible region and for the existence of minimizers of the problem, and thus for their relaxations, such as that of a pseudo B-stationary point to be defined later.
• The well-known ℓ 0 -function | • | 0 in sparsity estimation [28] can be written as:
| s | 0 = 1 ( 0,∞ ) (s) + 1 ( 0,∞ ) (−s) = 1 ( 0,∞ ) (|s|).(2)
This is a prominent applied instance of the open Heaviside function and provides much motivation for the theory developed in this paper. Conversely, the developed theory also provides a deeper understanding of the sparse optimization problem as a result of (??), any approximation of the open Heaviside function can immediately be specialized to the ℓ 0 -function.
• A term ψ(x) 1 [ 0,∞ ) (f (x)) involving the "closed" Heaviside function 1 [ 0,∞ ) (s) 1 if s ∈ [ 0, ∞ )
0 otherwise can be written as:
ψ(x) 1 [ 0,∞ ) (f (x)) = ψ(x) − ψ(x) 1 ( 0,∞ ) (−f (x)).
So the formulation (1) encompasses products of this kind that involves the closed Heaviside function. Subsequently, conditions imposed on the functions in (1) easily translate to corresponding conditions for functions involving the closed Heaviside function via the above relation.
In general, a locally Lipschitz function does not need to be directionally differentiable; however, for a univariate function f : (−ℓ, u) → R defined on an interval, if f is locally Lipschitz and monotone (i.e., nondecreasing or nonincreasing), then the one-sided derivatives:
f ′ (t; ±1) lim τ ↓0 f (t ± τ ) − f (t) τ , t ∈ (−ℓ, u)(3)
exist with signs determined by the monotonicity. For an integer N > 0, we let [N ] {1, · · · , N }.
Some Source Problems
The product functions in (1) are discontinuous piecewise functions; they include several interesting special cases, which we highlight below. These cases illustrate the versatility of the Heaviside functions in modeling a host of discontinuous functions in diverse contexts.
• Cost-efficient variable selection: This is an extension of the sparsity optimization problem in statistical estimation [28] in which there is a cost associated the collection of data in the modeling process [5,54,55] which we may formulate as either a soft penalty embedded in the objective function:
minimize x∈X⊆R n e(x) + λ n i=1 c i | x i | 0 ,(4)
or as a budget-type constraint:
minimize x∈X⊆R n e(x) subject to n i=1 c i | x i | 0 ≤ b,(5)
where e(x) is a loss function, λ > 0 is a given parameter, and the coefficients c i and the righthand constant b are positive. In addition to the application in modern-day sparse optimization, the term c i |x i | 0 is a classical modeling device in operations research applications as a set-up cost of an activity. Namely, a cost is incurred when there is a nonzero level of the ith activity and zero otherwise. The functional constraint expresses the available budget for the set-up or variableselection costs applicable in both the data or operational contexts. A cost associated with the level of the activity is included in either the objective term e(x) or the constraint X. One mathematical feature of the summation term in both formulations is worth noting: namely, all the coefficients associated with the ℓ 0 -functions are positive, thus, satisfying the sign condition mentioned before. These problems suggest that the multiplicative functions ϕ k (x) and φ ℓ (x) in the general problem (1) may be interpreted as costs in some applied models.
• Piecewise functions on complementary regions: Consider a simple example of a piecewise function which may or may not be continuous:
Ψ(x) = ψ 1 (x) if a ≤ f (x) ≤ b ψ 2 (x) if f (x) < a ψ 3 (x) if f (x) > b,(6)
for some scalars a and b satisfying −∞ ≤ a < b ≤ ∞. We then have
Ψ(x) = ψ 1 (x) 1 [ 0,∞ ) min( b − f (x), f (x) − a ) + ψ 2 (x) 1 ( 0,∞ ) (a − f (x)) + ψ 3 (x) 1 ( 0,∞ ) (f (x) − b) = ψ 1 (x) − ψ 1 (x) 1 ( 0,∞ ) max( f (x) − b, a − f (x) ) + ψ 2 (x) 1 ( 0,∞ ) (a − f (x)) + ψ 3 (x) 1 ( 0,∞ ) (f (x) − b),
whose validity is regardless of the continuity of Ψ on the sets f −1 (a) and f −1 (b). An interesting application of this class of piecewise function concerns the constrained optimization with (soft) penalty. Specifically, let's say that we wish to minimize a function ψ 1 (x) when the constraint f (x) ≥ 0 is satisfied and there is a penalty ψ 2 (x) when the constraint is not satisfied. This problem can be formulated as minimizing the combined objective:
ψ 1 (x) 1 [ 0,∞ ) (f (x)) + ψ 2 (x) 1 ( 0,∞ ) (−f (x))
. A special case of (6) is when the three component functions ψ i (x) are constants. More generally, a quantized function is a discontinuous step function that can very easily be described by a generalization of (6) with arbitrary (finite) number of mutually disjoint regions (intervals in the case of a univariate variable) within each of which the overall function is a constant. The references [39,52,53] have employed such quantized minimization problems for the training of deep neural networks.
• Best constraint selection: In contrast to the best variable selection problems (4) or (5), and yet so far at best minimally studied in the literature, the best constraint selection is a generalization of best variable selection and can be modeled using the summation:
K k=1 1 [ 0,∞) (f k (x)), where the family {f k (x) ≥ 0} K k=1
consists of the constraints to be selected. A related problem is when there is a cost c k (x) associated with the constraint f k (x) ≥ 0 being disrupted. This problem can be formulated as minimizing the weighted sum:
K k=1 c k (x) 1 ( 0,∞) (−f k (x))
or imposing a constraint defined by the sum.
Our work offers a pathway for the potential applications of best constraint selection in instances like these.
• On-off constraints by indicator variables: In the literature such as [6,30], such a constraint is f k (x) ≥ 0 if y k = 1 where y k ∈ {0, 1} is a binary variable that appears only in the constraint and for the sole purpose of turning on the constraint f k (x) ≥ 0. Equivalently, such an on-off constraint is equivalent to: f k (x) ≥ 0 if y k > 0 with y k restricted to be a continuous variable in the interval [0, 1]. In turn, the latter constraint is equivalent to y k f k (x) ≥ 0 with y k ∈ [0, 1], and instead of the constraint, a term 1 [ 0,∞) (y k f k (x)) can be added to the objective as part of the overall function to be minimized.
• Binary classifications by the sign function: In binary classification, it is customary to use the sign function to separate two classes. For instance, given a binary number σ = ±1 denoting classes A and B, respectively and with f (x) as the feature-dependent classification function, we classify the outcome from this function as A if f (x) > 0 and B if f (x) < 0. This classification scheme can by modeled by the composite indicator function: 1 ( 0,∞ ) (−σf (x)), which aims to count the number of misclassifications among the observational data. See [45] for a recent application of this formulation in the context of individualized decision making under uncertainty for medical treatment. Classification with margin is an extension that allows for minor errors; it classifies the outcome as A if f (x) ≥ ε and B if f (x) ≤ −ε, where ε > 0 is a small margin. This can be modeled by 1 ( 0,∞ ) (ε − σf (x)), which again counts the number of misclassifications.
• Products of indicators: It is clear that
1 [ 0,∞ ) (f (x)) 1 [ 0,∞ ) (g(x)) = 1 [ 0,∞ ) (min(f (x)
, g(x))); thus products of closed Heaviside functions can be combined into a single Heaviside function of the same kind via the use of the pointwise minimum operator. Less clear is the product of a closed and an open Heaviside function, say
1 [ 0,∞ ) (f (x)) 1 ( 0,∞ ) (g(x)
) . Nevertheless, we have
1 [ 0,∞ ) (f (x)) 1 ( 0,∞ ) (g(x)) = 1 ( 0,∞ ) (g(x)) 1 − 1 ( 0,∞ ) (−f (x)) = 1 ( 0,∞ ) (g(x)) − 1 ( 0,∞ ) (min(−f (x), g(x))),(x) = ψ 1 (x) if a ≤ f (x) < b. We see that ψ(x) = ψ 1 (x) 1 [ 0,∞ ) (f (x) − a) 1 ( 0,∞ ) (b − f (x)) = ψ 1 (x) 1 ( 0,∞ ) (b − f (x)) − 1 ( 0,∞ ) (a − f (x)) .
• Probabilistic functions and conditional expectations: The use of the Heaviside functions in equivalent formulations of probabilistic functions is well known; see [11] for a recent comprehensive study of a nonconvex nonsmooth approach for chance-constrained stochastic programs that is built on this fundamental formulation. The simplest example is the probabilistic function Pz(Ψ(x,z) ≥ 0), for some bivariate function Ψ(x, z). In the cited reference, we have considered a broad class of such functions defined as the difference of two convex functions each being the pointwise maximum of finitely many convex functions. An abstraction of such a difference-of-convex function Ψ(•, z) is a piecewise function of which the following bivariate extension of (6) is an example:
Ψ(x, z) ψ 1 (x, z) if a ≤ f (x, z) ≤ b ψ 2 (x, z) if f (x, z) < a ψ 3 (x, z) if f (x, z) > b.
Appendix B of the paper [10] contains many piecewise functions of the above kind arising from individualized decision-making under partial identification. Such a piecewise random functional gives rise to an expectation of piecewise functions such as Ez Ψ(x,z) with Ψ given above.
Additionally, consider the conditional expectation:
Ez φ(x,z) | f (x,z) ≤ 0 Ez φ(x,z) 1 ( −∞,0 ] (f (x,z)) Pz f (x,z) ≤ 0 = Ez φ(x,z) 1 ( −∞,0 ] (f (x,z)) Ez 1 ( −∞,0 ] (f (x,z))
, which again involves the closed Heaviside function. Conditional expectations have applications in modeling low probability-high consequence accidents [49] and in the semi-supervised structured classification problem [57]. The understanding and treatment of the deterministic problem (1) provides useful insights for studying stochastic programs with composite indicator functions such as the optimization with chance constraints involving discontinuous piecewise functions and conditional expectation functions that lead to fractional expectation functions, where the Heaviside functions are prominently present.
Equivalent Formulations: Global Minimizers
Before discussing stationarity solutions and local minimizers, we introduce classes of the problem (1) within which a (globally) optimal solution exists and there is an equivalence of the problem with several lifted formulations employing additional variables and special constraints. Equivalence means that there is a one-to-one correspondence between the optimal solutions of the problems and their optimal objective values are equal. The purpose of this section is twofold: (a) to show that the problem (1) has an optimal solution and is related to two well-studied problems in the literature under some sign restrictions on the family of functions {ϕ k , g k } K k=1 and {φ ℓ , h ℓ } L ℓ=1 , and (b) to support the sign conditions as a reasonable assumption to be imposed in subsequent sections. Since these equivalent formulations are for global minimizers whose computation is practically prohibitive if not impossible, and since they are not the focus in the later sections, we present the result below without the Heaviside constraint; i.e., for the following problem only:
minimize x∈X Φ(x) c(x) + K k=1 ϕ k (x) 1 ( 0,∞ ) (g k (x)).(7)
In the proof, we let (•) min denote the minimum objective value of the referenced problem; (t) ± max(±t, 0) be the nonnegative and nonpositive part of a scalar t; and let ⊥ denote the perpendicularity notation, which in the present context means the complementary slackness between the involved expressions.
Proposition 1. Let X be a compact set. Suppose that the functions c and {ϕ k , g k } K k=1 are continuous. The following two statements hold:
(A) Under the following sign restriction:
• for every k ∈ [K], the function ϕ k is nonnegative on the set X ∩ g −1 k (0), the problem (7) has an optimal solution; moreover, it is equivalent to:
-MPCC-1: minimize x ∈ X; s Φ MPCC1 (x, s) c(x) + K k=1 ϕ k (x) s k subject to 0 ≤ s k ⊥ ( g k (x) ) − ≥ 0, ∀ k ∈ [ K ] and 0 ≤ 1 − s k ⊥ ( g k (x) ) + ≥ 0, ∀ k ∈ [ K ].(8)
(B) Under the following strengthened sign restriction:
• for every k ∈ [K], the function ϕ k is nonnegative on the set X ∩ g −1 k ( −∞, 0 ], the problem (7) is equivalent to either one of the following two problems:
-MPCC-2: minimize x ∈ X; s Φ MPCC2 (x, s) c(x) + K k=1 ϕ k (x) s k subject to 0 ≤ 1 − s k ⊥ ( g k (x) ) + ≥ 0, ∀ k ∈ [ K ]
and s ∈ [ 0, 1 ] K (continuous variables);
(note the absence of the constraint s k ( g k (x) ) − = 0 compared to (8));
-on-off constraints:
minimize x ∈ X; z Φ on/off (x, z) c(x) + K k=1 ϕ k (x) (1 − z k ) subject to g k (x) ≤ 0 if z k = 1, ∀ k ∈ [K]
and z ∈ { 0, 1} K (binary variables).
Proof. We claim that the objective Φ is lower semicontinuous on X by showing that each product ϕ k (x) 1 ( 0,∞ ) (g k (x)) is lower semicontinuous on X under the sign assumption on the element functions {ϕ k } K k=1 . For this purpose, it suffices to show if ϕ and g are two continuous functions such that ϕ is nonnegative on X ∩ g −1 (0), then the level set:
L(α) { x ∈ X | ϕ(x) 1 ( 0,∞ ) (g(x)) ≤ α }
is closed for all scalars α ∈ R. Let {x ν } be a given sequence in L(α) converging to a limitx ∈ X.
There are 2 cases to consider:
• α ≥ 0: We must have for every ν, either g(x ν ) ≤ 0 or [ g(x ν ) > 0 and ϕ(x ν ) ≤ α ]. Then the limitx satisfies: either g(x) ≤ 0 or [ g(x) > 0 and ϕ(x) ≤ α ]. This shows thatx ∈ L(α), without requiring the sign restriction on ϕ.
• α < 0: We must have [ g(x ν ) > 0 and ϕ(x ν ) ≤ α ]. By the sign assumption on ϕ, the limitx must satisfy: g(x) > 0 and ϕ(x) ≤ α; sox ∈ L(α), completing the proof of the closedness of L(α).
Consequently, the problem (7) is a minimization problem of a lower semicontinous function on a compact set; thus it has an optimal solution. Omitting the proof of equivalence with the problem (8), we directly show the equivalence of the two problems (7) and (9). Let x ∈ X be arbitrary. Then the pair (x, s) is feasible to (9), where s k 1 ( 0,∞ ) (g k (x)) for all k ∈ [K]. Hence the minimum objective value of (7) is no less than that of (9).
Conversely, if (x, s) is feasible to (9), then Φ MPCC2 (x, s) ≥ Φ(x) by the sign restriction of ϕ k on X ∩ g −1 k ( −∞, 0 ]
. Therefore, equality holds and an optimal solution of one problem readily yields an optimal solution of the other. To show the equivalence of (7) and (10), let x ∈ X be arbitrary. Then the pair (x, z) is feasible to (10)
, where 1 − z k 1 ( 0,∞ ) (g k (x)) for all k ∈ [K]. Hence (7) min ≥ (10) min . Conversely, if (x, z) is feasible to (10), then Φ on/off (x, z) ≥ Φ(x).
Hence equality holds and there is a one-to-to correspondence between the optimal solutions of these two problems.
Being instances of a mathematical program with complementarity constraints (MPCC), the formulations (8) and (9) are similar to the equivalent formulations of an ℓ 0 -minimization problem used in [22]. These two MPCC's (8) and (9) are special instances of a mathematical program with vanishing constraints for which there is an extensive literature; see [2,14,32,33] which contain many more references. With an emphasis toward more general constraints, these references have paid significant attention to constraint qualifications [24] and their consequences for stationarity conditions and regularization methods. In contrast, our study bypasses such complementarity constraint qualifications and aims to analyze a kind of stationary solutions defined directly on the discontinuous piecewise problem (1). The formulation (10) is one with on-off constraints described by indicator variables [6,30]. The method of proof of the equivalence of the two problems (9) and (10) is fairly elementary; the same methodology is also discussed and used, e.g., in [36,41].
The principal purpose of presenting Proposition 1 is to highlight the two sign assumptions in (A) and (B). They delimit the applicability of the theory in the rest of the paper. The sign conditions can be constructively verified (if desired) when the functions ϕ k are convex and g k are affine (for (A)) or convex (for (B)). In general, these assumptions can be formulated equivalently as the optimum objective values of the minimization problems being nonnegative:
minimize x∈X ϕ k (x) subject to g k (x) = 0 | minimize x∈X ϕ k (x) subject to g k (x) ≤ 0.
respectively. These are convex programs under stated stipulations of g k and ϕ k .
Pseudo Bouligand Stationarity
With the discontinuity of the (open) Heaviside function it is not easy to derive transparent necessary conditions for a local minimizer of the problem (1), let alone computing it. Although there are various one-sided directional derivatives (such as those of the Dini kind), and more generally, the subderivatives in modern variational analysis [47] that one may apply to the objective function Φ(x), these derivatives are defined for general functions; in particular, their specializations to functions such as Φ(x) in (1) do not immediately yield useful insights about the problem without carefully unwrapping the details of the derivatives. Instead, our approach herein is based on elementary onesided directional derivatives and basic optimization theory. One immediate benefit of our approach is that it handles constraints with the Heaviside functions at much ease, unlike the variational approach which requires the constraints to be converted to extended-valued functions embedded in the objective. Another important point to make is that we aim to connect the theory with computations; namely, we want to ensure two goals of the defined solution concepts: (i) they are computationally achievable, and (ii) they are not over-relaxed. As the background for the subsequent Definition 3, we provide a result [27, Proposition 3] that was stated for a global minimizer in the reference, which we broaden to allow for a local minimizer.
Proposition 2. Let X ⊆ R n , θ : O ⊇ X → R,
and γ > 0 be given. Ifx is a global (local) minimizer of the problem:
minimize x∈X θ(x) + γ n i=1 | x i | 0 ,(11)
thenx is a global (local, respectively) minimizer of the problem:
minimize x∈X θ(x) subject to x A 0 (x) = 0,(12)
where
A 0 (x) i ∈ [n] |x i = 0 .
Restricted to problems with θ convex, a vectorx that is a global minimizer of (12) is termed a "pseudo-minimizer" of (11) in the reference. It turns out that such minimizers are the points of attraction of sequences of stationary solutions of the "folded concave" approximations of the ℓ 0problem; see [27,Proposition 5]. The main idea of the problem (12) to deal with the discontinuous function | • | 0 is to transfer, or "pull down" the discontinuity of the objective to a constraint. We plan to apply this idea to the problem (1); nevertheless the resulting "pulled-down" extension of (12) will have nonconvex objectives and constraints. This necessitates us to recall the definition of a Bouligand stationary (B-stationary in short) solution of a B-differentiable program. For an abstract optimization problem: minimize
x∈ X θ(x), where X is a closed set in R n and θ : O → R is a B-differentiable function, a vectorx ∈ X is a Bouligand stationary solution of θ on X [12, Definition 6.1.1] if θ ′ (x; v) ≥ 0 for all v ∈ T ( X;x), where T ( X;x) v ∈ R n ∃ {x ν } ⊂ X converging tox and {τ ν } ↓ 0 such that v = lim ν→∞ x ν −x τ ν is the tangent cone of X atx. If X is additionally convex, B-stationarity reduces to d-stationary ("d" for directional), which is: θ ′ (x; x −x) ≥ 0 for all x ∈ X.
For the problem (1), we define the following six index sets corresponding to a givenx ∈ X:
K > (x) { k ∈ [K] | g k (x) > 0 }; K = (x) { k ∈ [K] | g k (x) = 0 } K < (x) { k ∈ [K] | g k (x) < 0 } L > (x) { ℓ ∈ [L] | h ℓ (x) > 0 }; L = (x) { ℓ ∈ [L] | h ℓ (x) = 0 } L < (x) { ℓ ∈ [L] | h ℓ (x) < 0 }.
With these index sets, we define the "pulled-down", or "pseudo stationarity" problem atx:
minimize x Φ(x;x) c(x) + k ∈ K>(x) ϕ k (x) subject to x ∈ X; ℓ ∈ L>(x) φ ℓ (x) ≤ b g k (x) ≤ 0 ∀ k ∈ K = (x) ∪ K < (x) K ≤ (x) g k (x) ≥ 0 ∀ k ∈ K > (x) h ℓ (x) ≤ 0 ∀ ℓ ∈ L = (x) ∪ L < (x) L ≤ (x)
and
h ℓ (x) ≥ 0 ∀ ℓ ∈ L > (x) feasible set denoted S ps (x).(13)
Quite different from the three problems in Proposition 1, the above problem is obtained by pulling the indicator functions out of the objective function and the functional constraint and imposing constraints restricting the variable x according to those defined by the functions {g k , h ℓ } and satisfied by the vectorx on hand. Specialized to the ℓ 0 -optimization problem (11), the above pulled-down problem reduces to (12). Clearly, ifx is a globally optimal solution to (1) and provided that ϕ k is nonnegative on
X ∩ g −1 k (0) for k ∈ K > (x) and φ ℓ is nonnegative on X ∩ h −1 ℓ (0) for ℓ ∈ L > (x)
, then x is globally optimal for (13); however, the converse is generally not true because S ps (x) is at best only a subset of the feasible set of (1). This failed equivalence is not surprising for the main reason that (13) is defined at a givenx; this definition is for the purpose of answering the question: what is a necessary condition for a given feasible vector of (1) to be its local minimizer?
The following definition specifies a pseudo solution of (1) as a fixed point of the self-defined "locmin" or "B-stationarity" mapping. Note that withx given, problem (13) is a standard nonlinear program; thus, a B-stationary point of the problem is a vector
x ∈ S ps (x) such that Φ(•;x) ′ ( x; v) ≥ 0 for all v ∈ T (S ps (x); x).
Definition 3. A vectorx ∈ X is said to be a
• pseudo B-stationary point of (1) ifx is a B-stationary point of the pulled-down problem (13).
• pseudo local minimizer of (1) ifx is a local minimizer of the pulled-down problem (13).
Besides its dictionary meaning of being not real, the adjective "pseudo" carries the hiddent meaning that the concepts are defined in terms of an auxiliary problem induced by the candidate solution on hand. The following result shows that the above pseudo conditions are necessary for a local minimizer of (1), thus providing a first step in computing a promising candidate for a local minimum of the problem. (a)x is a local minimizer of (1).
(b)x is a pseudo local minimizer of (1).
(c)x is a pseudo B-stationary point of (1).
Proof. It suffices to show that the local minimizer of problem (1) must be a pseudo local minimizer. By the continuity of the functions {g k } K k=1 , there exists a neighborhood N ofx such that for all x ∈ N , it holds that: g k (x) < 0 for all k ∈ K < (x) and g k (x) > 0 for all k ∈ K > (x); and similarly for the {φ ℓ , h ℓ }-functions. We may restrict the neighborhood N so thatx is a minimizer of Φ on X ∩ N .
Hence if x ∈ N is feasible to (13), then x is also feasible to (1) because
L ℓ=1 φ ℓ (x)1 (0,∞) (h ℓ (x)) = ℓ∈L>(x) φ ℓ (x). Thus we have Φ(x;x) = Φ(x) ≥ Φ(x) = Φ(x;x)
showing thatx is a local minimizer of the problem (13); thus (b) holds.
An important point of Proposition 4 is that the two pseudo conditions are necessary for a local minimizer of (1). With the two approaches described in the two later sections and with the aid of a host of existing algorithms for practically implementing the approaches (as mentioned in the closing of the Introduction), the computation of a pseudo B-stationary solution can be accomplished by iterative algorithms for large classes of functions. This is in contrast to the computation of a pseudo local minimizer which in general is a daunting, if not impossible task. It is therefore natural to ask whether there are nonconvex nondifferentiable classes of problems for which a pseudo B-stationary point must be pseudo locally minimizing. It turns out that the answer is affirmative based on the convex-like property defined as follows; see [11,Section 4.1]. A function f : R n → R is said to be convex-like near a vectorx if there exists a neighborhood N ofx such that
f (x) ≥ f (x) + f ′ (x; x −x), ∀ x ∈ N .
A very broad class of convex-like functions consists of the following 3-layer composite functions:
f (x) ϕ • θ • ψ(x),(14)
where ϕ : R → R is piecewise affine and nondecreasing; θ : R → R is convex, and ψ : R n → R is piecewise affine; see [11,Lemma 10]. In particular, piecewise affine functions and convex functions are convex-like near any point. Under the convexity-like conditions, we aim to specialize [11,Proposition 9] to the problem (13). For this purpose, letx ∈ S ps (x). We then have the inclusion:
T (S ps (x);x) ⊆ v ∈ T (X;x) ℓ∈L>(x) φ ′ ℓ (x; v) ( ≤ 0) g ′ k (x; v) ≤ 0, ∀ k ∈ K = (x) h ′ ℓ (x; v) ≤ 0, ∀ ℓ ∈ L = (x) L(S ps (x);x),(15)
where the notation ( ≤ 0) means that this constraint is vacuous if the functional constraint holds as
a strict inequality atx; i.e., if L ℓ=1 φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) = ℓ ∈ L>(x) φ ℓ (x) < b.
The equality of the leftand right-hand cones in (15) is the Abadie constraint qualification (ACQ) for the set S ps (x) at the member vectorx. Sufficient conditions for this CQ to hold are known; in particular, either one of the following two sets of conditions yields the ACQ (see e.g. [42, Section 4.1]):
• (piecewise polyhedrality) all the functions {g k } k∈K=(x) and {h ℓ } ℓ∈L=(x) , (and {φ ℓ } ℓ∈L>(x) too if the functional constraint is binding atx) are piecewise affine; or
• (directional Slater) there exists a vector v ∈ T (X;x) such that ℓ∈L>(x) φ ′ ℓ (x; v) ( < 0 ) g ′ k (x; v) < 0, ∀ k ∈ K = (x) h ′ ℓ (x; v) < 0, ∀ ℓ ∈ L = (x),
where (< 0) has the same meaning as (≤ 0) when the functional constraint is not binding atx. We have the following result which follows readily from [11,Proposition 9].
Proposition 5. Under the blanket assumption of (1), suppose that c,
{g k } k∈K=(x) , {ϕ k } k∈K>(x) , {h ℓ } ℓ∈L=(x) , and {φ ℓ } ℓ∈L>(x)
, are all convex-like nearx, which is a pseudo B-stationary solution of (1). If the ACQ holds for the set S ps (x) atx, thenx is a pseudo local minimizer of (1).
In terms of the larger cone L(S ps (x);x), it follows that a feasible vectorx of (1) is a pseudo Bstationary solution if the following implication holds:
v ∈ L(S ps (x);x) ⇒ Φ(•;x) ′ (x; v) ≥ 0; (16) or equivalently, if 0 ∈ argmin v ∈ L(Sps(x);x) Φ(•;x) ′ (x; v). When the functions c, {g k } k∈K=(x) , {ϕ k } k∈K>(x) ,
{h ℓ } ℓ∈L=(x) , and {φ ℓ } ℓ∈L>(x) are of the difference-of-convex (dc) kind, then the latter minimization problem in v (withx given) is a dc constrained dc program that has been studied in [12,Section 6.4]. Under the ACQ for the set S ps (x) atx, the implication (16) is necessary and sufficient for pseudo B-stationarity. The upshot of this discussion is that with (or without) the ACQ, a feasible vector can in principle be checked for pseudo B-stationarity by methods existed in the literature; nevertheless, these methods can not be applied to compute a pseudo B-stationary solution. This is the task in the rest of the paper.
The pseudo B-stationarity definition can be phrased in an equivalent way, which when strengthened, yields a sufficient condition of a local minimizer of the problem (1). The latter condition is related to the formulations (8), (9) and (10) albeit with some obvious differences. First, the auxiliary variables s and z are pulled outside of these formulations; thereby eliminating the products in the objective functions of these problems. More interestingly, the result below highlights the difference between the necessary conditions (in Proposition 4) and the sufficient conditions for a local minimum. For the former, the condition is the existence of (binary) "multipliers"; for the latter, the condition is "for all" such multipliers.
Proposition 6. Let c, {ϕ k , g k } K k=1 , and {φ ℓ , h ℓ } L ℓ=1 be B-differentiable
functions on the open set O containing the closed set X. Letx ∈ X be given. The following three statements hold.
(a) A necessary condition forx to be a local minimizer of (1) is that there exist (finite) families of binary multipliers
{ξ k } k∈K=(x) ⊂ {0, 1} |K=(x)| and {µ ℓ } ℓ∈L=(x) ⊂ {0, 1} |K=(x)| such thatx is a local minimizer of the problem: minimize x Φ(x;x) c(x) + k ∈ K>(x) ϕ k (x) subject to x ∈ X; ℓ ∈ L>(x) φ ℓ (x) ≤ b ( 1 − ξ k ) g k (x) ≤ 0 ∀ k ∈ K = (x) g k (x) ≤ 0 ∀ k ∈ K < (x) g k (x) ≥ 0 ∀ k ∈ K > (x) ( 1 − µ ℓ ) h ℓ (x) ≤ 0 ∀ ℓ ∈ L = (x) h ℓ (x) ≤ 0 ∀ ℓ ∈ L < (x)
and
h ℓ (x) ≥ 0 ∀ ℓ ∈ L > (x) denoted as S ps (x; ξ, µ),
same as S ps (x) except for the constraints indexed by
K = (x) × L = (x).(17)
(b) Conversely, suppose that in a neighborhood ofx, the products ϕ k [g k ] + and φ ℓ [h ℓ ] + are nonnegative for all pairs (k, ℓ) in K = (x) × L = (x). Then a sufficient condition forx to be a local minimizer of (1) is that for all
{ξ k } k∈K=(x) ⊂ {0, 1} |K=(x)| and {µ ℓ } ℓ∈L=(x) ⊂ {0, 1} |L=(x)| ,
x is a local minimizer of (17).
(c) Alternatively, suppose that ϕ k (x) and φ ℓ (x) are nonnegative for all (k, ℓ) in K = (x)× L = (x). Then a sufficient condition forx to be a local minimizer of (1) is that for all
{ξ k } k∈K=(x) ⊂ {0, 1} |K=(x)| and {µ ℓ } ℓ∈L=(x) ⊂ {0, 1} |L=(x)| ,x is a local minimizer of minimize x Φ ξ ≥ (x;x) c(x) + k ∈ K>(x) ϕ k (x) + k ∈ K=(x) ξ k ϕ k (x) extra term with multiplier subject to x ∈ X; ℓ ∈ L>(x) φ ℓ (x) + ℓ ∈ L=(x) µ ℓ φ ℓ (x) extra term with multiplier ≤ b ( 1 − ξ k ) g k (x) ≤ 0 ∀ k ∈ K = (x) g k (x) ≤ 0 ∀ k ∈ K < (x) g k (x) ≥ 0 ∀ k ∈ K > (x) ( 1 − µ ℓ ) h ℓ (x) ≤ 0 ∀ ℓ ∈ L = (x) h ℓ (x) ≤ 0 ∀ ℓ ∈ L < (x)
and
h ℓ (x) ≥ 0 ∀ ℓ ∈ L > (x) .(18)
Proof. The necessary condition in (a) is clear because we can let ξ k = 0 = µ ℓ for all (k, ℓ) ∈ K = (x) × L = (x). For the sufficiency, suppose thatx is a local minimizer of (17) for all tuples {ξ k } k∈K=(x) and {µ ℓ } ℓ∈L=(x) as stated. We first note thatx must be feasible to (1) because
L ℓ=1 φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) = ℓ ∈ L>(x) φ ℓ (x).
Since the family of problems (17) is finite, there exists a neighborhood N ofx such that for all
{ξ k } k∈K=(x) ⊂ {0, 1} |K=(x)| and {µ ℓ } ℓ∈L=(x) ⊂ {0, 1} |L=(x)| , if x ∈ N is feasible to (17), then Φ(x;x) ≥ Φ(x;x) = Φ(x)
. Without loss of generality, we may assume that this neighborhood is such that for all x ∈ N , it holds that ϕ k (x) [g k (x)] + ≥ 0 and φ ℓ (x) [h ℓ (x)] + ≥ 0 for all (k, ℓ) ∈ K = (x) × L = (x); g k (x) < 0 for all k ∈ K < (x) and g k (x) > 0 for all k ∈ K > (x); and similarly for the {φ ℓ , h ℓ }-functions. Let x ∈ N be feasible to (1). For each pair (k, ℓ) ∈ K = (x) × L = (x), let ξ k 1 ( 0,∞ ) (g k (x)) and µ ℓ 1 ( 0,∞ ) (h ℓ (x)). To show that x is feasible to (17) for this pair (ξ, µ), it suffices to verify the functional constraint and ( 1 − ξ k ) g k (x) ≤ 0 and ( 1 − µ ℓ ) h ℓ (x) ≥ 0 for all (k, ℓ) ∈ K = (x) × L = (x). The latter is clear. For the former, we have
b ≥ L ℓ=1 φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) = ℓ∈L>(x) φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) + ℓ∈L=(x) φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) ≥ ℓ∈L>(x) φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)) because φ ℓ (x) [h ℓ (x)] + ≥ 0 for ℓ ∈ L = (x) = ℓ∈L>(x) φ ℓ (x).
Hence it follows that
Φ(x) = c(x) + k∈K>(x) ϕ k (x) 1 ( 0,∞ ) (g k (x)) + k∈K=(x) ϕ k (x) 1 ( 0,∞ ) (g k (x)) ≥ Φ(x,x) because ϕ k (x)[g k (x)] + ≥ 0 for k ∈ K = (x) ≥ Φ(x;x) = Φ(x),
as desired. Finally, the proof of statement (c) is very similar to that of (b). Omitting the details, we simply note thatx, being feasible to (18), must be feasible to (17), and thus to (1).
Remarks. To more precisely connect the pull-down problem (13) with the problem (17) for various choices of the binary pairs (ξ, µ), we note that the givenx is a local minimizer of the former if and only ifx is a local minimizer of the latter for some (ξ, µ). In view of statement (b) in Proposition 6, which involves for all (ξ, µ), it is natural to ask the following question: supposex is a local minimizer of (13), are there sufficient conditions that will ensurex to be a local minimizer of (1)? Since local optimality involves a neighborhood ofx wherein the signs of ϕ k (x)[g k (x)] + and φ ℓ (x)[h ℓ (x)] + are relevant, an answer to the question would essentially reduce to statement (b) in Proposition 6.
Incidentally, the difference between parts (b) and (c) in Proposition 6 is in their respective assumptions on the functions (ϕ k , g k ) and (φ ℓ , h ℓ ) for (k, ℓ) ∈ K = (x) × L = (x) and the two resulting problems (17) and (18).
The Epigraphical Approach
Guided by the sign assumption in case (A) of Proposition 1, which has two parts for the problem (1),
• for every k ∈ [K], each ϕ k is nonnegative on the set X ∩ g −1 k (0); and • for every ℓ ∈ [L], each φ ℓ is nonnegative on the set X ∩ h −1 ℓ (0); we present in this section a constructive approach for computing a pseudo B-stationary point of (1). The approach is based on an epigraphical formulation of the problem that lifts it to a higher dimension. We recall that the epigraph of a function f on the closed set S ⊆ O is the set
epi(f ; S) ( t, x ) ∈ R × S | t ≥ f (x) .
In what follows, we describe the tangent cone of the epigraph of a discontinuous product function ψ1 ( 0,∞ ) (f ); the description not only provides insights for the demonstration of the epigraphical approach to successfully accomplish the computational goal, but is also of independent interest as it relates to some existing results in the literature. A basic result of this kind for a locally Lipschitz function is available from [8, Theorem 2.4.9 part (a)]; but it is not applicable to the discontinuous Heaviside function. An advanced result for an arbitrary function, and for the indicator function in particular, can be found in [47,Theorem 8.2] which is based on the notion of subderivatives. Rather than going through the calculation of the latter derivatives for the product function of interest, we present an elementary derivation that exposes the epigraph of ψ1 ( 0,∞ ) (f ) as the union of two closed sets and highlights the consequence of the sign condition of ψ on the zero set of f .
S ∩ f −1 (0). Let π(x) ψ(x) 1 ( 0,∞ ) (f (x)). Then, epi(π; S) = ( t, x ) ∈ R × S | t ≥ ψ(x), f (x) ≥ 0 denoted E 1 ∪ R + × ( S ∩ f −1 ( −∞, 0 ] ) denoted E 2 .(19)
Thus for any pair (t, x) ∈ epi(π; S),
T (epi(π; S); (t, x)) = T (E 1 ; (t, x)) ∪ T (E 2 ; (t, x)),(20)
where T (E i ; (t, x)) ∅ if (t, x) ∈ E i for i = 1, 2. Moreover, the following three statements (a), (b), and (c) hold:
(a) (t, x) ∈ epi(π; S) if and only if x ∈ S and
min max( ψ(x) − t, −f (x) ), max( f (x), −t ) dc in (x, t) if ψ and f are dc ≤ 0;(21)
The function on the left side is piecewise affine in (x, t) if ψ and f are piecewise affine.
(b) For a pair (t,x) ∈ epi(π; S) witht = π(x), it holds that
• if f (x) > 0, then T (epi(π; S); (t,x)) = (dt, v) ∈ R × T (S;x) | dt ≥ ψ ′ (x; v) ; • if f (x) < 0, then T (epi(π; S); (t,x)) = R + × T (S;x); • if f (x) = 0 < ψ(x), then T (epi(π; S); (t,x)) ⊆ ( dt, v ) ∈ R + × T (S;x) | f ′ (x; v) ≤ 0(22)
with equality holding if the set {x ∈ S | f (x) ≤ 0} satisfies the ACQ atx;
• if f (x) = 0 = ψ(x), then T (epi(π; S); (t,x)) ⊆ ( dt, v ) ∈ R × T (S;x) | dt ≥ ψ ′ (x; v), f ′ (x; v) ≥ 0 ( dt, v ) ∈ R + × T (S;x) | f ′ (x; v) ≤ 0 ; (23) moreover, if ψ ′ (x; •) is nonnegative on T (S;x) ∩ (f ′ (x; •)) −1 (0), then the right-hand union in (23) is equal to epi ψ ′ (x; •) 1 ( 0,∞ ) (f ′ (x; •)); T (S;x) ; lastly, if the two setsS + x ∈ S | f (x) ≥ 0 andS − x ∈ S | f (x) ≤
(c) For a pair (t,x) ∈ epi(π; S) witht > π(x), it holds that
• if f (x) = 0 or f (x) = 0 = ψ(x), then T (epi(π; S); (t,x)) = R × T (S;x); • if f (x) = 0 < ψ(x) =t, then T (epi(π; S); (t,x)) = R × T (S;x) ift > ψ(x) ⊆ R × (T (S;x) ∩ (f ′ (x; •)) −1 ( −∞, 0 ]) ift < ψ(x); • if f (x) = 0 < ψ(x) =t, then T (epi(π; S); (t,x)) ⊆ ( dt, v ) ∈ R × T (S;x) | dt ≥ ψ ′ (x; v), f ′ (x; v) ≥ 0 ( dt, v ) ∈ R × T (S;x) | f ′ (x; v) ≤ 0 .
Proof. By definition of the epigraph, we have
epi(π; S) = ( t, x ) ∈ R × S | t ≥ ψ(x), f (x) > 0 ∪ R + × ( S ∩ f −1 ( −∞, 0 ] ) ;
thus epi(π; S) ⊆ E 1 ∪E 2 . Conversely, let (t, x) ∈ E 1 be such that f (x) = 0. Then t ≥ ψ(x) ≥ 0; hence (t, x) ∈ E 2 ⊆ epi(π; S). Thus, (19) holds. With the definition that T (E i ; (t, x)) ∅ if (t, x) ∈ E i for i = 1, 2, the equality (20) is clear. Statement (a) holds because of the following equivalence:
t ≥ ψ(x) and f (x) ≥ 0 or t ≥ 0 and f (x) ≤ 0 ⇔ [ max( ψ(x) − t, −f (x)) ≤ 0 ] or [ max( f (x), −t ) ≤ 0 ] ⇔ min max( ψ(x) − t, −f (x) ), max( f (x), −t ) ≤ 0.
To prove (b), suppose f (x) > 0. Then f (x) > 0 for all x sufficiently close tox. In this case, the equality T (epi(π; S); (t,x)) = (dt, v) ∈ R × T (S;x) | dt ≥ ψ ′ (x; v) can be proved as follows.
The inclusion ⊆ is straightforward to prove. For the "⊇" inclusion, let (dt, v) be such that v ∈ T (S;x) and dt ≥ ψ ′ (x; v). Let v = lim ν→∞ x ν −x τ ν for some sequence {x ν } ⊂ S converging tox and some sequence of scalars {τ ν } ↓ 0. Suppose dt > ψ ′ (x; v). Thent + τ ν dt > ψ(x ν ) for all ν sufficiently large. Thus (t + τ ν dt, x ν ) ∈ epi(π; S) for all such ν. This shows that (dt, v) ∈ T (epi(π; S); (t,x)) in this case. Suppose that dt = ψ ′ (x; v). Then for every ε > 0, (dt + ε, v) ∈ T (epi(π; S); (t,x)).
Since the tangent cone is closed, we deduce (dt, v) ∈ T (epi(π; S); (t,x)), completing the proof of the claimed equality of the two cones.
The second case where f (x) < 0 can be similarly argued. For the third case where f (x) = 0 < ψ(x), we havet = 0 and (t,x) ∈ E 2 \ E 1 . So the desired inclusion (22) and the equality under the stated CQ both hold easily.
Consider the last case where f (x) = 0 = ψ(x) so that (t,x) ∈ E 1 ∩ E 2 . It is easy to prove that T (E i ; (t,x)) for i = 1, 2 is contained in the two sets on the right-hand side of (23), respectively. If
ψ ′ (x; •) is nonnegative on T (S;x) ∩ (f ′ (x; •)) −1 (0)
, the claim that the right-hand union in (23) is equal to epi ψ ′ (x; •) 1 ( 0,∞ ) (f ′ (x; •); T (S;x)) can be proved similarly to that of the equality (19).
Finally, under the two additional ACQs, the equality (24) holds by the respective representation of T (E i ; (t,x)). The proof of statement (c) is similar and omitted.
Summarizing the various cases in the above proposition, we conclude that there are three "constraint qualifications" (in place if needed) to be satisfied by a pair (t,x) ∈ epi(π; S) in order for the tangent cone T (epi(π; S); (t,x)) to have an exact representation:
(i) ψ ′ (x; •) is nonnegative on T (S;x) ∩ (f ′ (x; •)) −1 (0);
(ii) the setS + x ∈ S | f (x) ≥ 0 satisfies the ACQ atx; and
(iii) the setS − x ∈ S | f (x) ≤ 0 satisfies the ACQ atx.
Moreover, refining the above analysis, it is easy to derive sufficient conditions for the set epi(π; S) = (t, x) ∈ R × S | (21) holds to satisfy the ACQ at the pair (π(x),x).
The penalized epigraphical formulation
In addition to the challenges associated with the Heaviside functions, the functional constraint also complicates the analysis and the design of computational algorithms for the problem (1). There are two ways to address this constraint: one is a direct treatment as a hard constraint; the other is by a soft penalty with the goal of recovering the satisfaction of the constraint and achieving the stationarity of the problem. In this section and Section 8, we adopt the latter penalty approach as it offers a unified treatment with the composite Heaviside functions occurring in the objective only. Throughout this subsection, the functions ϕ k and φ ℓ are nonnegative on X ∩ g −1 k (0) and X ∩ h −1 ℓ (0), respectively.
As a first step toward a computationally tractable formulation of (1), we make the substitutions
t k = ϕ k (x)1 ( 0,∞ ) (g k (x)) and s ℓ = φ ℓ (x)1 ( 0,∞ ) (h ℓ (x)), ∀ (k, ℓ) ∈ [K] × [L],(25)
and relax these definitional equalities to inequalities to obtain the epigraphical constraints:
(t k , x) ∈ epi(π ϕ k ; X) and (s ℓ , x) ∈ epi(π φ ℓ ; X),
where π ϕ k (x) ϕ k (x) 1 ( 0,∞ ) (g k (x)) and π φ ℓ (x) φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)).
We also relax the functional constraint by penalizing it in the objective. These maneuvers lead to the following penalty problem defined for a given parameter λ > 0:
minimize x ∈ X; t; s Φ λ (x, t, s) c(x) + K k=1 t k Φ(x) in epi-form +λ max L ℓ=1 s ℓ − b, 0 constraint residual fnc. in epi-form subject to ( t k , x ) ∈ epi(π ϕ k ; X) and ( s ℓ , x ) ∈ epi(π φ ℓ ; X), ∀ (k, ℓ) ∈ [K] × [L].(26)
We remark that the s-variables and the corresponding constraints (s ℓ , x) ∈ epi(π φ ℓ ; X) are not needed for the problem (7) where the functional constraint is absent.
By part (a) of Proposition 7, problem (26) is equivalent to the following problem with the epigraphical constraints exposed:
minimize x ∈ X; t; s Φ λ (x, t, s) c(x) + K k=1 t k + λ max L ℓ=1 s ℓ − b, 0 subject to min max( ϕ k (x) − t k , −g k (x) ), max( g k (x), −t k ) ≤ 0, ∀ k ∈ [K]
and
min max( φ ℓ (x) − s ℓ , −h ℓ (x) ), max( h ℓ (x), −s ℓ ) ≤ 0, ∀ ℓ ∈ [L].(27)
The latter formulation is the computational workhorse to derive a B-stationary solution of (26) via the epigraphical approach. In what follows, we show that under a suitable directional conditions on the functions φ ℓ in the functional constraint, a finite value of λ exists such that a B-stationary solution of (26) is a B-stationary solution of (13), hence a pseudo B-stationary solution of (1). The demonstration is carried out in 2 steps, with the first step being the recovery of the equalities of the t-variables in (25). This step is accomplished by the following lemma. Proof. Indeed, let v ∈ T ( X(ȳ);x) be given such that v = lim ν→∞ x ν −x τ ν for some sequence {x ν } in X(ȳ) converging tox and some sequence {τ ν } ↓ 0. It then follows that (v, 0) ∈ T ( X; (x,ȳ)) and
θ(•,ȳ) ′ (x; v) = lim ν→∞ θ(x ν ,ȳ) − θ(x,ȳ) τ ν = θ ′ ((x,ȳ); (v, 0)) ≥ 0,
where the two equalities hold by the B-differentiability of θ.
Let x; {t k } K k=1 ; {s ℓ } L ℓ=1(28)
be a B-stationary tuple of (26). By Lemma 8, it follows that the tuple
{t k } K k=1 ; {s ℓ } L ℓ=1
is a B-stationary solution of the problem:
minimize t; s K k=1 t k + λ max L ℓ=1 s ℓ − b, 0 subject to t k ≥ ϕ k (x) 1 ( 0,∞ ) (g k (x)), ∀ k ∈ [K] and s ℓ ≥ φ ℓ (x) 1 ( 0,∞ ) (h ℓ (x)), ∀ ℓ ∈ [L].
This is a trivial convex piecewise linear program in the (t, s)-variables with lower-bound constraints only. It can easily be seen thatt k = π ϕ k (x) for all k ∈ thatsl > π φ ℓ (x); or (ii)s ℓ = π φ ℓ (x) for all ℓ ∈ [L]. The following analysis addresses both cases. In case (i), it follows that L ℓ=1 π φ ℓ (x) < b, thus in particularx is feasible to (1). We will return to complete this case in the main Theorem 10. For now, we consider the second case where both sets of equalities in (25) hold atx. We next show that the vectorx in the B-stationary tuple (28) must be a B-stationary solution of the following problem defined with respect to a givenx ∈ X and without the auxiliary variables (t, s):
minimize x∈X Φ λ (x;x) c(x) + k∈K>(x) ϕ k (x) + λ max ℓ∈L>(x) φ ℓ (x) − b, 0 subject to g k (x) ≤ 0 ∀ k ∈ K ≤ (x) g k (x) ≥ 0 ∀ ℓ ∈ K > (x) h ℓ (x) ≤ 0 ∀ ℓ ∈ L ≤ (x) and h ℓ (x) ≥ 0 ∀ ℓ ∈ L > (x)
including x ∈ X, the set of these constraints is denoted by S ps (x) and is equal to S ps (x) without the functional constraint (29) Let dx be a tangent vector of S ps (x) atx. There exist sequences {x ν } →x and {τ ν } ↓ 0 such that x ν ∈ S ps (x) for all ν and dx = lim
ν→∞ x ν −x τ ν . Define dt k ϕ ′ k (x; dx) if k ∈ K > (x) 0 otherwise, and ds ℓ φ ′ ℓ (x; dx) if ℓ ∈ L > (x) 0 otherwise.
We claim that ( dt k , dx ) ∈ T (epi(π ϕ k ; X); (t k ,x)) and ( ds ℓ , dx ) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)),
∀ (k, ℓ) ∈ [K] × [L].
This is obviously true for all k ∈ K > (x) ∪ K < (x) and ℓ ∈ L > (x) ∪ L < (x), by the first two subcases of Proposition 7(b). For an index k ∈ K = (x); we have g k (x ν ) ≤ 0 by (29). With t ν k π ϕ k (x ν ) = 0, recallingt k = π ϕ k (x) = 0, we obtain dt k = 0 = lim ν→∞ t ν k −t k τ ν . Therefore, ( dt k , dx ) belongs to T (epi(π ϕ k ; X); (t k ,x)) for all k ∈ [K]. Similarly, we can prove ( ds ℓ , dx ) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)) for all ℓ ∈ [L]. Furthermore, it is easy to see that Φ ′ λ ((x,t,s); (dx, dt, ds)) = Φ λ (•;x) ′ (x, dx). This completes the proof thatx is a B-stationary solution of (29).
So far, we have not imposed any constraint qualifications; for the last step in concluding thatx is Bstationary solution of (13), we need the last condition imposed in Theorem10 below. This condition is the one stated in [12, Theorem 9.2.1] specialized to the problem (29); we refer to this reference for a brief history of the condition in the theory of exact penalization. For the problem (7) this extra condition is not needed; therefore we obtain a constructive approach for obtaining a pseudo B-stationary solution of this problem, which has no constraint involving the Heaviside functions, under only the sign conditions on the ϕ k functions (and the blanket B-differentiability assumption of the problem). For the sake of clarity in the last step of the proof of Theorem 10, we state the following simple lemma but omit its proof.
Lemma 9. Let f : O → R be B-differentiable atx. Let f max (x) max(f (x), 0). Then f ′ max (x; v) ≤ max f ′ (x; v), 0 , ∀ v ∈ R n .
Theorem 10. Under the blanket assumption of problem (1), assume that ϕ k and φ ℓ are nonnegative on X ∩ g −1 k (0) and X ∩ h −1 ℓ (0), respectively, for all (k, ℓ) ∈ [K] × [L], and that c and each ϕ k are Lipschitz continuous on X with Lipschitz constants Lip c and Lip ϕ , respectively. If x,t,s is a B-stationary tuplet of (26) corresponding to a λ satsfying
λ > Lip c + K Lip ϕ ,(30)
thenx is a pseudo B-stationary solution of the problem (1), provided that there exists a vector v ∈ T ( S ps (x);x) with unit length satisfying:
ℓ∈L>(x) φ ′ ℓ (x;v) ≤ −1.
Proof. Continuing the above analysis of the case where (25) holds atx, we suppose thatx fails the functional constraint; i.e.,
L ℓ=1 π φ ℓ (x) > b. Then we have 0 ≤ Φ λ (•;x) ′ (x,v) = c ′ (x;v) + k∈K>(x) ϕ ′ k (x;v) + λ ℓ∈L>(x) φ ′ ℓ (x;v) ≤ ( Lip c + K Lip ϕ ) − λ.
This contradicts the condition (30). Hencex is feasible to (1). To show thatx is B-stationary for (13), there are two cases to consider:
ℓ∈L>(x) φ ℓ (x) = b or ℓ∈L>(x) φ ℓ (x) < b.
Consider the former case first. Let v ∈ T (S ps (x);x). Then v ∈ T ( S ps (x);x) and
ℓ∈L>(x) φ ′ ℓ (x; v) ≤ 0. We have 0 ≤ Φ λ (•;x) ′ (x, v) = c ′ (x; v) + k∈K>(x) ϕ ′ k (x; v),(31)
establishing the B-stationarity ofx for the problem (13). In the latter case where
ℓ∈L>(x) φ ℓ (x) < b,
we have T (S ps (x);x) = T ( S ps (x);x) and the same expression (31) also holds.
What remains to be proved is the case where the B-stationary tuple (28) of (26) is such that
ℓ∈L>(x) φ ℓ (x) = L ℓ=1 π φ ℓ (x) < L ℓ=1s ℓ ≤ b
andsl > π φ ℓ (x) for at least onel. Let dx ∈ T (S ps (x);x) be arbitrary. There exist sequences {x ν } →x and {τ ν } ↓ 0 such that x ν ∈ S ps (x) ⊆ S ps (x) for all ν and dx = lim ν→∞ x ν −x τ ν . As we have already
shown, (dt k , dx) ∈ T (epi(π ϕ k ; X); (t k ,x)) for all k ∈ [K], where dt k ϕ ′ k (x; dx) if k ∈ K > (x) 0
otherwise. For an arbitrary scalar M > 0, define a vector ds M as follows:
ds M ℓ φ ′ ℓ (x; dx) if ℓ ∈ L > (x) ands ℓ = π φ ℓ (x), index set denoted L = > (x) −M ifs ℓ > π φ ℓ (x) 0 otherwise.
As before, we have (ds M ℓ , dx) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)) for all ℓ ∈ [L] except possibly whens ℓ > π φ ℓ (x). We show that the latter exception can be removed. Indeed, with M fixed but arbitrary, for an index ℓ of the latter kind, there are 2 cases to consider:
(i) h ℓ (x) = 0: then (−M, dx) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)) by part (c) of Proposition 7.
(ii) h ℓ (x) = 0: thens ℓ > π φ ℓ (x) = 0 ≥ h ℓ (x ν ), where the last inequality holds because x ν ∈ S ps (x). Thus,s ℓ − τ ν M > π φ ℓ (x ν ) = 0 for all ν sufficiently large and (−M, dx) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)) follows.
We have thus completed the proof that (ds M ℓ , dx) ∈ T (epi(π φ ℓ ; X); (s ℓ ,x)) for all ℓ ∈ [L]. We have
L ℓ=1 ds M ℓ = ℓ∈L = > (x) φ ′ ℓ (x; dx) − M | { ℓ :s ℓ > π φ ℓ (x) } | ≥ 1 .
By choosing M sufficiently large, we have L ℓ=1 ds M ℓ ≤ 0. By Lemma 9, it follows that
0 ≤ Φ λ ((x,t,s); (dx, dt, ds M )) ≤ c ′ (x; dx) + k∈K>(x) ϕ ′ k (x; dx) + λ max L ℓ ds M ℓ , 0 = c ′ (x; dx) + k∈K>(x) ϕ ′ k (x; dx),
showing thatx is a B-stationary solution of (13), as desired.
A Digression: Approximations of the (open) Heaviside Function
The lifted formulation (26), or its computational workhorse (27), requires the auxiliary variables t and s but has the advantage of leading directly to a pseudo B-stationary point of the original problem (1) under some mild conditions as stated in Theorem 10. Of independent interest, the next approach is based on approximations of the open Heaviside function and leads to approximated problems in the x-variable only. In this section, we digress from the discussion of the problem (1) and present a focused discussion on two approaches to construct such approximations and show how they are related to each other. One construction is based on truncation [11] and the other is based on the classical work of [18] on mollifiers, or smoothing [7].
(A2) θ(t, δ) = 0 ∀ t ≤ −θ(δ) 1 ∀ t ≥ θ(δ) for all δ > 0.
For subsequent purposes, we further stipulate that for all δ > 0,
Truncation derived approximations
Let θ : R × R ++ → R be such that there exist end-point functions θ and θ : R ++ → R + satisfying (A0) and
(T1) θ(t, δ) ≤ 0 ∀ t ≤ −θ(δ) ≥ 1 ∀ t ≥ θ(δ)
for all δ > 0;
(T2) lim To obtain the p-approximation function θ from θ, let
T [ 0,1 ] (t)
min max(t, 0), 1 = max min(t, 1), 0 = max(t, 0) − max(t − 1, 0), t ∈ R be the truncation operator to the range [ 0, 1 ] and define the composite function:
θ tr (t, δ) T [ 0,1 ] θ(t, δ) , (t, δ) ∈ R × R ++ .(33)
We formally state that the truncated function θ is a p-approximation of the Heaviside function in the first part of the next proposition. The truncation function was used in [11] as a unification scheme for many approximations of chance constraints in stochastic programs; see the cited reference for many prior works on the latter subject. Part (b) of the result provides easy sufficient conditions for (T1) and (T3) to hold; this part is the bridge between the truncation approach discussed herein and the nonifier approach to be discussed in the next subsection. Part (c) presents a broad family of approximating functions by truncation that generalize the perspective functions studied extensively in convex analysis [4,9,31,46] and employed extensively for integer programs in recent years; see e.g. [25]. The approximating functions in the last part of the proposition yield the family of folded concave approximations of the ℓ 0 function [12, Subsection 3.1.4]; these include the smoothly clipped absolute deviation scad function [20]; the minimax concave penalty mcp function [13,56]; the capped ℓ 1 -function [38, Section 5], and others. As shown in [1], all these functions are of the difference-ofconvex kind and not differentiable at the origin as they are approximations of the ℓ 0 -function that is discontinuous there.
p-approximates 1 ( 0,∞ ) and satisfies (A3).
(d) If θ 1 (t, δ) and θ 2 (t, δ) both p-approximate the open Heaviside function 1 ( 0,∞ ) , then their sum θ 1 (t, δ) + θ 2 (−t, δ) p-approximates the ℓ 0 -function | t | 0 .
Proof. For (a), we need to show that (T1) and (T3) imply (A1) and (A2) and that (T3) implies (A3). Clearly, (T2) implies (A1) for t = 0. For a t * > 0, we have t * ≥ θ(δ) for all δ > 0 sufficiently small. Thus, θ tr (t * , δ) = 1 = 1 ( 0,∞ ) (t * ) by (T1) and the definition of truncation, which is (A1) at t * . Similarly, (A1) also holds for t * < 0. Thus the pointwise convergence condition (A1) holds for all t ∈ R. Clearly (A2) holds by truncation. For (A3), we have θ tr (•, δ) ′ (t; ±1) = T ′ [ 0,1 ] ( θ(t, δ); θ(•, δ) ′ (t; ±1)).
Since T ′ [ 0,1 ] (t; 1) ≥ 0 ≥ T ′ [ 0,1 ] (t; −1) for all t ∈ R, we readily obtain (A3) from (T3). If θ is nondecreasing, then (T1 ′ ) clearly implies (T1); moreover, this function is directionally differentiable and by the definition (3), we have θ(•, δ) ′ (t; 1) ≥ 0 ≥ θ(•, δ) ′ (t; −1) for all t; statement (b) thus holds. For statement (c), define θ(δ) −m(δ) q(δ) and θ(δ) m(δ) (1 − q(δ)). By the limit on the function m in (34), we obtain lim δ↓0 θ(δ) = 0 = lim δ↓0 θ(δ). Finally, with θ(t, δ) ψ q(δ) + t m(δ) , condition (T2) holds also by the limits on q(δ) in (34). The last statement (d) is obvious by (2).
To illustrate the function θ in part (c) of the above proposition and the role of condition (T2) consider a common approximation of the Heaviside function 1 ( 0,∞ ) by the truncated hinge loss functions [45,51]:
T h (t, δ) 1 2δ max(t + δ, 0) − max(t − δ, 0) for δ > 0. Notice that T h (0, δ) = 1 2
for all δ; thus this approximation function fails condition (T2) and as a result does not "recover" the Heaviside function as δ ↓ 0. When this function T h (•, δ) is employed in sampled discretization of a probability function, as in the cited references:
Pz(f (x,z) > 0) = Ez[1 ( 0,∞ ) (f (x,z))] ≈ 1 N N s=1 T h (f (x, z s ), δ),
where {z s } N s=1 is a sample batch of size N of the random variablez, the gap between the indicator function and its approximation at the origin is less important because of the common assumption that Pz(f (x,z) = 0) = 0. Nevertheless, in a deterministic context, such a gap can be significant because the set f −1 (0) is typically of most interest for the composite function 1 ( 0,∞ ) (f (x)). Part of the reason for the gap is due to the symmetry of the function T h (•, δ) with respect to t = 0. To recover the Heaviside function exactly as δ ↓ 0, we may consider the following modified hinge loss function:
T h (t, δ) min max t δ + √ δ + √ δ 1 + √ δ , 0 , 1 , for 1 ( 0,∞ ) (t),(35)
which is derived from ψ being the identity function, q(δ) = √ δ 1 + √ δ and m(δ) = δ + √ δ, both satisfying the limits (34).
Nonifiers induced approximations
We present the other approach for deriving approximations for the Heaviside function based on "averaged functions" as defined in [18, Definition 3.1].
Definition 13. Given a locally integrable function f : R n → R and a family of bounded mollifiers
{ψ(•, δ) : R n → R + ; δ ∈ R + } that satisfy R n ψ(z, δ) dz = 1, supp ψ(•, δ) { z ∈ R n | ψ(z, δ) > 0 } ⊆ ρ δ B, for some {ρ δ } ↓ 0 as δ ↓ 0,
where B is a unit Euclidean ball in R n , the associated family {f ψ (•, δ)} δ≥0 of averaged functions is given by
f ψ (x, δ) R n f (x − z) ψ(z, δ) dz = R n f (z) ψ(x − z, δ) dz.
As noted in [18], ψ(•, δ) is a probability density function whose support tends to zero as δ ↓ 0; moreover, ψ(•, δ) does not need to be continuous. Focus of the references [7,18] has been on mollifiers ψ(•, δ) that lead to smooth (i.e., continuously differentiable) averaged functions f ψ (•, δ). Two sets of conditions ensure the latter property: (i) f is continuous with certain special choices of the family {ψ(•, δ)} (see [18,Proposition 3.11]), and (ii) the mollifiers ψ(•, δ) are continuously differentiable (see [18,Proposition 3.9]). Since we are most interested in nonsmooth averaged functions, we coin the term "nonifier" for ψ with the intention that the induced averaged functions f ψ (•, δ) are not necessarily differentiable. In what follows, we derive these averaged functions for the Heaviside function and show how they are related to those obtained from the previous truncation approach.
To begin, we take a bivariate function ψ : R × R ++ → R + such that ψ(•, δ) is integrable on R for every δ > 0 with
θ ψ (t, δ) = ∞ 0 ψ(t − s, δ) ds = t −∞ ψ(s, δ) ds, ∀ ( t, δ ) ∈ R × R ++ = min(t,ψ(δ)) −ψ(δ) ψ(s, δ) ds = 1 if t ≥ ψ(δ) t −ψ(δ) ψ(s, δ) ds if −ψ(δ) ≤ t ≤ ψ(δ) 0 if t ≤ −ψ(δ).
Defining the cumulative distribution function:
θ ψ (t, δ) t −ψ(δ) ψ(s, δ) ds, ∀ (t, δ) ∈ R × R ++ ,(36)
we see that θ ψ (•, δ) is equal to its own truncation; i.e., θ ψ (t, δ) = T [ 0,1 ] ( θ ψ (t, δ)) = θ ψ (t, δ); moreover, θ ψ (•, δ) is nondecreasing (because ψ(•, δ) is nonnegative) and satisfies condition (T1 ′ ). The following lemma pertains to conditions (T2) and (T3) for the function θ ψ (•, δ); no continuity of ψ(•, δ) is needed.
Lemma 14. The following two statements hold:
• If the one-sided limits ψ(t±; δ) lim τ ↓0
ψ(t ± τ, δ) exist, then θ ψ (•, δ) is B-differentiable at t with θ ψ (•, δ) ′ (t; ±1) = ± ψ(t±; δ), respectively;(37)
thus condition (T3) (and (A3)) hold for θ ψ (•, δ); i.e., θ ψ (•, δ) ′ (t; 1) ≥ 0 ≥ θ ψ (•, δ) ′ (t; −1).
• If lim sup Proof. To show the locally Lipschtiz continuity of θ ψ (•, δ) near t, let ε andτ be positive scalars such that τ ∈ [ 0,τ ] ⇒ | ψ(t ± τ, δ) − ψ(t±, δ) | ≤ ε.
Let t 1 and t 2 be two scalars in [ −τ ,τ ]. Consider first the case where both t 1 and t 2 are on the same side of t. We can write t 1 = t + τ 1 and t 2 = t + τ 2 , withτ ≥ τ 1 ≥ τ 2 ≥ 0.
θ ψ (t 1 , δ) − θ ψ (t 2 , δ) = t 1 t 2 ψ(s, δ) ds = t 1 t 2 ψ(s, δ) − ψ(t+, δ) ds + ψ(t+, δ) ( t 1 − t 2 ). Hence, | θ ψ (t 1 , δ) − θ ψ (t 2 , δ) | ≤ ( ε + ψ(t+, δ) ) ( t 1 − t 2 ).
If t 1 an t 2 are on opposite sides of t, then can write t 1 = t + τ 1 and t 2 = t − τ 2 with τ 1 and τ 2 both in the interval [ 0,τ ]. It follows that
θ ψ (t 1 , δ) − θ ψ (t 2 , δ) = t t 2 ψ(s, δ) ds + t 1 t ψ(s, δ) ds = t t 2 ψ(s, δ) − ψ(t−, δ) ds + ψ(t−, δ) ( t − t 2 ) + t 1 t ψ(s, δ) − ψ(t+, δ) ds + ψ(t+, δ) ( t 1 − t ).
Hence,
| θ ψ (t 1 , δ) − θ ψ (t 2 , δ) | ≤ ( ε + ψ(t−, δ) ) ( t − t 2 ) + ( ε + ψ(t+, δ) ) ( t 1 − t ) ≤ ε + max ψ(t−, δ), ψ(t+, δ) [ ( t − t 2 ) + ( t 1 − t ) ] = ε + max ψ(t−, δ), ψ(t+, δ) ( t 1 − t 2 ).
The locally Lipschitz continuity of θ ψ (•, δ) near t follows. Since θ ψ (•, δ) is nondecreasing, the Bdifferentiability of θ ψ (•, δ) at t follows. For the two limits in (37), we prove only for the negative direction; i.e., θ ψ (•, δ) ′ (t; −1) = −ψ(t−; δ), as the proof for the plus direction is similar (and a little more straightforward). We have
θ ψ (•, δ) ′ (t; −1) + ψ(t−, δ) = lim τ ↓0 θ ψ (t − τ, δ) − θ ψ (t, δ) + τ ψ(t−, δ) τ = lim τ ↓0 t−τ t ψ(s, δ) − ψ(t−, δ) ds τ ≤ lim τ ↓0 sup s∈[ t−τ,t ] | ψ(s, δ) − ψ(t−, δ) | = 0.
For the second statement, we have
θ ψ (0, δ) = 0 −ψ(δ) ψ(s, δ) ds ≤ ψ(δ) sup s∈R ψ(s, δ) ≤ ψ(δ) ψ(δ) + ψ(δ) ψ(δ) + ψ(δ) sup t∈R ψ(t, δ) = ψ(δ)/ψ(δ) 1 + ψ(δ)/ψ(δ) ψ(δ) + ψ(δ) sup t∈R ψ(t, δ).
Thus lim δ↓0 θ ψ (0, δ) = 0 as desired.
In summary, starting from a nonifier ψ(•, δ) satisfying the conditions in Lemma 14, the cumulative distribution function θ ψ (•, δ) yields a nondecreasing p-approximation function of the open Heaviside function satisfying conditions (A1), (A2), and (A3).
We next consider the reverse; i.e., we are given a bivariate function θ(t, δ) with θ(•, δ) satisfying the conditions in part (b) of Proposition 12 and also (T2). Since θ(•, δ) is B-differentiable, it is almost everywhere differentiable. Let ψ(•, δ) be any integrable function such that ψ(t, δ) = θ(•, δ) ′ (t) for almost all t ∈ [ −θ(δ), θ(δ) ]. Defining ψ(•, δ) to be zero outside the latter interval, we deduce that ψ(•, δ) is a well-defined nonifier; moreover
θ(t, δ) = t −θ(δ) θ(•, δ) ′ (s) ds = t −θ(δ) ψ(s, δ) ds = θ ψ (t, δ).
Combining the two parts of the analysis, we conclude that the p-approximating functions for the open Heaviside function obtained from the truncation approach coincide with those from the nonifier approach satisfying some mild properties.
Example 15. The truncated hinge loss function T h (t, δ/2) = 1 δ max(t + δ/2, 0) − max(t − δ/2, 0)
is the averaged function derived form the "symmetric" one-dimensional Steklov mollifier [18, Defini-
tion 3.8]: ψ(t, δ) = 1/δ if | t | ≤ δ/2 0 otherwise
, whose support is the interval −δ/2, δ/2 . As mentioned before, the truncated hinge loss T h (•, δ/2) fails to p-approximate the open Heaviside function because it violates condition (A1). More interesting is the fact there do not exist nonifiers for this function that satisfy the conditions in Lemma 14. Nevertheless, by considering the asymmetric
Steklov function: ψ a (t, δ) = 1 ψ(δ) + ψ(δ) if t ∈ −ψ(δ), ψ(δ)
Convergence to Pseudo B-stationary Solutions
We consider the approximation of a pseudo B-stationary solution of the problem (1) by a combination of penalization of the functional constraint and approximation of the composite Heaviside functions:
minimize x∈X Φ λ (x, δ) c(x) + K k=1 ϕ k (x) θ ϕ k (g k (x), δ) denoted ϕ(x, δ) +λ max L ℓ=1 φ ℓ (x) θ φ ℓ (h ℓ (x), δ) − b denoted φ(x, δ) , 0 ,(38)
where for each pair (k, ℓ) ∈ [K] × [L], θ ϕ k (•, δ) and θ φ ℓ (•, δ) are p-approximations of the Heaviside function 1 ( 0,∞ ) with support in the interval [ −θ ϕ k (δ), θ ϕ k (δ) ] and [ −θ φ ℓ (δ), θ φ ℓ (δ) ], respectively, that shrink to zero when δ ↓ 0; in particular, (A1), (A2), and (A3) in Definition 11 are satisfied by these approximating functions. With the functions c and each ϕ k being Lipschitz continuous on X with Lipschtiz constants Lip c and Lip ϕ , respectively, we take λ satisfying (30).
One of the conditions that we will impose in the analysis is a sign condition, labelled (C2) below, on the directional derivatives of the functions g k and h ℓ . The lemma below shows that this will hold if these are convex piecewise affine functions.
Lemma 16. Let f : R n → R be a convex piecewise affine function. For every vectorx, there exists a neighborhood N ofx such that for all v ∈ R n ,
f ′ (x; v) ≤ 0 ⇒ f ′ (x; v) ≤ 0, ∀ x ∈ N .
Proof. Since f is convex piecewise affine, it can be written as the pointwise maximum of finitely many affine functions [12,Proposition 4.4.6]; i.e.,
f (x) = max 1≤i≤I (a i ) ⊤ x + b i , ∀ x ∈ R n ,
for some positive integer I, n-vectors
{a i } I i=1 , and scalars {b i } I i=1 .
For any x, let
A(x) i ∈ [I] | f (x) = (a i ) ⊤ x + b i
be the maximizing index set of the affine pieces of f . It then follows that for the givenx, there exists a neighborhood N such that A(x) ⊆ A(x) for all x ∈ N . Since
f ′ (x; v) = max i∈A(x) (a i ) ⊤ v + b i , ∀ (x, v) ∈ R n × R n ,
the desired conclusion of the lemma follows readily.
Let {δ ν } be a sequence of positive scalars converging to zero. Let {x ν } be a corresponding sequence of d-stationary points of Φ λ (•, δ ν ) on X so that Φ λ (•, δ ν ) ′ (x ν ; v) ≥ 0 for all v ∈ T (X; x ν ). Suppose that {x ν } converges to the limit x * . In what follows, we show that x * is a pseudo B-stationary point of (1) by verifying two things under the condition (30) on λ: (a) x * is feasible to (1), and (b) the implication (16), which we restate below in terms of the vector x * on hand:
v ∈ L(S ps (x * );
x * ) ⇒ Φ(•; x * ) ′ (x * ; v) ≥ 0.(39)
Letting σ k (x; v) sgn(g ′ k (x; v)) with sgn(0) defined to be zero, we write the directional derivative of ϕ(•, δ ν ) at x ν as the sum of 3 terms:
ϕ(•, δ ν ) ′ (x ν ; v) = K k=1 ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ) + ϕ k (x ν ) | g ′ k (x ν ; v) | (θ ϕ k (•, δ ν )) ′ (g k (x ν ); σ k (x ν ; v)) = T ϕ > (x ν ; v) + T ϕ = (x ν ; v) + T ϕ < (x ν ; v), where T ϕ > (x ν ; v) = k∈K>(x * ) ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ) + ϕ k (x ν ) | g ′ k (x ν ; v) | (θ ϕ k (•, δ ν )) ′ (g k (x ν ); σ k (x ν ; v)) T ϕ = (x ν ; v) = k∈K=(x * ) ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ) + ϕ k (x ν ) | g ′ k (x ν ; v) | (θ ϕ k (•, δ ν )) ′ (g k (x ν ); σ k (x ν ; v)) T ϕ < (x ν ; v) = k∈K<(x * ) ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ) + ϕ k (x ν ) | g ′ k (x ν ; v) | (θ ϕ k (•, δ ν )) ′ (g k (x ν ); σ k (x ν ; v)) .
Consider the sum T ϕ > (x ν ; v). For each k ∈ K > (x * ), we have g k (x * ) > 0. Hence g k (x ν ) > θ ϕ k (δ ν ) for all ν sufficiently large; this yields, by (A1):
θ k (g k (x ν ), δ ν ) = 1 and θ ′ k (•, δ ν ) ′ (g k (x ν ); σ k (x ν ; v)) = 0. Thus, T ϕ > (x ν ; v) = k∈K>(x * ) φ ′ k (x ν ; v).
Similarly, we can show that T ϕ < (x ν ; v) = 0 for all ν sufficiently large. The rest of the proof is divided into three parts.
• To analyze the term T ϕ = (x ν ; v) and the corresponding term in φ(•, δ ν ) ′ (x * ; v), we need assumptions (C1) and (C2):
(C1):
There exists an open neighborhood N * such that on X ∩ N * , the functions {ϕ k } k∈K=(x * ) and { φ ℓ } ℓ∈L=(x * ) are nonnegative; this is a pointwise sign condition related to those in Proposition 1, demanding in particular the nonnegativity of these functions in a region around x * and possibly outside the respective sets
X ∩ g −1 k ( −∞, 0 ] and X ∩ h −1 ℓ ( ∞, 0 ]. (C2): The implications g ′ k (x * ; v) ≤ 0 ∀ k ∈ K = (x * ) v ∈ T (X; x * ) ⇒ g ′ k (x; v) ≤ 0 for all k ∈ K = (x * ) and all x ∈ N * (40) h ′ ℓ (x * ; v) ≤ 0 ∀ ℓ ∈ L = (x * ) v ∈ T (X; x * ) ⇒ h ′ ℓ (x; v) ≤ 0 for all ℓ ∈ L = (x * ) and all x ∈ N *(41)
are motivated by Lemma 16 which provides sufficient conditions for them to hold.
Under (C1) and (C2), we deduce that if v satisfies the left-hand condition in (40), then by (32) applied to θ ϕ k , we obtain, for all ν sufficiently large and all k ∈ K = (x * ),
ϕ k (x ν ) | g ′ k (x ν ; v) | θ ϕ k (•, δ ν ) ′ (g k (x ν ); σ k (x ν ; v)) ≤ 0
Hence, for such ν, we have
T ϕ k;= (x ν ; v) ≤ ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ).
Consequently, we deduce, for all ν sufficiently large,
g ′ k (x * ; v) ≤ 0 ∀ k ∈ K = (x * ) v ∈ T (X; x * ) ⇒ ϕ(•, δ ν ) ′ (x ν , v) ≤ k∈K>(x * ) ϕ ′ k (x ν ; v) + k∈K=(x * ) ϕ ′ k (x ν ; v) θ ϕ k (g k (x ν ), δ ν ).(42)
Similarly, we also have, for all ν sufficiently large,
h ′ ℓ (x * ; v) ≤ 0 ∀ ℓ ∈ L = (x * ) v ∈ T (X; x * ) ⇒ φ(•, δ ν ) ′ (x ν , v) ≤ ℓ∈L>(x * ) φ ′ ℓ (x ν ; v) + ℓ∈L=(x * ) φ ′ ℓ (x ν ; v) θ φ k (h ℓ (x ν ), δ ν ).(43)
• To establish the feasibility of x * for the problem (1), we postulate three more assumptions: as we will see below, assumption (C3) ensures the objective recovery of the sequence {x ν }; (C4) is the same as that in Theorem 10 and (C5) is the Clarke regularity on several key functions.
(C3) For all k ∈ K = (x * ) and ℓ ∈ L = (x * ), the limits lim ν→∞ θ ϕ k (g k (x ν ), δ ν ) = 0 and lim ν→∞ θ φ ℓ (h ℓ (x ν ), δ ν ) = 0 (44) are functional consistency requirements of the sequence {x ν } in the following sense. These limits ensure that for all pairs (k, ℓ),
lim ν→∞ θ ϕ k (g k (x ν ), δ ν ) = 1 ( 0,∞ ) (g k (x * )) and lim ν→∞ θ φ ℓ (h ℓ (x ν ), δ ν ) = 1 ( 0,∞ ) (h ℓ (x * )).
Together with the same limits for k ∈ K > (x * ) ∪ K < (x * ) and ℓ ∈ L > (x * ) ∪ L < (x * ), we deduce in particular that
lim ν→∞ c(x ν ) + K k=1 ϕ k (x * ) θ ϕ k (g k (x ν ), δ ν ) = Φ(x * ),
which is reasonable to postulate in order for x * to be a stationarity point of some kind for the objective function of the original problem (1). If the approximation function θ k is chosen as in part (c) of Proposition 12, i.e. if
θ k (t, δ) T [ 0,1 ] ψ k q k (δ) + t m k (δ)
where the functions ψ k , q k , and m k are as specified in the proposition, then the limit (44) holds if lim ν→∞ g k (x ν ) m k (δ ν ) = 0. Admittedly, this is a condition on the sequence of iterates {x ν } relative to the sequence of parameters {δ ν } that needs to be addressed from the source of the iterates, e.g., as generated by an iterative algorithm. Regrettably, the design of such algorithms is beyond the scope of the present work but will be the focus of a subsequent computational study which will be guided by this high-level background result. The limit on the sequence {θ φ ℓ (h ℓ (x ν ), δ ν )} has to do with the satisfaction of the functional constraint and facilitates the demonstration of the desired pseudo B-stationarity of the limit x * ; this will become clear in what follows. Subsequently, we will establish a weaker stationarity property of the limit x * without (C3).
(C4) For any x ∈ X such that ℓ∈L>(x) φ ℓ (x) > b, there exists a vectorv ∈ T ( S ps (x); x) with unit length satisfying:
ℓ∈L>(x) φ ′ ℓ (x;v) ≤f ′ (z ν ; v) ≤ f ′ (x; v), ∀ v ∈ R n .
Condition (C4) requires that for any x ∈ X that is infeasible to problem (1), the constraint function ℓ∈L>(x) φ ℓ (•) has a descent direction at x. This condition is in the same spirit as a classical one since the early days of the theory of exact penalty methods [44] for the recovery of feasibility (and hence stationarity) in a penalized problem; see also [12, Proposition 9.2.2 (a)]. As in problems without the Heaviside function, condition (C4) can be related to the concept of weak-sharp minima [23] properly extended. We give a brief discussion of the connection in an Appendix. The Clarke regularity in (C5) holds in particular if the considered functions are the composition of a convex function with a smooth mapping [47,Exercise 10.25].
Continuing the analysis, assume for the sake of contradiction that
L ℓ=1 φ ℓ (x * ) 1 ( 0,∞) (h ℓ (x * )) > b.
Then for all ν sufficiently large,
φ(x ν , δ ν ) = L ℓ=1 φ ℓ (x ν ) θ ϕ ℓ (h ℓ (x ν ), δ ν ) − b > 0. Hence, for the vector v in (C4), we have 0 ≤ Φ λ (•, δ ν ) ′ (x ν ;v) = c ′ (x ν ;v) + ϕ(•, δ ν ) ′ (x ν ;v) + λ φ(•, δ ν ) ′ (x ν ;v) ≤ c ′ (x ν ;v) + k∈K>(x * ) ϕ ′ k (x ν ;v) + k∈K=(x * ) ϕ ′ k (x ν ;v) θ ϕ k (g k (x ν ), δ ν ) + λ ℓ∈L>(x * ) φ ′ ℓ (x ν ;v) + ℓ∈L=(x * ) φ ′ ℓ (x ν ;v) θ φ k (h ℓ (x ν ), δ ν ) ≤ ( Lip c + K Lip ϕ ) + λ ℓ∈L>(x * ) φ ′ ℓ (x ν ;v) + Lip φ ℓ∈L=(x * ) θ φ k (h ℓ (x ν ), δ ν ) ,(45)
where Lip φ is a local Lipschitz constant of φ ℓ near x * . Taking the limit ν → ∞ and using (44), we obtain, by the Clarke regularity of the functions {φ ℓ } ℓ∈L>(x * ) at x * ,
0 ≤ Lip c + K Lip ϕ + λ ℓ∈L>(x * ) lim sup ν→∞ φ ′ ℓ (x ν ;v) ≤ Lip c + K Lip ϕ + λ ℓ∈L>(x * ) φ ′ ℓ (x * ;v) ≤ Lip c + K Lip ϕ − λ,
This contradiction completes the feasibility proof of x * for (1).
• Finally, to complete the proof of the desired implication (39), suppose that x * satisfies the functional constraint as an equality; i.e., In the other case where L ℓ=1 φ ℓ (x * ) 1 ( 0,∞) (h ℓ (x * )) < b, we must have φ(x ν , δ ν ) < b for all ν sufficiently large. Thus, max φ(•, δ ν ), 0 ′ (x ν ; v) = 0 for all such ν. Consequently, in either case, it follows that 0 ≤ lim sup (42), (C3), and Clarke regularity, establishing the right-hand side of (39). We have thus proved the following main result of this section.
φ ′ ℓ (x ν ; v) + ℓ∈L=(x * ) φ ′ ℓ (x ν ; v) θ φ ℓ (h ℓ (x ν ), δ ν ), 0 by (43) ≤ max lim sup ν→∞ ℓ∈L>(x * ) φ ′ ℓ (x ν ; v) + Lip φ ℓ∈L=(x * ) θ φ ℓ (h ℓ (x ν ), δ ν ) , 0 ≤ max ℓ∈L>(x * ) φ ′ ℓ (x * ; v),ν→∞ c ′ (x ν ; v) + ϕ(•, δ ν ) ′ (x ν ; v) ≤ c ′ (x * ; v) + k∈K>(x * ) ϕ ′ k (x * ; v), by
Theorem 17. Under the blanket assumption of problem (1), let c and each ϕ k be Lipschitz continuous on X with Lipschitz constants Lip c and Lip ϕ , respectively. Let {δ ν } be a sequence of positive scalars converging to zero and for each ν, let x ν be a d-stationary point of Φ λ (•, δ ν ) on X with λ satisfying (30). Suppose lim ν→∞ x ν = x * . Under assumptions (C1)-(C5), it holds that x * is a pseudo B-stationary solution of (1).
Without (C3), the desired pseudo B-stationarity of the limit x * as defined in Definition 3 is in jeopardy. By strengthening (C4) and (C5), we can still establish a weak pseudo B-stationarity property of x * . The two strengthened conditions are:
(C4 ′ ) There exists a vectorv ∈ T ( S ps (x * ); x * ) with unit length satisfying:
ℓ∈L>(x) φ ′ ℓ (x * ;v) + ℓ∈L=(x) max φ ′ ℓ (x * ;v), 0 ≤ −1.(46)
While more demanding than (C4), this strengthened condition is still in the spirit of the common requirements in the theory of exact penalization (see conditions (a) and (b) in [12, Theorem 9.2.1]).
(C5 ′ ) In addition to those in (C5), the functions {ϕ k } k∈K=(x * ) , and {φ ℓ } ℓ∈L=(x * ) are also Clarke regular at x * .
all ℓ ∈ L > (x * ), we have
ℓ∈L>(x * ) φ ℓ (x * ) + ℓ∈L=(x * ) µ * ℓ φ ℓ (x * ) ≥ L ℓ=1
φ ℓ (x * ) 1 ( 0,∞ ) (h ℓ (x * )), thus x * is feasible to (1). Finally, the proof that x * is a B-stationary solution of (47) is similar to that in the last part of the proof of Theorem 17.The details are not repeated.
An illustrative example
We use the following slight modification of the function (6) to illustrate the sign assumptions in Theorems 10 and 17 on the simplified problem: minimize x∈X Ψ(x), where
Ψ(x) = ψ 1 (x) if a ≤ f (x) < b ψ 2 (x) if f (x) < a ψ 3 (x) if f (x) ≥ b.(48)
Similar to the previous derivation, we can write
Ψ(x) = ψ 3 (x) + ψ 1 (x) − ψ 3 (x) 1 ( 0,∞ ) (b − f (x)) + ψ 2 (x) − ψ 1 (x) 1 ( 0,∞ ) (a − f (x)).
Given a vectorx ∈ X, there are five pseudo stationarity problems (13) Clearly, not all stationary solutions of the above five problems are local minimizers of Ψ on X; nevertheless, they provide candidate minimizers. The sign conditions in Theorem 10 require that:
• [f (x) = a, x ∈ X] ⇒ ψ 2 (x) ≥ ψ 1 (x); and
• [f (x) = b, x ∈ X] ⇒ ψ 1 (x) ≥ ψ 3 (x).
Under these conditions, which basically stipulate that the function Ψ can not rise up at a point of discontinuity, the epi-hypographical approach will produce a pseudo B-stationary solution of Ψ on X. In contrast, the pointwise sign stipulations at x * in condition (i) of Theorem 17 require that
• f (x * ) = a ⇒ ψ 2 (x) ≥ ψ 1 (x) in a neighborhood of x * , and
• f (x * ) = b ⇒ ψ 1 (x) ≥ ψ 3 (x) in a neighborhood of x * .
Condition (40) becomes:
• there exists a neighborhood N ofx such that
f ′ (x * ; v) ≥ 0 v ∈ T (X; x * ) ⇒ f ′ (x; v) ≥ 0 ∀ x ∈ N .
Along with the objective consistency condition of the iterates, the above sign stipulations provide sufficient conditions for x * to be a pseudo B-stationary solution of Ψ on X.
Concluding Remarks
At the completion of the paper, the authors are grateful to receive a preprint [48] in which the author introduces two concepts of consistent approximations in composite optimization; see Definition 2.2 therein. The setting of this reference is the minimization of an objective that is the sum of an extended-valued constraint indicator function and a composite function h • F (x) where h is a convex extended-valued function and F is a vector function whose components are locally Lipschitz continuous functions. While in principle, the consistency concepts defined therein can be extended to any optimization problem without bother of its structure, the challenge is twofold: (a) how to define the approximations, and (b) establishing the consistency of the derived approximations. Although the problem (38) is an approximation of (1), it is rather doubtful if the theory in the reference could be applied to our context. For one thing, a product ψ1 ( 0,∞) (f ) is very different from the composite family h • F with h and F as stated. As one can see, our analysis makes extensive use of the product form and properties of the approximations of the Heavisde functions, in particular requiring assumptions that are akin to such a structure. Hopefully, our work will provide a motivation to extend the theory of consistent approximations to broader composite classes of discontinuous functions.
To apply Proposition 19 to condition (C4), we rewrite the various index sets in S ps (x) more generally. Specifically, for any two triplets of index sets K {K > , K = , K < } and L {L > , L = , L < } with the former partitioning {1, · · · , K} and the latter partitioning {1, · · · , L}, define the closed set
S(K, L) x ∈ X g k (x) ≤ 0 ∀ k ∈ K < ∪ K = g k (x) ≥ 0 ∀ k ∈ K > h ℓ (x) ≤ 0 ∀ ℓ ∈ L < ∪ L = h ℓ (x) ≥ 0 ∀ k ∈ L > .
and consider the optimization problem:
minimize x∈S(K,L) f K,L (x) max ℓ∈L> φ ℓ (x) − b, 0 .
Assume that each function φ ℓ is bounded below on X. Then a very loose sufficient condition for (C4) to hold is that for all pairs (K, L), the minimum value of f K,L on S(K, L) is zero and the pair f K,L , S(K, L) satisfies condition (c) of Proposition 19 with δ = 1. We believe that it may be possible to tighten this sufficient condition and derive a result similar to the proposition for condition (C4); such details are beyond the scope of this work and best left for a separate investigation.
where 1 (0,∞) is the "open" Heaviside function given by1 ( 0,∞ ) (s) 1 if s ∈ ( 0, ∞ )0 otherwise and the following holds [the polyhedrality of X is not needed in several results; it is stated as a blanket assumption primarily to avoid the use of advanced constraint qualifications in the context of tangent cones]:Blanket assumption: X is a polyhedron contained in the open subset O of R n and c, { ϕ k , g k } K k=1 , and { φ ℓ , h ℓ } L ℓ=1 are B-differentiable (where B is for Bouligand) functions from O → R. By definition, a function ψ : O → R is B-differentiable [12, Definition 4.1.1] atx ∈ O if ψ is locally Lipschitz continuous atx (i.e., Lipschitz continuous in an open neighborhood ofx) and directionally differentiable there; i.e., the elementary one-sided directional derivative
which is the difference of two open Heaviside functions. An example where a product of an open and a closed Heaviside function may occur is a modification of the piecewise function (6): say ψ
Proposition 4 .
4Let c, {ϕ k , g k } K k=1 , and {φ ℓ , h ℓ } L ℓ=1 be B-differentiable functions on the open set O containing the closed set X. Among the following three statements for a vectorx ∈ X, it holds that (a) ⇒ (b) ⇒ (c):
Proposition 7 .
7Let ψ and f be B-differentiable functions on the open set O that contains a closed set S. Suppose that ψ is nonnegative on
0 satisfy the ACQ atx, then T (epi(π; S); (t,x)) = epi ψ ′ (x; •) 1 ( 0,∞ ) (f ′ (x; •)); T (S;x) .
Lemma 8 .
8Let θ : O x × O y → R be a B-differentiable bivariate variable on the open set O x × O y containing the closed set X ⊆ R n+m . If (x,ȳ) is a B-stationary point of θ on X, thenx is a B-stationary point of θ(•,ȳ) on X(ȳ) { x | (x,ȳ) ∈ X }. A similar statement holds for the y-variable.
[K], but similar equalities are not guaranteed for the s-variables. In fact, there are two possibilities: (i) L ℓ=1s ℓ ≤ b and there existsl ∈ [L] such
Definition 11 .
11We say that the bivariate functions θ : R × R ++ → [ 0, 1 ] p-approximate ("p" for pointwise) the (open) Heaviside function 1 ( 0,∞ ) if there exist endpoint functions θ and θ : R ++ → R t, δ) = 1 ( 0,∞ ) (t) for all t ∈ R; [this condition allows us to define θ(t, 0)1 ( 0,∞ ) (t),thereby extending the domain of definition of θ to R × R + ]; and
(
A3) the function θ(•, δ) is B-differentiable on [ −θ(δ), θ(δ) ] with the directional derivatives satisfyingθ(•, δ) ′ (t; 1) ≥ 0 ≥ θ(•, δ) ′ (t; −1), ∀ t ∈ [ −θ(δ), θ(δ) ].(32)By (A2), condition (A3) implies that θ(•, δ) ′ (t; 1) ≥ 0 ≥ θ(•, δ) ′ (t; −1) for all t ∈ R. Clearly, ifθ p-approximates the Heaviside function 1 ( 0,∞ ) , then so does the function ψ(variable t being scaled by the positive univariate function m : R ++ → R ++ satisfying lim sup δ↓0 m(δ) < ∞. In the next two subsections, we present two ways to obtain p-approximiations of the Heaviside function satisfying the B-differentiability requirement.
(
T3) θ(•, δ) is B-differentiable on an open interval containing [ −θ(δ), θ(δ) ] with θ(•, δ) ′ (t; 1) ≥ 0 ≥ θ(•, δ) ′ (t; −1), ∀ t ∈ [ −θ(δ), θ(δ) ].
Proposition 12 .
12The following statements hold: (a) If θ satisfies (T1), (T2), and (T3), then its truncation function θ tr p-approximates the Heaviside function 1 ( 0,∞ ) and satisfies (A3). (b) If θ(•, δ) is nondecreasing and satisfies: (T1 ′ ) θ(−θ(δ), δ) = 0 and θ(θ(δ), δ) = 1 for all δ > 0, then (T1) holds; if additionally θ(•, δ) is locally Lipschitz, then (T3) holds. (c) If ψ : R → R is a nondecreasing B-differentiable function satisfying ψ(0) = 0 and ψ(1) q : R ++ → [ 0, 1 ] and m : R ++ → R
dt = 1 and there exist end-point functions ψ and ψ : R ++ → R + satisfying condition (A0), i.e.δ), such that supp(ψ(•, δ)) ⊆ [ −ψ(δ), ψ(δ) ]. It then follows that
T2) holds for θ ψ (•, δ).
end bounds ψ(δ) and ψ(δ) satisfying the conditions in Lemma 14, we can derive a host of modified truncated hinge loss functions, such as (35), that p-approximate the open Heaviside function.
(x * ) 1 ( 0,∞) (h ℓ (x * )) = b.Let v ∈ L(S ps (x * ); x * ) be arbitrary.
ℓ (x * ; v) ≤ 0 is a stipulation on v.
atx for the problem minimize x∈X Ψ(x) depending on the value of f (x); these problems are• [ minimize x∈X ψ 1 (x) ] if f (x) ∈ ( a, b ); • [ minimize x∈X ψ 2 (x) ] if f (x) ∈ ( −∞, a ); • [ minimize x∈X ψ 3 (x) ] if f (x) ∈ ( b, ∞ ); • [ minimize x∈X ψ 1 (x) subject to f (x) ≥ a ] if f (x) = a; • [ minimize x∈X ψ 3 (x) subject to f (x) ≥ b ] if f (x) = b.
−1; and (C5) the functions c, {ϕ k } k∈K>(x * ) , and {φ ℓ } ℓ∈L>(x * ) are Clarke regular [8] at x * . In the context of a B-differentiable function f : O → R (which these functions are), Clarke regularity at a vectorx ∈ O means that for all sequences {z ν } converging tox, it holds that lim supν→∞
Acknowledgements. The authors are grateful to the referees for their insightful comments that have helped to improve the presentation of the manuscript.Proposition 18. Under the blanket assumption of problem(1), let c and each ϕ k be Lipschitz continuous on X with Lipschitz constants Lip c and Lip ϕ , respectively. Let {δ ν } be a sequence of positive scalars converging to zero and for each ν, let x ν be a d-stationary point of Φ λ (•, δ ν ) on X with λ satisfying(30). Suppose lim ν→∞ x ν = x * . Under assumptions (C1), (C2), (C4 ′ ), and (C5 ′ ),x * is feasible to(1)and there exist scalars1 ]such that x * is a B-stationary solution of:Proof. Belonging to the interval [ 0, 1 ], the sequences, which we may assume, without loss of generality are limits of the displayed sequences, respectively. We claim that x * satisfies the functional constraint in(47). Assume otherwise. ThenThe strengthened condition(46)and the Clarke regularity condition (C5 ′ ) then yield a contradiction by letting ν → ∞. This shows thatAppendix: (C4) and weak sharp minima. Consider the optimization problemwhere f is a continuous function bounded below on the closed set S. Let The problem(49), or the pair (f, S), is said to have weak sharp minima if there exists a constant η > 0 such thatThe definition of weak sharp minima was introduced in the Ph.D. thesis of Ferris[23]. Extensive discussion of this property and its role in optimization can be found in [19, Section 6.5]. The proposition below contains 3 statements. Statement (a) asserts the existence of weak sharp minima for(49). Statement (b) is known as Takahashi condition[50]in nonlinear analysis; it implies in particular the existence of a global minimizer to the optimization problem in question. Statement (c) is the key to connecting condition (C4) to weak sharp minima. Proof of the implication (b) is by the renowned Ekeland's variational principle[15]; proof of the proposition can be found in[19,Section 6.5]; see also[12,Section 8.5.4].Proposition 19. Let S be a closed set in R n and f be a continuous real-valued function defined and bounded below on S. Let f inf denote the infimum value of f on S. The two statements (a) and (b) are equivalent and either statement is implied by statement (c):(a) The pair (f, S) has weak sharp minima; i.e., F min = ∅ and for some constant c > 0,(b) There exists a constant c > 0 such that for each x ∈ S with f (x) > f inf , there exists a vector y ∈ S distinct from x such that(c) Suppose that f is B-differentiable on S and there exists a scalar δ > 0 such that for every x ∈ S with f (x) > f inf , a vector d ∈ T (S; x) with unit (Euclidean) length exists satisfying f ′ (x; d) ≤ −δ.Moreover, if f is a convex function and S is a convex set, then (b) implies (c); so all three statements (a), (b), and (c) are equivalent.
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[
"An intuitive introduction to the concept of spatial coherence",
"An intuitive introduction to the concept of spatial coherence"
] |
[
"H J Rabal \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n",
"N L Cap \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n",
"E Grumel \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n",
"M Trivi \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n",
"La Plata -Cic \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n",
"Uid Optimo \nCentro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina\n"
] |
[
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina",
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina",
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina",
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina",
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina",
"Centro de Investigaciones Ópticas (CONICET\nDepartamento Ciencias Básicas\nFacultad de Ingeniería\nUniversidad Nacional de la Plata\nGonnet, La Plata 1897P.O. Box 3Argentina"
] |
[] |
The concept of spatial coherence is usually hard to be understood the first time that it is studied. We propose here a fully intuitive geometric description that does not contain mathematical difficulties and permits to understand how a Young´s Fringes system is obtained with a source not spatially coherent. It is based in a very simple experiment that permits the detection of spatial coherence in a scene. Experimental results are shown.
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[
"https://export.arxiv.org/pdf/1408.3820v1.pdf"
] | 59,477,139 |
1408.3820
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7e6e94148dea443837fcff35280bc988a9a5a783
|
An intuitive introduction to the concept of spatial coherence
H J Rabal
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
N L Cap
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
E Grumel
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
M Trivi
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
La Plata -Cic
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
Uid Optimo
Centro de Investigaciones Ópticas (CONICET
Departamento Ciencias Básicas
Facultad de Ingeniería
Universidad Nacional de la Plata
Gonnet, La Plata 1897P.O. Box 3Argentina
An intuitive introduction to the concept of spatial coherence
optical coherencespatial coherenceVan Cittert -Zernike Theorem
The concept of spatial coherence is usually hard to be understood the first time that it is studied. We propose here a fully intuitive geometric description that does not contain mathematical difficulties and permits to understand how a Young´s Fringes system is obtained with a source not spatially coherent. It is based in a very simple experiment that permits the detection of spatial coherence in a scene. Experimental results are shown.
INTRODUCTION
The concept of spatial coherence is usually difficult to understand. An introductory approach on the coherence subject was developed by M. L. Calvo [1].
The rigorous study of the spatial coherence has been developed by Van Cittert and Zernike [2]. It states that the normalized degree of coherence is the Fourier transform of the intensity distribution for uncorrelated emitters.
When the interference phenomenon is studied, it usually starts with the calculation of the irradiance in an observation plane due to the superposition of two waves.
In the classical calculation of the field due to plane waves coming from two sources, if they are coherent (they keep a constant phase difference) and monochromatic (have the same frequency), two facts that, when taken rigorously result to be only one as if the waves are strictly monochromatic their phase difference is effectively constant, then, the waves can be described as [3]: (1) Where I 0 are the irradiance of the two waves and 12 is the phase difference between them. We assumed that the electrical fields of both waves are parallel.
) cos cos . 2 t r k t r k E E v I t t E E v
The phase difference 12 in the case of incoherent sources is not constant but changes so fast and ordinary detectors cannot detected the interference phenomenon. The irradiance due to two such sources is the sum of the irradiances of each of them.
Nevertheless, when we observe a light source through a pupil composed by two thin slits, we find a fringe system. Why does it happen? How can we justify it?
How is it possible that two (or more) incoherent elements of a source could give rise to nonzero visibility fringes stable in time in spite of the fact that their relative phases are fluctuating randomly?
Each idealized point element of it produces high contrast fringes, but different elements are not supposed to give rise to stable interference patterns.
Then, the only possibility for high visibility to subsist when both point source elements are present should be that the source points are separated by such a distance that makes the individual fringe systems to coincide.
Then, emphasis is exerted in the fact that the existence of measurable fringes visibility is due to the superposition in consonance of multiple fringe systems originating in different source elements that are incoherent between them.
This idea, originally used to calculate visibility in times before the VCZ Theorem was stated , can be exemplified by using two very small and close pinholes very near to the eyes and observing through them outdoor scenarios (see Figure 1). Even if the available light is not monochromatic, fringes can be observed in luminance discontinuities, such as edges, wires or poles, images of the Sun in dew drops or cylindrical surfaces also show fringes with visibility high enough to be discerned. It is easy, then, to figure out that low or zero visibility in extended sources is due to the superposition of shifted fringes systems.
We suggest here a description using elementary trigonometric identities to explain how a Young
Fringes system can be obtained from a source constituted by incoherent point sources. The visibility in the fringes with a compound source is found as the coincidence of several shifted fringes systems coming each from every single point. These are added on an intensity basis.
This approach leads in a natural way to the same result for the visibility as the Van Cittert-Zernike
Theorem for any arbitrary source distribution.
We use a simple experiment to illustrate this proposal. It is consisting in the observation of a scene through a card with two very small and very close slits.
SIMPLE EXPERIMENT WITH A NATURAL SCENE
If we observe a point like light source with an optical system limited by two parallel slit apertures, it can be observed that in the image there is Young´s Fringes fringe pattern.
If the source is composed by several incoherent emitters, the observable irradiance are too fast to be detected in the optical range, their average is zero and no fringes are observed.
Nevertheless, if this optical system is pointed to any natural scene, it can be that it appears covered with fringes. As an example, in Figure 1 we show a natural scene (a backyard) though an optical system (the camera) limited by two thin parallel slits. The corresponding Young Fringes that can be observed in the irradiance discontinuities.
How we can solve this contradiction? Why we find fringes?
To look for the reason we are going to consider very simple sources and to ask what happens with the fringes that they give rise.
TWO MUTUALLY NON COHERENT POINT LIGHTSOURCES
In Figure 2, A and B in plane π represent two quasi monochromatic point sources separated by a distance X 0 . They have the same wavelength λ and the same irradiance I 0 . Narrow slits P 1 and P 2 are separated a distance d. The lens L, with focal distance f 0 conjugates the planes and ´. The distance z is much bigger than d distances and the distance z´ ~ f 0 .
For each quasi monochromatic source corresponds in plane to the light distribution found in a Young´s Fringes experiment.
The irradiance distributions I A and I B in plane due to the sources A and B respectively can be described as:
x f d I x I A 0 2 0 cos 4 (2) 0 0 2 0ć os 4 X x f d I x I B (3)
The visibility of the fringes system produced by I A and I B , when both overlap, is:
' ' . cos 0 z d X I I I I V MIN MAX MIN MAX (4)
The visibility depends on the separation d between P 1 and P 2 (points that are used in the correlation in the Van Cittert Zernike´s Theorem) as well as the relation between ' ' 0 Z X and the wavelength. Imposing a 2 (or integer multiples of it) shift between both fringes systems, they will be in consonance and the visibility of the composed system will be a maximum. It is due to coincidence of the fringes and not to interference between light coming from the different sources.
CONTINUOUS SOURCE DISTRIBUTIONS
Following the same line of reasoning as before, if there are N discrete point sources with irradiance Ii, located at points xí, the intensity distribution in plane Q-Q´ results:
i W N i i x x I x I 2 1 cos
As is described in [4], for the case of a continuous quasi monochromatic incoherent intensity distribution source, the visibility of the Young fringes systems becomes:
' ' cos ) ' ( 2 dx x x x I x I x x A B x 2 cos 2 1 with ' ' 2 cos ) ' ( cos dx x x I A x and ' ' 2 sin ) ' ( sin dx x x I A x ' ) ' ( dx x I B
where I(x´) is the density of irradiance per unit length.
Then, the maximum and the minimum irradiances will be, respectively:
2 ' ) ' ( 2 ' ' 2 sin ) ' ( ' ' 2 cos ) ' ( 2 / 1 2 2 dx x I dx x x I dx x x I V x x where 0 f d w x (6)
If it, as shown in equation 5, is the modulus of the normalized Fourier Transform of the source distribution, it could be expected that complementary sources would give rise to similar fringes, but contrast reversed. Figure 3 shows the result of the experiment when the source is a uniform distribution covered with a thin slit. Young´s Fringes can be seen against the uniform background. Intuitively, it could be thought as the dark slit covering a fringes system that would cancel the fringes generated by a set of points in the background. If this cancellation is prevented by the slit, the fringes can be observed and are contrast reversed with respect to those of the complementary screen.
If P is the transmittance in intensity function of a certain binary screen, then P =1-P describes the transmittance of its complementary screen. The visibility of the Young´s Fringes obtained with the latter, as a function of the distance d between slits, consists in a delta distribution in the origin of frequencies minus the Fourier Transform of P. The change of sign indicates contrast reversal. So, Babinet´s Principle holds but with a minor change in its interpretation. Figure 4 shows the results obtained using as object a curved slit and its complementary.
CONCLUSIONS
When a single point-like source is observed though a two narrow slits, we obtain a Young fringes pattern, where the contributions of each aperture are added on a field basis.
For the spatial extended light sources, high visibility Young fringes can still be observed if every point of it gives rise to a fringes system that coincides with the produced by the others. In this case the addition of these elementary contributions is in intensity.
For the systems to coincide and obtaining a good visibility result there should not exist source points very near that spoil the others visibility. This only happens when the source exhibits spatial discontinuities (i.e. When the Fourier Transform is not a Dirac´s Delta distribution). It is the presence of source discontinuities that gives rise to the visibility predicted by the Van Cittert-Zernike Theorem.
This is also true when the source is a uniform field and there are isolated discontinuities (dilute dark object on bright uniform field). So that visibility behaves as fulfilling a Babinet like complementarity property, although giving fringes that are contrast reversed between a mask and its complementary.
A Holodiagram description of some of these phenomena can be found in [4], where this approach is extended to somewhat more complex sources distributions.
When the source distribution is not entirely contained in a plane perpendicular to the optic axis, the calculation turns out to be a little more involved but still can be described using this geometric approach. [4].
The visibility is given by the modulus of the normalized Fourier Transform of the intensity density distribution of the source. This is the result of the Van Cittert Zernike´s Theorem. Visibility of the fringes depends on the distance d (coordinates difference) between points P1 and P2 and not in their actual position in front of the lens. Notice that only elementary calculus and trigonometric identities are used in this description and at every step a clear understanding is easily maintained of their meaning, To add another experimental verification, a V shaped object (a transparence mask with two bright convergent slits) can be used and observed through the double parallel slits Young´s experiment to observe how the separation of the points of the V produce fringes systems that add their effects or cancel them according to their distance (see Figure 3) giving a periodic (chromatic, as it depends also in wavelength) variation of visibility. Does visibility fulfill a Babinet Principle? If this approach is correct, then, given a mask M, its complementary mask M should hide those fringe contributions to the image that are themselves complementary to those given by M.
Figure 1 :Figure 2 :
12Small departures in experimental visibility observation can be expected if the slits are no A natural scene showing Young´s fringes (slits horizontal) Young fringes pattern obtained from two point sources mutually non coherent.
Figure 3 :
3Fringes produced by a V shaped double slit used as incoherent object and using two parallel slits in front of the objective, as in Young´s experiment. Two different V were included and a vertical single slit in the middle for comparison. Notice the periodic variation and contrast reversals in visibility in the vertical direction in the Vs.
Figure 4 :
4The fringes obtained with an object in the shape of a curved slit and its complementary one showing Babinet´s Principle.
Figure 1 AFigure 2 .
12natural scene showing Young´s fringes (slits horizontal) Young fringes pattern obtained from two point sources mutually non coherent.
Figure 3 :
3Fringes produced by a V shaped double slit used as incoherent object and using two parallel slits in front of the objective, as in Young´s experiment. Two different V were included and a vertical single slit in the middle for comparison. Notice the periodic variation and contrast reversals in visibility in the vertical direction in the Vs.
Figure 4 :
4The fringes obtained with an object in the shape of a curved slit and its complementary one showing Babinet´s
Principle.
ACKNOWLEDGEMENTSThis research was supported by grants of University of La Plata, ANPCyT, CONICET and CICPBA (Argentina)
Investigación y Ciencia. M L Calvo, Coherencia, Spanish) and Calvo, M. L. MayoCalvo, M. L. Coherencia óptica. Investigación y Ciencia. Pp 66-73. Mayo 1995. (In Spanish) and Calvo, M. L. http://e-ico.org/node/255
Principles of Optics, (4th. M Born, E Wolf, Pergamon Press508Oxford UKBorn M. and Wolf E. Principles of Optics, (4th.ed.) Pergamon Press, Oxford UK pp. 508. (1970).
. E Hecht, A Zajac, Optics, Addison Wesley PublishingReading MAHecht E. and Zajac, A. Optics, Addison Wesley Publishing, Reading MA, (1974)
The holodiagram in a geometrical approach to the calculation of fringe visibility. H Rabal, Rabal, H. "The holodiagram in a geometrical approach to the calculation of fringe visibility".
. Optik. 113Optik, 113, 260-266, (2002)
|
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"Segmentation in large-scale cellular electron microscopy with deep learning: A literature survey",
"Segmentation in large-scale cellular electron microscopy with deep learning: A literature survey"
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"Anusha Aswath \nComputer Science and Artificial Intelligence\nBernoulli Institute of Mathematics\nUniversity Groningen\nGroningenThe Netherlands\n\nDept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands\n",
"Ahmad Alsahaf \nDept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands\n",
"Ben N G Giepmans \nDept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands\n",
"George Azzopardi \nComputer Science and Artificial Intelligence\nBernoulli Institute of Mathematics\nUniversity Groningen\nGroningenThe Netherlands\n"
] |
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"Computer Science and Artificial Intelligence\nBernoulli Institute of Mathematics\nUniversity Groningen\nGroningenThe Netherlands",
"Dept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands",
"Dept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands",
"Dept. Biomedical Sciences of Cells and Systems\nUniversity Groningen\nUniversity Medical Center Groningen\nGroningenThe Netherlands",
"Computer Science and Artificial Intelligence\nBernoulli Institute of Mathematics\nUniversity Groningen\nGroningenThe Netherlands"
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Automated and semi-automated techniques in biomedical electron microscopy (EM) enable the acquisition of large datasets at a high rate. Segmentation methods are therefore essential to analyze and interpret these large volumes of data, which can no longer completely be labeled manually. In recent years, deep learning algorithms achieved impressive results in both pixel-level labeling (semantic segmentation) and the labeling of separate instances of the same class (instance segmentation). In this review, we examine how these algorithms were adapted to the task of segmenting cellular and sub-cellular structures in EM images. The special challenges posed by such images and the network architectures that overcame some of them are described. Moreover, a thorough overview is also provided on the notable datasets that contributed to the proliferation of deep learning in EM. Finally, an outlook of current trends and future prospects of EM segmentation is given, especially in the area of label-free learning.
| null |
[
"https://export.arxiv.org/pdf/2206.07171v3.pdf"
] | 257,834,222 |
2206.07171
|
3839deed7befd3fc35032fe86b35b4a35879b94a
|
Segmentation in large-scale cellular electron microscopy with deep learning: A literature survey
Anusha Aswath
Computer Science and Artificial Intelligence
Bernoulli Institute of Mathematics
University Groningen
GroningenThe Netherlands
Dept. Biomedical Sciences of Cells and Systems
University Groningen
University Medical Center Groningen
GroningenThe Netherlands
Ahmad Alsahaf
Dept. Biomedical Sciences of Cells and Systems
University Groningen
University Medical Center Groningen
GroningenThe Netherlands
Ben N G Giepmans
Dept. Biomedical Sciences of Cells and Systems
University Groningen
University Medical Center Groningen
GroningenThe Netherlands
George Azzopardi
Computer Science and Artificial Intelligence
Bernoulli Institute of Mathematics
University Groningen
GroningenThe Netherlands
Segmentation in large-scale cellular electron microscopy with deep learning: A literature survey
Electron microscopysegmentationsupervisedunsuperviseddeep learningsemanticinstance
Automated and semi-automated techniques in biomedical electron microscopy (EM) enable the acquisition of large datasets at a high rate. Segmentation methods are therefore essential to analyze and interpret these large volumes of data, which can no longer completely be labeled manually. In recent years, deep learning algorithms achieved impressive results in both pixel-level labeling (semantic segmentation) and the labeling of separate instances of the same class (instance segmentation). In this review, we examine how these algorithms were adapted to the task of segmenting cellular and sub-cellular structures in EM images. The special challenges posed by such images and the network architectures that overcame some of them are described. Moreover, a thorough overview is also provided on the notable datasets that contributed to the proliferation of deep learning in EM. Finally, an outlook of current trends and future prospects of EM segmentation is given, especially in the area of label-free learning.
Introduction
Electron microscopy (EM) is widely used in life sciences to study tissues, cells, subcellular components and (macro) molecular complexes at nanometer scale. Two-dimensional (2D) EM aids in diagnosis of diseases, but routinely it still depends upon biased snapshots of areas of interest. Automated pipelines for collection, stitching and open access publishing of 2D EM have been pioneered for transmission EM (TEM) images (Faas et al., 2012) as well as scanning TEM (STEM) (Sokol et al., 2015) for acquisition of areas up to 1mm 2 at nanometer-range resolution. Nowadays, imaging of large areas at high resolution is entering the field as a routine method and is provided by most EM manufacturers. We term this nanotomy, for nano-anatomy (Ravelli et al., 2013;de Boer et al., 2020;Dittmayer et al., 2021). The large-scale images allow for open access world-wide data sharing; see nanotomy.org 1 for more than 50 published studies and the accessible nanotomy data. (topleft) and snapshots from this total map at higher zoom showing several cellular, subcellular and macromolecular structures as indicated and annotated. Note the information density of these maps: millions of subcellular structures of a kind can be present per dataset (de Boer et al., 2020). Full access to digital zoomable data at full resolution is via http://www.nanotomy.org.
A typical nanotomy dataset has a size of 5-25GB at 2.5nm pixel size. Nanotomy allows scientists to pan and zoom through different tissues or cellular structures, Fig. 1. Large-scale 2D EM provides unbiased data recording to discover events such as pathogenesis of diseases and morphological (shape and texture) changes at the subcellular level. Moreover, nanotomy allows for the quantification of subcellular hallmarks. With state-of-the-art 2D EM technology, such as multibeam scanning EM (Eberle et al., 2015;Ren and Kruit, 2016), up to 100 times faster acquisition and higher throughput allows for imaging of tissue-wide sections in the range of hours instead of days. For a side-by-side example of single beam versus multibeam nanotomy, see de Boer and Giepmans (2021). Given the automated and faster image acquisition in 2D EM a data avalanche (petabyte range per microscope/month) is becoming a reality.
Automated large-scale three-dimensional (3D) or volume EM (vEM), which creates stacks of images, is also booming (Peddie and Collinson, 2014;Titze and Genoud, 2016;Peddie et al., 2022). The faster acquisition of 3D EM for serial-sectioning transmission EM (ssTEM) and serial block-face scanning EM (SBF-SEM) technologies can also lead to accumulation of petabytes of data. For instance, a complete brain volume of an adult fruit fly was imaged by ssTEM (Zheng et al., 2018), which covered a single neuron cell in a volume of 1mm 3 or 10, 000 voxels and required 100 TB. Additionally, manual annotation is not practical due to the size of 3D EM datasets. An example by Heinrich et al. (2021) shows that one person needed two weeks to manually label a fraction (1 µm 3 ) of a whole-cell volume containing tons of instances of various types of organelles, whereas the whole cell could take 60 person-years. Whole-cell cryo-electron tomography (cryo-ET) has also advanced the capabilities of 3D EM to investigate the structure of cellular architecture and macromolecular assemblies in their native environment. The core data acquisition techniques of such 2D and 3D EM technologies are listed in Table 1.
This increase in the scale and acquisition speeds of EM data accelerated the development of compatible methods for automatic analysis, especially in the areas of semantic and instance segmentation. Semantic segmentation classifies the pixels of an image into semantically meaningful categories, e.g. nuclei and background, while instance segmentation focuses on separating individual instances within the same class; e.g. the delineation of apposed mitochondria.
In the past, traditional image analysis methods as well as shallow learning algorithms 2 have been used for the segmentation of EM images, for instance using statistical analysis of pixel neighborhoods (Kylberg et al., 2012), eigenvector analysis (Frangakis and Hegerl, 2002), watershed and hierarchical region merging (Liu et al., 2012(Liu et al., , 2014, superpixel analysis and shape modeling (Karabag et al., 2019), and random forest (Cao et al., 2019). However, the past few years marked a dominance of deep learning (DL) in this domain, similarly to the trends of segmentation in light microscopy and other medical imagining modalities Litjens et al., 2017). Compared to traditional image analysis and machine learning with handcrafted features, deep learning segmentation reduces or removes the need for domain knowledge of the specific imaged sample to extract relevant features .
The popularity of DL segmentation led to the development of DL plug-ins for many of the routinely used biomedical image analysis software tools like CellProfiler (Carpenter et al., 2006), ImageJ (Schindelin et al., 2012), Weka (Arganda-Carreras et al., 2017), and Ilastik (Berg et al., 2019), which had previously been limited to traditional image processing methods or shallow learning. Moreover, it led to the development of specialized tools that enable biologists to train and use DL networks with the aid of graphical user interfaces (von Chamier et al., 2021;Belevich and Jokitalo, 2021).
We review the recent progress of automatic image segmentation in EM, with a focus on the last six years that marked significant progress in both DL-based semantic and instance segmentation, while also giving an overview of the main DL architectures that enabled this progress.
The manuscript is organized as follows: Section 2 describes the literature search strategy used for this review. Section 3 presents the benchmark datasets, which have been key for the progress of the segmentation methods. Section 4 lays the background about the main neural network architectures for 2D and 3D segmentation of EM datasets. Sections 5 and 6 review the papers that propose new methodologies for semantic and instance segmentation with different DL approaches. These are followed by Section 7, which describes the evaluation metrics used in the reviewed papers. Finally, Section 8 provides an outlook of the overall progress of this field along with a discussion on future prospects. Table 1: Main large-scale EM techniques. More information is given in the MyScope website 2 and the reviews by Peddie and Collinson (2014), Titze and Genoud (2016) and Kievits et al. (2022). The last row shows example 2D images and 3D stacks of such technologies except STEM, an example of which is shown in Fig. 1.
2D EM
Data acquisition technique
Transmission Electron Microscopy (TEM) A widefield electron beam illuminates an ultra-thin specimen and transmitted electrons are detected on the other side of the sample. The structure that is electron dense appears dark and others appear lighter depending on their (lack of) scattering.
Scanning Electron Microscopy (SEM)
The raster scanning beam interacts with the material and can result in backscattering or the formation of secondary electrons. Their intensity reveals sample information.
Scanning Transmission Electron Microscopy (STEM) SEM on ultrathin sections and using a detector for the transmitted electrons.
3D EM
Serial section TEM (ssTEM) or SEM (ssSEM) Volume EM technique for examining 3D ultrastructure by scanning adjacent ultrathin (typical 60-80nm) sections using TEM or SEM, respectively.
Serial Block-Face scanning EM (SBF-SEM) The block face is scanned followed by removal of the top layer by a diamond knife (typical 20-60nm) and the newly exposed block face is scanned. This can be repeated thousands of times.
Focused Ion Beam SEM (FIB-SEM) Block face imaging as above, but sections are repeatedly removed by a focused ion beam that has higher precision than a knife (typically down to 4nm), making it suitable for smaller samples.
Cryo-electron tomography (Cryo-ET) It captures a series of 2D projection images of a flash-frozen specimen from different angles, and then uses computational reconstruction methods to generate a 3D model or tomogram.
TEM 2D
Cryo-ET 2D SEM 2D ssSEM volume -2D sections SBF-SEM volume FIB-SEM (Ciresan et al., 2012) (Chen et al., 2017d) (Kasthuri et al., 2015) (Abdollahzadeh et al., 2021) (Lucchi et al., 2011)
Strategy of literature search
Our survey strategy is motivated by the following questions:
• Which datasets are accessible for EM analysis, what are their challenges and what role do they play in DL research? • How is EM image (semantic and instance) segmentation being addressed by fully/semi/un/self-supervised DL pipelines?
To answer these questions, the following search query was used in Pubmed, Web of Science, and Google Scholar on words in titles (TI) only, restricted to 2017-2022: TI=((electron microscopy OR EM) AND (segmentation OR semantic OR instance OR supervised OR unsupervised OR selfsupervised OR semi-supervised)), and title or abstracts containing (deep learning, segmentation, electron microscopy) on Google Scholar. Results from the query that were outside the scope of this study, such as deep learning in material sciences and methods based on traditional image processing (pre-DL era), were excluded. The forward and backward snowballing technique was then used to compile the final list of 38 papers. Fig. 2 summarizes this collection of 38 papers in terms of learning technique (fully supervised or not), segmentation type (semantic or instance), application (2D or 3D) and the underlying modeling backbone. Before reviewing these papers, we discuss the key EM datasets and describe the evolution of DL architectures, which are two crucial components that have been permitting the progress of EM segmentation analysis. Figure 2: Categorization of the 38 papers reviewed in this survey. The papers are first categorized on the learning paradigm (fully vs. semi/un/self-supervised) and on the segmentation type (semantic vs. instance). Each quadrant shows the distributions of applications (2D vs. 3D) and DL backbones (U-Net vs. FCN vs. Other) of the papers that use the corresponding learning and segmentation approaches.
Collections of key EM datasets
Collections of labeled and unlabeled EM images have played a significant role in advancing DL research for EM segmentation, and some were associated with notable segmentation competitions and challenges. This section provides the details of all collections used by the 38 papers in this survey. Table 2 reports the main properties of these datasets and below is an in-depth discussion of their characteristics and the challenges they address. The discussion is categorized according to the EM modality used to acquire the datasets.
Serial section TEM and SEM datasets
Serial-section transmission or scanning EM (ssTEM or ssSEM) is used for studying synaptic junctions and highly-resolved membranes in neural tissues. Advances in microscopy techniques in serial section EM have enabled the study of neurons with increased connectivity in complex mammalian tissues (such as mice and humans) and even whole brain tissues of smaller animal models, like the fruit fly and zebrafish. This imaging approach visualizes the generated volumes in a highly anisotropic manner, i.e. the x-and y-directions have a high resolution, however, the z-direction has a lower resolution, as it is reliant on serial cutting precision.
The Drosophila larve dataset (#1) 3 of the ISBI 2012 challenge was the first notable EM dataset for automatic neuronal segmentation, featuring two volumes with 30 sections each. The main challenge of that dataset is to develop algorithms that can accurately segment the neural structures present in the EM images. The success of deep neural networks as pixel classifiers in the ISBI 2012 challenge (Ciresan et al., 2012) paved the way for deep learning in serial section EM segmentation. Recently, a connectome of an entire brain of a Drosophila fruit fly has been published by Winding et al. (2023), and will serve as a new resource for various follow-up works.
The CREMI3D dataset (#2) consists of three large and diverse sub-volumes of neural tissue along with ground truth annotations for training and evaluation purposes, and was part of a competition at the MICCAI 2016 conference. The dataset comes from a full adult fly brain (FAFB) volume and contains 213 teravoxels. It was imaged at the synaptic resolution to understand the functioning of brain circuits (connectomics) and its goal was to segment neurons, synapses, and their pre-post synaptic partners. The CREMI3D dataset is part of the FlyEM project and since its inception, it has been used to evaluate various image analysis methods for neural circuit reconstruction, including DL approaches such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs).
The SNEMI3D dataset (#3) consists of a volume of 100 ssSEM images of the neural tissue from a mouse cortex. It is a subset of the largest mouse neocortex dataset imaged by Kasthuri et al. (2015) using an automated ssSEM technique and hence is also known as the Kasthuri dataset. The dataset was created as part of the ISBI 2013 challenge on segmentation of neural structures in EM images. The main challenge of this dataset is to develop algorithms that can accurately segment the neuronal membranes present in the EM images and reconstruct a 3D model of the tissue. This is a difficult task due to the large size of the dataset and the complexity of the neural structures, namely axons, dendrites, synapses, and glial cells.
The Kasthuri++ and Lucchi++ (#4, #9) datasets were introduced by Casser et al. (2018) with corrected annotations of Kasthuri and Lucchi. The Kasthuri dataset, which is used for dense reconstructions of the neuronal cells was corrected for the jaggedness between inter-slice components as they were not accurate. The Lucchi dataset is a FIB-SEM dataset used for the segmentation of mitochondria in the mouse neocortex. It was corrected for consistency of all annotations related to mitochondrial membranes, as well as to rectify any categorization errors in the ground truth. The Xiao (#5) dataset for mitochondria segmentation was collected from a rat brain by Xiao et al. (2018a) using advanced ssSEM technology. Automated cutting was used to produce 31 sections, each with an approximate thickness of 50 nm for segmenting mitochondria. The ground truth dataset was prepared through 2D manual annotation and image registration of serial-section images, which was made publicly available for accelerating neuroscience analysis.
Mito-EM (#6) introduced the largest mammalian mitochondria dataset from humans (MitoEM-H) and adult rats (MitoEM-R). It is about 3600 times larger than the standard dataset for mitochondria segmentation (Lucchi) containing mitochondria instances of at least 2000 voxels in size. Complex morphology such as mitochondria on a string (MOAS) connected by thin microtubules or instances entangled in 3D were captured using ssSEM. The MitoEM dataset was created to provide a comprehensive view of the ultrastructure of mitochondria and to facilitate a comparative study of mitochondrial morphology and function in rats and humans.
The NucMM dataset (#7) (Lin et al., 2021) contains two fully annotated volumes; one that contains almost a whole zebrafish brain with around 170,000 nuclei imaged using ssTEM; and another that contains part of a mouse visual cortex with about 7,000 nuclei imaged using micro-CT. Micro-CT or micro-computed tomography uses X-rays to produce 3D images of objects at low resolution and hence is not a part of this review. The large-scale nuclei instance segmentation dataset from ssTEM covers 0.14mm 3 of the entire volume of the zebrafish brain at 4 × 4 × 30 nm/voxel. As most of the nuclei segmentation datasets are from light microscopy at the µm scale, the dataset was downsampled to 512 × 512 × 480 nm/voxel.
FIB-SEM datasets
FIB-SEM generates datasets that are ideal for automated connectome tracing and for examining brain tissue at resolutions lower than 10 × 10 × 10 nm. The method can produce sections with a thickness of 4 nm, but the volumes are typically smaller in comparison to other techniques, due to their high z-resolutions.
The Lucchi dataset (#8) is an isotropic FIB-SEM volume imaged from the hippocampus of a mouse brain, and it has the same spatial resolution along all three axes. This dataset has now become the de facto standard for evaluating mitochondria segmentation performance. Efforts to expand FIB-SEM to larger volumes were made by Takemura et al. (2015) who compiled the FIB-25 (#10) dataset by reconstructing the synaptic circuits of seven columns in the eye region of a Drosophila's brain. FIB-25 contains over 10,000 annotated neurons, including their synaptic connections, and is one of the most comprehensive EM datasets of the Drosophila brain to date. It was created to provide a detailed map of the neural circuits in the Drosophila brain and to facilitate the study of neural connectivity and information processing. The dataset is publicly available and can be accessed through the FlyEM project website.
Enhanced FIB-SEM techniques have also enabled high-throughput and reliable long-term imaging for large-scale EM (10 3 to 3 × 10 7 µm 3 ), such as the OpenOrganelle atlas (#11) of 3D whole cells and tissues of Xu et al. (2021). The datasets for the 3D reconstruction of cells were made open-source under the OpenOrganelle repository for exploring local cellular interactions and their intricate arrangements.
Other FIB-SEM datasets include ones requiring a high-resolution analysis of 3D organelles in important tissues of the heart muscle and urinary bladder. Cardiac mitochondria (#12) is a FIB-SEM dataset introduced to segment mitochondria in cardiomyocytes (Khadangi et al., 2021b). The FIB-SEM technique was needed to better characterize diffusion channels in mitochondria-rich muscle fibers. Isotropic voxels at 15 nm resolution were imaged according to the set of experiments performed by Glancy et al. (2015). The UroCell (#13) from FIB-SEM was imaged by Mekuč et al. (2020) to focus on mitochondria and endolysosomes and was further extended to Golgi apparatus and fusiform vesicles. The dataset is unique as it is publicly available for further analysis of the epithelium cells of the urinary bladder, where the organelles form an important component in maintaining the barrier between the membrane of the bladder and the surrounding blood tissues.
SBF-SEM datasets
Connectomics research was also based on popular datasets imaged using SBF-SEM (Helmstaedter et al., 2013;Briggman et al., 2011). Imaging using SBF-SEM produces anisotropic sections but does not need image registration and avoids missing sections in comparison to serial-sectioning TEM/SEM, as the technique images the sample intact on a block surface. Such a technique also enabled imaging large volumes for studying the organization of neural circuits and cells across hundreds of microns through millimeters of neurons in a z-stack.
The Perez dataset (#14) (Perez et al., 2014) involved the acquisition of 1283 serial images from the hypothalamus's suprachiasmatic nucleus (SCN), a small part of the mouse brain, to produce an image stack with tissue dimensions approximately measuring 450,000 µm 3 . The large acquired volume was downsampled from 3.8 to 7.8 nm/pixel in the x − y resolution to scale up the processing of these tetra-voxel-sized SBF-SEM images. It was introduced for the automatic segmentation of mitochondria, lysosomes, nuclei, and nucleoli in brain tissues.
SegEM (#15) introduced an EM dataset acquired using SBF-SEM from the mouse somatosensory cortex (Berning et al., 2015). The images in the SegEM dataset are provided with corresponding segmentation labels for dendrites, axons, and synapses. The labels were generated using a semiautomated approach which involved a combination of skeleton annotations and machine learning algorithms to trace long neurites accurately. Since then, SegEM has been used for benchmarking popular models like flood-filling networks that test the efficiency of algorithms on volume-spanning neurites.
The Guay dataset (#16) is a fully annotated dataset of platelet cells from two human subjects and was designed for dense cellular segmentation (Guay et al., 2021). It has also been used for large-volume cell reconstruction along with mitochondria, nuclei, lysosomes, and various granules inside the cells.
The Axon dataset (#17) is a collection of SBF-SEM images of white matter tissue from rats, captured at a lower resolution of 50 nm/pixel (Abdollahzadeh et al., 2021). The low-resolution image stack of 130000 µm 3 was enough to resolve structures like myelin, myelinated axons, mitochondria, and cell nuclei. A wide field of view employing low-resolution SBF-SEM stacks was considered important for quantifying metrics such as myelinated axon tortuosity, inter-mitochondrial distance, and cell density.
CDeep3M proposed two new datasets from SBF-SEM and cryo electron tomography (cryo-ET) for automatic segmentation. The first one, CDeep3M-S (#18), is a large SBF-SEM dataset for membrane, mitochondria, and synapse identification from the cerebellum and lateral habenula of mice. Imaged at 2.4 nm pixel size, a cloud implementation of the latest architecture for anisotropic datasets was used to segment structures such as the neuronal membrane, synaptic vesicles mitochondria, and nucleus in brain tissues. The second dataset, CDeep3M-C (#21), was from cryo-ET and is explained further in subsection 3.4.
The EMPIAR-10094 dataset (#19) consists of EM images of cervical cancer "HeLa" cells imaged using SBF-SEM. The dataset is imaged at 8192 × 8192 pixels over a total of 518 slices, and consists of different HeLa cells distributed in the background of the embedding resin. The dataset has been made publicly available with no labels and has mostly been used for delineating structures such as plasma membranes and nuclear envelopes.
Unlabeled datasets, such as CEM500K, from various unrelated experiments and EM modalities for solving the segmentation of a particular structure seem promising. The CEM500k (#20) is an EM unlabeled dataset containing around 500,000 images from various unrelated experiments and different EM modalities for cellular EM. The images from different experiments were standardized to 2D images of size 512×512 pixels with pixel resolutions ranging from 2 nm in datasets from serial section EM and ∼20 nm for SBF-SEM. The dataset was further filtered by removing duplicates and low-quality images in order to provide robustness to changes in image contrast and making it suitable for training modeling techniques.
Cryo-ET datasets
Electron tomography (ET) is used to obtain 3D structures of EM sections using the tilt-series acquisition technique. Cryo-ET does so at cryogenic temperatures to image vitrified biological samples. Attempts for segmentation on cryo-ET can be found by Moussavi et al. (2010) and in the review of Carvalho et al. (2018). The identification of macromoleular structures is beyond the scope of this review. Cryo-ET presents challenges in visualizing and interpreting tomographic datasets due to two main factors. Firstly, sample thickness increases as the tilt angle increases, leading to an artifact known as the "missing wedge" and reduced resolution in the z-direction. Secondly, vitrified biological samples are sensitive to electron dose, resulting in a low signal-to-noise ratio and difficulties in distinguishing features of interest from background noise. As the resolution capacity of TEM decreases with the increase in sample thickness, focused ion beam (FIB) milling can be used to obtain a high-resolution tomogram. Cryo-FIB SEM is an evolving technology for cellular imaging that is rapidly being used in recent years. This is mainly attributable to its ability to image larger specimens that may be too thick for cryo-ET, such as whole cells or tissues.
CDeep3M-C (#21) is a cryo-ET dataset for the segmentation of vesicles and membranes from the mouse brain (Haberl et al., 2018a). At a voxel size of 1.6 nm, it was used to digitally recreate a tiny section (approximately 1.5 × 1.5 × 1.5µm 3 ) of a high-pressure frozen tissue. The final volume was built from 7 sequential tomograms (serial sections), each created by tilting a sample every 0.5 • in an electron beam from −60 • to + 60 • . The cellular cryo-ET dataset (#22) was acquired at low magnification for annotation and qualitative cellular analysis of organelles like mitochondria, vesicles, microtubules, and plasma membrane (Chen et al., 2017d). The PC12 cell line was reconstructed using 30 serial sections imaged at 850 × 850 × 81 pixel size at 2.8 nm resolution. The tomograms of platelets and cyanobacteria utilized in that work are from previously published datasets (Wang et al., 2015;Dai et al., 2013).
Background of backbone deep learning networks for EM semantic and instance segmentation
The rapid progress of DL methods, in particular CNNs, has had a great impact on advancing segmentation of EM images, as well as other medical images of various modalities (Litjens et al., 2017;Shen et al., 2017), including light microscopy (Xing et al., 2017;Liu et al., 2021). Deep learning in EM analysis has also been addressed in the reviews by Treder et al. (2022) and Ede (2021). The former gives a broad overview of different EM applications in both physical and life sciences and the latter provides a practitioner's perspective focused on the hardware and software packages to perform DL-based EM analysis. In contrast, this review provides an in-depth view of (Long et al., 2015) and U-Net (Ronneberger et al., 2015). Each of the fully connected (fc) layers in FCN and convolutional layers in U-Net are followed by the nonlinear activation function ReLU and max pooling. In FCN, the fully connected layers are then converted to convolutional layers via the 'fc to conv' component. The last layer uses a softmax function to assign a probability class score to each pixel. The FCN decoder includes an upsampling component that is linearly combined with the low-level feature maps in the third convolutional layer of the encoder. The sizes of these feature maps are four times less than the size of the input image I (denoted by I/4). Finally, there is a direct upsampling from I/4 to the original size of I followed by softmax for classification. The symmetrical U-Net architecture shares the features maps in the encoder with the decoder path together with skip connections.
fully/semi/self/un-supervised deep learning methods for the semantic and instance segmentation in (sub)cellular EM. This section covers the main milestones in the progression of network architectures and their key attributes, which are necessary to put in context the 38 papers that are reviewed in this work. Semantic segmentation in EM images is the identification of objects or subcellular organelles in such a way that each pixel is mapped to a specific class. This is different than instance segmentation, which refers to the process of dividing an image into multiple segments, each corresponding to a unique object or instance. Instance segmentation is particularly important in the study of cellular structures and their interactions, as it allows for the identification and quantification of individual objects in large-scale datasets.
The CNN designed by Ciresan et al. (2012), for instance, was used for the semantic segmentation of neuronal membranes in stacks of EM images. The images were segmented by predicting the label of each local region or patch covered by a convolutional filter in a sliding window approach and introduced max-pooling layers instead of sub-sampling layers. As indicated by Arganda-Carreras et al. (2015), it led to winning the ISBI 2012 neuronal segmentation challenge 4 .
Despite its success, the method suffered from two major limitations -firstly, the sliding window approach was slow due to the redundancy of processing large overlaps between adjacent patches, and secondly, there was a trade-off between the size of the patches (context) and localization accuracy. Since the network's depth is an important factor for a larger receptive field (the size of the viewing field from which the network receives information), larger patches require deeper networks. Localization ability, however, decreases with deeper networks due to downsampling by the many max pooling layers and the use of smaller patches allows the network to see only a little context. Improvements in the semantic segmentation of EM images continued with the development of the Fully Convolutional Network (FCN) (Long et al., 2015) and the U-Net architecture (Ronneberger et al., 2015), Fig. 3. The concept of expanding a CNN to handle inputs of any size using fully convolutional layers instead of fully connected ones helped evolve dense predictions for segmentation. A skip architecture was introduced to make use of a feature spectrum that merges deep, coarse, semantic information with shallow, fine, appearance information. The U-Net extended an FCN network with a U-shaped topology to optimize the tradeoff between localization and context. The contracting path (encoder) captures a larger context using the downsampled features and the expanding path (decoder) upsamples features to their original size with the same number of layers making it a symmetric or U-shaped network. The skip connections between the encoder-decoder layers bypass some of the neural network layers and as a result, an alternative and shorter path is provided for backpropagating the error of the loss function, which contributed to avoiding the vanishing gradient problem (Krizhevsky et al., 2012). Increased connectivity in the upsampling path within FCNs and the consideration of multi-level contexts were key to improving semantic segmentation (Badrinarayanan et al., 2017;Drozdzal et al., 2016).
DeepLab is another family of semantic segmentation networks, which have the ability to achieve robustness for different scales without increasing computational complexity (Chen et al., 2014(Chen et al., , 2017b(Chen et al., ,c, 2018. DeepLab architectures are based on FCNs but extended with the use of dilated (or atrous) convolutions, which were originally proposed by Yu and Koltun (2016), and imagelevel features. The atrous dilations are used within Atrous Spatial Pyramid Modules (ASPP), which perform multi-scale feature extraction by using multiple atrous convolutions with different dilation rates. As a backbone network, the latest DeepLab architecture, namely DeepLab v3+, uses the Residual Neural Network (ResNet) to produce image-level feature maps. The module performs parallel convolution on the feature map obtained from the ResNet backbone and outputs multiple feature maps, which are then concatenated and fed into the next layer. This allows the network to capture features of multiple scales, which is crucial for tasks like semantic segmentation. ResNet is notable for its ability to overcome the vanishing gradient problem and the degradation issue, simultaneously (He et al., 2016). This breakthrough was attributable to the introduction of residual connections, which allow the network to learn residual functions, or the difference between the desired output and the current output, rather than the full function. This helps the network to learn more effectively and avoid overfitting.
The 3D segmentation of neuronal stacks was set as a challenge in ISBI 2013. The major challenges in analyzing volume EM datasets are misalignments or missing sections due to serial sectioning, and volume anisotropy due to different resolutions in different directions. Specifically, it refers to the situation where the resolution in the z-axis (the depth dimension) is lower than the resolution in the x − y plane.
There are three typical approaches for the analysis of 3D volumes. The first involves 2D segmentation of each image in the stack, followed by 3D reconstructions based on clustering techniques, that may range from basic watershed to complex graph cuts algorithms. The second approach is based on 3D CNNs, which can learn representations of volumetric data that include 3D spatial context. One example of such 3D CNNs is the 3D U-Net by Ç içek et al. (2016), which was inspired by the original U-Net that uses local and larger contextual information. It was then extended into the V-Net model by Milletari et al. (2016) by adding residual stages. The HighRes3DNet is another 3D CNN based on the FCN architecture, with dilated and residual convolutions, and has been successful in obtaining accurate segmentations of neuronal mitochondria . In terms of performance, both HighRes3DNet and V-Net have achieved state-of-the-art results on several medical image segmentation benchmarks. However, HighRes3DNet has been shown to have better performance on tasks involving high-resolution and multi-modal medical images, while V-Net has been shown to be more efficient in terms of computational resources and memory usage. A variant of the 3D network is the hybrid 2D-3D methodology as proposed by Lee et al. (2015) for the segmentation of anisotropic volumes. They utilize only 2D convolutions in the initial layers that downsample the input feature maps with high x − y resolution (independent of the z-axis) until they are roughly isotropic to be efficiently processed by 3D convolutions.
Graph analysis is the third approach for 3D segmentation. Graph-based methods typically involve partitioning a graph into regions or clusters based on properties such as color or intensity values, edge strength, or other image features such as shape. These methods often use graph theory algorithms, like graph cuts or minimum spanning trees, to identify regions that are distinct from one another. This may be coupled with structure-based analysis that uses certain geometrical properties to identify boundaries between objects. Global shape descriptors were used to learn the connectivity of 3D super voxels by Lucchi et al. (2013) for segmentation using graph-cuts, addressing issues with local statistics and distracting membranes. Turaga et al. (2010) suggested how CNNs can be used for directly predicting 3D graph affinities based on a structured loss function for neuronal boundary segmentation. The proposed loss function assigned scores to the edges between adjacent pixels based on their likelihood of belonging to same or different regions and also penalized their assignment for achieving incorrect predictions that violate the underlying structure of the image.
Instance segmentation involves classifying each pixel/voxel of a given image/volume to a particular class along with assigning a unique identity to pixels/voxels of individual objects. Instance segmentation using deep learning can be divided into proposal-based (top-down) and proposal-free (bottom-up) approaches. Proposal-based approaches such as RCNN, FastRCNN, and FasterRCNN are two-stage detection networks that use a deep neural network for feature extraction (encoder) and region proposals for the segmentation of objects of interest, followed by bounding box regression and classification to obtain instance segmentation (Liu et al., 2020b). Mask-RCNN (He et al., 2017) is a popular choice for generic object instance segmentation built upon FasterRCNN, which uses a branch of the network to predict a binary mask for each object instance. Top-down instance segmentation has also been accomplished using recurrent networks with attention mechanisms, either by extracting visual characteristics and producing instance labels one item at a time or by guiding the formation of bounding boxes followed by a segmentation network (Ren and Zemel, 2017;Ghosh et al., 2019). The Flood Filling Network (FFN) uses this concept to obtain individual object masks directly from raw image pixels (Januszewski et al., 2018) and has also been used for EM segmentation as reviewed below.
The other approach is known as proposal-free, which aims to combine semantic and instance segmentation in a bottom-up approach. This was the strategy taken by Chen et al. (2017a), where the prediction of contours/edges of objects along with semantic masks were incorporated into FCNs in a multi-task learning approach. Both contour/edge maps and semantic masks were then fused to obtain the instance segmentation maps. Other approaches use boundary-aware instance information (e.g. the distance between object boundaries or the amount of overlap between objects) to fuse edge features with intermediate layers of the network (Bai and Urtasun, 2017; Oda et al., 2018).
Semantic instance segmentation is another family of techniques that addresses instance segmentation with semantic-based approaches. Instead of inferring for each pixel the probability of belonging to a certain class, they infer the probability of belonging to a certain instance of a class. In fact, De Brabandere et al. (2017) proposed a discriminative loss function in this regard and demonstrated that it is superior than the cross-entropy and Dice loss function for instance segmentation. The discriminative loss function consists of three terms: a segmentation term, which penalizes incorrect class predictions; a boundary term, which penalizes incorrect boundary predictions; and a regularization term, which encourages smoothness in the predicted masks.
Fully supervised methods
Fully supervised methods use annotated images (training data) to learn computational models that can segment structures in unseen images from similar distributions (test data). The training set is used by the algorithm to determine the model's parameters in such a way as to maximize the model's generalization ability. Table 3 summarizes the 33 papers (of the 38) that have used supervised learning for the semantic and instance segmentation of (sub) cellular structures.
End-to-end learning -semantic segmentation
End-to-end learning is a machine learning approach where a single model learns to perform a task without relying on pre-defined intermediate steps or features. Instead, the model is trained to map the input data directly to the desired output, in a single end-to-end process. End-to-end learning has become increasingly popular in recent years due to advances in deep learning, which allow the creation of models with large numbers of layers that can learn complex representations of data. These models are trained using backpropagation, a method for updating the weights of the model based on the error generated by a given loss function between the predicted output and the true output, which allows the model to improve its performance over time.
The 16 papers that fall within this category are focused on the semantic segmentation of two main cellular structures, namely NM -neuronal membranes (8 papers) and M -mitochondria (5 papers). Other structures include N -nuclei, NE -nuclear envelopes, and L -lysosome.
Neuronal membrane segmentation refers to the process of identifying and separating the neuronal membrane from other structures in an EM image. Segmenting neuronal membranes in EM volumes helps partition an image into distinct regions that represent different neuronal cells and processes. It is essential for studying the function of neurons along with their synaptic connections for understanding the different signaling pathways in the brain. Digital reconstruction or tracing of 3D neurons depends on the accuracy of neuronal membrane segmentation as discontinuities could lead to merge and split errors which in turn affect the reconstruction.
Similarly, mitochondria segmentation is the process of identifying and separating mitochondria, a type of organelle found in eukaryotic cells, from other structures in an EM image. Mitochondria segmentation is a challenging task due to the variability in their size, shape, and distribution within cells. Accurately segmenting mitochondria in 2D and 3D is important for studying the structure and function of these organelles, as well as investigating their role in various diseases.
Below we categorize the proposed approaches based on their underlying 2D or 3D CNN architectures. Table 3: The list of 33 (out of 38) papers reviewed in this work that are based on fully supervised learning frameworks with 2D and 3D CNN architectures applied to both semantic and instance segmentation. The abbreviation Org. stands for the studied organelle/s. The Type (2D and/or 3D) column indicates the type of methods used and problems addressed. The studies that are marked as both 2D and 3D use a 2D backbone method coupled with some post-processing operations for 3D reconstruction. The other studies that are flagged as 2D or 3D only, use 2D or 3D only backbones to address 2D or 3D problems, respectively. The numbers in the Datasets column serve as correspondences to the identifiers in Table 2, and the definitions of the performance metrics are presented in Section 7.
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Org.
Approaches based on 2D CNNs
Successes of DL networks for segmentation in EM were achieved using 2D architectures with deep contextual networks. Those networks typically had FCN or U-Net as backbones. Deeper contextual networks have generally produced better 2D segmentations that mostly allowed doing away with multi-step post-processing for obtaining 2D segmentations and 3D reconstructions.
Residual Deconvolutional Networks (RDN) by Fakhry et al. (2017) are based on a combination of residual connections, which allow for the efficient training of deep networks, and deconvolutional layers in the decoder of 2D U-Net, which help to recover spatial information lost during downsampling. The proposed method was evaluated on the ISBI 2012 (Drosophila VNC) and 2013 (SNEMI3D) benchmark datasets and compared to several state-of-the-art segmentation methods. The results demonstrated that RDNs were superior in terms of segmentation accuracy and required a simple post-processing step such as watershed to segment/reconstruct neural circuits. Oztel et al. (2017) proposed using a median filtering approach to incorporate 3D context for the reconstruction of mitochondria from the output of 2D segmentations. An FCN was used for delineating mitochondria from the background followed by median filtering along the z direction in the volume of images. This z-filtering allows the removal of spurious strokes and the recovery of regions of interest when sufficient adjacent slices contain the missed component.
The deep contextual residual network (DCR) by Xiao et al. (2018b) is an extension of FCN with residual blocks and multi-scale feature fusion. They used the summation based skip connections which fuse high-level details from output of deconvolutions in the decoder and low-level information from ResNet encoder. The proposed post-processing method with a multi-cut approach and 3D contextual features proved important to reduce discontinuities (boundary splits or merges), which in turn helped to reduce false positives and false negatives in various 2D sections. DCR outperformed several state-of-the-art segmentation methods on the ISBI 2012 dataset.
Advanced networks for different tasks may be too computationally demanding to run on affordable hardware, leading users to modify macro-level design aspects. Examples of such modifications include downsampling input images and reducing network size or depth to ensure compatibility with computer hardware constraints. Casser et al. (2018) introduced a fast mitochondria segmentation method using a reduced number of layers and lightweight bilinear upsampling instead of transposed convolutions in the decoder of U-Net. Moreover, they introduced a novel data augmentation method that generates training samples on the fly by randomly applying spatial transformations to the original images, which leads to increased training efficiency and robustness to variations in image quality. The authors also incorporate a post-processing step based on z-filtering to reconstruct 3D mitochondria. The proposed approach was evaluated on several EM datasets and achieved state-of-the-art performance in terms of segmentation accuracy and speed.
Data augmentation is a technique that is mostly used in machine learning and computer vision to increase the size and diversity of a training set. This is the case with most of the papers that are reviewed here. The process involves applying various transformations or modifications to the existing data in order to create new, but similar, instances of the data. It is particularly useful in cases where the size of the available dataset is limited, as it allows the model to learn from a larger and more diverse set of data without requiring additional data collection efforts. It can also help prevent overfitting and improve the robustness of the model by exposing it to a wider range of data variations. Moreover, test-time augmentation has also been proven effective to average out noise in predictions but at the cost of time complexity (Lee et al., 2017;Zeng et al., 2017;Xiao et al., 2018a;Yuan et al., 2021).
A residual encoder module with ASPP for multi-scale contextual feature integration was in-vestigated by Jiang et al. (2019). The decoder module included the fusion of previous low-level features and high-level features from the output of ASPP, followed by bi-linear upsampling to obtain the segmentation map. They achieved better performance compared to the baseline, U-Net, and Deeplab v3+ for the segmentation of cell bodies and cell nuclei.
The Dense-UNet model was proposed by Cao et al. (2020) as an extension of the popular U-Net architecture that incorporates densely connected blocks within the U-Net's skip connections. The densely connected blocks help to improve gradient flow and feature reuse, which leads to better feature representation and higher segmentation accuracy. Besides its outstanding results on the ISBI 2012 challenge, the model turned out to be highly robust to variations in noises and artifacts of neuronal membrane images, requiring no further post-processing.
FusionNet is a fully residual U-Net architecture that combines different levels of feature representations by fusing the output of multiple sub-networks with different receptive fields. It includes a residual learning framework along with deconvolutional layers to improve the training convergence and segmentation accuracy. The study by Quan et al. (2021) showed that an integrated multi-stage refinement process using four concatenated FusionNet units can effectively eliminate the requirement for any proofreading 5 A novel data augmentation strategy was also proposed by Spiers et al. (2021), which simulates realistic variations in the EM images to improve the robustness of their 2D CNN for the semantic segmentation of nuclear envelopes. The proposed approach based on 2D U-Net achieved high segmentation accuracy and can be used to extract meaningful biological information from the segmented nuclear envelope, such as the distribution of nuclear pores. Their model was run on each axis after transposing the stack, and the resulting three orthogonal predictions were merged to produce the ultimate segmentation.
Chen et al. (2017d) used a 2D CNN with only four layers for the segmentation of membranes, mitochondria, vesicles, and microtubules in cryo-ET. The architecture of the CNN layers was optimized to capture a large context by utilizing 15 × 15 pixel kernels in the first two layers. This design allowed for the use of a single max-pooling layer to downsample the output to half the input resolution, which aids in distinguishing intricate details of structures such as single (vesicle, microtubule) or double membrane (plasma membrane, mitochondria). A CNN for each of the four structures was trained with a few sections of the tomogram containing structures of interest. Automated segmentation was required for subsequent sub-tomogram classification and averaging for the determination of in-situ structures for the molecular components of interest.
Approaches based on 3D CNNs
Similar to 2D deep architectures, a 3D CNN consists of multiple layers of filters, including convolutional, pooling, and activation layers, to learn spatial features from the input data. The filters scan the input volume at different locations and orientations to identify features that are relevant for segmentation. The key difference between 2D and 3D CNNs is the inclusion of an additional depth dimension in the input data. This allows the network to capture the spatial and depth relationships between adjacent slices in the volume. Due to the large amount of data and computational resources required for training 3D CNNs, such methods are typically used in high-end computing environments, such as specialized workstations or cloud computing platforms. Hybrid 2D-3D architectures have also been investigated that try to find the right trade-off between high computational demand and effectiveness.
In this review, there are three approaches that adopted complete 3D CNN architectures in a fully supervised way. The first is the work by Cheng and Varshney (2017) who proposed a 3D CNN for the segmentation of mitochondria in volumetric data. The authors also propose a novel data augmentation technique that uses stochastic sampling in the pooling layers to generate realistic variations in the feature space. In their thorough investigation, they conclude that the 3D CNNs outperform their 2D counterparts with a high statistical significance. The improvement was mainly attributable to the introduced augmentations as well as to the factorized convolutions which also permitted high efficiency, which was also proven useful in FIB-SEM (isotropic) volumes. Mekuč et al. (2020) also presented a 3D CNN-based method for the segmentation of mitochondria and endolysosomes in volumetric EM. The proposed method is based on the HighRes3DNet architecture, but it has the filters in the first layer constrained to having zero mean, and called it HighRes3DZMNet. The zero mean layer made the neural network robust to changes in the brightness of the volume inputs. The network is trained using the UroCell dataset for jointly segmenting mitochondria and endolysosomes due to similar morphologies of these biological structures. The method was also applied to segment mitochondria in the Lucchi++ dataset to achieve state-of-theart segmentation results for FIB-SEM volumes. Heinrich et al. (2021) also relied on a 3D CNN for the segmentation of 35 organelle classes in cells from FIB-SEM volumes. The multi-channel 3D U-Net was trained on 28 volumes from the open-source OpenOrganelle collection covering four different cell types. They investigated how one segmentation model that is trained with samples of all 35 organelles compares with more specific models that are trained with subsets of semantically-related organelle classes, such as the endoplasmic reticulum (ER) and its associated structures, namely ER exit sites, ER membrane, and ER lumen. It turned out, that the single model that is trained by all classes outperforms the more specific ones. This is attributable to the richer diversity in the training set which resulted in a model with better generalization abilities.
Hybrid 2D-3D approaches were adopted for the segmentation of volume datasets in order to reduce the computational cost of 3D convolutions in certain layers and achieve better convergence. Their main application lies in the ability to segment anisotropic volumes for efficiently processing their 3D context. For instance, both anisotropic and isotropic EM volumes could be processed using hybrid 2D-3D network architectures that include 3 × 3 × 1 convolutions instead of 3 × 3 × 3 to modify them to 2D ones. Xiao et al. (2018a) was the first to introduce a fully residual hybrid 2D-3D network with deep supervision to improve mitochondria segmentation. For reducing the number of parameters, 3D convolutions were used only in the first and last layers of a 3D U-Net. A deeply supervised strategy was proposed by injecting auxiliary branches into the initial layers of the decoder for avoiding the vanishing gradients problem. The complexity of the network allowed it to use a simple connected component analysis method for 3D reconstruction across both anisotropic and isotropic volume EM datasets. Lee et al. (2017) adapted the hybrid 2D-3D model of Turaga et al. (2010) to predict 3D affinity maps for the segmentation of neuronal membranes in 3D volumes. The proposed CNN model incorporated multi-slice inputs along with long-range affinity-based auxiliary supervision in both the z-and x−y directions. They utilized a hybrid 2D-3D U-Net for segmenting anisotropic volumes and post-processing with a simple mean-affinity agglomeration strategy for segmenting neuronal regions. The proposed affinity supervision simulates the use of boundary maps with different thicknesses in the DeepEM3D (Section 5.3), outperforming it in the SNEMI3D competition.
A structured loss that favors high affinities between 3D voxels was used to obtain topologically correct segmentations by Funke et al. (2018). The affinity predictions were accurate enough to be used with a simple agglomeration to efficiently segment both isotropic and anisotropic (CREMI, FIB, and SegEM) data, outperforming methods with more elaborate post-processing pipelines. Bailoni et al. (2022) used signed graphs to anticipate both attractive and repulsive forces among 3D voxels, enabling graph prediction through a 3D U-Net, in a manner similar to the method proposed by Funke et al. (2018). Building on the concept of long-range affinities for boundary detection, Heinrich et al. (2018) used neighboring context to predict voxel-wise distance maps through regression loss instead of probabilities. Those distance predictions, when thresholded, generated precise binary segmentations for synapses. Such distance prediction maps with simple thresholding allowed scaling the prediction at high-throughput speeds (3 megavoxels per second) for a full adult fly brain volume of 50 teravoxels in size.
End-to-end learning -instance segmentation
End-to-end learning approaches are also the most popular ones for instance segmentation, which requires the delineation of each instance within the same class of structures. This is particularly important for classes of structures that tend to be apposed with each other, such as mitochondria.
CNN-based methods for instance segmentation were grouped into two categories by Wei et al. (2020): top-down and bottom-up. Top-down methods typically utilize region proposal networks followed by precise delineation in each region. Conversely, bottom-up approaches aim to predict a binary segmentation mask, an affinity map, or a binary mask with instance boundary followed by several post-processing steps to distinguish instances. Due to the undefined scale of bounding boxes in EM images, bottom-up approaches have been the preferred methodology for 2D and 3D instance segmentation.
The delineation of neuronal membrane does not require binary labels to distinguish one type of neuron from the other. There is no interesting semantics involved, unlike distinguishing a subcellular structure from other irrelevant structures or backgrounds followed by delineation to obtain individual instances. This type of segmentation is also referred to as image partitioning, as it divides the entire image into different neuronal parts based on its membranes. Such partitioning allows for the reconstruction of individual neuronal structures using post-processing. Figure 4 shows examples of semantic and instance segmentation of mitochondria along with an illustration of neuronal 3D reconstruction after image partitioning.
Approaches based on 2D CNNs
The only top-down approach from the reviewed works in this paper is the one proposed by Liu et al. (2020a). They introduced a pipeline that complements Mask-RCNN. In particular, they proposed a mechanism that refines undersegmented mitochondria in the output of Mask-RCNN, by iteratively enhancing the field of view that preserves the previous segmentation states. They systematically demonstrated that their approach outperformed competing methods that rely on U-Net, FFN, and Mask-RCNN in instance segmentation of mitochondria. Shape prior turned out to be important for some techniques to improve the quality of instance segmentation. Shape prior refers to the incorporation of prior knowledge about the expected shape or structure of an object of interest into segmentation algorithms. For example, Yuan et al. (2021) proposed the Hive-Net CNN, which was designed to overcome the challenges posed by the high variability in mitochondria shapes and sizes, as well as the presence of other cellular structures in the images. The network consists of multiple view-specific sub-networks that process different views of the image, and a centerline-aware hierarchical ensemble module that combines the outputs of the sub-networks to generate the final segmentation result. The centerline-aware module uses a new type of loss function that encourages the network to learn the morphology of mitochondria and to segment them along their centerlines. The proposed network was evaluated on two publicly available datasets, and an ablation study concluded that the centerline-aware module and the view-specific sub-networks were critical for achieving high segmentation accuracy.
Shape information has also been exploited by the hierarchical encoder-decoder network (HED-Net) for the instance segmentation of mitochondria . That strategy leveraged the shape information available in the manual labels to train the model more effectively. Instead of relying solely on the ground truth label maps for model training, an additional subcategory-aware supervision was introduced. That was achieved by decomposing each manual label map into two complementary label maps based on the ovality of the mitochondria. The resulting three-label maps were used to supervise the training of the HED-Net. The original label map was used to guide the network to segment all mitochondria of varying shapes, while the auxiliary label maps guided the network to segment subcategories of mitochondria with circular and elliptic shapes, respectively. The experiments conducted on two publicly available benchmarks show that the proposed HED-Net outperforms state-of-the-art methods.
The inclusion of apriori knowledge about shape in segmentation algorithms contributes to increased specificity as they become more selective in delineating the structures of interest and keep false positives to a minimum. They can also improve generalization ability especially when the training data is limited. Methods that use shape priors, however, are more structure-specific, and therefore different methods may need to be designed for the segmentation of distinct organelles.
Approaches based on 3D CNNs
The largest instance segmentation dataset for mitochondria (MitoEM) proposed by Wei et al. (2020) benchmarks the dataset by proposing a 3D U-Net. It is trained with binary masks and contours using two separate decoders, followed by a marker-controlled watershed to obtain instance segmentations, and is called U3D-BC +MW for short. Wei et al. (2020) introduced two networks, MitoEM-R and MitoEM-H, citing variations in sizes, shapes, and noise content for serial sections from rat and human samples. The MitoEM-R network can generalize on the human dataset as the rat samples have complex mitochondrial morphologies. The simpler U3D-BC +MW method was shown to be more effective than FFNs, as they were not able to capture the fine geometry of mitochondria with complex shapes or in close contact to each other.
The DeepACSON approach by Abdollahzadeh et al. (2021), which was proposed for the instance segmentation of axons and nuclei in 3D volumes, is supported by a postprocessing method that relies on shape features. To correct for topological errors, a cylindrical shape decomposition algorithm is used as a postprocessing step to identify any erroneously detected axons and to correct under-segmented ones at their cross-overs. The circularity of the cell nucleus is corrected using the level-set based geometric deformable model, which approximates the initial shape of the object with a curve. This is then adjusted to minimize an energy function associated with the curve when it fits perfectly to the object's boundaries. Energy functions enable the inclusion of shape information, whether it is a vague concept like smoothness constraints or a precise idea like shape constraints (strict adherence to a particular shape).
Nuclei instance segmentation on a large-scale EM dataset was proposed by Lin et al. (2021). Their network, U3D-BCD, was inspired by the U3D-BC above but involved the additional learning of a signed Euclidean distance map along with foreground masks and instance contours to capture the structure of the background for segmentation. To locate the seeds for object centers, their methods starts by thresholding the predictions to identify markers with high foreground probability and distance value, but low contour probability. Next, the marker-controlled watershed transform algorithm is applied with the predicted distance map and seeds to generate masks. This approach has two advantages over the U3D-BC model , which also utilizes marker-controlled watershed transform for decoding. Firstly, the consistency among the three representations is leveraged to locate the seeds, which is more robust than the U3D-BC method which relies on only two predictions. Secondly, it uses the smooth signed distance map in the watershed decoding process, which is more effective in capturing instance structure than the foreground probability map used in U3D-BC. Li et al. (2022) addressed 3D mitochondria instance segmentation with two supervised deep neural networks, namely ResUNet-H and ResU-Net-R, for the rat and human samples on the MitoEM dataset, respectively. Both networks produce outputs in the form of a semantic and instance boundary masks. Due to the increased difficulty of the human sample, Res-UNet-H has an additional decoder path to separately predict the semantic mask and instance boundary, while Res-UNet-R has only one path. Once the semantic mask and instance boundary are obtained, a seed map is synthesized, and the mitochondria instances are obtained using connected component labeling. To enhance the networks' segmentation performance, a simple but effective anisotropic convolution block is designed, and a multi-scale training strategy is deployed. The MitoEM dataset has sparsely distributed imaging noise, with the human sample having a stronger subjective noise level than the rat sample. To reduce the influence of noise on segmentation, an interpolation network was utilized to restore the regions with noise, which were coarsely marked by humans. Besides mitochondria instance segmentation, the proposed method was demonstrated to have superior performance for mitochondria semantic segmentation. Mekuč et al. (2022) extended their previous approach based on the HighRes3DZMNet by postprocessing steps with active contours to separate apposing mitochondria and thus achieving instance segmentation. By means of experiments on the extended UroCell dataset they demonstrated that this new approach is more effective than the U3D-BC +MW method.
Ensemble learning
Ensemble learning methods combine outputs of multiple algorithms or models to obtain better predictive performance in terms of accuracy and generalization. Pixel-or voxel-wise averaging and the majority or median voting are among the main aggregation methods.
An ensemble technique was in fact investigated by Zeng et al. (2017) for the segmentation of neuronal membranes in the brain volumes. They trained several variations of their DeepEM3D network, which could process different numbers of input slices and inputs with varying thicknesses of object boundaries. The DeepEM3D network extended the FCN architecture by introducing a hybrid network with 3D convolutions in the first two layers to enable integrating anisotropic information in the early stages, and 2D layers afterwards. DeepEM3D employed inception and residual modules, multiple dilated convolutions, and combined the result of three models that integrated one, three, and five consecutive serial sections. Employing an ensemble strategy for enhancing boundaries (by maximum superposition) within the probability maps generated by these models proved essential for performing with near-human accuracy in the SNEMI3D challenge.
CDeep3M is a cloud implementation of DeepEM3D to segment various anisotropic SBF-SEM and cryo-ET datasets (Haberl et al., 2018b). Trained by a few sub-volumes of the cryo-ET tomogram, the resulting network was able to segment vesicles and membranes with high accuracy in other tomograms. The network implementation proved efficient for segmenting large-volume EM datasets such as SBF-SEM making it easier to analyze enormous amounts of imaging data.
The strengths of the ensemble paradigm was also confirmed by Guay et al. (2021) for the segmentation of cytoplasm, mitochondria, and four types of granules in platelet cells. They demonstrated that the best segmentation performance (in terms of intersection over union) was achieved by combining the output of the top k performing weak classifiers, with each such classifier learned by a small portion of the training data. Similar to above, each model was a hybrid 2D-3D network used to segment anisotropic SBF-SEM volumes. They also highlighted that besides its effectiveness, their ensemble paradigm ensured better reproducibility of the results in comparison to individual models that were sensitive to initialization.
Multiple network outputs were also combined with a workflow for binary EM segmentation provided by the EM-stellar platform (Khadangi et al., 2021b). Unlike the above two approaches, Khadangi et al. (2021b) used the ensemble paradigm to aggregate the output of different types of networks, namely CDeep3EM (Haberl et al., 2018a), EM-Net (Khadangi et al., 2021a), PReLU-Net (He et al., 2015), ResNet, SegNet, U-Net, and VGG-16. A cross-evaluation using a heatmap of different evaluation metrics revealed that no single deep architecture performs consistently well across all segmentation metrics. This is why ensemble approaches have an edge over individual methods as they leverage the strengths of each underlying model as was demonstrated in the evaluation of two different datasets for mitochondria segmentation in cardiac and brain tissue.
Transfer learning
Transfer learning is a framework that adapts the knowledge acquired from one dataset to another, and is generally used when an application has an insufficient amount of training samples. A pre-trained model is fine-tuned, usually in the final layers, with the training samples of a new dataset. This technique was used by Mekuč et al. (2020) for the segmentation of mitochondria and endolysosomes from the background in EM images. Since mitochondria and endolysosomes share similar texture and mitochondria are more in abundance a binary segmentation model was first learned to segment mitochondria from the background. Subsequently, transfer learning was used to adapt the learned model for the segmentation of endolysosomes too. This was achieved by freezing all layers of the network except for the last one, which was fine-tuned by a smaller training set that included examples of endolysosomes. This approach is a demonstration how transfer learning can be used when the availability of a certain structure is limited.
Fine-tuning a pre-trained network comes with the risk of overfitting to the few labeled training examples of the new dataset or application. This challenge has opened up new research avenues, namely few-shot learning and domain adaptation. The former can be a meta-learning approach that "learns to learn" from a given pre-trained model when conditioned on a few training examples (referred to as the support set) to perform well on new queries passed through a fixed feature extractor (Shaban et al., 2017).
Few-shot learning was the focus of the work by Dietlmeier et al. (2019), who proposed a few-shot hypercolumn-based approach for mitochondria segmentation in cardiac and outer hair cells. The idea behind hypercolumn feature extraction was to extract features from different levels of a pretrained CNN and combine them to form a single, high-dimensional feature representation for each pixel. The VGG-16 model pre-trained on the ImageNet dataset was used to extract hypercolumns, which were then passed through a linear regressor for actively selecting features. Only 20 labeled patches (2 %-98% train-test split) were used from a FIB-SEM stack for training a gradient-based boosting classifier (XGBoost). They showed how high segmentation accuracy on the Drosophila VNC dataset could be achieved by actively selecting features and learning using far less training data and even by using a single training sample (single-shot).
Domain adaptation is another form of transfer learning, where the source to target datasets share the same labels (classes) but have a different data distribution. Changes in data distribution can be due to slightly different experimental parameters during EM imaging or due to the imaging of different tissue types or body locations. Bermúdez-Chacón et al. (2018) proposed the two-stream U-Net architecture, where the weights are related, yet different for each of the two domains, for supervised training on a few target labels. Only 10% of labeled target data was required for domain adaptation to achieve state-of-the-art performance when compared to a U-Net trained on a fully annotated dataset.
Configurability and Reproducability
A key challenge in designing CNNs is the determination of the right architecture for the problem at hand. This has motivated research effort in what are known self-configurable networks that can automatically determine certain design choices. A self-configurable network is thus a type of artificial neural network that is capable of dynamically adapting its structure and parameters based on the input data and task concerned. This concept was used by Isensee et al. (2019), who proposed the no-new-Net (nnU-Net) framework that consists of a 2D U-Net, 3D U-Net and a cascade of two 3D U-Nets. Self-configuration based on cross-validation was used to automatically determine some hyperparameters, such as the patch size, batch size and number of pooling operations. While it was shown to be very effective in various semantic segmentation problems in medical image benchmark datasets, its generalization ability in EM datasets has yet to be evaluated comprehensively.
An experimental study by Franco-Barranco et al. (2022) uncovered substantial reproducibility issues of different networks proposed for mitochondria segmentation in EM data. Additionally, it distinguished the impact of innovative architectures from that of training choices (such as pre-processing, data augmentation, output reconstruction, and post-processing strategies) by conducting multiple executions of the same configurations. Their systematic analysis enabled the identification of stable and lightweight models that consistently deliver state-of-the-art performance on publicly available datasets.
Semi-, un-and self-supervised methods
Semi-supervised and unsupervised learning are two types of machine learning methods, whose main difference between them is the amount of labeled data they use to train the model.
Unsupervised learning is a type of machine learning that deals with finding patterns and relationships in unlabeled training data. In this case, the algorithm learns to identify patterns and Table 4: The list of 5 (out of 38) papers reviewed in this work and that are based on semi-, un-and self-supervised learning frameworks along with co-relative light and electron microscopy (CLEM) as discussed in Section 4.2). The abbreviation Org. stands for the studied organelle/s. The Type (2D and/or 3D) column indicates the type of methods used and problems addressed. The studies that are marked as both 2D and 3D use a 2D backbone method coupled with some post-processing operations for 3D reconstruction. The other studies that are flagged as 2D or 3D only, use 2D or 3D only backbones to address 2D or 3D problems, respectively. The numbers in the Datasets column serve as correspondences to the identifiers in Table 2, and the definitions of the performance metrics are presented in Section 7.
Citation
Org JI 3D U-Net Self-supervised learning, fine-tuning relationships in the data by clustering or grouping similar data points together. Semi-supervised learning, on the other hand, is a combination of supervised and unsupervised learning. It uses both labeled and unlabeled data to train the model. The labeled data is used to train the model on specific tasks, while the unlabeled data is used to help the algorithm learn patterns and relationships in the data (Zhu and Goldberg, 2009). In self-supervised learning, a model is trained on a dataset with labels that are automatically generated from the data itself. The goal is to learn useful representations of the data that can be used for downstream tasks, such as segmentation.
A common strategy for semi-supervised learning is to use label propagation through self-training. The process begins by training a classifier on labeled samples and then classifying the unlabeled samples. A selection of these samples based on an active selection strategy or learned classifier is then added to the training set and the process is repeated multiple times (Cheplygina et al., 2019). This can be performed either inductively or transductively. The former refers to training a model on unseen targets to add new information to the previously trained model so that it can generalize on new unseen data, and the latter to training a model based on a select subset of labeled and unlabeled data to be able to predict correctly on a limited set of seen targets.
A semi-supervised approach was proposed by Takaya et al. (2021) for the segmentation of neuronal membranes. They called their approach 4S that stands for sequential semi-supervised segmentation. It was based on the fact that adjacent images in a volume are strongly correlated. The goal of their method is to have a model that can only generalize to the next few slices instead of to the whole volume. This was achieved by starting with a few labeled slices that are used to train the first model. Then, in an iterative approach the model was used to infer the segmentation maps of a small set of subsequent images and the resulting segmentation maps were used as pseudolabels to retrain the model. Label propagation from labeled to the available unlabeled data was performed by predicting pseudo labels on the subsequent sections which represent the same targets and whose predictions could be included in the next round of model training as ground truth labels. It allowed the training to weigh the most recent inputs heavily unlike transfer learning where the goal is to generalize well on all use cases of the unlabeled dataset.
Another semi-supervised method was introduced by Wolny et al. (2022) for the segmentation of mitochondria and neuronal membranes. In contrast to the above, their goal was to train a model with a few manually annotated images, which can generalize for the whole dataset. In particular, they used a training set with a combination of positive labeled data and unlabeled data of positive and negative instances. As there is no direct supervision on the unlabeled part of the image, an embedding consistency term was introduced by training two networks on different data-augmented versions of each pixel. This was coupled with a push-pull loss function that they proposed to enforce constraints between different instances. It was realized by using anchor projections in the embedding space of a point in each instance to derive a soft label based on the set of surrounding pixels in the projected space. The instance segmentation was then achieved by grouping the pixel embeddings. This semi-supervised method is notable for a good tradeoff between segmentation performance and effort in manual annotation.
Unsupervised learning was explored by Bermúdez-Chacón et al. (2019), who investigated the unsupervised domain adaptation strategy for mitochondria segmentation to demonstrate how a model trained on one brain structure (source: mouse striatum) could be adapted to another brain structure (target: mouse hippocampus). Labeled data was only available to train the model on the source dataset (striatum). Visual correspondences were then used to determine pivot locations in the target dataset to characterize regions of mitochondria or synapses. These locations were then aggregated through a voting scheme to construct a consensus heatmap, which guided their model adaptation in two ways: a) optimizing model parameters to ensure agreement between predictions and their sets of correspondences, or b) incorporating high-scoring regions of the heatmap as soft labels in other domain adaptation pipelines. These unsupervised techniques yielded high-quality segmentations on unannotated volumes for mitochondria and synapses, consistent with results obtained under full supervision, without the need for new annotation effort.
In the case of severe domain shifts such as from a FIB-SEM to an ssSEM dataset as investigated by Peng et al. (2020), adversarial learning may be used for domain adaptation in different tissues of various species. Adversarial learning is a machine learning paradigm that trains a model with an adversarial loss function that encourages the model to learn domain-invariant features. Peng et al. (2020) combined the geometrical cues from annotated labels with visual cues latent in images of both the source and target domains using adversarial domain adaptive multi-task learning. Instead of manually-defined shape priors, they learned geometrical cues from the source domain through adversarial learning, while jointly learning domain-invariant and discriminative features. By doing so, the model learned features that were useful for both source and target domains, and could perform well on the target domain despite having only labeled data in the source domain. The method was evaluated extensively on three benchmarks under various settings through ablations, parameter analysis, and comparisons, demonstrating its superior performance in segmentation accuracy and visual quality compared to state-of-the-art methods.
Contrastive learning is a self-supervised paradigm where a model is trained to learn useful representations of input data by contrasting similar and dissimilar samples. The basic idea is to take a set of positive pairs (e.g., two different augmentations of the same image) and a set of negative pairs (e.g., two images containing different types of objects), and train the model to assign higher similarity scores to positive pairs and lower similarity scores to negative pairs. This results in a model that captures the underlying structure of the data and can be used for downstream tasks like classification, object detection, and semantic segmentation. Conrad and Narayan (2021) used contrastive learning, specifically moment contrast, (He et al., 2020), to learn useful feature representation from the unlabeled CEM500K dataset followed by transfer learning on given datasets. The heterogeneity of CEM500k coupled with the unsupervised initialization of a segmentation model contributed to achieving state-of-the-art results on six benchmark datasets that concern different types of organelles.
Segmentation evaluation metrics
Segmentation methods are evaluated by measuring the extent of overlap between the ground truth (GT) and prediction (PR) segmentation maps.
For semantic segmentation, all GT connected components are considered as one object, and similarly all PR connected components are treated as one object. This reduces the problem to binary classification. Typical performance measures include Accuracy, Precision and Recall and their harmonic mean (also called F1-score or Dice similarity coefficient (DSC)), the Jaccard Index (JI), also known as the Intersection over Union (IoU), and the Conformity coefficient Chang et al. (2009), Fig. 5.
Accuracy = (T P + T N )/(T P + F P + F N + T N ) P recision (P ) = T P/(T P + F P ) Recall (R) = T P/(T P + F N ) F 1 (or DSC) = 2P R/(P + R) JI (or IoU ) = T P/(T P + F P + F N ) Conf ormity = 1 − (F N + F P )/T P(1)
where TP, FP, FN, and TN are the number of true positives, false positives, false negatives, and true negatives at pixel level. The Accuracy measure is a ratio of all correctly classified pixels to all pixels, Precision is the ratio of all true positive pixels to the number of positive predictions made by the algorithm, and Recall (Sensitivity or True Positive Rate) is the ratio of all true positive pixels to the number of all positive pixels in the ground truth. The JI (or IoU) and DSC measure the similarity between the predicted class labels and the true class labels, while the Conformity coefficient measures the ratio of the number of misclassified pixels to the number of true positive pixels subtracted from 1. A negative Conformity value indicates that the number of misclassified pixels is higher than the true positive ones, and vice-versa. Each of these metrics has its own strengths and weaknesses, and the choice of metric depends on the specific requirements of the classification task and the goals of the analysis. For example, accuracy is a simple and a good global measure but it is only suitable when the class distribution is balanced.
For the segmentation of partitions, such as neuronal structures, the preservation of the topology is more important than the pixel-based accuracy. For instance, a prediction that oversegments (e.g. splitting the delineation of a neuron in two or more partitions) should be penalized more than a prediction of displaced, shrunk or expanded segments. Metrics such as the Rand Index (RI), Warping error (WE), and variation of information (VOI) take into account the topological errors in neuronal membrane segmentation. RI measures the similarity between the PR and GT segmentation maps, by calculating the sum of pairs of pixels that are both in the same object and both in different objects out of the total combination of pixel pairs in both GT and PR maps (Unnikrishnan et al., 2007;Arbelaez et al., 2010). The complement of the RI (i.e. 1 -RI) is known as the Rand Error (RE). The adapted rand error (ARAND) was the evaluation metric used in the SNEMI 3D challenge and is given as 1 -the maximal F -score of the RI. The maximal F -score is achieved when the precision and recall are at their optimal trade-off. The RI provides a score ranging from 0 to 1, with a value of 1 indicating perfect matching between two objects. Other metrics that were part of the ISBI 2012 challenge are the WE and pixel error (PE). The WE is a segmentation metric that penalizes topological disagreements, i.e: the number of splits and mergers required to obtain the desired segmentation. On the other hand, the PE is defined as the ratio of the number of pixel locations at which the GT and PR labelings disagree. While expanding, shrinking, or translating a boundary between two neurons does not affect the WE, they incur a large PE.
The variation of information (VOI) quantifies the distance between PR and GT objects by measuring the amount of information that is lost or gained when one segmentation is transformed into the other (Arbelaez et al., 2010). The VOI between the GT and PR components is the sum of two conditional entropies: the first one, H(P R|GT ), is a measure of over-segmentation, the second one, H(GT |P R), a measure of under-segmentation. These measures are referred to as the VOI split or merge error, respectively. The VOI and ARAND were also combined to form the CREMI score by first taking the sum of the VOI split and VOI merge and then combining the result with ARAND using geometric mean.
Evaluations of segmentation quality were most accurately reflected by the normalized versions of the RI and VOI, which are denoted by V rand and V inf o , respectively (Arganda-Carreras et al., 2015). Given a pair of GT and PR segmentation maps, V rand provides a weighted harmonic mean of the split and merge errors, where the split error is the probability of two selected pixels belonging to the same segment in PR given that they belong to the same segment in GT, and vice versa for the merge error. In pixel pair classification, the split and merge scores can be seen as representing precision and recall, respectively, for identifying whether the pixels belong to the same object (true positives) or different objects (true negatives). When the split and merge errors are weighed equally it is known as the Rand F -score. Similarly, V inf o is given by the mutual harmonic mean of the information-theoretic split and merge scores, which defines how much information in PR is provided by GT and vice versa.
For the evaluation of object detection, where different connected components are treated as different objects, the above measures are also applicable. The main difference is the way a true positive is considered. In object detection a PR region is considered a TP if it overlaps with more than a given threshold (e.g. 50%) a GT component in terms of IoU, otherwise it is a FP. The unmatched GT components are then considered as FNs. A popular metric in object detection is average precision (AP), which is essentially the area under the precision-recall curve that is determined by systematically changing the detection threshold. The default AP measure uses a 50% IoU overlap threshold, but other variations of the AP can be used depending on how strict the evaluation must be. The term AP-α denotes the average precision at a given IoU threshold α. The higher the α the stricter the evaluation is. In problems with more than two classes, the mean AP (mAP) can be used to aggregate all the APs of all the classes involved by taking their average.
Instance segmentation requires more detailed measures to quantify the segmentation mask accuracy along with the detection performance. Metrics such as the aggregated Jaccard index (AJI) and the Panoptic Quality (PQ), which were originally proposed by Kumar et al. (2017) and Kirillov et al. (2019), respectively, have also been used in EM Yuan et al., 2021) to evaluate instance segmentation algorithms more comprehensively. See Fig. 5 for an example. AJI = N j=1 |GT j ∩ P R j * | N j=1 |GT j ∪ P R j * | + i∈F P |P R i | P Q = j∈T P JI(GT j , P R j * ) |T P | × |T P | |T P | + 1 2 |F P | + 1 2 |F N |
where N is the number of GT regions, and j * is the index of the connected region in P R that is matched with the largest overlapping region (in terms of JI) with ground truth segment GT j ; FP is the set of false positive segments in P R without the corresponding ground truth regions in GT , FN is the set of false negative segments in GT that have been left unmatched with any regions in P R and TP is the set of all matched regions in GT and P R with at least 50% overlap in JI. The symbol |.| indicates the cardinality of a given set. A GT component can only be used once to match with a PR component. In case there are multiple PR components overlapping the same GT component, the GT component will only be matched with the PR component having the largest IoU. The AJI is an object-level performance metric which measures the ability of a segmentation algorithm to accurately identify and delineate individual objects within an image. It takes into account both the segmentation quality and the accuracy of object identification. For problems where many GT regions are apposing or in very close proximity with each other (e.g. mitochondria in 2D EM), there is a high risk that one PR region overlaps multiple GT regions. Such cases are overpenalised by the AJI measure. Overpenalization is prevented to happen with PQ because the matching of PR and GT regions are only valid when they overlap with more than 50% in JI.
Discussion and open challenges
Convolutional neural networks have become the standard choice for automatic feature extraction and segmentation in EM data. The most notable backbone networks are FCN and U-Net. Their use of deeper contextual network architectures is essential for accurate 2D prediction and by extension for 3D reconstruction. To produce dense predictions, early methods used a stack of successive convolutions followed by spatial pooling. Consecutive methods upsample high-level feature maps and combine them with low-level feature maps to restore crisp object boundaries and global information during decoding. To extend the receptive field of convolutions in the initial layers, numerous techniques have advocated the use of dilated or atrous convolutions. Recent works use spatial pyramid pooling to gather multi-scale contextual information in order to acquire global information in upper levels. More specialized architectures came into prominence to solve certain problems of anisotropy using hybrid 2D-3D networks and have now become the de facto for anisotropic EM datasets. The extension of 3D networks for graph-based affinity labeling proved useful as they can efficiently model structures across volume stacks to avoid several postprocessing steps for 3D reconstruction.
The advent of state-of-the-art deep neural networks led to a saturation of segmentation performance on small datasets, such as the ones from Ciresan et al. (2012) and Lucchi et al. (2011). Despite the focus in many studies on improving network architectures, a lot of the differences in performance can in fact be attributed to variations in preprocessing, data augmentation, and postprocessing (Isensee et al., 2019;Franco-Barranco et al., 2022). It is therefore likely that a focus on these areas in the near future will lead to new milestones in EM segmentation performance. Many of these developments, whether in network design or in postproceesing, have been in fully supervised segmentation, a technique that is greatly limited by the availability and quality of annotated data. While the number of large-scale annotated EM datasets has dramatically risen in recent years, the published datasets are not always precisely annotated, and are often composed of crude masks built semi-automatically using pre-trained networks and proofreading.
This limiting scarcity of annotation in EM is due to the complexity of the produced images, their large scale, and the ways in which the annotations are obtained. Manual annotations can either be performed by one or a few domain experts, or they could be performed collaboratively by large groups in a crowd-sourced manner. Expert annotations are more accurate and time consuming. For instance, the large-scale connectomics project required extensive labeling and proofreading (Plaza and Funke, 2018). Crowd-sourced approaches on the other hand require additional organizational efforts, specialized software, and instruction of the participants (Spiers et al., 2021).
In addition to manual annotation, EM images can be labeled using specialized imaging modalities that target specific structures in the sample. For instance, CLEM (Correlative light electron microscopy) is used to label structures targeted with fluorescent probes at (sub)cellular scales (de Boer et al., 2015;Drawitsch et al., 2018;Heinrich et al., 2021). Other EM modalities that could assist in annotation include energy dispersive X-ray spectroscopy (EDX), electron energy loss spectroscopy (EELS), cathodoluminescence (CL), and secondary ion mass spectroscopy at the nanoscale (NanoSIMS) (Pirozzi et al., 2018). These methods reduce the bias in human annotation, but may require longer sample preparation, specialized equipment, or additional image processing to produce segmentations.
The scarcity of annotated EM data could also be addressed algorithmically by relying on semisupervised and unsupervised learning techniques. These techniques are able to segment EM images with minimal or no annotations, that can scale to larger datasets with varied structures. Few-shot learning for segmentation has shown promising results in natural images (Dong and Xing, 2018;Tao et al., 2020), and the use of transformers in segmentation could prove useful for large-scale EM data in the future (Dosovitskiy et al., 2021;Zheng et al., 2021).
These prospects of label-free segmentation highlight the importance of collecting unlabeled yet relevant segmentation datasets, like the curation of unlabeled heterogeneous mitochondria images in CEM500K. Such datasets have shown how unlabeled pre-training using self-supervision could pave the way for breakthroughs in EM segmentation.
Complex datasets with challenges such as MitoEM, NucMM, or even unlabeled datasets such as CEM500K led to deep generalist models rather than specialized networks but still have a long way to go as sub-cellular image segmentation using large-scale EM is yet to explore both challenges in biological research and computer vision.
Acknowledgement
Bai, M., Urtasun, R., 2017. Deep watershed transform for instance segmentation, in: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5221-5229.
Bailoni, A., Pape, C., Hütsch, N., Wolf, S., Beier, T., Kreshuk, A., Hamprecht, F.A., 2022. Gasp, a generalized framework for agglomerative clustering of signed graphs and its application to instance segmentation, in: de Boer, P., Giepmans, B.N., 2021. State-of-the-art microscopy to understand islets of langerhans: what to expect next? Immunology and Cell Biology 99, 509-520.
Figure 1 :
1Large-scale EM ('nanotomy') of a section of human pancreas. Overview of a single large-scale EM
A
-Axons, CC -Canalicular channel, CH -Chromatin, CN -Centrosome, CP -Cytoplasm, D -Dendrites, EN -Endoplasmic Reticulum, ERES -Endoplasmic Reticulum Exit Site, G -Golgi, GC -Glial cells, GN -Granules, L -Lysosome, LD -Lipid Droplet, M -Mitochondria, MT -Microtubule, My -Myelin. N -Nucleus, NE -Nuclear Envelope, NM -Neuronal membrane, NP -Nuclear Pore, Nu -Nucleolus, PM -Plasma Membrane, R -Ribosome, S -Synapse, SP -Synaptic partners, V -Vesicle.
Figure 3 :
3Encoder-decoder networks of the original works on FCN
Figure 4 :
4Example of (top row) semantic and instance segmentation of mitochondria and (bottom row) neuronal membrane segmentation followed by 3D reconstruction of neuronal objects from a volumetric EM image. (a) Raw EM 2D section extracted from a FIB-SEM volume of a mouse kidney from the OpenOrganelle jrc mus-kidney dataset 6 . (b, c) Ground truth labels for semantic and instance segmentation. The instance segmentation map identifies each individual mitochondria with a unique color. (d) Raw EM 2D section extracted from the SNEMI3D (#3) dataset for the task of neuronal membrane segmentation and reconstruction. (e) The ground truth map of the neuronal membrane segmentation, which is used to partition the image completely. (f ) 3D reconstruction of selected neuronal structures that pass through the given 2D section from adjacent sections of the EM volume. The information from multiple images is used to create a 3D reconstruction through various post-processing methods, such as clustering, watershed, or graph-based methods.
Figure 5 :
5Common performance metrics for segmentation methods. For semantic segmentation, the overall overlap of the ground truth (GT) mask with the prediction (PR) is compared without differentiating between objects of the foreground class. As to instance segmentation, each GT component is matched with only one PR component, the one with which it has the largest intersection. In the above example, the GT component 'c' overlaps with two PR components, 'A' and 'B', but is matched only with 'B' due to a larger overlap. The Aggregated Jaccard Index (AJI) is the ratio of the sum of all intersections of the matched pairs of GT and PR components to the sum of the unions of such pairs plus the sum of all pixels in the unmatched GT and PR components. The Panoptic Quality (PQ) captures both semantic and instance segmentation performance. The former is the sum of all IoUs between the matched GT and PR components divided by the number of matched components (TPs), and the latter is the number of TPs divided by the number of TPs plus half of the FPs and FNs together. The symbol |.| indicates the cardinality of the set concerned.
Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11645-11655. Belevich, I., Jokitalo, E., 2021. Deepmib: user-friendly and open-source software for training of deep learning network for biological image segmentation. PLoS computational biology 17, e1008374. Berg, S., Kutra, D., Kroeger, T., Straehle, C.N., Kausler, B.X., Haubold, C., Schiegg, M., Ales, J., Beier, T., Rudy, M., et al., 2019. Ilastik: interactive machine learning for (bio) image analysis. Nature Methods 16, 1226-1232. Bermúdez-Chacón, R., Altingövde, O., Becker, C., Salzmann, M., Fua, P., 2019. Visual correspondences for unsupervised domain adaptation on electron microscopy images. IEEE transactions on medical imaging 39, 1256-1267. Bermúdez-Chacón, R., Márquez-Neila, P., Salzmann, M., Fua, P., 2018. A domain-adaptive twostream U-Net for electron microscopy image segmentation, in: 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), IEEE. pp. 400-404. Berning, M., Boergens, K.M., Helmstaedter, M., 2015. Segem: efficient image analysis for highresolution connectomics. Neuron 87, 1193-1206.
Table 2 :
2Key datasets from studies that perform high-resolution automated (volume) EM segmentation using deep learning. The abbreviations of the (sub) cellular structures are defined in the legend.# Dataset
Acquisition Region
Pixel/Voxel
size (nm)
Pixels
Labeled (sub) cellular
structures
Public repository
1 ISBI 2012/
Drosophila
VNC
ssTEM
Nervous cord
(Drosophila)
4 × 4 × 50
512 × 512 × 30
NM
https://imagej.net/events/
isbi-2012-segmentation-challenge
2 MICCAI 2016/
CREMI3D
ssTEM
Adult fly brain
(Drosophila)
4 × 4 × 40
1250 × 1250 × 125
NM, S, SP
https://cremi.org
3 ISBI 2013/
SNEMI3D /
Kasthuri
ssSEM
Neocortex
(Mouse)
3 × 3 × 30
1024 × 1024 × 100
NM
https:
//snemi3d.grand-challenge.org/
4 Kasthuri++
ssSEM
Neocortex
(Mouse)
3 × 3 × 30
1643 × 1613 × 85
M, NM
https://casser.io/connectomics
5 Xiao
ssSEM
Cortex (Rat)
2 × 2 × 50
8624 × 8416 × 20
M
http://95.163.198.142/MiRA/
mitochondria31/
6 MitoEM
ssSEM
Cortex(Human,
rat)
8 × 8 × 30
4096 × 4096 × 1000
M
https:
//mitoem.grand-challenge.org/
7 NucMM
ssSEM
Whole brain
(Zebrafish)
4 × 4 × 30
1450 × 2000 × 397
N
https:
//nucmm.grand-challenge.org/
8 Lucchi / EPFL
Hippocampus
FIB-SEM
Hippocampus
(Mouse)
5 × 5 × 5
1024 × 768 × 165
M
https://www.epfl.ch/labs/
cvlab/data/data-em/
9 Lucchi++
FIB-SEM
Hippocampus
(Mouse)
5 × 5 × 5
1024 × 768 × 165
M
https://casser.io/connectomics
10 FIB-25
FIB-SEM
Optic lobe
(Drosophila)
8 × 8 × 8
520 × 520 × 520
N, S
http://research.janelia.org/
FIB-25/FIB-25.tar.bz2
11 OpenOrganelle
FIB-SEM
Interphase HeLa,
Macrophage,
T-cells
8 × 8 × 8
Varying sizes
CN, CH, EN, ER, ERN,
ERES, G, LP, L, MT,
NE, NP, Nu, N, PM, R, V
https:
//openorganelle.janelia.org
12 Cardiac
mitochondria
FIB-SEM
Heart muscle
(Mouse)
15 × 15 × 15
1728 × 2022 × 100
M
http://labalaban.nhlbi.nih.gov/
files/SuppDataset.tif
13 UroCell
FIB-SEM
Urothelial cells
(Mouse)
16 × 16 × 15
5 subvolumes of
256 × 256 × 256
G, L, M, V
https://github.com/
MancaZerovnikMekuc/UroCell
14 Perez
SBF-SEM
Brain (Mouse)
7.8 × 7.8 × 30
16000 × 12000 × 1283
L, M, Nu, N
https://www.sci.utah.edu/
releases/chm v2.1.367/
15 SegEM
SBF-SEM
Mouse cortex
11 × 11 × 26
279 volumes of
100 × 100 × 100
NM
https://segem.rzg.mpg.de/
webdav/SegEM challenge/
16 Guay
SBF-SEM
Platelets (Human)
10 × 10 × 50
800 × 800 × 50
Cell, CC, CP, GN, M
https://leapmanlab.github.io/
dense-cell/
17 Axon
SBF-SEM
White matter
(Mouse)
50 × 50 × 50
1000 × 1000 × 3250
A, M, My, N
http://segem.brain.mpg.de/
challenge/
18 CDeep3M-S
SBF-SEM
Brain (Mouse)
2.4 × 2.4 × 24
16000 × 10000 × 400
M, NM, Nu, V
https:
//github.com/CRBS/cdeep3m
19 EMPIAR-10094
SBF-SEM
HeLa cells
10 × 10 × 50
8192 × 8192 × 517
Unlabeled
http://dx.doi.org/10.6019/
EMPIAR-10094
20 CEM500K
All of the
above
20 regions (10
organisms)
2 × 2 × 2 to
20 × 20 × 20
224 × 224 × 496544
Unlabeled
https://www.ebi.ac.uk/empiar/
EMPIAR-10592/
21 CDeep3M-C
Cryo-ET
Brain (Mouse)
1.6×1.6×1.6
938 × 938 × 938
NM, V
https:
//github.com/CRBS/cdeep3m
22 Cellular
Cryo-ET
Cryo-ET
PC12 cells
2.8×2.8×2.8
938 × 938 × 938
L, M, PM, V
https://www.ebi.ac.uk/emdb/
EMD-8594
Private indicates that the dataset used is not publicly available.Type Datasets
Performance
Backbone
Main methodological components
2D 3D
metrics
End-to-end learning -semantic segmentation
Fakhry et al. (2017)
NM
1, 3
RE, WE, PE
2D U-Net
Residual blocks, deconvolutions
Oztel et al. (2017)
M
1
Acc, P, R, F1, JI
2D FCN
Block processing, Z-filtering
Chen et al. (2017d)
MT, M, PM, V
22
No evaluation
2D FCN
A CNN architecture with four layers
Xiao et al. (2018b)
NM
1
V rand , V inf o
2D FCN
Residual blocks, multi-level features
Casser et al. (2018)
M
4, 9
Acc, P, R, JI
2D U-Net
Few parameters, light-weight model
Jiang et al. (2019)
N
Private *
JI, Acc
2D FCN
Residual, atrous, multi-level fusion
Cao et al. (2020)
NM
1
V rand
2D U-Net
Dense blocks, summation-skip
Quan et al. (2021)
NM
1
V rand , V inf o
2D U-Net
Residual, summation-skip, multi-stage
Spiers et al. (2021)
NE
19
P, R, F1
2D U-Net
Tri-axis prediction
Cheng and Varshney (2017)
M
8
P, R, JI
3D U-Net
Factorised convolutions
Lee et al. (2017)
NM
3
RE
3D U-Net
3D graph affinity, hybrid 2D-3D, residual
Xiao et al. (2018a)
M
5, 8
JI, DSC
3D U-Net
Hybrid 2D-3D, residual, auxiliary supervision
Funke et al. (2018)
NM
2, 10, 15
V inf o , CREMI
3D U-Net
3D graph affinity prediction
Heinrich et al. (2018)
NM
2
CREMI
3D U-Net
Signed distance regression map, hybrid 2D-3D
Mekuč et al. (2020)
M, L
13
TNR, R, DSC
3D FCN
HighRes3DZMNet,
zero-mean,
resid-
ual/atrous
Heinrich et al. (2021)
Many
10
DSC
3D U-Net
Multi-class segmentation
Bailoni et al. (2022)
NM
2
ARAND
3D U-Net
Signed 3D graph affinity prediction
End-to-end learning -instance segmentation
Liu et al. (2020a)
M
8
Acc, P, R, JI, DSC Mask-RCNN
Recursive network, multiple bounding boxes
Yuan et al. (2021)
M
4, 8
JI, DSC, AJI, PQ
2D U-Net
Hierarchical view ensemble module, multi-
task
Luo et al. (2021)
M
4, 8
JI, DSC, AJI, PQ
2D U-Net
Residual blocks, two-stage, shape soft-labels
Wei et al. (2020)
M
6, 8
JI, AP-75
3D U-Net
Mask, contour prediction, watershed
Abdollahzadeh et al. (2021)
A, N
17
V inf o , ARAND
3D U-Net
Shape-based postprocessing
Lin et al. (2021)
N
7
AP-50, AP-75, AP
3D U-Net
Hybrid 2D-3D module, residual blocks
Li et al. (2022)
M
6
JI, DSC, AP
3D FCN
Hybrid 2D-3D module, multi-scale
Mekuč et al. (2022)
M
13
TPR, TNR, JI, DSC 3D U-Net
HighRes3DzNet, geodesic active contours
Ensemble learning -semantic segmentation
Zeng et al. (2017)
NM
3
RE
3D U-Net
Hybrid 3D-2D, residual/inception/atrous
Haberl et al. (2018b)
NM, M, N, V
18, 21
A, P, R, F-1
3D U-Net
Hybrid 3D-2D, residual/inception/atrous
Guay et al. (2021)
C, M, GN
16
Mean JI
3D U-Net
Hybrid 2D-3D, spatial pyramids
Khadangi et al. (2021b)
M
12, 18
Acc, TPR, TNR,
F1, JI, V rand , V inf o
2D U-Net
Ensemble of different networks
Transfer learning -semantic segmentation
Dietlmeier et al. (2019)
M
1
Acc, P, F1
VGG
Few shot, hypercolumn features, boosting
Bermúdez-Chacón et al. (2018) M
Private *
JI
2D U-Net
Deep domain adaptation, two-stream U-Net
Configurable networks -semantic segmentation
Isensee et al. (2019)
NM
2
Acc, P, F1
2D, 3D U-Net nn U-Net, self-configuring method
Franco-Barranco et al. (2022)
M
4, 8
JI
2D, 3D U-Net Stable networks, blended output, z-filtering
*
Shallow learning in this context refers to supervised machine learning with hand-crafted features, or traditional unsupervised techniques such as PCA and clustering.
https://myscope.training/
#n refers to the entry n inTable 2.
https://imagej.net/events/isbi-2012-segmentation-challenge
Proofreading refers to the manual validation of segmented (manual or automatic) image data.
https://open.quiltdata.com/b/janelia-cosem-datasets/tree/jrc mus-kidney/
www.rug.nl/research/fse/themes/dssc/
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Cdeep3m-plug-and-play cloud-based deep learning for image segmentation. M G Haberl, C Churas, L Tindall, D Boassa, S Phan, E A Bushong, M Madany, R Akay, T J Deerinck, S T Peltier, Nature methods. 15Haberl, M.G., Churas, C., Tindall, L., Boassa, D., Phan, S., Bushong, E.A., Madany, M., Akay, R., Deerinck, T.J., Peltier, S.T., et al., 2018a. Cdeep3m-plug-and-play cloud-based deep learning for image segmentation. Nature methods 15, 677-680.
Cdeep3m-plug-and-play cloud-based deep learning for image segmentation. M G Haberl, C Churas, L Tindall, D Boassa, S Phan, E A Bushong, M Madany, R Akay, T J Deerinck, S T Peltier, Nature methods. 15Haberl, M.G., Churas, C., Tindall, L., Boassa, D., Phan, S., Bushong, E.A., Madany, M., Akay, R., Deerinck, T.J., Peltier, S.T., et al., 2018b. Cdeep3m-plug-and-play cloud-based deep learning for image segmentation. Nature methods 15, 677-680.
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Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. K He, X Zhang, S Ren, J Sun, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionHe, K., Zhang, X., Ren, S., Sun, J., 2015. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in: Proceedings of the IEEE international conference on computer vision, pp. 1026-1034.
Deep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionHe, K., Zhang, X., Ren, S., Sun, J., 2016. Deep residual learning for image recognition, in: Pro- ceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778.
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Connectomic reconstruction of the inner plexiform layer in the mouse retina. M Helmstaedter, K L Briggman, S C Turaga, V Jain, H S Seung, W Denk, Nature. 500Helmstaedter, M., Briggman, K.L., Turaga, S.C., Jain, V., Seung, H.S., Denk, W., 2013. Connec- tomic reconstruction of the inner plexiform layer in the mouse retina. Nature 500, 168-174.
nnu-net: Breaking the spell on successful medical image segmentation. F Isensee, J Petersen, S A Kohl, P F Jäger, K H Maier-Hein, arXiv:1904.081281arXiv preprintIsensee, F., Petersen, J., Kohl, S.A., Jäger, P.F., Maier-Hein, K.H., 2019. nnu-net: Breaking the spell on successful medical image segmentation. arXiv preprint arXiv:1904.08128 1, 2.
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[] |
[
"EXPECTED VALUES OF L-FUNCTIONS AWAY FROM THE CENTRAL POINT",
"EXPECTED VALUES OF L-FUNCTIONS AWAY FROM THE CENTRAL POINT"
] |
[
"Chantal David ",
"Patrick Meisner "
] |
[] |
[] |
We compute the expected value of Dirichlet L-functions defined over Fq[T ] attached to cubic characters evaluated at an arbitrary s ∈ (0, 1). We find a transition term at the point s = 1 3 , reminiscent of the transition at the point s = 1 2 of the bound for the size of an L-function implied by the Lindelöf hypothesis. We show that at s = 1 3 , the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.
| null |
[
"https://export.arxiv.org/pdf/2305.15120v1.pdf"
] | 258,865,802 |
2305.15120
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451299d6a2d8bb15ebd05207af892bd89f1076cc
|
EXPECTED VALUES OF L-FUNCTIONS AWAY FROM THE CENTRAL POINT
24 May 2023
Chantal David
Patrick Meisner
EXPECTED VALUES OF L-FUNCTIONS AWAY FROM THE CENTRAL POINT
24 May 2023
We compute the expected value of Dirichlet L-functions defined over Fq[T ] attached to cubic characters evaluated at an arbitrary s ∈ (0, 1). We find a transition term at the point s = 1 3 , reminiscent of the transition at the point s = 1 2 of the bound for the size of an L-function implied by the Lindelöf hypothesis. We show that at s = 1 3 , the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.
where g is the dimension of Θ χ . That is, it will be bounded for s above 1 2 , while the bound grows exponentially as s decreases below 1 2 . The bound at s = 1 2 depends only on the genus and is believed to grow polynomially in g on average.
Conrey, Farmer, Keating, Rubinstein and Snaith [CFKRS] developed a recipe for conjecturing moments of L-functions over Q at s = 1 2 . Andrade and Keating [AK2012] then adapt the recipe to L-functions over F q (T ) and conjecture that
1 |M 2 (2g)| χ∈M2(2g) L( 1 2 , χ) k = P k (g) + o(1) (1.4)
where M 2 (2g) is the set of primitive quadratic characters of genus 2g and P k is an explicit polynomial of degree k(k+1) 2 . They prove this for k = 1, while Florea [Fl2017, Fl2017, Fl2017] confirms it in the cases k = 2, 3, 4, improving the quality of error terms from the number field case (for k = 4, only the leading three terms of the polynomial P k of degree 10 can be obtained).
In addition to the papers mentionned above, many papers have been published on moments of quadratic characters at the central point both in number fields and function fields, including [AK2012, Di2019, DW2021, Ju1981, KS2000, So2000, Yo2009]. Moments of higher order characters have been less studied however, there has been some recent interest, including [BaYo2010, Di2004, FHL2003, Lu2004, DFL2022]). In particular, the first author, with Florea and Lalin [DFL2022], showed that as g → ∞
1 |M 1 3 (g)| χ∈M 1 3 (g) L( 1 2 , χ) = A q + o(1) (1.5)
where A q is an explicit constant, depending only on q, and M 1 3 (g) is the set of primitive cubic characters of genus g with the added condition that if q ≡ 1 mod 3, then χ| F * q = χ 3 , a fixed, nontrivial cubic character of F * q . Notice that the expected value at s = 1 2 is bounded, contrary to the case of quadratic characters. This can be explained by the fact that quadratic characters form a symplectic family, while cubic characters form a unitary family.
In this paper, we are interested in what happens to values such as in (1.5) for arbitrary values of 0 < s < 1 and for the family H(g) = χ F := · F 3 : genus of χ F is g and F is square-free (1.6) where · F 3 is the cubic-residue symbol modulo F (as defined in Section 2.1). Theorem 1.1. Let q ≡ 1 mod 6 and ǫ > 0. Then for any ǫ < s < 1 − ǫ not equal to 1 3 , we have 1 |H(3g)| and ζ q (s) is defined by (2.6). If s = 1 3 , we get
1 |H(3g)| χ∈H(3g) L( 1 3 , χ) = C q g + 1 q − 1 + P deg(P ) |P | + 2 |P | 3 + 2|P | 2 − 1 + O(1) where C q = P 1 − 1 |P |(|P | + 1) ,
where the product runs over all prime polynomials P = P (T ) ∈ F q [T ], and the error terms do not depend on q.
To our knowledge, this is the first result in the literature computing the first moment at s = 1/3 for any family of cubic Dirichlet characters, over function fields or number fields. Based on the random matrix model of Section 1.2, we could speculate that a similar asymptotic would hold for the the first moment at s = 1/ℓ for families of Dirichlet characters of order ℓ, at least for ℓ prime, but there is no hope to prove such a result with the current knowledge, as only very partial results are known for the residues of the Dirichelt series of ℓ-th order Gauss sums, which are shrouded in mystery.
Remark 1.2. When q is fixed and g → ∞, we should write the second result as
1 |H(3g)| χ∈H(3g) L( 1 3 , χ) = C q g (1 + o(1)) ,
since the constants are the same size as the error. But we will need the more precise expression for the q-limit result of the next theorem. Moreover, we believe there should be an explicit constant appearing in the shape of a prime sum. We write the result as above to illustrate this fact but make no claims that the prime sum we write is correct. Determining the correct prime sum would involve improving Lemma 4.8.
Remark 1.3. The values L(s, χ) are not real, but the sum is since L(s, χ) + L(s, χ) is real (when s is). When s = 1 2 , we recover a result similar to the main results of [DFL2022], but for a different family, as they considered the full families. Computations for thin families of cubic characters as (1.6) are typically easier, but maybe paradoxically, computing the dual term for the thin family presented new challenges, as one needs to consider weighted averages of Gauss sums. The case s = 1 3 is specially challenging as one needs to compute exactly the main term and the dual term with error terms good enough to see the cancellation. In [DFL2022], the authors attempted to see the corresponding cancellation of the secondary term for s = 1 2 which is achieved in this paper (compare Propositions 3.1 and 4.1) for the full families, but the cancellation occurred inside the error term.
Comparing Theorem 1.1 to (1.3), we see similar phenomenon occurring, except with a transition at s = 1 3 . When s > 1 3 , the error term decays with g and we get an explicit constant. At s = 1 3 , the error term stops decaying, but the main term has a pole, resulting in a contribution of roughly a constant times one-third of the genus (since the characters in H(3g) have genus 3g). When s < 1 3 , the error term begins growing exponentially, making the main term no longer a main term.
The main technique in proving Theorem 1.1 is to use the approximate functional equation to write the L-function as a principal sum and a dual sum, and we average each of them over the family. In typical applications where one computes the moments at s = 1 2 , the main term will come from the principal sum and the oscillations of the sign of the functional equation will make the dual sum smaller.
In fact, for s > 1 3 , we find that the principal sum has M (s) as a main term as well as a secondary term that decays. However, this secondary term has a pole at s = 1 3 and starts growing exponentially for s < 1 3 . While for the dual sum we find a main term which is exactly the negative of the secondary term of the principal sum. Analyzing the poles at s = 1 3 of each of the principal and dual sum then yields the result for s = 1 3 .
The principal sum is a straight forward double character sum and we apply standard number theory techniques to compute it. However, the dual sum comes with an extra factor of the sign of the functional equation. To handle this, we then need to compute exactly the average of cubic Gauss sums coming from the functional equation. Over number fields, such averages were first considered by [Pa1978] following the ideas of Kubota [Ku1969, Ku1971] involving "metaplectic forms" (see also [KP1984]), and over function fields, by Hoffstein [Ho1992] and Patterson [Pa2007]. The precise formulas that are needed to evaluate the average of cubic Gauss sums were developed in [DFL2022], building on [Ho1992, Pa2007]. We will follow the notation of [DFL2022] closely and rely on their results when computing the dual sum in Section 4. In particular, we will extend the main result of [DFL2022] in order to average Gauss sums weighted by Euler products (see Proposition 4.7).
Finally, we would like to bring attention to the transition of E s (g) at s = 2 3 . This is completely analogous with what happens to the main term at s = 1 3 and is not too surprising since the functional equation relates the value at 1 3 to the value at 2 3 . With this, it seems reasonable that the true behaviour of E 2 3 (g) should be a constant times g, and not g 2 . However, it is not immediately clear what to expect for the true behaviour beyond 2 3 . It is very natural to ask if Theorem 1.1 would hold over number fields, i.e averaging over a similar thin family of cubic characters over Q(ξ 3 ), where ξ 3 is a primitive third root of unity. The exact estimates for average of cubic Gauss sums of [DFL2022] and the present paper would have to be developed over number fields, which should not be a problem. We are also using the Riemann Hypothesis which is true over function fields (or the Lindelöf Hypothesis, see Lemma 3.2) to estimate the principal sum, but this could presumably be avoided by using the cubic large sieve, as in [Lu2004] and [BaYo2010]. For the thin family of cubic characters over Q(ξ 3 ), it was show by the first author and Güloglu [DG2022] that it is possible to break the "trivial" support of the Fourier transform in the one level-density (under GRH) to obtain a positive proportion of non-vanishing of L-functions of cubic characters, by bounding averages of Gauss sums over primes, which seems encouraging. For the full families of cubic characters, a (much smaller) positive proportion of non-vanishing can be found by using different techniques, based on the mollified moments, which was done over function fields for the non-Kummer family in [DFL2021] and over Q(ξ 3 ) for the Kummer family in [GoYe2023+] (under GRH).
We are not aware of any result like Theorem 1.1 over number fields.
1.2. Random Matrix Model. Another reason why looking at statistics when s = 1 2 is interesting, is that we see that when δ χ = 1 the q-dependence on the right-hand side of (1.2) disappears, and
L( 1 2 , χ) = det(1 − Θ χ )
. That is, we may apply the Katz-Sarnak [KS1999] philosophy which states that the Frobenii of a family of L-functions should equidistribute in some compact matrix Lie group.
Specifically, if f is any continuous class function and F g is a "nice" family of L-functions of fixed genus g, then the Katz-Sarnak philosophy predicts that there is some compact matrix Lie group, G g 1 , such that
lim q→∞ 1 |F g | L∈Fg f (Θ L ) = (1 + o(1)) Gg f (U )dU. (1.7)
where dU is the Haar measure and the o(1) term vanishes as g tends to infinity.
Applying this with the continuous class function f (U ) = det(1 − U ) k allows us to predict that the moments of L-functions in "nice" families at s = 1 2 should behave like the moments of the characteristic polynomial of a matrix in a compact matrix Lie group at 1, and
lim q→∞ 1 |F g | L∈Fg L( 1 2 ) k = (1 + o(1)) Gg det(1 − U ) k dU.
This framework can further help to explain the recipe in [CFKRS] and the ensuing conjecture in (1.4). That is, it is known that the Frobenii attached to quadratic characters are symplectic and that
USp(2g) det(1 − U ) k dU = Q k (g)
for some explicit polynomials of degree k(k+1) 2 , which coincides with the conjecture of Andrade and Keating [AK2012].
Many results ([BCDGL2018, CP2019, DG2022, Me2019, Me2023+]) suggest that the compact matrix Lie group attached to cubic characters is the group of unitary matrices. This is further reinforced by (1.5) and Theorem 1.1 with the observation that
lim q→∞ A q = lim q→∞ M ( 1 2 ) = U(g) det(1 − U )du = 1.
However, this framework can not, a priori, help to explain the transition term at s = 1 3 since the right hand side of
L( 1 3 , χ) = det(1 − q 1/6 Θ χ ) (1.8)
is not independent of q and, hence, can not be identified with a single continuous class function f of the unitaries as q grows. Tr(U j ) λj where λ = 1 λ1 2 λ2 · · · is the partition consisting of λ 1 ones, λ 2 twos, etc. These form a basis for continuous class functions of the unitary matrices and so it would be enough to prove (1.7) for f = P λ for all λ.
Towards this, the second author in [Me2021+] proved a partial result towards (1.7) for some family of cubic characters F 3 (N ). Specifically, for all λ such that |λ| := jλ j < N we have
lim q→∞ 1 |F 3 (N )| χ∈F3(N ) P λ (Θ χ ) = U(N ) P λ (U )dU. (1.9)
One unsatisfying aspect of (1.9) is that the right hand side is always 0. The main goal of [Me2021+] was to find an appropriate normalization for the left hand side to be non-zero and see how that could effect the random matrix interpretation. Indeed, it was shown that for |λ| < 3N
4 [Me2021+, Theorem 1.1 with r = 3] lim q→∞ 1 |F 3 (N )| χ∈F3(N ) q |λ| 6 P λ (Θ χ ) = U(N ) P λ (U )det(1 − ∧ 3 U )dU (1.10) where det(1 − ∧ 3 U ) := 1≤i1<i2<i3≤N (1 − x i1 x i2 x i3 )
and the x i are the eigenvalues of U .
While extending (1.9) to all λ would give us a result like (1.7) it is not immediately clear what extending (1.10) for all λ would give us. One possible interpretation is that, by the definition of P λ , we have q |λ| 6 P λ (Θ χ ) = P λ (q 1 6 Θ χ ) and so an extension of (1.10) would imply the following conjecture.
Conjecture 1.4. Let F (N ) be a "nice" family of cubic characters defined over F q [T ] and with conductor bounded by N , and let f be a continuous class function. Then
lim q→∞ 1 |F (N )| χ∈F (N ) f (q 1 6 Θ χ ) = (1 + o(1)) U(N ) f (U )det(1 − ∧ 3 U )dU
where the o(1) tends to 0 as N tends to infinity.
Notice that the result of [Me2021+] implies that Conjecture 1.4 is true for the family of F 3 (N ) and all continuous class functions in the span of {P λ : |λ| < 3N/4}. Similar techniques would prove similar results for the family H(3g) as well. However, we see by (1.8) that
L( 1 3 , χ) = f (q 1 6 Θ χ ) where f (U ) = det(1 − U ) is not in the span of {P λ : |λ| < 3N/4}
. Regardless of this, we can use Theorem 1.1 to prove Conjecture 1.4 in this case.
Theorem 1.5. Conjecture 1.4 is true with the family H(3g) and the continuous class function f (U ) = det(1 − U ). Specifically,
lim q→∞ 1 |H(3g)| χ∈H(3g) L( 1 3 , χ) = 1 + O 1 g U(3g) deg(1 − U )det(1 − ∧ 3 U )dU.
We see here another "natural" reason to restrict to H(3g). That is, we are aiming to mimic the connection between quadratic characters and symplectic matrices. Since symplectic matrices only exist in even dimensions it seems "natural" that whatever analogous connection can be made would be most prevalent in dimensions divisible by 3.
1.4. Structure of the paper. We present in Section 2 some background on Lfunctions and cubic characters over function fields, and we describe the thin family of cubic characters that we are using. In particular, we generalize the approximate functional equation of [DFL2022] from s = 1 2 to general s. Using the approximate functional equation, the average value can be written as a principal sum and a dual sum as in (2.21). We compute the principal sum in Section 3, the dual sum in Section 4, and using those estimates, the proof of theorems 1.1 and 1.5 are given in Section 5.
2. Background on L-functions 2.1. Cubic characters over F q [T ]. We denote by M the set of monic polynomials in F q [t], and by M d , repectively M ≤d , the set of monic polynomials in F q [t] of degree d, respectively of degree ≤ d.
Let q be a prime power with q ≡ 1 mod 6. We fix once and for all an isomorphism Ω from the cube roots of unity in F * q to µ 3 = {1, ξ 3 , ξ 2 3 }, the cube roots of unity in C * , where ξ 3 = e 2πi/3 . We then denote by χ 3 the cubic residue symbol of F * q given by
χ 3 (a) = Ω(a (q−1)/3 ), for all a ∈ F * q . (2.1) For each prime P ∈ F q [t]
, we define the cubic residue symbol of conductor P
χ P : F q [t]/(P ) → µ 3 as follows: for a ∈ F q [t], if P | a, then χ P (a) = 0, and otherwise, χ P (a) = α, where α ∈ µ 3 is such that a q deg P −1 3 ≡ Ω −1 (α) mod P.
There are then 2 cubic characters of conductor P , χ P and χ 2 P = χ P . We extend the definition to F ∈ M by multiplicativity. Writing F = P e1 1 . . . P es s where the P i are distinct primes and the e i are positive integers, we define
χ F (a) = χ P1 (a) e1 . . . χ Ps (a) es . (2.2)
Then, χ F is a cubic character of conductor rad(F ) = P 1 . . . P s . Conversely, all the primitive cubic characters of conductor P 1 . . . P s are given by χ e1 P1 . . . χ es Ps with 1 ≤ e i ≤ 2, and there are 2 s such characters.
We say that a cubic character χ is even if χ| F * q = χ 0 , the trivial character, and that χ is odd if χ| F * q = χ 3 or χ 2 3 . We define δ χ = 1 when χ is odd 0 when χ is even.
The best classification of cubic characters is by genus. From the Riemann-Hurwitz formula (Theorem 7.16 of [Ro2002]), we compute
g = deg cond(χ) − 2 + δ χ ,
and we denote by M 3 (g) the set of primitive cubic characters over F q [t] of genus g. M 3 (g) is naturally divided in three disjoint subsets M 0 3 (g), M 1 3 (g), M 2 3 (g), depending on the restriction of χ over F q , and we define for j = 0, 1, 2,
M j 3 (g) = {χ ∈ M 3 (g) : χ| F * q = χ j 3 } (2.3)
where we identify χ 0 3 = χ 0 . In particular, we get
M 3 (g) = M 0 3 (g) ∪ M 1 3 (g) ∪ M 2 3 (g).
Using the observation that q n −1 3 ≡ n(q−1) 3 mod q − 1, we get that if a ∈ F * q , then for any prime P ,
a q deg(P ) −1 3 = a deg(P )(q−1) 3 ≡ Ω −1 χ deg(P ) 3 (a) mod P.
Extending this multiplicatively, we find that
χ F | F * q = χ deg(F ) 3
. Hence, if we define
M j 3 (d) := {F ∈ F q [T ] : F cube-free, deg(rad(F )) = d, deg(F ) ≡ j mod 3} then we have M 0 3 (g) = {χ F : F ∈ M 0 3 (g + 2)} and for j = 1, 2, M j 3 (g) = {χ F : F ∈ M j 3 (g + 1)} Let H(d) := {F ∈ F q [T ] : F square-free, deg(F ) = d}.
For g ≡ 0 mod 3, let
H(g) := {χ F : F ∈ H(g + 1)} ⊂ M 1 3 (g) ⊂ M 3 (g) while if g ≡ 1 mod 3, let H(g) := {χ F : F ∈ H(g + 2) ∪ H(g + 1)} ⊂ M 0 3 (g) ∪ M 2 3 (g) ⊂ M 3 (g)
. Somewhat surprisingly, if g ≡ 2 mod 3, then we find that there are no elements of M 3 (g) with a square-free discriminant. Hence we set H(g) = ∅ in this case.
We see that there is a natural bijection from H(3g) to H(3g + 1), which is reminiscent of the family usually considered for quadratic characters M 2 (2g) which comes with a natural bijection to H(2g + 1). From this point of view, the family H(3g) is a "natural" extension of the quadratic family.
As the functional equation of the L-functions depends on the parity of the character, it will be useful to distinguish them. Thus, we define
H e (g) := {χ ∈ H(g) : χ is even} and H o (g) := {χ ∈ H(g) : χ is odd}.
Notice that the even characters are exactly those in M 0 3 (g) so that if g ≡ 1 mod 3, H e (g) = ∅. Now, the standard square-free sieve tells us that for d ≥ 1
|H(d)| = q d ζ q (2) = q d − q d−1 , (2.4) so that we get |H e (g)| = 0 g ≡ 1 mod 3 |H(g + 2)| g ≡ 1 mod 3 = 0 g ≡ 1 mod 3 q g+2 − q g+1 g ≡ 1 mod 3 and |H o (g)| = |H(g + 1)| g ≡ 2 mod 3 0 g ≡ 2 mod 3 = q g+1 − q g g ≡ 2 mod 3 0 g ≡ 2 mod 3
from which we may conclude that
|H(g)| = |H e (g)| + |H o (g)| = q g+1 − q g g ≡ 0 mod 3 q g+2 − q g g ≡ 1 mod 3 0 g ≡ 2 mod 3 . 2.2. Functional Equation.
The affine zeta function over F q [t] is defined by
Z q (u) = f ∈M u deg f = P 1 − u deg P −1 = 1 (1 − qu) (2.5)
for |u| < q −1 . The right-hand side provides an analytic continuation to the entire complex plane, with a simple pole at u = 1/q with residue −1/q. We also define
ζ q (s) = Z q (q −s ). (2.6)
Replacing in (2.4), we express the size of H(g) in terms of values of ζ q (s):
|H(g)| = q g+1 ζq(2) g ≡ 0 mod 3 q g+2 ζq(3) g ≡ 1 mod 3 0 g ≡ 2 mod 3.
Let χ be a primitive cubic Dirichlet character as defined in Section 2.1, and let h ∈ M be its conductor. We define the L-function in the u-variable as
L(u, χ) = F ∈M χ(F )u deg(F ) so that L(s, χ) = L(q −s , χ), where L(s, χ) is defined in (1.1).
If χ is even, we have that L(1, χ) = 0, and we define the completed L-function
L C (u, χ) = L(u, χ) (1 − u) 1−δχ . (2.7)
Let g be the genus of the character χ. It follows from the Weil conjectures [We1948] that L C (u, χ) is a polynomial of degree g and it satisfies the functional equation
L C (u, χ) = ω(χ)( √ qu) g L C 1 qu , χ (2.8)
where ω(χ) is the sign of the functional equation. To give a formula for ω(χ), we need to define the Gauss sums of characters over F q [t].
We first start with Gauss sums for characters over F * q . If χ is a non-trivial character of F q , we define
τ (χ) := a∈F * q χ(a)e 2πitr Fq /Fp (a) p .
Then, τ (χ) = τ (χ) and |τ (χ)| = q 1/2 , and we denote the sign on the Gauss sum by
ǫ(χ) := q −1/2 τ (χ).
We extend the definition to trivial characters by defining ǫ(χ 0 ) = 1.
To define the Gauss sums of general characters over F q [t], we define the exponential over F q (t) as follows: for any a ∈ F q ((1/t)), we have e q (a) = e 2πitr Fq /Fp (a 1 )
p ,
where a 1 is the coefficient of 1/t in the Laurent expansion of a. We then have the usual properties: e q (a + b) = e q (a)e q (b) and e q (a) = 1 for a ∈ F q [t]. Also, if a, b, h ∈ F q [t] with a ≡ b mod h, then e q (a/h) = e q (b/h). For χ a primitive cubic character of modulus h over F q [t], the Gauss sum of χ is
G(χ) = a mod h χ(a)e q a h .
It is not hard to show that G(χ) = G(χ) and |G(χ)| = q deg h/2 .
ω(χ) = ǫ(χ deg h 3 ) G(χ) q deg h/2
where χ 3 is the cubic character defined in (2.1). We then have
ω(χ) = ω(χ)
and |ω(χ)| = 1.
The following result generalizes [DFL2022, Proposition 2.4] which gives the approximate functional equation when s = 1 2 . Proposition 2.2 (Approximate functional equation). Let χ be a primitive character of genus g, and A be a positive integer. If χ is odd, then
L(q −s , χ) = f ∈M ≤A χ(f ) |f | s + ω(χ)(q 1/2−s ) g f ∈M ≤g−A−1 χ(f ) |f | 1−s (2.9)
If χ is even, then
L(q −s , χ) = 1 1 − q 1−s f ∈M ≤A+1 χ(f ) |f | s − q 1−s f ∈M ≤A χ(f ) |f | s + 1 1 − q s ω(χ) q (s−1/2)g ζ q (2 − s) ζ q (s + 1) f ∈M ≤g−A χ(f ) |f | 1−s − q s f ∈M ≤g−A−1 χ(f ) |f | 1−s (2.10)
Proof. Since L(u, χ) is a polynomial of degree g + 1 − δ χ , we may write
L(u, χ) = g+1−δχ n=0 a n (χ)u n (2.11) where a n (χ) = f ∈Mn χ(f ). (2.12) Similarly, for L C (u, χ) we write L C (u, χ) = g n=0 b n (χ)u n .
(2.13) Substituting (2.13) into (2.8) and comparing coefficients we get b n (χ) = ω(χ)q n−g/2 b g−n (χ).
(2.14)
Applying (2.7), we can write a n (χ) in terms of b n (χ) such that for n = 0, . . . , g a n (χ) = b n (χ)
δ χ = 1 b n (χ) − b n−1 (χ) δ χ = 0 (2.15) while if δ χ = 0 then a g+1 (χ) = −b g (χ)
. Reversing (2.15), we can write b n (χ) in term of a n (χ) such that b n (χ) = a n (χ)
δ χ = 1 n m=0 a m (χ) δ χ = 0 (2.16)
for n = 0, . . . , g.
Finally, if δ χ = 0, then we may apply (2.14) to (2.15) to obtain for any 0 ≤ A ≤ g,
a g−A (χ) = b g−A (χ) − b g−A−1 (χ) = ω(χ)q g/2−A b A (χ) − ω(χ)q g/2−A−1 b A+1 (χ) = ω(χ)q g/2−A b A (χ) − ω(χ)q g/2−A−1 (a A+1 (χ) + b A (χ)) .
Rearranging and taking conjugates we then obtain
b A (χ) q (A+1)s = 1 q − 1 ω(χ) q g/2 a g−A (χ)q (A+1)(1−s) + a A+1 (χ) q (A+1)s (2.17)
where we have used 1/ω(χ) = ω(χ).
Replacing A with g − A − 1, s with 1 − s and χ with χ we obtain
b g−A−1 (χ) q (g−A)(1−s) = 1 q − 1 ω(χ) q g/2 a A+1 (χ)q (g−A)s + a g−A (χ) q (g−A)(1−s) . (2.18)
Splitting the sum in (2.13) at an arbitrary point A, and applying (2.14), we obtain
L C (u, χ) = A n=0 b n (χ)u n + g n=A+1 b n (χ)u n = A n=0 b n (χ)u n + ω(χ) g n=A+1 q n−g/2 b g−n (χ)u n = A n=0 b n (χ)u n + ω(χ)( √ qu) g g−A−1 n=0 b n (χ) q n u n .
If δ χ = 1, then L(u, χ) = L C (u, χ) and a n (χ) = b n (χ) and so we get the odd approximate function equation
L(q −s , χ) = A n=0 a n (χ) q sn + ω(χ)(q 1/2−s ) g g−A−1 n=0 a n (χ) q (1−s)n = f ∈M ≤A χ(f ) |f | s + ω(χ)(q 1/2−s ) g f ∈M ≤g−A−1 χ(f ) |f | 1−s . (2.19)
If δ χ = 0, then we get L(u, χ) = (1 − u)L C (u, χ) and we get
L(q −s , χ) = A n=0 b n (χ) q sn (1 − q −s ) + ω(χ)q (1/2−s)g g−A−1 n=0 b n (χ) q (1−s)n (1 − q −s ) = A n=0 b n (χ) q sn 1 − q −s + ω(χ) ζ q (2 − s) ζ q (s + 1) q g/2 q sg g−A−1 n=0 b n (χ) q (1−s)n 1 − q −(1−s) .
Expanding the (1 − q −s ) term in the first series and applying (2.15) and (2.17) with the observation that b 0 (χ) = a 0 (χ), we obtain
A n=0 b n (χ) q sn 1 − q −s = b 0 (χ) + A n=1 b n (χ) − b n−1 (χ) q sn − b A (χ) q (A+1)s = b 0 (χ) + A n=1 a n (χ) q sn − 1 q − 1 ω(χ) q g/2 a g−A (χ)q (A+1)(1−s) + a A+1 (χ) q (A+1)s = f ∈M ≤A χ(f ) |f | s − 1 q − 1 ω(χ) q g/2 a g−A (χ)q (A+1)(1−s) + a A+1 (χ) q (A+1)s .
Likewise, expanding the (1 − q −(1−s) ) and applying (2.15) and (2.18), the second series becomes
f ∈M ≤g−A−1 χ(f ) |f | 1−s − 1 q − 1 ω(χ) q g/2 a A+1 (χ)q (g−A)s + a g−A (χ) q (g−A)(1−s) .
Combining everything, when δ χ = 0, we obtain the even approximate function equation
L(q −s , χ) = f ∈M ≤A χ(f ) |f | s + ζ q (2 − s) ζ q (s + 1) ω(χ) q (s−1/2)g f ∈M ≤g−A−1 χ(f ) |f | 1−s + 1 1 − q 1−s a A+1 (χ) q (A+1)s + 1 1 − q s ζ q (2 − s) ζ q (s + 1) ω(χ) q (s−1/2)g a g−A (χ) q (g−A)(1−s) . (2.20)
where we note that
− 1 q − 1 1 + ζ q (2 − s) ζ q (s + 1) q s = 1 1 − q 1−s and − 1 q − 1 1 + ζ q (s + 1) ζ q (2 − s) q 1−s = 1 1 − q s . Finally, rewriting a A+1 (χ) q A+1 s = f ∈M ≤A+1 χ(f ) |f | s − f ∈M ≤A χ(f ) |f | s and a g−A (χ) q (g−A)(1−s) = f ∈M ≤g−A χ(f ) |f | 1−s − f ∈M ≤g−A−1 χ(f ) |f | 1−s
we obtain the even approximate functional equation.
Principal and Dual
Terms. We see that since H(3g) = H o (3g), we always have the functional equation
L(q −s , χ) = f ∈M ≤A χ(f ) |f | s + ω(χ)q ( 1 2 −s)3g f ∈M ≤3g−A−1 χ(f ) |f | 1−s .
Further, it will be convenient for computations if we split at an integer that is divisible by 3. Therefore, we define the principal sum where M (s) and C q are as an in Theorem 1.1. If s = 1 3 , then
P s (3g, 3A) := χ∈H(3g) f ∈M ≤3A χ(f ) |f | s and the dual sum D s (3g, 3A) = q ( 1 2 −s)3g χ∈H(3g) ω(χ) f ∈M ≤3g−3A−1 χ(f ) |f | 1−s ,P 1 3 (3g, 3A) |H(3g)| = C q A + 1 + P deg(P ) |P | 2 + |P | − 1 +O ǫ q (ǫ−1)A +q A−(1−ǫ)(3g+1) +q (2+ǫ)A− 1 2 (3g+1)
3.2. Error Term. Since we are assuming q ≡ 1 mod 6, we have by cubic reciprocity (Theorem 3.5 of [Ro2002]) that
χ F (f ) = f F 3 = F f 3 = χ f (F ).
Therefore, since H(3g) = {χ F : F ∈ H(3g + 1)} we may rewrite
P s (3g, 3A) = f ∈M ≤3A 1 |f | s F ∈H(3g+1) χ f (F ).
We now wish to compute the innermost sum. To do this, we consider the generating series
P(u; f ) := ∞ d=0 F ∈H(d) χ f (F )u d .
This series converges for |u| < q −1 .
Lemma 3.2. If f is not a cube, then
P(u; f ) = L(u, χ f ) L(u 2 , χ f ) .
In particular, P(u; f ) can be analytically extended to the region |u| < q −1/4−ǫ . Moreover, if Γ 1 = {u : |u| = q −1/2 }, then for any ǫ > 0,
max u∈Γǫ |P(u, f )| ≪ q ǫ deg(f )
Proof. Since we are summing over all square-free polynomials, we get the Euler product
P(u; f ) = P 1 + χ f (P )u deg(P ) = P 1 − χ 2 f (P )u 2 deg(P ) 1 − χ f (P )u deg(P ) = L(u, χ f ) L(u 2 , χ f )
where we have used the fact that χ f is a cubic character and thus χ 2 f = χ f . Now, since χ f is non-trivial (f is not a cube), L(u, χ f ) and L(u 2 , χ f ) are analytic for all u, and P(u; f ) is analytic for |u| < q −1/4−ǫ , since this region does not contain the zeroes of L(u 2 , χ f ). Furthermore, since again χ f is not trivial, for |u| = q −1/2 , we have bounds
L(u, χ f ) ≪ ǫ q ǫ deg f L(u 2 , χ f ) ≫ ǫ q −ǫ deg f1 |H(d)| f ∈M ≤A f = 1 |f | s F ∈H(d) χ f (F ) ≪ ǫ q (1−s+ǫ)A− 1 2 d .
Proof. We first show that when f is not a cube,
F ∈H(d) χ f (F ) ≪ ǫ q ǫ deg(f ) q 1 2 d . (3.1)
With Γ 1 as in the previous lemma, we get that P(u;f ) u d+1 is meromorphic in the region bounded by Γ 1 with a pole only at u = 0. Hence,
1 2πi Γ1 P(u; f ) u d+1 du = Res u=0 P(u; f ) u d+1 = F ∈H(d) χ f (F ),
and 1 2πi Γ1
P(u; f ) u d+1 du ≪ ǫ max u∈Γ1 P(u; f ) u d+1 ≪ ǫ q ǫ deg(f ) q 1 2 d
which completes the proof of (3.1). Applying this result, we then get that
f ∈M ≤A f = 1 |f | s F ∈H(d) χ f (F ) ≪ ǫ q 1 2 d f ∈M ≤A f = |f | (ǫ−s) ≪ q 1 2 d n≤A q n q n(ǫ−s) ≪ q 1 2 d q (1−s+ǫ)A
and the result follows from the fact that |H(d)| = q d ζq(2) .
Main Term.
In the case that f = h 3 is a perfect cube, then
χ f (F ) = F f 3 = F h 3 3 = 1 (F, f ) = 1 0 (F, f ) = 1 .
Hence, in the case that f is a perfect cube, we get that
F ∈H(d) χ f (F ) = |{H(d, f )}| where H(d, f ) = {F ∈ H(d) : (F, f ) = 1}
. Therefore, we consider the generating series
Q(u; f ) = ∞ d=0 |H(d, f )|u d = (F,f )=1 µ 2 (F )u deg(F ) = P ∤f (1 + u deg(P ) ) = P |f (1 + u deg(P ) ) −1 P 1 − u 2 deg(P ) 1 − u deg(P ) = P |f (1 + u deg(P ) ) −1 Z q (u) Z q (u 2 ) = P |f (1 + u deg(P ) ) −1 1 − qu 2 1 − qu
We see that Q(u; f ) can be meromorphically extended to the region |u| < 1 with a simple pole at u = q −1 . Thus, if we consider the contour
Γ 2 = u : |u| = q −ǫ ,
we get that
1 2π Γ2 Q(u; f ) u d+1 du = Res u=0 Q(u; f ) u d+1 + Res u=q −1 Q(u; f ) u d+1
We see that
Res u=0 Q(u; f ) u d+1 = |H(d; f )|
and for d ≥ 1,
Res u=q −1 Q(u; f ) u d+1 = lim u→q −1 (u − q −1 ) u d+1 P |f (1 + u deg(P ) ) −1 1 − qu 2 1 − qu = − P |f 1 + 1 |P | −1 q d − q d−1 .
Now, for u ∈ Γ 2 , we find that
|Q(u; f )| = P |f (1 − u deg(P ) ) −1 1 − qu 2 1 − qu ≪ P |f (1 − q −1 ) −1 1 + q 1−2ǫ 1 − q 1−ǫ ≪ ǫ (1 + q −1 ) deg(f )
and we conclude that
H(d; f ) = P |f 1 + 1 |P | −1 |H(d)| + O (1 + q −1 ) deg(f ) q ǫd .
Lemma 3.4. For any ǫ > 0 let s > ǫ. Then if s = 1 3 , we get
1 |H(d)| f ∈M ≤3A f = 1 |f | s F ∈H(d) χ f (F ) = M (s)+C q q (1−3s)A 1 − q 3s−1 +O ǫ q (ǫ−3s)A + q A−(1−ǫ)d
where M (s) and C q are as in the Theorem 1.1. If s = 1 3 , then we get
1 |H(d)| f ∈M ≤3A f = 1 |f | s F ∈H(d) χ f (F ) = C q A + 1 + P 1 |P | 2 + |P | − 1 + O ǫ q (ǫ−1)A + q A−(1−ǫ)d .
Proof. Indeed, if we write f = h 3 , then we get
f ∈M ≤3A f = 1 |f | s F ∈H(d) χ f (F ) = h∈M ≤A |H(d, h)| |h| 3s = h∈M ≤A 1 |h| 3s P |h 1 + 1 |P | −1 |H(d)| + O ǫ (1 + q −1 ) deg(h) q ǫd . (3.2)
First, we compute the error term and find
h∈M ≤A (1 + q −1 ) deg(h) q ǫd |h| 3s = q ǫd m≤A q + 1 q 3s m ≪ q ǫd+A (3.3)
For the main term, we define the generating series
G s (v) := h∈M P |h 1 + 1 |P | −1 v deg(h) |h| 3s .
Expanding it as an Euler product, we see that
G s (v) = P 1 + 1 + 1 |P | −1 ∞ k=1 v q 3s k deg(P ) (3.4) = P 1 − v q 3s deg(P ) + 1 − 1 |P |+1 v q 3s deg(P ) 1 − v q 3s deg(P ) = Z q v q 3s P 1 − v deg(P ) |P | 3s (|P | + 1)
Thus we see that that G s (v) can be meromorphically continued to the region |v| ≤ q 3s−ǫ with a simple pole when v = q 3s−1 .
Therefore, if s > ǫ and s = 1/3, and we define Γ 3 = {v : |v| = q 3s−ǫ }, then we get that
1 2πi Γ3 G s (v) 1 − v dv v A+1 = Res v=0 G s (v) (1 − v)v A+1 + Res v=q 3s−1 G s (v) (1 − v)v A+1 + Res v=1 G s (v) (1 − v)v A+1 .
Expanding 1 1−v as a Taylor series we get that
Res v=0 G s (v) (1 − v)v A+1 = ∞ n=0 Res v=0 G s (v) v A−n+1 = h∈M ≤A 1 |h| 3s P |h 1 + 1 |P | −1 = 1 |H(d)| f ∈M ≤3A f = 1 |f | s F ∈H(d) χ f (F ) + O ǫ q A q (ǫ−1)d
using (3.2) and (3.3). We also have
1 2πi Γ3 G s (v) 1 − v dv v A+1 ≪ max v∈Γ3 Gs(v) 1−v q (3s−ǫ)A ≪ ǫ q (ǫ−3s)A .
So, we have obtained the error terms of the lemma, and it remains to compute the two other residues. By (3.4), we see that
Res v=q 3s−1 G s (v) (1 − v)v A+1 = − q (1−3s)A 1 − q 3s−1 P 1 − 1 |P |(|P | + 1) while Res v=1 G s (v) (1 − v)v A+1 = −ζ q (3s) P 1 − 1 |P | 3s (|P | + 1) = −M (s)
which converges because s > ǫ, s = 1 3 . This completes the proof for s = 1 3 . Now, if s = 1 3 , we get the same residue at 0 and the same error terms (replacing s by 1 3 ), but we now have a double pole at v = 1, and it remains to compute the residue at v = 1.
Denoting
K(v) := P 1 − v deg(P ) |P |(|P | + 1) we obtain that Res v=1 G 1 3 (v) (1 − v)v A+1 = lim v→1 d dv K(v) v A+1 = −K(1)(A + 1) + K ′ (1)
The result now follows from the fact that C q = K(1) and that
K ′ (v) K(v) = d dv log(K(v)) = − P deg(P )v deg(P )−1 |P |(|P | + 1) − v deg(P ) .
Proof of Proposition 3.1. We can now combine Lemmas 3.3 (with d = 3g + 1 and replacing A by 3A) and 3.4 (with d = 3g + 1) to prove Proposition 3.1.
The Dual Term
Contributions of the Dual Sum.
This section is devoted to proving contribution of the dual terms.
Proposition 4.1. For any ǫ > 0 and s < 1 − ǫ, we have that if s = 1 3 , then
D s (3g, 3A) |H(3g)| = −C q q (1−3s)A 1 − q 3s−1 + O ǫ q 3(1−s+ǫ)A q (2−ǫ)g + q ( 3 4 +ǫ)g q 3(s+ 1 4 +ǫ)A + q ( 1 3 −s)(3g+1) E s (3g, 3A)
where C q is as in Theorem 1.1 and
E s (3g, 3A) = 1 s < 2/3 (g − A) 2 s = 2/3 (g − A)q (3s−2)(g−A) s > 2/3. If s = 1 3 , then D 1 3 (3g, 3A) |H(3g)| = C q g − A − 1 3 q + 2 q − 1 + P deg(P ) |P | 3 + 2|P | 2 − 1 + O ǫ q (2+ǫ)A q (2−ǫ)g + q ( 3 4 +ǫ)g q ( 7 4 +ǫ)A + 1 .
Remark 4.2. As in Theorem 1.1, when q is fixed and g − A → ∞, we could write the second result as
D 1 3 (3g, 3A) |H(3g)| = C q (g − A) (1 + o(1)) .
We keep it as is to give evidence of an explicit constant appearing in the shape of a prime sum.
4.2.
Applying Results of [DFL2022]. Let χ F be any cubic character as defined by (2.2). This is then a character modulo F , but not necessarily primitive. We define the generalized cubic Gauss sum
G(V, F ) := a mod F χ F (a)e q aV F . (4.1)
It is important to notice that if χ F has conductor F ′ with deg(F ′ ) < deg(F ) then G(1, F ) = G(χ F ). Conversely, if F is, say square-free, then we do have that G(1, F ) = G(χ F ).
D s (3g, 3A) = q ( 1 2 −s)3g f ∈M ≤3g−3A−1 1 |f | 1−s F ∈H(3g+1) (F,f )=1 χ F (f )ω(χ F ) = ǫ(χ 3 ) q 3gs+ 1 2 f ∈M ≤3g−3A−1 1 |f | 1−s F ∈H(3g+1) (F,f )=1 G(f, F ).
Finally, we state another result that can be found in Lemma 2.12 of [DFL2022].
Lemma 4.4. If (F 1 , F 2 ) = 1, then
G(V, F 1 F 2 ) = χ 2 F1 (F 2 )G(V, F 1 )G(V, F 2 ) = G(V F 2 , F 1 )G(V, F 2 ).
Moreover, if there exists a prime P such that P 2 |F and P ∤ V , then
G(V, F ) = 0.
The second part of Lemma 4.4 implies that if F ∈ M 3g+1 is not square-free and (F, f ) = 1 then G(f, F ) = 0. Therefore, in the formula of D s (3g, 3A) above, we may remove the condition that F is square-free, and write
D s (3g, 3A) = ǫ(χ 3 ) q 3gs+ 1 2 f ∈M ≤3g−3A−1 1 |f | 1−s F ∈M3g+1 (F,f )=1 G(f, F ). (4.2)
Proposition 4.5 (Proposition 3.1 of [DFL2022]). Let f = f 1 f 2 2 f 3 3 with f 1 and f 2 square-free and coprime. We have, for any ǫ > 0
F ∈M d (F,f )=1 G(f, F ) = δ f2=1 ρ(d; f ) G(1, f 1 ) |f 1 | 2/3 q 4d/3 ζ q (2) P |f 1 + 1 |P | −1 +O ǫ δ f2=1 q ( 1 3 +ǫ)d |f 1 | 1 6 + q d |f | 1 4 +ǫ where ρ(d; f ) = 1 d + deg(f ) ≡ 0 mod 3 τ (χ3) q 1/3 d + deg(f ) ≡ 1 mod 3 0 d + deg(f ) ≡ 2 mod 3 .
Proof. In [DFL2022, Proposition 3.1], the second error term is of the form
1 2πi |u|=q −σ ψ(f, u) u d du u
for any 2 3 < σ < 4 3 . In that region, we have the convexity bound ψ(f, u) ≪ |f | Using the above proposition in (4.2), we write
D s (3g, 3A) = M T s (3g, 3A) + O ǫ (ET s (3g, 3A)) where M T s (3g, 3A) := ǫ(χ 3 )q ( 4 3 −s)3g+ 5 6 ζ q (2) f ∈M ≤3g−3A−1 δ f2=1 ρ(1; f )G(1, f 1 ) |f | 1−s |f 1 | 2/3 P |f 1 + 1 |P | −1
and ET s (3g, 3A) := 1
q 3gs+ 1 2 f ∈M ≤3g−3A−1 1 |f | 1−s δ f2=1 q ( 1 3 +ǫ)(3g+1) |f 1 | 1 6 + q 3g+1 |f | 1 4 +ε ,
where we use the fact that |ǫ(χ 3 )| = 1.
Bounding the Error Term.
Lemma 4.6. For any ǫ > 0 and s ≥ 0, we get that
ET s (3g, 3A) ≪ q ( 15 4 +ǫ)g−3(s+ 1 4 +ǫ)A
Proof. For the second sum of ET s (3g, 3A), we have
q 3g+1 q 3gs+ 1 2 f ∈M ≤3g−3A−1 |f | 1 4 +ǫ |f | 1−s = q 3g(1−s)+ 1 2 n≤3g−3A−1 q (s+ 1 4 +ǫ)n ≪ q ( 15 4 +ǫ)g−3(s+ 1 4 +ǫ)A.
For the first sum of ET s (3g, 3A), if f 2 = 1, then we can write f = f 1 f 3 3 where f 1 is square-free and f 3 is anything. Hence, setting B = 3g − 3A − 1, we get
q (s−1/6)n1 1 s < 2/3 B−n1 3 s = 2/3 q (s− 2
3 )(B−n1) s > 2/3, and working case by case with trivial bounds, it is straightforward to see that this error term is also bounded by q ( 15 4 +ǫ)g−3(s+ 1 4 +ǫ)A .
Extending Proposition 4.5.
Since there is a factor of δ f2=1 in the sum over f in M T s (3g, 3A), we can write f = f 1 f 3 3 where f 1 ∈ H(n 1 ) and f 3 ∈ M n3 . Moreover, since G(1, f ) = 0 unless f is square-free, we may extend the sum over H(n 1 ) to the sum over M n1 . Hence, we can rewrite
M T s (3g, 3A) = ǫ(χ 3 )q ( 4 3 −s)3g+ 5 6 ζ q (2) n1+3n3≤3g−3A−1 ρ(n 1 + 1; 1) q (1−s)(n1+3n3) × f1∈Mn 1 f3∈Mn 3 G(1, f 1 ) |f 1 | 2/3 P |f1f3 1 + 1 |P | −1
We now extend Proposition 4.5 to get an estimate when G(1, F ) is multiplied by an Euler product.
Proposition 4.7. For any values a P such that |a P | = O 1 |P | , we have for any
H ∈ M, F ∈Mn G(1, F ) |F | 2/3 P |F P ∤H (1 − a P ) = ρ(n; 1)q 2n/3 ζ q (2) P ∤H 1 − a P |P | + 1 + O q n/3 .
Proof. We first expand the Euler product as
P |F P ∤H (1 − a P ) = D|F (D,H)=1 µ(D)a D ,
where
a D := P |D a P .
Notice that by hypothesis on a P , have a D = O 1 |D| which we will use often. Expanding the Euler product like this, we obtain The idea now is to use the work of [DFL2022] (specifically Proposition 4.5 which is [DFL2022, Proposition 3.1]) to evaluate this innermost sum. However, we notice that if D ∈ M n , then F ∈ M 0 and hence F = 1, and Proposition 4.5 won't apply (the main term and the error term are the same size), so we must treat this case separately. That is, we note that Hence, if deg(D) < n then by applying Proposition 4.5 we get (since D is always square-free) that
F ∈M n−deg(D) G(1, F D) =G(1, D) F ∈M n−deg(D) (F,D)=1 G(D, F ) =G(1, D)ρ(n − deg(D); D) G(1, D) |D| 2/3 q 4(n−deg(D))/3 ζ q (2) P |D 1 + 1 |P | −1 + O ǫ G(1, D) q ( 1 3 +ǫ)(n−deg(D)) |D| 1 6 + q (n−deg(D)) |D| 1 4 +ǫ = ρ(n; 1) ζ q (2) q 4n/3 |D| P |D 1 + 1 |P | −1 + O ǫ q ( 1 3 +ǫ)n |D| ǫ + q n |D| 1 4 −ǫ .
Hence, summing the main term over D, we find that
D∈M<n (D,H)=1 µ(D)a D |D| P |D 1 + 1 |P | −1 = D∈M (D,H)=1 µ(D)a D |D| P |D 1 + 1 |P | −1 + O 1 q n = P ∤H 1 − a P |P | + 1 + O 1 q n ,
and summing the error term over D, we find that
D∈M<n (D,H)=1 a D q ( 1 3 +ǫ)n |D| ǫ + q n |D| 1 4 −ε ≪ q n .
Replacing in (4.3), this finishes the proof.
Applying the proposition, we obtain
f1∈Mn 1 f3∈Mn 3 G(1, f 1 ) |f 1 | 2/3 P |f1f3 1 + 1 |P | −1 = f3∈Mn 3 P |f3 1 + 1 |P | −1 f1∈Mn 1 G(1, f 1 ) |f 1 | 2/3 P |f1 P ∤f3 1 − 1 |P | + 1 = ρ(n 1 ; 1)q 2n1/3 ζ q (2) P 1 − 1 (|P | + 1) 2 f3∈Mn 3 P |f3 1 + 1 |P | + 1 −1 + O ǫ q n1/3+n3
Now, with the observation that ρ(m + 3n; 1) = ρ(m; 1) and
q 2n1/3 = f1∈Mn 1 1 |f 1 | 1/3
we can write n1+3n3≤3g−3A−1 ρ(n 1 + 1; 1)ρ(n 1 ; 1)q 2n1/3 q (1−s)(n1+3n3)
f3∈Mn 3 P |f3 1 + 1 |P | + 1 −1 = n≤3g−3A−1 ρ(n + 1; 1)ρ(n; 1) q (1−s)n f1f 3 3 ∈Mn 1 |f 1 | 1/3 P |f3 1 + 1 |P | + 1 −1 .
Using the definition of ρ(d; f ), we find that ǫ(χ 3 )q 5/6 ρ(n + 1; 1)ρ(n; 1) = q 3|n 0 3 ∤ n Hence we can write Lemma 4.9. Let C q is as in Theorem 1.1. For any ǫ > 0 and s < 1 − ǫ, if s = 1 3 , we have
M T s (3g, 3A) = M M T s (3g, 3A) + O(M ET s (3g, 3A)) where M M T s (3g, 3A) := q ( 4 3 −s)3g+1 ζ 2 q (2) P 1 − 1 (|P | + 1) 2 m≤g−A−1 1 q (1−s)3mM M T s (3g, 3A) = −C q q 3g+1 ζ q (2) q (1−3s)A 1 − q 3s−1 +O ǫ q (1+ǫ)g+3(1−s+ǫ)A + q ( 4 3 −s)3g+1 E s (3g, 3A) while if s = 1 3 , we have M M T s (3g, 3A) = C q q 3g+1 ζ q (2) g − A − 1 3 + 1 q − 1 + P deg(P ) |P | 3 + 2|P | 2 − 1 +O ǫ q (1+ǫ)g+(2+ǫ)A + q 3g+1
Proof. We first note that we can rewrite the series in M M T s (3g, 3A) as
m≤g−A−1 f1f 3 3 ∈M3m 1 |f 1 | 4 3 −s |f 3 | 3(1−s) P |f3 1 + 1 |P | + 1 −1 . (4.4)
We consider the generating series
D(v) := f1,f3∈M 1 |f 1 | 4 3 −s |f 3 | 3(1−s) P |f3 1 + 1 |P | + 1 −1 v deg(f1f 3 3 ) = f1∈M v deg(f1) |f 1 | 4 3 −s f3∈M P |f3 1 + 1 |P | + 1 −1 v 3 deg(f3) |f 3 | 3(1−s) , and we compute f1∈M v deg(f1) |f 1 | 4 3 −s = f1∈M v q 4 3 −s deg(f1) = ζ q v q 4 3 −s = 1 1 − q s− 1 3 v and f3∈M P |f3 1 + 1 |P | + 1 −1 v 3 deg(f3) |f 3 | 3(1−s) = P 1 + 1 + 1 |P | + 1 −1 v q 1−s 3 deg(P ) 1 − v q 1−s 3 deg(P ) = P 1 − v q 1−s 3 deg(P ) P 1 − v 3 deg(P ) |P | 3(1−s) |P | + 2 = ζ q v 3 q 3(1−s) P 1 − v 3 deg(P ) |P | 3(1−s) (|P | + 2)
.
Let K s (v) := P 1 − v 3 deg(P ) |P | 3(1−s) (|P | + 2)
which is an analytic function on the region |v| < q 1−s . Then
D(v) = K s (v) (1 − q 3s−2 v 3 )(1 − q s− 1 3 v) , (4.5)
can be meromorphically extended to the region |v| ≤ q 1−s−ǫ with poles at v = q 1 3 −s and v = ξ j 3 q 2 3 −s for j = 0, 1, 2 where ξ 3 is a primitive root of unity. Notice that K s (v) is uniformly bounded for |v| ≤ q 1−s−ǫ .
Therefore, as long as
s < 1 − ǫ, if we set Γ 4 = {v : |v| = q 1−s−ǫ }, we get that if s = 1 3 , 2 3 then 1 2πi Γ4 D(v) 1 − v 3 dv v 3(g−A−1)+1 = Res v=0 D(v) (1 − v 3 )v 3(g−A−1)+1 + Res v=q 1 3 −s D(v) (1 − v 3 )v 3(g−A−1)+2 + 2 j=0 Res v=ξ j 3 q 2 3 −s D(v) (1 − v 3 )v 3(g−A−1)+1 + Res v=ξ j 3 D(v) (1 − v 3 )v 3(g−A−1)+1
By the same computation as in Lemma 3.4 we have that (4.4) is exactly the residue at v = 0 and the integral over Γ 4 is bounded by q (s−1+ǫ)(3(g−A−1)+1) . Multiplying by
q ( 4 3 −s)3g+1 ζ 2 q (2) P 1 − 1 (|P | + 1) 2 , (4.6)
the residue at s = 0 gives M M T s (3g, 3A), and the contribution of the integral over Γ 4 is bounded by
q ( 4 3 −s)3g q 3(s−1+ǫ)(g−A) = q (1+ǫ)g+3(1−s+ǫ)A . (4.7)
So, it remains to compute the contribution (to M M T s (3g, 3A)) of the other residues.
For v = q 1 3 −s , we have Res v=q 1 3 −s D(v) (1 − v 3 )v 3(g−A−1)+1 = lim v=q 1 3 −s −q 1 3 −s K s (v) (1 − q 3s−2 v 3 )(1 − v 3 )v 3(g−A−1)+1 = − K s (q 1 3 −s ) 1 − q −1 q (3s−1)(g−A−1) 1 − q 1−3s .
We now observe that ζ q (2) = 1 1−q −1 and
P 1 − 1 (|P | + 1) 2 K s (q 1 3 −s ) = P 1 − 1 (|P | + 1) 2 1 − 1 |P | 2 (|P | + 2) = P 1 − 1 |P |(|P | + 1) = C q .
Therefore, multiplying the residue at v = q 1 3 −s by (4.6), the contribution of the residue to M M T s (3g, 3A) is
C q q ( 4 3 −s)3g+1 ζ q (2) q (3s−1)(g−A−1) 1 − q 1−3s = −C q q 3g+1 ζ q (2) q (1−3s)A 1 − q 3s−1
which gives the main term of M M T s (3g, 3A) for s = 1 3 . We now use the fact that K s (v) is invariant under multiplication of v by cube roots of unity to get that the sum the residues at v = ξ j 3 q 2 3 −s will be First we need to write the functions det(1 − U ) and det(1 − ∧ 3 U ) in a common basis. For any infinite tuple (λ j ) of non-negative integers with only finitely many non-zero entries we may write the partition λ = ∞ j=1 j λj (5.2) consisting of λ j copies of j.
K s (q 2 3 −s ) 1 − q 2−3s q (3s−2)(g−A−1) 2 j=0 Res v=ξ j 3 q 2 3 −s ξ −j 3 q s− 2 3 (1 − ξ j 3 q 1 3 )(1 − q 3s−2 v 3 ) = K s (q 2 3 −s ) 1 − q 2−3s q (3s−2)(g−A−1) 2 j=0 lim v→ξ j 3 q 2 3 −s ξ −j 3 q s− 2 3 (v − ξ j 3 q 2 3 −s ) (1 − ξ j 3 q 1 3 )(1 − q 3s−2 v 3 ) = − K s (q 2 3 −s ) 1 − q 2−3s q (3s−2)(g−A−1) 2 j=0 lim v→ξ j 3 q 2 3 −s 1 (1 − ξ j 3 q 1 3 ) i =−j (1 − q s− 2 3 ξ i 3 v) = K s (q
The Newton identities then tells us for any U ∈ U (N ), we can write
det(1 − U ) = |λ|≤N (−1) ℓ(λ) z λ P λ (U )
where ℓ(λ) = λ j is the length of λ and
P λ (U ) = ∞ j=1
Tr(U j ) λj and z λ = ∞ j=1 j λj λ j !.
A result of Diaconis and Evans then shows that these P λ (U ) form an orthogonal basis.
Theorem 5.1 (Theorem 2.1 from [DE2001]). For any partitions λ, µ with min(|λ|, |µ|) ≤ N , we have U(N ) P λ (U )P µ (U )dU = δ λµ z λ where δ λµ is the indicator function λ = µ.
So it remains to write det(1 − ∧ 3 U ) in the basis formed by P λ (U ). For this, we will need some new notation.
If λ, µ are partitions, written as in (5.2), then define the product λ · µ as λ · µ = ∞ j=1 j λj +µj , and by repeating this process, we can define positive integer powers of partitions
λ k = ∞ j=1 j kλj .
We denote the the zero partition as 0 = ∞ j=1 j 0 and so we obtain λ · 0 = λ and λ 0 = 0. Finally, for any positive integer k, we then define kλ = ∞ j=1 (kj) λj the partition consisting of λ j copies of kj. Now, let P 3 denote the set of partitions of 3 and consider a tuple of non-negative integers (a jµ ) j,µ where µ runs over the elements of P 3 , j runs over the positive integers and only finitely many of the a jµ are non-zero. Then, for any such tuple we define the partition λ(a jµ ) = To compute the sum over the (a jµ ) j,µ , we consider the generating series and Theorem 1.5 then follows from this and (5.1), using N = 3g.
Setup and Main Result. Let q = p a be a prime power and consider the ring of polynomials F q [T ] consisting of polynomials with coefficients in the finite field F q . Denote χ as a Dirichlet character of F q [T ] and M the set of monic polynomials in F q [T ]. Then define the L-function attached to χ as Hypothesis implies that there exists a conjugacy class of unitary matrices Θ χ , called the Frobenius class, such that L(s, χ) = (1 − q −s ) 1−δχ det(χ = 0 or 1 depending on the parity of χ. From here we can impose some trivial bound on the size of L(s, χ), and we have as q tends to infinity |L(s,
1. 3 .
3Weighted Random Matrix Model. An important set of continuous class functions are the mixed trace functions P λ (U ) := ∞ j=1
Contributions of the Principal Sum. This section is devoted to proving contribution of the principal sum.
Proposition 3 . 1 .
31For any ǫ > 0 and s > ǫ, we have if s = 13 P s (3g, 3A) |H(3g)| = M (s)+C q q (1−3s)A 1 − q 3s−1 +O ǫ q (ǫ−3s)A +q A−(1−ǫ)(3g+1) +q (1−s+ǫ)3A− 1 2 (3g+1)
for any ǫ > 0. The first bound is the Lindelöf hypothesis [BCDGL2018, Theorem 5.1] and the second bound is proven in [DFL2022, Lemma 2.6].Lemma 3.3. For any ǫ > 0, we have
For any (f, F ) = 1 and V , we haveχ F (f )G(V, F ) = G(f V, F ).Applying Lemmas 2.1 and 4.3, with the observation that χ f (F ) = 0 if (F, f ) = 1, we get that
1 2 ( 3 2
13−σ)+ǫ from [DFL2022, Proposition 3.11]. Taking σ = 1, we get the result.
1, F D) = G(D, F )G(1, D) (F, D) = 1 0 (F, D) = 1 .
Lemma 4 . 8 ..
48For any value of A, we get thatM ET s (3g, 3A) ≪ q ( 4 3 −s)(3g+1) E s (3g, 3A)Proof. Indeed, we have M ET s (3g, 3A) = q ( Computing the Main Dual Term.
..
Proof of Theorem 1.1. By (2.21) and Propositions 3.1 and 4.1, we have for any A = cg with 0 < c < 1, and ǫ < s < 1 − ǫ but s = ) + q (1−3s)g E s (3g, 3A) Optimizing the error term, we chose the cut off that equalizes the first two term error terms. Hence, we chose A = 3 5 g and conclude that for s Proof of Theorem 1.5. We see from the definition of C q that lim q→∞ C q = 1 and by the prime polynomial theorem we get to 0 as q tends to ∞. Hence we get from Theorem 1.1 − U )det(1 − ∧ 3 U )dU to see that it matches.
have used the fact that if λ = λ(a jµ ) then
F
(det(1 − U )det(1 − ∧ 3 U )dU =
Gg will be a subgroup of U (g) and will typically be one of U (g), U Sp(g), SO(g), SOeven(g), SO odd (g).
and, similarly, the sum over residues at v = ξ j 3 will be.Furthermore, we see that K s (q 2 3 −s ) is independent in s so that for any s = 2 3 , we have that multiplying the sum of the residues at v = ξ j 3 q 2 3 −s by (4.6), we get a contribution bounded bySimilarly, for s < 1 − ǫ, we have that K s (1) is bounded and so the contribution from the residues at v = ξ j 3 will be bounded, and multiplying by (4.6), this gives a contribution bounded by q ( 4 3 −s)3g+1 . This completes the proof for s = 1 3 , 2 3 . For s = 2 3 , the residue at v = q s− 1 3 remain the same but we now get double poles at v = ξ j 3 . In this case we getFor s = 1 3 , the residues at v = ξ j 3 q s− 2 3 , j = 0, 1, 2 and ξ j 3 , j = 1, 2 remain the same but we now have a a double pole at v = 1 with residue(1)K 1 3 (1) Multiplying by (4.6), we get a contribution of(1), and we finish the proof with the observation thatWe can now combine Lemmas 4.6, 4.8 and 4.9 to prove Proposition 4.1.
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Patrick Meisner, address: [email protected] of Mathematical Sciences. Gothenburg, Sweden EmailChalmers University of TechnologyPatrick Meisner: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden Email address: [email protected]
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[] |
[
"A RELATION BETWEEN TWO DIFFERENT FORMULATIONS OF THE BERRY'S CONJECTURE",
"A RELATION BETWEEN TWO DIFFERENT FORMULATIONS OF THE BERRY'S CONJECTURE"
] |
[
"Alba García-Ruiz "
] |
[] |
[] |
The Random Wave Conjecture of M. V. Berry is the heuristic that eigenfunctions of a classically chaotic system should behave like Gaussian random fields, in the large eigenvalue limit. In this work we collect some definitions and properties of Gaussian random fields, and show that the formulation of the Berry's conjecture proposed using local weak limits is equivalent to the one that is based on the Benjamini-Schramm convergence. Finally, we see that both these formulations of the Berry's property imply another property known as inverse localization that relates high energy eigenfunctions and solutions to the Euclidean Helmholtz equation.
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[
"https://export.arxiv.org/pdf/2305.14906v1.pdf"
] | 258,866,064 |
2305.14906
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73b1b08b1bbb0b9ec076ab62623e3c94924b3af5
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A RELATION BETWEEN TWO DIFFERENT FORMULATIONS OF THE BERRY'S CONJECTURE
24 May 2023
Alba García-Ruiz
A RELATION BETWEEN TWO DIFFERENT FORMULATIONS OF THE BERRY'S CONJECTURE
24 May 2023
The Random Wave Conjecture of M. V. Berry is the heuristic that eigenfunctions of a classically chaotic system should behave like Gaussian random fields, in the large eigenvalue limit. In this work we collect some definitions and properties of Gaussian random fields, and show that the formulation of the Berry's conjecture proposed using local weak limits is equivalent to the one that is based on the Benjamini-Schramm convergence. Finally, we see that both these formulations of the Berry's property imply another property known as inverse localization that relates high energy eigenfunctions and solutions to the Euclidean Helmholtz equation.
Introduction
In his influential papers [8,9] Berry gave a heuristic description of the behavior of high-energy wave-functions of quantum chaotic systems. He suggested that highfrequency eigenfunctions of the Laplacian in geometries where the geodesic flow is sufficiently chaotic should, in some sense, at the wavelength scale, behave like an isotropic Gaussian field Ψ Berry with covariance function
(1.1) E [Ψ Berry (x)Ψ Berry (y)] = S d−1 e 2πi(x−y)θ dω d−1 (θ) = c d J Λ (|x − y|) |x − y| Λ ,
where dω d−1 is the uniform measure in the (d − 1)-sphere, J Λ is the Λ-th Bessel function of the first kind with Λ = d−2 2 and c d > 0 is a constant such that we have E |Ψ(x)| 2 = 1.
This ambiguous comparison between a deterministic system and a stochastic field is known as the Random Wave Model (RWM). The RWM was first introduced in the study of chaotic quantum billiards on flat domains and a weaker version of it was proved in the context of random regular graphs [5]. The RWM has led to many conjectures concerning L p norms, semi-classical measures or volume and topology of nodal domains of chaotic eigenfunctions. Several of these conjectures have been addressed numerically ( [16], [3], [4], [6]) or experimentally ( [18], [7], [20]).
However, there is no agreement on how Berry's conjecture should be formulated rigorously because the idea of a sequence of deterministic objects having a random limit can be interpreted in different ways. The reader can for instance refer to [27], [28], [24] and [19] for different mathematical perspectives on Berry's conjecture. Some of these conjectures focus only on the values of the eigenfunctions, viewing (M, d Vol M ) as a probability space and each eigenfunction as a random variable.
These formulations do not provide any insight into the properties of the nodal set, number of nodal domains and other related questions. Some other formulations, as the one in [10], have led to groundbreaking results concerning the nodal sets of monochromatic waves satisfying the RWM (see, for instance, [26]).
In [21], [22] and [1] one can find two different formulations of the Random Wave Conjecture that takes into account both the shape of the eigenfunction and its distribution of values. Although both formulations are based on different notions (the one proposed in [21] is related to what the author calls local weak limits, defined using a covering by charts of the manifold, and the one from [1] makes heavy use of the Benjamini-Schramm convergence), the idea behind is similar: for a random x ∈ M , the eigenfunction ψ n Exp x (·/ √ λ n ) should behave like a random monochromatic wave as long as the geodesic flow on (M, g) is chaotic. In this work we show that both formulations of the Berry conjecture are equivalent in the context of a compact manifold M . As a consequence of the equivalence in compact manifolds, we are able to ensure that the local weak limit formulation does not depend on the choice of charts. Note that, doing the necessary identifications, in the graph theory literature it can be seen that Benjamini-Schramm limit and local weak limit are the same notions.
In the final section, we will introduce a related notion, the one of the inverse localization property, and show that, with any of these formulations, the Berry property implies the inverse localization. Roughly speaking, we say that a compact manifold M satisfies the inverse localization property if we can approximate an arbitrary solution of the Helmholtz equation on R d using high energy eigenfunctions of the manifold. For the time being, all the known examples of manifolds satisfying this property do have a high multiplicity, that seems to be key to construct the approximating eigenfunction. However, the version of the inverse localization that we get from the Berry property is in some sense stronger: there exists a sequence of eigenfunctions associated to different eigenvalues such that any solution to Helmholtz equation can be approximated by one of these. Notice that the sequence does not depend on the solution to Helmholtz equation chosen and that for any eigenvalue we are considering just one eigenfunction associated to it, so the degeneracy of eigenvalues plays no role here.
Organization of the paper. In Section 2 the properties asked to M are specified and some needed definitions are given. In Section 3, local weak limit and BS convergence are defined and the two different formulations of the Berry property are stated. The main theorem of the paper is as follows. of open subsets with some extra conditions (specified at the beginning of Section 2), then Berry's property in the local weak limit form (given in definition 3.2) is equivalent to Berry's property in BS form (as defined in 3.9).
Sections 4 and 5 are devoted to prove both implications of this theorem. Finally, in Section 6, the inverse localization property is introduced and it is proved that the Berry property implies the former. To conclude, some comments are made around this idea.
Starting definitions and notation. Gaussian fields
In all what follows, we consider a compact connected Riemannian manifold of dimension d without boundary, (M, g). dx is the volume measure on M and we will denote by ∆ the Laplace-Beltrami operator on the manifold. An easy application of the classical spectral theorem for compact manifolds ensures that there exists (ψ n ) n an orthonormal basis of L 2 (M ) that consists of functions on M such that ∆ψ n + λ n ψ n = 0 and ψ n 2 2 = Vol(M ). Assume that the eigenvalues are ordered in non-decreasing order. We may furthermore suppose that we have a finite family {U m } mmax m=1 of open subsets with the following properties: For each x ∈ M , we will denote by exp x : T x M → M the exponential map at x induced by the metric g on M . Moreover, given x and y in M , we will denote by dist(x, y) the Riemannian distance between x and y.
• M ⊂ mmax m=1 U m , • for any m 1 = m 2 we have U m1 ∩ U m2 = ∅
Unless otherwise stated, the spaces C ∞ (M ) and C ∞ (R d ) will be equipped with the topology of uniform convergence of derivatives on compact sets. We will also define in C ∞ (R d ) a distance d given by the Fréchet structure of C ∞ (R d ). This means that we first define the class of semi-norms
(2.1) f k,n = sup f (k) (x) : x ∈ B(0, n)
and then the distance given by
(2.2) d(f, g) = ∞ n=0 ∞ k=0 2 −k−n f − g k,n 1 + f − g k,n .
With this distance, we recover the smooth topology in C ∞ (R d ) and we can use it to define balls B(f, ǫ). Notice that C ∞ (R d ) with this distance is a locally compact metric space. Moreover, when we speak of probability measures on these spaces, we will assume that they are equipped with the Borel σ-algebra. Our next goal is to introduce the field Ψ Berry . It is a stationary Gaussian field on R d whose spectral measure is the uniform measure on the unit sphere S d . We first recall some definitions:
Definition 2.1 (Smooth random field). A smooth random field (on R d ) is a map X from a probability space (Ω, B, P ) to C ∞ (R d ) that is measurable, where C ∞ (R d )
is considered with the topology of convergence of all derivatives over all compact sets. Notice that for any n ∈ N and any t 1 , . . . , t n ∈ R d , the vector (X(t 1 ), X(t 2 ), . . . , X(t n )) is a random vector. Definition 2.2 (Gaussian random field). A Gaussian random field is a (smooth) random field where all the finite dimensional distributions F t1,...,t k (x 1 , . . . , x k ) = P (X(t 1 ) x 1 , . . . , X(t k ) x k ) are multivariate normal distributions for any choice of k and {t 1 , . . . , t k }. Since multivariate normal distributions are completely specified by expectations and covariances, to determine a Gaussian random field X t it suffices to specify m(t) := E{X t } and C(t, s) := Cov{X t , X s } in an appropriate way. If m ≡ 0 we say that the Gaussian field is centered. Definition 2.3 (Stationary and isotropic random field). We say that a Gaussian random field is stationary if C(t, s) depends only on t − s and m(t) = m is constant. An isotropic Gaussian random field is a stationary Gaussian random field whose covariance function depends on the distance alone, i.e. C(t, s) = C(τ ) where τ = dist(t, s).
Definition 2.4 (Law and equivalence of random fields). For a random field X, we define its law as the probability measure µ X : B(C ∞ (R d )) → R given by µ X = P X −1 : B(R) → R, such that for any Borel set A ∈ B(C ∞ (R d )), µ X (A) = P (X −1 (A)). Here P is the probability we have in the probability space (Ω, B, P ) where X is defined.
We say that two random fields X 1 and X 2 are equivalent if they have the same law, i.e. µ X 1 = µ X 2 . In the sequel we will always identify fields which are equivalent and we will speak indifferently of the field and its law.
As explained for instance in Section A.11 of [23] there is a bijection between smooth centered Gaussian fields on R d and positive definite functions defined on R d × R d . Also, recall that, by Bochner's theorem [11, Section 2.1.11], for any finite Borel complex measure µ on R d , its Fourier transformμ can be used to define a continuous positive definite function K(x, y) =μ(x − y). If, in addition, µ is compactly supported, its Fourier transform is of class C ∞ , and gives rise to a unique smooth Gaussian field X on R d (up to law equivalence). In this case, we call µ the spectral measure of X.
Definition 2.5 (Ψ Berry , the random monochromatic wave). We call random isotropic monochromatic wave, and denote by Ψ Berry , the unique stationary Gaussian field on R d whose spectral measure is the uniform measure on the unit sphere S d . This field Ψ Berry : R d → R is uniquely defined as the centered stationary Gaussian random field, with covariance function
(2.3) E [Ψ Berry (x)Ψ Berry (y)] = S d−1 e 2πi(x−y)θ dω d−1 (θ),
where dω d−1 is the uniform measure on S d−1 .
Let us consider the space F P := f ∈ C ∞ (R d ), s.t. − ∆f = f . Then it is easy to check that Ψ Berry is almost surely an element of F P . This way, if A ⊂ F P is a Borel set, P (Ψ Berry ∈ A) is well-defined and
(2.4) µ Berry : B(F P ) → R + 0 A → P (Ψ Berry ∈ A),
is a Borel measure on F P ⊂ C ∞ (R d ). In other words, the probability measure µ Berry defined on C ∞ (R d ) is supported on F P . In what follows we will work with this measure µ Berry and with Ψ Berry indistinguishably.
Statement of the conjectures
We first consider the formulation of the conjecture proposed in [22]:
Recall that we have a finite family {U m } mmax m=1 of open subsets on which there exists a family of vector fields (V m 1 , . . . , V m d ) forming an orthonormal frame of the tangent bundle. For a given x ∈ U m , we define the function Exp x : R d → R
as Exp x (y) := exp x d j=1 y j V m j (x) . Let p be a random point in U m chosen uniformly with respect to the volume measure dx. For each n, let φ n p ∈ C ∞ R d be the random field defined by φ n p (y) := ψ n Exp p (y/ √ λ n ) . It is a random element of C ∞ (R d ). Notice that the definition of φ n p depends on the chart U m chosen. Definition 3.1 (Convergence in law). Let µ Berry be the probability measure on C ∞ (R d ) associated to Ψ Berry . We say that φ n p (y) converges in law as a random field towards Ψ Berry in the frame U if, for any continuous, bounded functional
F : C ∞ (R d ) → R, we have (3.1) 1 Vol(U ) U F φ n p dp − −−− → n→∞ E µBerry [F ] = C ∞ (R d ) F dµ Berry .
Then the conjecture as formulated in [22] can be stated as follows.
Definition 3.2 (Formulation of Berry's conjecture in the local weak limit form).
We have the Berry property in the local weak limit form if for any U m , as λ n → ∞ when n → ∞, the family φ n p (y) converges in law as a random field towards the Gaussian field Ψ Berry . . . , m max , and so definition 3.2 does depend a priori on them. However, if it holds for one choice of sets U m and frames V m , then it also holds for any other choice. This can be seen, for example, as a consequence of the equivalence between this formulation and the one using Benjamini-Schramm convergence that we prove here.
On the other hand, we consider the formulation of the conjecture introduced in [1]. For the d-dimensional Riemannian manifold M = (M, g), let M n = (M, g n ) denote the rescaling of M by the factor √ λ n , i.e. we change only the metric by multiplying every distance by √ λ n . A property of this M n is that if φ : M → R is an eigenfunction of the Laplacian on M associated to an eigenvalue λ, then the very same function is also an eigenfunction on M n with eigenvalue λ ′ = λ/λ n .
Just like it is done in [1], we start by considering the space M d of pointed, connected, complete Riemannian manifolds of dimension d up to pointed isometries, with its smooth topology. The reader should see [2, §A.1] for a precise definition of this topology, however the philosophy is as follows: two pointed manifolds (M, p) and (N, q) are close if there exist two compact subsets of M and N containing large neighborhoods of the base points p and q respectively that are diffeomorphic via a map φ, that is close in the C ∞ metric to an isometry.
Another way of seeing this is the following: a sequence of pointed Riemannian manifolds, let us say (M n , p n ), converges in the C ∞ metric towards (M, p) if for every radius R > 0, there exists a sequence of maps f n : B M (p, R) → M n with f n (p) = p n such that the Riemannian metric f * g n on the metric ball B M (p, R) inside M , pulled back from the Riemannian metric g n on M n , converges to the restriction of the Riemannian metric g on M in C ∞ -topology. It can be proved that the smooth topology on M d is induced by a Polish topology, i.e. the space M d is separable and completely metrizable; see [2, §A.2] for a proof of this result. The space M d is not compact but Cheeger's compactness theorem can be used to show that the subspace consisting of pointed manifolds (M, p) with uniformly bounded geometry is a compact subspace; see [2, §A.1] for the proof. The next step is to construct a measure associated to (M n , p). Pushing forward the normalized Riemannian volume measure under the following map one obtains a probability measure µ M on M d .
(3.2) M → M d , p → (M, p)
We now recall a common notion of convergence for these measures and use it to state the definition of BS convergence. Beware that some authors refer to the topology of the previous definition as weak topology. Following [1], let us now explain a particularity of the measures thus obtained, though we will not use it in the sequel.
Let
T 1 M d be the space of isometry classes of unit tangent bundles T 1 p M, p, v , where v ∈ T 1 p M . The geodesic flows on each T 1 p M combine to give a continuous flow on T 1 M d , (3.3) σ t : T 1 M d → T 1 M d
What is more, in each fiber T 1 p M of the following map it one can define a (Liouville) measure ω M,p induced by the Riemannian metric on M .
(3.4) T 1 M d → M d ; (M, p, v) → (M, p)
Any measure µ on M d can then be lifted to a measure µ on T 1 M d defined by the equation d µ = ω M,p dµ. The Liouville measure on the unit tangent bundle of a Riemannian manifold is invariant under the geodesic flow and, similarly, the measure µ M is invariant under the flow g t .
Using the same notation as in [2], we say that a measure µ on M d is unimodular if µ is invariant under (3.3). We also refer to [2] for other characterizations of unimodularity. Then we have just seen the following remark, that we will not use in the proofs.
Remark 3.6. The weak * limit of µ Mn is a unimodular probability measure on M d .
In general, around a randomly chosen point, M n has no reason to be similar to a given manifold for large n. Instead, the limiting object is a unimodular probability measure on M d that precisely encodes how the manifold, for large n, looks like near randomly chosen base points. This perspective is studied in great details in [2]. By work of Cheeger and Gromov (see, for example [25,Chapter 10]), the subset of M d consisting of pointed manifolds (M, p) with bounded geometry is compact. Here, bounded geometry means that the sectional curvatures of M , and all of their derivatives, are uniformly bounded, and the injectivity radius at the base point p is bounded away from zero. Compact manifolds are examples of manifolds with bounded geometry. By the Riesz representation theorem and Alaoglu's theorem, this implies that the set of unimodular probability measures supported on manifolds with bounded geometry is weak * compact, since unimodularity is a weak * closed condition.
One can similarly define the BS-convergence of a sequence {M n } ∞ n=1 of manifolds with associated functions φ n : M n → R. We follow again [1]. Consider the space
(3.5) E d = (M, p, φ) M connected, complete d-manifold, p ∈ M, φ : M → R smooth pointed isometries equipped with its smooth topology where [M, p, φ] is close to [N, q, ψ]
if there are compact subsets of M and N containing large radius neighborhoods of p and q respectively, that are diffeomorphic via a map D that is C ∞ -close to an isometry and that also satisfies that φ and ψ • D are C ∞ close. As seen in [1,Proposition 8], the topological space E d has a compatible structure of a Polish space (i.e. a complete, separable metric space).
We can now equip the topological space E d with the σ−algebra B, generated by its open sets, and define a probability measure on E d as a σ−additive function B → [0, 1] that maps the whole E d to 1. Similarly to what we did before, we can recall the following notion of convergence: Definition 3.7 (Convergence in the weak* topology). A sequence of probability measures µ n on E d is said to converge in the weak * topology towards a probability measure µ if for each bounded, continuous real function F on E d we have
(3.6) lim n→∞ µ n (F ) = µ(F ).
For any M and φ : M → R, smooth map, we can consider the push forward of the normalized Riemannian volume measure under the map
(3.7) M → E d , p → [M, p, φ]
and obtain a probability measure µ M,φ on E d . As we did with M d , we shall denote by T 1 E d the space of isometry classes of tangent bundles with a function
T 1 M, p, v, φ where v ∈ T 1 p M .
Here again it comes equipped with a continuous (geodesic) flow and any measure µ on E d can be lifted to a measure µ on T 1 E d using the volume form on the fiber. However, the measure µ Mn,φn is not invariant under the geodesic flow unless φ n is constant on M n . Let us try to write this otherwise. We recall from the introduction that the probability measure µ Berry associated to the random field Ψ Berry is supported on functions u such that −∆u = u. In other words, Ψ Berry is almost surely an eigenfunction of eigenvalue 1 of the Laplacian and, in particular, Ψ Berry is almost surely a smooth function on R d .
Moreover, notice that, as a Riemannian metric is infinitesimally Euclidean, it follows from the definition that, as n → ∞, the sequence (M n ) BS-converges to R d . Precisely, because of the weak * compactness, µ Mn converges to a unimodular measure on M d . By the rescaling, M n is closer to R d whenever n → ∞, so the limit towards which µ Mn converges should be the Dirac measure at [R d , 0] ∈ M d . Note that R d being homogeneous, the limit measure does not depend on a particular choice of base point. We loosely say that (M n ) BS-converges towards R d . Also remember that if φ : M → R is an eigenfunction of the Laplacian on M with eigenvalue λ, then the very same function is an eigenfunction on M n with eigenvalue λ ′ = λ/λ n .
We consider now the sequence inside E d given by in R d . In the case of the sequence we are considering this is precisely G([M n , p, ψ n ]) = G(ψ n (Exp p (·/ √ λ n ))), where we can use the orthonormal basis around p because λ n → ∞ and G can be seen as a functional on C ∞ (R d ). Notice that we are making an abuse of notation by using the same letter G for both a functional on E d and C ∞ (R d ). Then, using the exponential map Exp, µ Mn,ψn induces a measure on C ∞ (R d ), call it ν Mn,ψn whose limit (which we will assume that exists) is supported in functions satisfying the Helmholtz equation ∆ψ + ψ = 0, denoted by F P . This can be seen in the following lemma.
Lemma 3.10. The limit lim n→∞ ν Mn,ψn is a probability measure supported in F P .
Proof. Let us proceed by contradiction and assume the opposite. Suppose that there exists a function f ∈ C ∞ (R d ) which is not in F P but it is in the support of the limit measure lim n→∞ ν Mn,ψn , meaning that for any ǫ > 0 it is true that lim n→∞ ν Mn,ψn (B(f, ǫ)) > 0. Here the ball B is constructed using the distance d given by the Fréchet structure of C ∞ (R d ) defined in (2.2), and has positive mass asymptotically.
Recall that C ∞ (R d ) with this distance is a locally compact metric space and that F P is closed in the topology of smooth convergence of derivatives on compact sets. Then we can find a small ǫ 0 > 0 such that dist B(f, ǫ 0 ), F P c ǫ0 > 0 and also
(3.9) lim inf n→∞ Vol p ∈ M : φ n p ∈ B(f, ǫ 0 ) c ′ ǫ0 > 0.
On the one hand, it can be easily found a lower positive bound for d(φ n p , −∆φ n p ). Just notice that the map g → d(g, −∆g) is continuous and strictly positive on the ball B(f, ǫ 0 ) (because it is not in F P ). Therefore, it is bounded from below by some positive constant ǫ 1 . In particular, as far as we consider the points p ∈ M such that φ n p ∈ B(f, ǫ 0 ), we have d(φ n p , −∆φ n p ) ǫ 1 > 0. Notice that this bound does not depend on n, for any n large enough.
On the other hand, it can be easily seen that for those points p with φ n p in B(f, ǫ 0 ), the set of φ n p is bounded in C ∞ (R d ) and so all the derivatives of φ n p are uniformly bounded in n. Also notice that (1)·d(φ n p , 0). In the second last equality, we have used that the distance is a series of seminorms, so that d(u, v) = d(u − v, 0). Finally, in the last part we have used that (∆ − ∆ Mn )φ n p k,m = C k,m / √ λ n . We get this expression thanks to the bound of all the derivatives of φ n p and the explicit form of ∆ Mn depending on the metric g n = √ λ n · g. To conclude we recall that, in particular, the eigenfunctions φ n p are uniformly bounded in n for the points p we are considering. As a consequence, d(∆φ n p , −φ n p ) = o n→∞ (1). This contradict the positive bound already found, and so we can assure that such an f does not exists, concluding that µ Berry is supported in F P .
(3.10) d(∆φ n p , −φ n p ) = d(∆φ n p , ∆ Mn φ n p ) = d((∆−∆ Mn )φ n p , 0) = o n→∞
The set of linear combinations of products of these two kinds of functions is a subset of C(E d ) that separates points, i.e., given distinct points α, β ∈ E d , there exists a linear combination of products of functions of this type such that
i F i (α)G i (α) = i F i (β)G i (β)
. By mean of the Stone-Weierstrass theorem, we get to know that this set is dense is the space of bounded, continuous functionals on E d . Therefore, to study the limit of a measure in the set E d , it is enough to study it over products of F and G. The sum can be taken outside of the limit using linearity of the integral.
Then we can conclude that any weak limit of µ Mn,ψn is supported in
(3.11) [R d , p, ψ] ∈ E d : p ∈ R d , ∆ψ = −ψ = [R d , 0, ψ] ∈ E d : ∆ψ = −ψ .
This set is equivalent to F P/ ∼ where f, g ∈ F P satisfy f ∼ g if, and only if, there exists an origin preserving isometry S such that f = g • S. We therefore loosely identify such a weak limit with a random field on R d .
All considered, we can now reformulate the conjecture proposed in [1] as follows: µ Mn,φn converges to µ Berry in the weak * topology or, what is the same, for each bounded, continuous real functional F of C ∞ (R d ) which is invariant under origin preserving isometries, we have Here F is only defined in C ∞ (R d ) but as both µ Berry and lim n→∞ ν Mn,φn are supported in [R d , p, ψ] ∈ E d : p ∈ R d , ∆ψ = ψ , we only care about these points. As we are only considering classes up to pointed isometries, we notice that, for any isometry S of R d that preserves the origin S(0) = 0, then it should be true that F (ψ) = F (ψ • S), this is, we only consider functionals that are invariant up to origin preserving isometries.
In the next two Sections we prove both implications of this theorem. First, in Section 4 we show that the conjecture of definition 3.2 implies the conjecture given by 3.9. Then, in Section 5, we prove the reciprocal, that the conjecture of definition 3.9 implies the conjecture given by 3.2.
Local weak limit formulation implies BS formulation
We start by noticing the following: let U ⊂ M be a Borel set of positive Lebesgue measure, and suppose that we have the Berry's conjecture with the local formulation on U . Then for any Borel set U ′ ⊂ U of positive Lebesgue measure, we also have the conjecture on U ′ . A proof of this fact can be found in [21] for the case of a domain of M = R d and can be easily modified for our case. This means that if we consider M covered by a family of open sets where the Berry property holds, then for any open subset with an orthonormal frame we would also have the Berry property in that subset.
We continue by considering the map F (H n (p))dp.
Let us now split this integral using the partition introduced at the beginning of Section 2: F (H n (p))dp.
As said in the previous section, it is enough to study products of functions of the two mentioned types:
(4.4) µ Mn,ψn (F · G) = 1 Vol(M ) mmax m=1 Um F ([M n , p])G(φ n p )dp.
Then,
(4.5) 1 Vol(M ) mmax m=1 Um F ([M n , p])G(φ n p )dp = mmax m=1 Vol(U m ) Vol(M ) 1 Vol(U m ) Um F ([M n , p])G(φ n p )dp .
Let us consider first the case when F ≡ c, some constant. Using, by hypothesis, that the Berry property is satisfied with the local weak limit formulation, we can assert that
(4.6) lim n→∞ ν Mn,ψn (c · G) = mmax m=1 Vol(U m ) Vol(M ) lim n→∞ 1 Vol(U m ) Um c · G(φ n p )dp = (4.7) mmax m=1 Vol(U m ) Vol(M ) E µBerry [c · G] = E µBerry [c · G] = µ Berry (c · G).
The next step is to notice that, since F depends only on the manifold and G is a bounded functional, we have for any m,
(4.8) Um F ([M n , p])G(φ n p )dp − Um F ([R d , 0])G(φ n p )dp (4.9) C · sup p∈M F ([M n , p]) − F ([R d , 0]) = C · ǫ n − −−− → n→∞ 0 Therefore, (4.10) lim n→∞ |µ Mn,φn (F · G) − µ Berry (c · G)| lim n→∞ C·ǫ n |µ Mn,φn (c · G) − µ Berry (c · G)| = 0, and so (4.11) lim n→∞ µ Mn,ψn (F · G) = lim n→∞ µ Mn,ψn (c · G) = lim n→∞ ν Mn,ψn (c · G) = µ Berry (c · G),
for any G bounded, continuous functional on C ∞ (R d ). c · G is also any bounded, continuous functional on C ∞ (R d ). Notice that here we have again made an abuse of notation by using the same letter G for a certain class of functionals on E d and for functionals on C ∞ (R d ). This concludes the proof of this implication.
BS formulation implies Local weak limit formulation
For the other implication we assume that lim n→∞ ν Mn,ψn (F ) = µ Berry (F ), for any bounded, continuous real functional F of C ∞ (R d ) which is invariant under origin preserving isometries. This means that (5.1)
lim n→∞ ν Mn,φn , F = mmax m=1 Vol(U m ) Vol(M ) lim n→∞ 1 Vol(U m ) Um F (φ n p )dp = µ Berry , F ,
when tested against bounded, continuous real functionals F of C ∞ (R d ) which are invariant under origin preserving isometries.
First, we will see that, up to a subsequence, for any m, there exists a limit µ m such that for any functional G (which is not necessarily invariant w.r.t. isometries) we have:
(5.2) ∃ lim n→∞ 1 Vol(U m ) Um G(φ n p )dp = µ m (G).
To prove this let us define the probability measure on the space of C ∞ (R d ), µ n m given by
(5.3) µ n m , G = 1 Vol(U m ) Um G(φ n p )dp,
for any G bounded, continuous functional; and recall Prokhorov's theorem for this particular case: the collection {µ n m } ∞ n=1 is tight if, and only if, its closure is sequentially compact in the space of probability measures on C ∞ (R d ), i.e., up to a subsequence, there exists the limit lim n→∞ µ n m = µ m . The only thing that is left to obtain (5.2) is to check that {µ n m } ∞ n=1 is tight but this is essentially [21,Lemma 1], where this measures are shown to be tight in the bigger space of C k functions. The aim now is to prove that this limit is equal to µ Berry for any 1 m m max .
As can be seen in [21], this limit µ m is translation invariant. We recall the proof of this fact here, but first we need to introduce few definitions.
Let
y ∈ R d . For any f ∈ C ∞ (R d ), we define τ y f ∈ C ∞ (R d ) by (τ y f )(·) = f (y+·). If F is now a functional on C ∞ (R d ), we define τ y F by (τ y F )(f ) = F (τ y f ) for all f ∈ C ∞ (R d ). Finally, if µ is a measure on C ∞ (R d ),
we define τ y µ by τ y µ, F = µ, τ y F for all bounded, continuous functionals F . Lemma 5.1. Let µ m be a limit as considered in this section. Then for any y ∈ R d , we have τ y µ m = µ m .
Proof. Notice that ∃ lim n→∞ µ n m = µ m . We now compute, for any y ∈ R d and F bounded, continuous functional,
(5.4) µ n m , τ y F = 1 Vol(U m ) Um F (τ y φ n p )dp = 1 Vol(U m ) Um F ψ n Exp p · + y √ λ n dp = (5.5) 1 Vol(U m ) Um F ψ n Exp Exp p (y/ √ λn) · √ λ n + O 1/ λ n dp = (5.6) 1 Vol(U m ) Um F φ n p dp + O 1/ λ n + O Vol O m ∆(O m − y/ λ n ) = (5.7) µ n m , F + o n→∞ .
Here O m is the Euclidean domain that goes to U m by Exp p , O m ∆(O m − y/ √ λ n ) is the symmetric difference between O m and O m − y/ √ λ n and its volume goes to zero with n, since
Vol O m ∩ (O m − y/ √ λ n ) − −−− → n→∞ 0.
In the second line we have used that fact that F is a bounded, continuous functional and we are using geodesic coordinates in a small neighborhood of the origin.
Taking the limit n → ∞, we obtain µ m (τ y F ) = µ m (F ) for any F , so that τ y µ m = µ m .
It is easy to check that (F P, B(F P ), µ Berry ) is a probability space on which R d acts by the translations τ as measure-preserving transformations. By the Fomin-Grenander-Maruyama theorem [23, Section B], µ Berry is ergodic for the action of the translations, which means that for every set A ∈ B(F P ) satisfying µ Berry ((τ y A) ∆A) = 0, either µ Berry (A) = 0, or µ Berry (A) = 1.
It can also be proved, see for instance [12], that ergodic measures are the extreme points of the set of the action invariant measures. This means that µ Berry can not be expressed as a strict convex combination of two different translation invariant probability measures. It can be generalized by induction to any finite-length convex combination ans it is also true for general integration as can be see by the following lemma.
Lemma 5.2. If µ is a Borel measure that is ergodic with respect to translations and we have a decomposition µ = Y v y dm(y), where Y is a measurable set, v y are all translations invariant probability measures and the integral is defined using the measure dm of Y . Then v y = µ for m-a.e. y.
Proof. Let T (C ∞ ) be the set of all continuous, bounded functionals in C ∞ (R d ). If both sets
(5.8) F ∈T (C ∞ ) {y ∈ Y, F, µ < F, v y } and (5.9) F ∈T (C ∞ )
{y ∈ Y, F, µ > F, v y } have measure zero then we have finished. Assume otherwise. Without loss of generality, we have a continuous functional F such that
(5.10) A = {y ∈ Y, F, µ < F, v y } satisfies 0 < m(A). Notice that if m(A) = 1 then we can not have µ = Y v y dm(y), so 0 < m(A) < 1. Now, we have that for all G ∈ T (C ∞ ), (5.11) G, µ = m(A) · G, 1 m(A) A v y dm(y) + m(A c ) · G, 1 m(A c ) A c v y dm(y) .
This gives a representation of µ as a convex sum of two translation invariant probability measures. By ergodicity of µ, each of the measures is in fact µ. But this is a contradiction since integrating the first measure against F is strictly bigger than F, µ . This conclude the proof.
Consider now the set Ω = ω : R d → R d , ω is an isometry preserving the origin and its Haar measure dm. For any bounded continuous functional G, we consider the following
(5.12) G = 1 |Ω| Ω G(ω(·))dm(ω),
which is defined using the Haar measure for the integral.
Obviously, this functional is still bounded and continuous and now it is invariant by isometries that preserve the origin.
Similarly to what we did before, if µ is a measure on C ∞ (R d ), we define µ by µ, G = µ, G for all bounded, continuous functionals G. This way, we have that, for all m and any bounded continuous functional G This means that evaluating µ m against any functional is the same as evaluating µ m against an isometry invariant functional. If we sum along m and use the hypothesis, we get (5.14)
m Vol(U m ) Vol(M ) µ m , G = m Vol(U m ) Vol(M ) µ m , G = µ Berry ( G) = µ Berry (G).
It is easy to check that µ m preserves the property of being translation invariant: µ m , τ y G = µ m , τ y G = µ m , G = µ m , G . Also notice that the measure µ Berry is invariant with respect to isometries so that (5.15)
µ Berry ( G) = µ Berry 1 |Ω| Ω G(ω(·))dm(ω) = 1 |Ω| Ω µ Berry (G)dm(ω) = µ Berry (G).
Then, we have µ Berry as a linear combination of translation invariant measures. By using Fomin-Grenander-Maruyama theorem, we conclude that any µ m = µ Berry .
The last step is to consider
(5.16) µ Berry , G = µ m , G = 1 |Ω| Ω µ m , G(ω(·)) dm(ω) = 1 |Ω| Ω µ m • ω, G dm(ω) = 1 |Ω| Ω (µ m • ω)dm(ω), G ,
where µ m • ω stands for µ m • ω, G = µ m , G(ω(·)) . In this calculation we have µ Berry as a linear combination of translation invariant measure µ m • ω. Using lemma 5.2, this ensures that, for almost all ω ∈ Ω, µ m • ω = µ Berry . Notice that it is enough to have one ω such that µ m • ω = µ Berry because then we can compose with ω −1 and get, µ m = µ Berry • ω −1 = µ Berry , because µ Berry is invariant. This is the same as saying that φ n p converges in law as a random field towards Ψ Berry in U m . To conclude, just notice that this being true for any m is the formulation of Berry's conjecture that we aimed to obtain.
The inverse localization property
A related notion to the random wave model is the idea of inverse localization introduced in [15] and related works of the same authors. The local behavior of an eigenfunction on any compact Riemannian manifold associated to large eigenvalues over length scales of order 1/ √ λ is given by a solution to the Helmholtz equation,
(6.1) ∆h + h = 0.
A well known partial converse due to Hörmander [17] is that, given any ball B ⊂ R d and any fixed solution h to the Helmholtz equation on B, one can pick a sequence of approximate Laplace eigenfunctions on the manifold whose behavior on a ball of radius 1/ √ λ reproduces that of h modulo a small error, λ being the approximate frequency of the approximate eigenfunction. A slightly imprecise but very intriguing question is whether one can replace approximate eigenfunctions by bone fide eigenfunctions in this estimate. The inverse localization principle gives an answer for this question in certain particular cases.
This idea was first introduced in [14], where the authors proved the inverse localization property for Beltrami fields on the three-dimensional sphere and flat torus. To our best knowledge, the only known examples of manifold satisfying the inverse localization property are certain flat tori [13] and the round sphere S d and all Riemannian quotients thereof [15]. In these works, the definition of inverse localization is the following. Definition 6.1. (Inverse localization) A compact manifold M has the inverse localization property if for some p ∈ M , any ǫ > 0, any r ∈ N ∪ {0} and any h : R d → R solution to the Helmholtz equation (i.e. ∆h + h = 0 in the whole R d ), there exists a sequence of eigenvalues λ k − −−− → k→∞ ∞ and a sequence of associated eigenfunctions ψ k , ∆ M ψ k + λ k ψ k = 0, that for any k large enough satisfy
(6.2) ψ k Exp p · √ λ k − h C r (B) < ǫ.
Here φ k p := ψ k Exp p · √ λ k is the standard localization of ψ k and B is the unit Euclidean ball on the Euclidean space.
In this definition, the point around which we localize is fixed beforehand and does not depend on h (in fact, in all the known examples we can approximate any solution to the Helmholtz equation around any fixed point of the manifold). Moreover, in these examples, there are no restrictions on the eigenfunctions that can be used to approximate. However, the idea of inverse localization is imprecise and the rigorous definition can vary in these and other aspects, for example, the norm and domain of the approximation. The Berry property considered here implies a slightly different version of this inverse localization idea defined as follows. Remark 6.3. Notice that Definition 6.2 does not imply Definition 6.1 because the set N depends on h, while in the latter the point p is fixed. However, it is stronger in the following two aspects:
• While for definition 6.1 to be true it is enough to have the existence of a sequence of eigenvalues along which we can approximate, in definition 6.2 the limit needs to be true for all the sequences of eigenvalues going to infinity. • In definition 6.2, the eigenfunction associated to each eigenvalue is always the same. This means that it is possible to approximate any h using the same sequence of eigenfunctions. The same is not true for definition 6.1, where different eigenfunctions associated to the same eigenvalue can be needed to approach different solutions to the Helmholtz equation.
We recall that, by the local weak limit formulation, if M has the Berry property it is true that for any bounded, continuous functional G : C ∞ (R d ) → R it holds that
(6.4) E p (G(φ k p )) − −−− → k→∞ E(G(Ψ Berry )).
We can then assert the following Proposition 6.4. If (M, g) satisfies the Berry's property, then it also exhibits strong inverse localization.
Proof. Fix a small error ǫ > 0. Let χ ∈ C ∞ (R d ) be an even decreasing function that is equal to 1 on (0, ǫ/2) and is supported on (0, ǫ). Fix a monochromatic wave h, a natural number r ∈ N and take the functional F defined as
(6.5) F (f ) := χ f − h C r (B)
for any f ∈ C ∞ (R d ). Explicitly, we have that
(6.6) F (f ) = χ f − h C r (B) = 1 if f − h C r (B) < ǫ/2 Smooth in the middle 0 if f − h C r (B) > ǫ.
In particular,
(6.7) F (φ k p ) = χ φ k p − h C r (B) ,
for any k. This non-linear functional is continuous and bounded. Since the Berry property holds by assumption, we have
(6.8) E p (F (φ k p )) − −−− → k→∞ E(F (Ψ Berry )).
Notice that (6.9)
E(F (Ψ Berry )) = C r (B) F (f )dµ Berry (f ) = C r (B) χ f − h C r (B) dµ Berry (f ) = (6.10) {f ∈C r , f −h <ǫ/2} χ ( f − h ) dµ Berry (f )+ {f ∈C r ,ǫ/2 f −h ǫ} χ ( f − h ) dµ Berry (f ) (6.11) = µ Berry {f ∈ C r (B), f − h C r (B) ǫ/2} + {f ∈C r ,ǫ/2 f −h ǫ} χ ( f − h ) dµ Berry (f ).
The next step is to show that the set of functions that are ǫ/2-close to h in C r (B) metric has positive Berry measure. This is standard and can be deduced from [23, Section A7]. A brief sketch of a proof is as follows.
First, recall that any solution to the Helmholtz equation can be expanded as a Bessel-Fourier series as (6.12) h
(x) = ∞ l=0 d l m=1 c lm J l+ d 2 −1 (|x|) |x| d 2 −1 Y lm x |x| .
Here J ν denotes the Bessel function of order ν and {Y lm (ξ)} is a real-valued orthonormal basis of d-dimensional spherical harmonics; the order l means that the spherical harmonic is the restriction to the sphere S d−1 of a homogeneous harmonic polynomial of degree l, and d l is the multiplicity of this space.
On the other hand, it is known that the field Ψ Berry can be written as
(6.13) Ψ Berry (x) = ∞ l=0 d l m=1 a lm J l+ d 2 −1 (|x|) |x| d 2 −1 Y lm x |x| ,
where a lm are independent Gaussian variables. Since the topology we are considering in C ∞ (R d ) is precisely that of convergence of each coefficient in the expansion, we have the desired result because for any l ∈ N and any 1 m d l , P (a lm = c lm ) > 0.
Using this, we can then infer that E(F (Ψ Berry )) > 0. We conclude that E p (F (φ k p )) converges to a positive number when k → ∞, and hence for big enough k (and so, small enough 1/ √ λ k ) there must be a positive measure set of points on M for which F (φ k p ) > 0. This clearly implies, by the definition of F , that for those points and large values of k, we have that φ k p − h C r (B) < ǫ. This can be made true for any ǫ, so we have proved that (6.14) lim
k→∞ φ k p − h C r (B) = 0.
This is, of course, also true for any subsequence of λ k so indeed (6.15) lim
λ→∞ ψ λ Exp p · √ λ − h C r (B) = 0,
for a positive measure set of points p, which is precisely definition 6.2.
Regarding this result, there are a few remarks to be made.
• The set of admissible points for any k ∈ N, say N k ⊂ M , satisfies the stronger condition of being "asymptotically dense". Since the property is also true if we restrict ourselves by choosing p in an arbitrary open set, say p ∈ U ⊂ M , it is clear that the set lim k→∞ U ∩N k has positive measure for any arbitrary open set. This implies that for any ǫ > 0 and any point q ∈ M , the set lim k→∞ B(q, ǫ) ∩ N k = ∅, and therefore lim k→∞ N k is dense in M when taking the limit. • Our main interest is in the following easy application: Corollary 6.5. If (M, g) does not exhibit strong inverse localization, then it does not admit Berry's property. For example, as seen in [13], some tori (irrational ones) do not satisfy the inverse localization property (neither 6.1 nor 6.2) and, therefore, do not satisfy Berry's property.
• The inverse localization property can be applied to study nodal sets of eigenfunctions. Precisely, it is true the following result [13]:
Theorem 6.6. If a manifold M does satisfy 6.1 or 6.2, given any natural N and a collection of compact embedded hypersurfaces Σ j (1 j N ) of R d that are not linked, any positive integer r and any ε, there exists some R > 0 such that for all large enough n there is an eigenfunction ψ n with eigenvalue λ n such that the function (6.17) ψ n Exp p · √ λ n has at least N nodal components of the form (6.18) Σ n j := λ −1/2 n Φ n (c j Σ j + p j ) and at least N nondegenerate local extrema in the ball of radius R. Here c j > 0, p j ∈ R d , and Φ n is a diffeomorphism of R d which is close to the identity Φ n − id C r (R d ) < ε.
Therefore, using this formulation of the Berry's conjecture we can gain some knowledge in the topology of nodal set of eigenfunctions of manifolds satisfying the RWM.
• Definition 3.2 does also make sense if we consider M to be a manifold with boundary and (ψ n ) n to be Dirichlet eigenfunctions, and can be found in [21]. The inverse localization can also be defined in that context.
and • for any m, there exists a family of vector fields on U m , say (V m 1 , . . . , V m d ), forming an orthonormal frame of the tangent bundle T U m .
Remark 3.3. The construction of local weak limits depends on the choice of the partition {U m } mmax m=1 and frames {V m j } d j=1 , m = 1, 2, .
Definition 3 . 4 (
34Convergence in the weak* topology). A sequence of probability measures (µ n ) on M d converge to µ in the weak * topology if F dµ n → F dµ for every bounded, continuous function F : M d → R.
Definition 3.5 (BS-convergence of manifolds). A sequence (M n ) of compact connected complete Riemannian d-manifolds is convergent in the sense of Benjamini-Schramm, or just BS-converges, if the sequence µ Mn converges in the weak* topology of the set of all probability measures on M d .
Definition 3. 8 (
8BS-convergence of manifolds with functions). A sequence (M n , φ n ) where each M n is a compact connected complete Riemannian d−manifold and φ n : M n → R is smooth, is convergent in the sense of Benjamini-Schramm, or just BS-converges, if there exists a probability measure µ on E d such that the sequence µ Mn,φn converges to µ in the weak * topology.
Definition 3. 9 (
9Formulation of Berry's conjecture in BS form,[1]). Let M be a compact d-dimensional manifold and M n = (M, g n ) be given by the rescaling of M by √ λ n ; and let (ψ n ) n be an orthonormal basis of L 2 (M ) that consists of eigenfunctions associated to eigenvalues λ n . Then, we have the Berry property in BS form if (M n , ψ n ) BS converges to the isotropic monochromatic Gaussian random field with eigenvalue 1, Ψ Berry . This means that µ Mn,ψn converges to the probability measure µ Berry on the space of smooth functions on R d .
(3. 8 )
8S M,p = [M n , p, ψ n ] M n = (M, g n ), rescaling of M by √ λ n ; p ∈ M ; ψ n eigenfunction associated to λ n ∞ n=1 Let us pay attention to bounded, continuous functionals of two specific kinds: • F : E d → R depending only on [M, p]. Then F can be seen as a functional on M d and µ Mn,ψn (F ) ≡ µ Mn (F ). As we know that µ Mn converges to the Dirac measure at [R d , 0] ∈ M d , then lim n→∞ µ Mn (F ) = F [R d , 0] . • G : E d → R depending only on φ(exp p ) in a small neighborhood of 0
ν
Mn,φn (F ) = µ Berry (F ) = E µBerry [F ] = C ∞ (R d ) F dµ Berry .
(4. 1 )
1H n : M → E d , p → [(M, g n ), p, ψ n ], and applying a change of variables together with the definition of µ Mn,φn to write, for any bounded continuous function F in E d , (4.2) µ Mn,ψn (F ) = M F (H n (p))dµ Unif (p) = 1 Vol(M ) M
( 5 .
513) µ m , G = µ m , G .
Definition 6.2. (Strong inverse localization) A compact manifold M has the strong inverse localization property if for any eigenvalue λ there exist an eigenfunction ψ λ , ∆ M ψ λ + λψ λ = 0, such that for any r ∈ N ∪ {0} and any h : R d → R solution to the Helmholtz equation (i.e. ∆h + h = 0 in the whole R d ), there exists a positive measure subset of the manifold N ⊂ M which satisfies p ∈ N .
F
(Ψ Berry )).
Theorem 1.1. If M is a compact, connected Riemannian manifold covered by a finite family {U m }mmax
m=1
Email address: [email protected]
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[
"A TRACE FORMULA FOR RIGID VARIETIES, AND MOTIVIC WEIL GENERATING SERIES FOR FORMAL SCHEMES",
"A TRACE FORMULA FOR RIGID VARIETIES, AND MOTIVIC WEIL GENERATING SERIES FOR FORMAL SCHEMES"
] |
[
"Johannes Nicaise "
] |
[] |
[] |
We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on itsétale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme X of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series.When X is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f . When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x.If m J is prime to the characteristic exponent of k, we construct anétale cover E o J of E o J as follows: we can cover E o J by affine open formal subschemes U = Spf V of X such that π = uv mJ with u, v ∈ V and u a unit. We putThe restrictions of U over E o J glue together to anétale cover E o J of E o J . If X is a stf t formal R-scheme, then all these definitions coincide with the usual ones (see[31]). In particular, E i = E i is a regular k-variety, and N i is the length of the local ring of the k-scheme X s at the generic point of E i .
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10.1007/s00208-008-0273-9
|
[
"https://export.arxiv.org/pdf/math/0703026v2.pdf"
] | 15,958,634 |
math/0703026
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e01bc64a64ddc3aefd1a6231488b1641517256e0
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A TRACE FORMULA FOR RIGID VARIETIES, AND MOTIVIC WEIL GENERATING SERIES FOR FORMAL SCHEMES
26 Sep 2008
Johannes Nicaise
A TRACE FORMULA FOR RIGID VARIETIES, AND MOTIVIC WEIL GENERATING SERIES FOR FORMAL SCHEMES
26 Sep 2008
We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on itsétale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme X of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series.When X is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f . When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x.If m J is prime to the characteristic exponent of k, we construct anétale cover E o J of E o J as follows: we can cover E o J by affine open formal subschemes U = Spf V of X such that π = uv mJ with u, v ∈ V and u a unit. We putThe restrictions of U over E o J glue together to anétale cover E o J of E o J . If X is a stf t formal R-scheme, then all these definitions coincide with the usual ones (see[31]). In particular, E i = E i is a regular k-variety, and N i is the length of the local ring of the k-scheme X s at the generic point of E i .
Introduction
Let R be a complete discrete valuation ring, and denote by k and K its residue field, resp. its field of fractions. We assume that k is perfect. We fix a uniformizing parameter π in R and a separable closure K s of K, and we denote by K t and K sh the tame closure, resp. strict henselization of K in K s .
The main goal of the present paper is to establish a broad generalization of the trace formula in [31, 5.4], to the class of special formal R-schemes (i.e. separated Noetherian formal R-schemes X such that X/J is of finite type over R for each ideal of definition J; these are also called formal R-schemes of pseudo-finite type). For any special formal R-scheme X, Berthelot constructed in [8, 0.2.6] its generic fiber X η , which is a rigid variety over K, not quasi-compact in general. If X η is smooth over K, our trace formula relates the set of unramified points on X η to the Galois action on theétale cohomology of X η , under a suitable tameness condition.
The set of unramified points on X η is infinite, in general, but it can be measured by means of the motivic Serre invariant, first introduced in [29] and further refined in [32]. A priori, this motivic Serre invariant is only defined if X η is quasi-compact. However, using dilatations, we show that there exists an open quasi-compact rigid subvariety X of X η such that X(K sh ) = X η (K sh ). Moreover, the motivic Serre invariant of X depends only on X η , and can be used to define the motivic Serre invariant S(X η ) of X η . If we denote by X 0 the underlying k-variety of X (i.e. the closed subscheme of X defined by the largest ideal of definition), then this motivic Serre invariant takes values in a certain quotient of the Grothendieck ring The research for this article was partially supported by ANR-06-BLAN-0183. 1 of varieties over X 0 . Our trace formula states that the ℓ-adic Euler characteristic of S(X η ) coincides with the trace of the action of any topological generator of the tame geometric monodromy group G(K t /K sh ), on the graded ℓ-adic cohomology space ⊕ i≥0 H i (X η × K K t , Q ℓ ). As an auxiliary result, we compute Berkovich' ℓ-adic nearby cycles [7] associated to a special formal R-scheme whose special fiber is a strict normal crossings divisor.
Next, we generalize the theory of motivic integration of differential forms on formal schemes of finite type, to the class of special formal R-schemes, and we extend the constructions and result in [31] to this setting. In particular, if k has characteristic zero, we associate to any generically smooth special formal R-scheme its motivic volume, which is an element of the localized Grothendieck ring of X 0varieties M X0 . Moreover, if X is a regular special formal R-scheme of pure relative dimension m, we associate to any continuous differential form ω in Ω m+1 X/k (X) its Gelfand-Leray form ω/dπ, which is a section of Ω m Xη/K (X η ). This construction allows us to define for any regular special formal R-scheme X its Weil generating series, a formal power series over the localized Grothendieck ring M X0 whose degree d coefficient measures the set of unramified points on X η × K K(d). Here K(d) denotes the totally ramified extension K(( d √ π)) of K; the Weil generating series depends on the choice of π if k is not algebraically closed.
If X is a smooth irreducible k-variety, endowed with a dominant morphism f : X → Spec k[π], then we denote by X its π-adic completion. It is a regular special formal R-scheme. We show that, modulo normalization, its Weil generating series coincides with Denef and Loeser's motivic zeta function associated to f , and its motivic volume coincides with the motivic nearby cycles [21].
Finally, we study the analytic Milnor fiber F x of f at a closed point x of the special fiber X s = f −1 (0). This object was introduced in [30], and its points and etale cohomology were studied in [31]. In particular, if k = C, theétale cohomology of F x coincides with the singular cohomology of the topological Milnor fiber of f at x, and the Galois action corresponds to the monodromy. We will show that F x completely determines the formal germ of f at x: it determines the completed local ring O X,x with its R-algebra structure induced by the morphism f . The formal spectrum X x := Spf O X,x is a regular special formal R-scheme, and its generic fiber is precisely F x . The Weil generating series of X x coincides (modulo normalization) with the local motivic zeta function of f at x, and its motivic volume with Denef and Loeser's motivic Milnor fiber.
To conclude the introduction, we give a survey of the structure of the paper. In Section 2, we study the basic properties and constructions for special formal Rschemes: generic fibers, formal blow-ups, dilatations, and resolution of singularities. We also encounter an important technical complication w.r.t. the finite type-case: if X is a special formal R-scheme and ω is a differential form on X η , there does not necessarily exist an integer a > 0 such that π a ω is defined on X. We call forms which (locally on X) have this property, X-bounded differential forms; this notion is important for what follows.
In Section 3, we compute the ℓ-adic tame nearby cycles on a regular special formal R-scheme X whose special fiber is a tame strict normal crossings divisor (Proposition 3.3). We prove that any such formal scheme X admits an algebraizableétale cover (Proposition 3.2), and then we use Grothendieck's description of the nearby cycles in the algebraic case [1, Exp. I].
We generalize the theory of motivic integration to the class of special formal R-schemes in Section 4. In fact, using dilatations, we construct appropriate models which are topologically of finite type over R, and for which the theory of motivic integration was developed in [35] and [29]. Of course, we have to show that the result does not depend on the chosen model. In particular, we associate a motivic Serre invariant to any generically smooth special formal R-scheme X (Definition 4.5) and we show that it can be computed on a Néron smoothening (Corollary 4.14).
In Section 5, we construct weak Néron models for tame ramifications of regular special formal R-schemes whose special fiber is a strict normal crossings divisor (Theorem 5.1) and we obtain a formula for the motivic Serre invariants of these ramifications. We define the order of a bounded gauge form ω on the generic fiber X η of a smooth special formal R-scheme X along the connected components of X 0 , and we deduce an expression for the motivic integral of ω on X (Proposition 5.14).
The trace formula is stated and proven in Section 6.4 (Theorem 6.4). It uses the computation of the motivic Serre invariants of tame ramifications in Section 5, the computation of the tame nearby cycles in Section 3, and Laumon's result on equality of ℓ-adic Euler characteristics with and without proper support [28].
In Section 7, we consider regular special formal R-schemes X whose special fibers X s are strict normal crossings divisors, where char(k) = 0. We define the order of a bounded gauge form ω on the generic fiber X η along the components of X s , and we use this notion to compute the motivic integral of ω on all the totally ramified extensions of X η (Theorem 7.12). These values are used to define the volume Poincaré series for any generically smooth special formal R-scheme X and any bounded gauge form ω on X η , and we obtain an explicit expression for this series in terms of a resolution of singularities (Corollary 7.13). In particular, the limit of this series is well-defined, and does not depend on ω, and we use it to define the motivic volume of X (Definition 7.35). In Section 7.3, we introduce the Gelfand-Leray form ω/dπ associated to a top differential form ω on X over k (Definition 7.21), using the fact that the wedge product with dπ defines an isomorphism between Ω m Xη/K and (Ω m+1 X/k ) rig if X is a generically smooth special formal R-scheme of pure relative dimension m (Proposition 7. 19). If X is regular and ω is a gauge form, then ω/dπ is a bounded gauge form on X η and the volume Poincaré series of (X, ω/dπ) depends only on X, and not on ω; we get an explicit expression in terms of any resolution of singularities (Proposition 7.30). We call this series the Weil generating series associated to X. Now let X be a smooth, irreducible variety over k, endowed with a dominant morphism X → Spec k[t], and denote by X its t-adic completion. We show in Section 8 that the analytic Milnor fiber of f at a closed point x of the special fiber X s = f −1 (0) completely determines the formal germ of f at x (Proposition 8.7). Finally, in Section 9, we prove that the Weil generating series of X and F x coincide, modulo normalization, with the motivic zeta function of f , resp. the local motivic zeta function of f at x (Theorem 9.5 and Corollary 9.6), and we show that the motivic volumes of X and F x correspond to Denef and Loeser's motivic nearby cycles and motivic Milnor fiber (Theorem 9.7). This refines the comparison results in [31].
I am grateful to Christian Kappen for pointing out a mistake in an earlier version of this article.
Notation and conventions. Throughout this article, R denotes a complete discrete valuation ring, with residue field k and quotient field K, and we fix a uniformizing parameter π. Some of the constructions require that k is perfect or that k has characteristic zero; this will be indicated at the beginning of the section. For any field F , we denote by F s a separable closure. We denote by K sh the strict henselization of K, by R sh the normalization of R in K sh , and by K t the tame closure of K in K s . We denote by K t and K s the completions of K t and K s . We denote by p the characteristic exponent of k, and we fix a prime ℓ invertible in k. We say that R ′ is a finite extension of R if R ′ is the normalization of R in a finite field extension K ′ of K.
For any integer d > 0 prime to the characteristic exponent of k, we put K(d) := K[T ]/(T d − π). This is a totally ramified extension of degree d of K. We denote by R(d) the normalization R[T ]/(T d − π) of R in K(d). For any formal R-scheme X and any rigid K-variety X, we put X(d) := X × R R(d) and X(d) := X × K K(d). Moreover, we put X := X × K K t .
If S is a scheme, we denote by S red the underlying reduced scheme. A S-variety is a separated reduced S-scheme of finite type.
If F is a field of characteristic exponent p, ℓ is a prime invertible in F , and S is a variety over F , then we say anétale covering T → S is tame if, for each connected component S i of S, the degree of theétale covering T × S S i → S i is prime to the characteristic exponent p of F . We call a Q s ℓ -adic sheaf F on S tamely lisse if its restriction to each connected component S i of S corresponds to a finite dimensional continuous representation of the prime-to-p quotient π 1 (S i , s) p ′ , where s is a geometric point of S i . We call a Q s ℓ -adic sheaf F on S tamely constructible if there exists a finite stratification of S into locally closed subsets S i , such that the restriction of F to each S i is tamely lisse. If M is a torsion ring with torsion orders prime to p, then tamely lisse and tamely constructible sheaves of M -modules on S are defined in the same way.
If X is a Noetherian adic formal scheme and Z is a closed subscheme (defined by an open coherent ideal sheaf on X) we denote by X/Z the formal completion of X along Z. If N is a coherent O X -module, we denote by N /Z the induced coherent O d X/Z -module. We embed the category of Noetherian schemes into the category of Noetherian adic formal schemes by endowing their structure sheaves with the discrete (i.e. (0)-adic) topology. If X is a Noetherian adic formal scheme and J a coherent ideal sheaf on X, we'll write V (J ) for the closed formal subscheme of X defined by J .
For the theory of stf t formal R-schemes (stf t=separated and topologically of finite type) and the definition of the Grothendieck ring of varieties K 0 (V ar Z ) over a separated scheme Z of finite type over k, we refer to [31]. Let us only recall that L denotes the class of the affine line A 1 Z in K 0 (V ar Z ), and that M Z denotes the localized Grothendieck ring K 0 (V ar Z )[L −1 ]. The topological Euler characteristic
χ top (X) := i≥0 (−1) i dim H i (X × k k s , Q ℓ )
induces a group morphism χ top : M Z → Z. The definition of the completed localized Grothendieck ring M Z is recalled in [32, §4.1].
If V = ⊕ i∈Z V i is a graded vector space over a field F , such that V i = 0 for all but a finite number of i ∈ Z and such that V i is finite dimensional over F for all i, and if M is a graded endomorphism of V , then we define its trace and zeta function by
T r(M | V ) := i∈Z (−1) i T r(M | V i ) ∈ F ζ(M | V ; T ) := i∈Z (det(Id − T M | V i )) (−1) i+1 ∈ F (T ) Finally, if X is a rigid K-variety, we put H(X) := ⊕ i≥0 H i (X, Q ℓ )
where the cohomology on the right is Berkovich' ℓ-adicétale cohomology [5].
Special formal schemes
We recall the following definition: if A is an adic topological ring with ideal of definition J, then the algebra of convergent power series over A in the variables (x 1 , . . . , x n ) is given by
A{x 1 , . . . , x n } := lim ←− n≥1 (A/J n )[x 1 , . . . , x n ]
Definition 2.1. Let X be a Noetherian adic formal scheme, and let J be its largest ideal of definition. The closed subscheme of X defined by J is denoted by X 0 , and is called the reduction of X. It is a reduced Noetherian scheme.
This construction defines a functor (·) 0 from the category of Noetherian adic formal schemes to the category of reduced Noetherian schemes. Note that the natural closed immersion X 0 → X is a homeomorphism. Definition 2.2 (Special formal schemes [7], §1). A topological R-algebra A is called special, if A is a Noetherian adic ring and, for some ideal of definition J, the Ralgebra A/J is finitely generated.
A special formal R-scheme is a separated Noetherian adic formal scheme X endowed with a structural morphism X → Spf R, such that X is a finite union of open formal subschemes which are formal spectra of special R-algebras. In particular, X 0 is a separated scheme of finite type over k.
We denote by X s the special fiber X × R k of X. It is a formal scheme over Spec k. If X is stf t over R, then X s is a separated k-scheme of finite type, and X 0 = (X s ) red .
Note that our terminology is slightly different from the one in [7, § 1], since we impose the additional quasi-compactness condition.
Special formal schemes are called formal schemes of pseudo-finite type over R in [4]. We adopt their definitions ofétale, adicétale, smooth, and adic smooth morphisms [4, 2.6]. If X is a special formal R-scheme, we denote by Sm(X) the open formal subscheme where the structural morphism X → Spf R is smooth.
Berkovich shows in [7, 1.2] that a topological R-algebra A is special, iff A is topologically R-isomorphic to a quotient of the special R-algebra
R{T 1 , . . . , T m }[[S 1 , . . . , S n ]] = R[[S 1 , . . . , S n ]]{T 1 , . . . , T m }
It follows from [37,38] that special R-algebras are excellent, as is observed in [15, p.476].
Any stf t formal R-scheme is special. Note that a special formal R-scheme is stf t over R iff it is R-adic. If X is a special formal R-scheme, and Z is a closed subscheme of X 0 , then the formal completion X/Z of X along Z is special.
We say that a special formal R-scheme X is algebraizable, if X is isomorphic to the formal completion of a separated R-scheme X of finite type along a closed subscheme Z of its special fiber X s . In this case, we say that X/Z is an algebraic model for X. If X is stf t over R and Z = X s (i.e. X is isomorphic to the π-adic completion X of X), we simply say that X is an algebraic model for X. Finally, if I is the π-adic completion of a coherent O X -module I, we say that (X, I) is an algebraic model for (X, I).
If X is a special formal R-scheme, R ′ is a finite extension of R, and ψ is a section in X(R ′ ), then we denote by ψ(0) the image of Spf R ′ in X 0 .
2.1. The generic fiber of a special formal scheme. Berthelot explains in [8, 0.2.6] how to associate a generic fiber X η to a special formal R-scheme X (see also [16, §7]). This generic fiber X η is a separated rigid variety over K, not quasicompact in general, and is endowed with a canonical morphism of ringed sites sp : X η → X (the specialization map). This construction yields a functor (.) η from the category of special formal R-schemes, to the category of separated rigid K-varieties. We say that X is generically smooth if X η is smooth over K.
If Z is a closed subscheme of X 0 , then sp −1 (Z) is an open rigid subvariety of X η , canonically isomorphic to ( X/Z) η by [8, 0.2.7]. We call it the tube of Z in X, and denote it by ]Z[. We recall the construction of X η in the case where X = Spf A is affine, with A a special R-algebra, following [16, 7.1]. The notation introduced here will be used throughout the article.
Let J be the largest ideal of definition of A. For each integer n > 0, we denote by A[J n /π] the subalgebra of A ⊗ R K generated by A and the elements j/π with j ∈ J n . We denote by B n the J-adic completion of A[J n /π] (this is also the π-adic completion), and we put C n = B n ⊗ R K. Then C n is an affinoid algebra, the natural map C n+1 → C n induces an open embedding of affinoid spaces Sp C n → Sp C n+1 , and by construction, X η = ∪ n>0 Sp C n .
For each n > 0, there is a natural ring morphism A ⊗ R K → C n which is flat by [16, 7.1.2]. These morphisms induce a natural ring morphism
i : A ⊗ R K → O Xη (X η ) = ∩ n>0 C n Definition 2.3 ([27],2.3)
. A rigid variety X over K is called a quasi-Stein space if there exists an admissible covering of X by affinoid opens
X 1 ⊂ X 2 ⊂ . . . such that O X (X n+1 ) → O X (X n ) has dense image for all n ≥ 1.
A crucial feature of a quasi-Stein space X is that H i (X, F ) vanishes for i > 0 if F is a coherent sheaf on X, i.e. the global section functor is exact on coherent modules on X. This is Kiehl's "Theorem B" for rigid quasi-Stein spaces [27, 2.4].
Proposition 2.4. If X = Spf A is an affine special formal R-scheme, then X η is a quasi-Stein space.
Proof. Let J be the largest ideal of definition in A. Put X n = Sp C n ; then
X 1 ⊂ X 2 ⊂ . . .
is an affinoid cover of X η . Fix an integer n > 0, and let {g 1 , . . . , g s } be a set of generators of the ideal J n in A. Then by construction, X n consists exactly of the points x in X η such that |(g j /π)(x)| ≤ 1 for j = 1, . . . , s, by the isomorphism [16, 7.1.2] C n ∼ = C n+1 {T 1 , . . . , T s }/(g 1 − πT 1 , . . . , g s − πT s ) As Kiehl observes right after Definition 2.3 in [27], this implies that X η is quasi-Stein.
Let A be a special R-algebra, and X = Spf A. Whenever M is a finite A-module, we can define the induced coherent sheaf M rig on X η by
M rig | Sp (Cn) := (M ⊗ A C n ) ∼ Here (M ⊗ A C n ) ∼ denotes the coherent O Sp Cn -module associated to the finite C n -module M ⊗ A C n .
If X = Spf A is topologically of finite type over R, then X η is simply Sp (A⊗ R K), and M rig corresponds to the (A ⊗ R K)-module M ⊗ R K.
Lemma 2.5. If A is a special R-algebra and X = Spf A, the functor
(.) rig : (Coh X ) → (Coh Xη ) : M → M rig from the category (Coh X ) of coherent O X -modules to the category (Coh Xη ) of co- herent O Xη -modules, is exact.
Proof. This follows from the fact that the natural ring morphism A → C n is flat for each n > 0, by [16, 7.1.2]. Proposition 2.6. For any special formal R-scheme X, there exists a unique functor
(.) rig : (Coh X ) → (Coh Xη ) : M → M rig such that M rig | Uη = (M | U ) rig
for any open affine formal subscheme U of X. The functor (.) rig is exact. For any morphism of special formal R-schemes h : Y → X, and any coherent O X -module M , there is a canonical isomorphism (h * M ) rig ∼ = (h η ) * M rig . Moreover, if h is a finite adic morphism, and N is a coherent O Y -module, then there is a canonical isomorphism (h * N ) rig ∼ = (h η ) * (N rig ).
Proof. Exactness follows immediately from Lemma 2.5. It is clear that (.) rig commutes with pull-back, so let h : Y → X be a finite adic morphism of special formal R-schemes, and let N be a coherent O Y -module. We may suppose that X = Spf A is affine; then Y = Spf D with D finite and adic over A, and h * N is simply N viewed as a A-module. By [16, 7.2
.2], the inverse image of Sp C n ⊂ X η in Y η is the affinoid space Sp (D ⊗ A C n ), so both (h * N ) rig | Sp Cn and (h η ) * (N rig )| Sp Cn are associated to the coherent C n -module N ⊗ A C n .
Example 2.7. The assumption that h is finite and adic is crucial in the last part of Proposition 2.6. Consider, for example, the special formal R-scheme X = Spf R[[x]], and denote by h : X → Spf R the structural morphism. Choose a series (a n ) in K such that |a n | → ∞ as n → ∞, but with |a n | ≤ log n. Then the power series f = n≥0 a n x n in K[[x]] defines an element of O Xη (X η ) = (f η ) * (O X ) rig since it converges on every closed disc D(0, ρ) with ρ < 1, but it does not belong to
R[[x]] ⊗ R K = (h * O X ) rig .
Lemma 2.8. If X is a special formal R-scheme and M is a coherent O X -module, then the functor (.) rig induces a natural map of K-modules
i : M (X) ⊗ R K → M rig (X η )
and this map is injective. If M = O X then i is a map of K-algebras.
Proof. The map i is constructed in the obvious way: if X = Spf A is affine and m is an element of M (X), then the restriction of i(m) to Sp C n is simply the element m ⊗ 1 of M rig (Sp C n ) = M ⊗ A C n . The general construction is obtained by gluing.
To prove that i is injective, we may suppose that X = Spf A is affine; we'll simply write M instead of M (X). Let m be an element of M and suppose that i(m) = 0. Suppose that m is non-zero in M ⊗ R K, and let M be a maximal ideal in A ⊗ R K such that m is non-zero in the stalk M M . By [16, 7.1.9], M corresponds canonically to a point x of X η and there is a natural local homomorphism (A ⊗ R K) M → O Xη,x which induces an isomorphism on the completions, so i(m) = 0 implies that m vanishes in the M-adic completion of M M . This implies at its turn that m vanishes in M M since M M is separated for the M-adic topology [22, 7.3.5]; this contradicts our assumption. Corollary 2.9. If X is a special formal R-scheme and M is a coherent O X -module, then M rig = 0 iff M is annihilated by a power of π.
Proof. We may assume that X is affine, say X = Spf A. By Lemma 2.8, M (X) ⊗ R K = 0, so M is annihilated by a power of π. is an isomorphism, and fits in the commutative diagram
M (X) ⊗ R K − −−− → N (X) ⊗ R K M rig (X η ) − −−− → N rig (X η )
where the vertical arrows are injections and the horizontal arrows are isomorphisms.
Proof. We extend the morphism f to an exact sequence of coherent O X -modules
0 − −−− → ker(f ) − −−− → M f − −−− → N − −−− → coker(f ) − −−− → 0
Since (.) rig is an exact functor by Lemma 2.5, and (f ) rig is an isomorphism by assumption, ker(f ) rig and coker(f ) rig vanish, and hence, ker(f ) and coker(f ) are π-torsion modules, by Corollary 2.9. Since X is affine, the above exact sequence gives rise to an exact sequence of A-modules
0 − −−− → ker(f )(X) − −−− → M (X) f − −−− → N (X) − −−− → coker(f )(X) − −−− → 0
and by tensoring with K, we obtain the required isomorphism. The remainder of the statement follows from Lemma 2.8.
By [16, 7.1.12], there is a canonical isomorphism of O Xη -modules
Ω i Xη/K ∼ = Ω i X/R rig
for any special formal R-scheme X and each i ≥ 0.
2.2. Bounded differential forms.
Definition 2.11. Let X be a special formal R-scheme. For any i ≥ 0, we call an element ω of Ω i Xη /K (X η ) an X-bounded i-form on X η , if there exists a finite cover of X by affine open formal subschemes {U (j) } j∈I such that for each j ∈ I, ω| U (j) η belongs to the image of the natural map
Ω i X/R (U (j) ) ⊗ R K → Ω i Xη/K (U (j)
η ) By Lemma 2.8, this definition is equivalent to saying that ω belongs to the image of the natural map
(
Ω i X/R ⊗ R K)(X) → Ω i Xη/K (X η ) where Ω i X/R ⊗ R K
is a tensor product of sheaves on X. If X is stf t over R then any differential form on X η is X-bounded, by quasicompactness of X η . This is false in general: see Example 2.7 for an example of an unbounded 0-form. Lemma 2.12. If X = Spf A is an affine special formal R-scheme, and i ≥ 0 is an integer, then an element
ω of Ω i Xη/K (X η ) is X-bounded iff it belongs to the image of the natural map Ω i X/R (X) ⊗ R K → Ω i Xη/K (X η ) Proof. Since X is affine, (Ω i X/R ⊗ R K)(X) = Ω i X/R (X) ⊗ R K
Corollary 2.13. Let X be a special formal R-scheme, and let i ≥ 0 be an integer.
If ω is a X-bounded i-form on X η , then for any finite cover of X by affine open formal subschemes {U (j) } j∈I , and for each j ∈ I, ω| U (j) η belongs to the image of the natural map
Ω i X/R (U (j) ) ⊗ R K → Ω i Xη/K (U (j) η ) Lemma 2.14. Let X be a special formal R-scheme, such that X η is reduced. An element f of O Xη (X η ) is X-bounded iff it is bounded, i.e. iff there exists an integer M such that |f (x)| ≤ M for each point x of X η .
Proof. Since an element f of O X (X) satisfies |f (x)| ≤ 1 for each point x of X η by [16, 7.1.8.2], it is clear that an X-bounded analytic function on X η is bounded.
Assume, conversely, that f is a bounded analytic function on X η . To show that f is X-bounded, we may suppose that X = Spf A is affine and flat. Since the natural
map A ⊗ R K → O Xη (X η ) is injective, A ⊗ R K is reduced; since A is R-flat, A is reduced. If A is integrally closed in A ⊗ R K, then the image of the natural map A ⊗ R K → O Xη (X η )
coincides with the set of bounded functions on X η , by [16, 7.4.1-2]. So it suffices to note that the natural map A ⊗ R K → B ⊗ R K is bijective, where B is the normalization of A in A ⊗ R K (B is a special R-algebra since it is finite over A, by excellence of A; see [15]).
2.3.
Admissible blow-ups and dilatations. Let X be a Noetherian adic formal scheme, let J be an ideal of definition, and let I be any coherent ideal sheaf on X. Following the tf t-case in [10, §2], we state the following definition.
Definition 2.15 (Formal blow-up). The formal blow-up of X with center I is the morphism of formal schemes 16. Let X be a Noetherian adic formal scheme with ideal of definition J , let I be a coherent ideal sheaf on X, and consider the formal blow-up h : X ′ → X of X at I. (2) The blow-up morphism X ′ → X is adic and topologically of finite type. In particular, X ′ is a Noetherian adic formal scheme. (3) (Universal property) The ideal IO X ′ is invertible on X ′ , and each morphism of adic formal schemes g : Y → X such that IO Y is invertible, factors uniquely through a morphism of formal schemes Y → X ′ . (4) The formal blow-up commutes with flat base change: if f : Y → X is a flat morphism of Noetherian adic formal schemes, then
X ′ := lim −→ n≥1 Proj ⊕ ∞ d=0 I d ⊗ O X (O X /J n ) → X Proposition 2.X ′ × X Y → Y is the formal blow-up of Y at IO Y . (5) If
K is an open coherent ideal sheaf on X, defining a closed subscheme Z of X, then the formal blow-up of X/Z at I/Z is the formal completion along Z of the formal blow-up of X at I.
Proof. Point (1) follows immediately from the definition, and (2) follows from (1).
In (3) and (4) we may assume that X and Y are affine; then the result follows from the corresponding properties for schemes, using (1). Point (5) is a special case of (4).
Corollary 2.17. Let X be a special formal R-scheme with ideal of definition J , let I be a coherent ideal sheaf of X, and consider the admissible blow-up h : X ′ → X of X at I.
(1) The blow-up X ′ is a special formal R-scheme.
(2) If X is flat over R, then X ′ is flat over R.
Proof. Point (1) follows immediately from Proposition 2.16 (2). To prove (2), we may assume that X = Spf A is affine; then flatness of X ′ follows from Proposition 2.16(1), the fact that the scheme-theoretic blow-up of Spec A at I(X) is flat over R, and flatness of the completion morphism.
Let X be a special formal R-scheme, and let J be an ideal of definition. Let I be a coherent ideal sheaf on X, open w.r.t. the π-adic topology (i.e. I contains a power of π). We will call such an ideal sheaf π-open. We do not demand that I is open w.r.t. the J -adic topology on X. If I is π-open, we call the blow-up X ′ → X with center I admissible 1 .
We can give an explicit description of admissible blow-ups in the affine case, generalizing [10, 2.2].
A ′ i = A[ ξ 1 ξ i , . . . , ξ p ξ i ]/(f i ξ j ξ i − f j ) j=1,...,p A i = A ′ i /(f i − torsion) (here the ξ i /ξ j serve as variables, except when i = j). We write A i and A ′ i for the J-adic completions of A i , resp. A ′ i . Then A ′ i = A{ ξ 1 ξ i , . . . , ξ p ξ i }/(f i ξ j ξ i − f j ) j=1,...,p A i = A ′ i /(f i − torsion) and Spf A ′ i is the open formal subscheme of X ′ where f i generates IO X ′ . In partic- ular, {Spf A ′ i } i=1,...,p
is an open cover of X ′ . If, moreover, A is flat over R, then
A i = A ′ i /(f i − torsion) = A ′ i /(π − torsion) Proof.
The proof is similar to the stf t-case [10, 2.2]. First, we show that
A i = A ′ i /(f i − torsion). Since A i is a finite A ′ i -module, A i = A i ⊗ A ′ i A ′ i . Since A ′ i is Noetherian, A ′ i is flat over A ′ i , so A ′ i /(f i − torsion) ⊗ A ′ i A ′ i = A ′ i /(f i − torsion) Now, we show that π-torsion and f i -torsion coincide in A ′ i if A is R-flat. Since I is π-open in A, it contains a power of π. Since f i generates I A ′ i , the f i -torsion is contained in the π-torsion. But A i is R-flat since A is R-flat, so A i = A ′ i /(f i − torsion) is R-flat, i.e. has no π-torsion.
The remainder of the statement is clear. 19. Let X be a special formal R-scheme, and let h : X ′ → X be an admissible blow-up with center I. The induced morphism of rigid varieties h η : X ′ η → X η is an isomorphism. Proof. We may assume that X is affine, say X = Spf A, with J as largest ideal of definition. Let I = (f 1 , . . . , f p ) be a π-open ideal in A, and let h : X ′ → X be the blow-up of X at I. We define B n and C n as in Section 2.1, for n > 0.
Proposition 2.
We adopt the notation from Lemma 2.18. Since the admissible blow-up h is adic, and the induced morphism V (JO X ′ ) → V (J) is of finite type, it follows from [16, 7.2.2] that the restriction of h η : (Spf A i ) η → X η over Sp C n is given by h n,i : Sp C n ⊗ A A i → Sp C n 1 Contrary to the terminology used in [10] for the stf t case, our definition of admissible blow-up does not assume any flatness conditions on X.
for each i and each n. However, the natural map
C n ⊗ A A ′ i → C n ⊗ A A i is an isomorphism since the f i -torsion in A ′ i is killed if we invert π (because f i divides π in A ′ i , as f i generates the open ideal IA ′ i ).
Hence, {h n,1 , . . . , h 1,p } is nothing but the rational cover of Sp C n associated to the tuple (f 1 , . . . , f p ) (see [9, 8.2.2]); note that these elements generate the unit ideal in C n , since I contains a power of π, which is a unit in C n . Definition 2.20 (Dilatation). Let X be a flat special formal R-scheme, and let I be a coherent ideal sheaf on X containing π. Consider the admissible blow-up h : X ′ → X with center I. If U is the open formal subscheme of X ′ where IO X ′ is generated by π, we call U → X the dilatation of X with center I. Proposition 2.21. Let X be a flat special formal R-scheme, let I be a coherent ideal sheaf on X containing π, and let Z be a closed subscheme of X 0 . The dilatation of X/Z with center I/Z is the formal completion along Z of the dilatation of X with center I.
If, moreover, X is stf t over R and (X, I) has an algebraic model (X, I), then the dilatation of X/Z at I/Z is the formal completion along Z of the dilatation of X at I (as defined in [11, 3.2/1]).
Proof. This is clear from the definition. Proposition 2.22. Let X be a flat special formal R-scheme, and let I be a coherent ideal sheaf on X containing π. Let h : U → X be the dilatation with center I, and denote by Z the closed formal subscheme of X s defined by I. The dilatation U is a flat special formal R-scheme, and h s : U s → X s factors through Z. The induced morphism h η : U η → X η is an open immersion.
If g : V → X is any morphism of flat special formal R-schemes such that g s : V s → X s factors through Z, then there is a unique morphism of formal R-schemes i : V → U such that g = h • i.
If I is open, then U is stf t over R.
Proof. It is clear that h s factors through Z. The morphism U η → X η is an open embedding because U is an open formal subscheme of the blow-up X ′ of X at I, and X ′ η → X η is an isomorphism by Proposition 2.19. Since g s factors through Z, we have IO V = (π). In particular, by flatness of V, the ideal IO V is invertible, and by the universal property of the blow-up, g factors uniquely through a morphism i :
V → X ′ to the blow-up X ′ → X at I. The image of V in X ′ is necessarily contained in U since π generates IO V : if v were a closed point of V mapping to a point x in X ′ \ U, then we could write π = a · f in O X ′ ,x with f ∈ IO X ′ and a not a unit. Thus yields π = i * a · i * f in O V,v , but since i * f belongs to IO V , we have also i * f = b · π in O V,v , so π = c · π with c not a unit, and 0 = (1 − c) · π. Since 1 − c is invertible in O V,v , this would mean that π = 0 in O V,v , which contradicts flatness of V over R.
Finally, assume that I is open. Then the ideal IO U is open, and by definition of the dilatation, it is generated by π. This implies that IO U is an ideal of definition, and that U is stf t. Proposition 2.23. Let X be a flat special formal R-scheme, let U be a reduced closed subscheme of X 0 , and denote by U → X the completion map of X along U . If we denote by U ′ → U and X ′ → X the dilatations with center U 0 = U , resp. X 0 , then there exists a unique morphism of formal R-schemes U ′ → X ′ such that the square
U ′ − −−− → U X ′ − −−− → X commutes, and this morphism U ′ → X ′ is the dilatation of X ′ with center X ′ s × X0 U . Proof.
The existence of such a unique morphism U ′ → X ′ follows immediately from Proposition 2.22; to show that this is the dilatation with center X ′ s × X0 U , it suffices to check that U ′ → X ′ satisfies the universal property in Proposition 2.22. Let h : Z → X ′ be a morphism of flat special formal R-schemes such that h s :
Z s → X ′ s factors through X ′ s × X0 U .
This means that the composed morphism Z s → X 0 factors through U , and hence, Z → X factors through a morphism g : Z → U. Moreover, again by Proposition 2.22, g factors uniquely through a morphism f :
Z → U ′ .
On the other hand, let
f ′ : Z → U ′ be another morphism of formal R-schemes such that h is the composition of f ′ with U ′ → X ′ . Then the compositions of f and f ′ with U ′ → U coincide, so f = f ′ by the uniqueness property in Proposition 2.22 for the dilatation U ′ → U.
Berthelot's construction of the generic fiber of a special formal R-scheme can be restated in terms of dilatations. Let J be an ideal of definition of X, and for any integer e > 0, consider the dilatation
h (e) : U (e) → X with center (π, J e ). The formal R-scheme U (e) is stf t over R, and h (e)
η is an open immersion. By the universal property of the dilatation, h (e) can be decomposed uniquely as
U (e) h (e,e ′ ) − −−− → U (e ′ ) h (e ′ ) − −−− → X for any pair of integers e ′ ≥ e ≥ 0. Moreover, h (e,e ′ ) induces an open immersion h (e,e ′ ) η : U (e) η ֒→ U (e ′ )
η .
Lemma 2.24. The image of the open immersion
h (e) η : U (e) η → X η consists of the points x ∈ X η such that |f (x)| ≤ |π| for each element f of the stalk (J e ) sp(x) .
Proof. Let R ′ be a finite extension of R, and let ψ be a section in X(R ′ ). Then by the universal property in Proposition 2.22, ψ lifts to a section in U (e) (R ′ ) iff (J e , π)O Spf R ′ (the pull-back through ψ) is generated by π. If we denote by x the image of the morphism ψ η in X η , this is equivalent to saying that |f (x)| ≤ |π| for all f in (J e ) sp(x) .
Proposition 2.25. The set {U (e)
η | e > 0} is an admissible cover of X η .
Proof. Let Y = Sp B be any affinoid variety over K, endowed with a morphism of rigid K-varieties ϕ : Y → X η . We have to show that the image of ϕ is contained in U (e) η , for e sufficiently large. We may assume that X is affine, say X = Spf A. Since J is an ideal of definition on X, we have |f (x)| < 1 for any f ∈ J (X) and any x ∈ X η . Since J is finitely generated, and by the Maximum Modulus Principle [9, 6.2.1.4], there exists a value e > 0 such that for each element f in J (X) e and each point y of Y , |f (ϕ(y))| < |π|. By Lemma 2.24, this implies that the image of ϕ is
contained in U (e) η .
Hence, we could have defined X η as the limit of the direct system (U
(e) η , h (e,e ′ ) η )
in the category of rigid K-varieties.
Remark. If we assume that X = Spf A is affine and that J is the largest ideal of definition on X, then the dilatation U (e) → X is precisely the morphism Spf B n → X introduced at the beginning of Section 2.1. This can be seen by using the explicit description in Lemma 2.18.
2.4.
Irreducible components of special formal schemes. If X is a special formal R-scheme, the underlying topological space |X| = |X 0 |, and even the scheme X 0 , reflect rather poorly the geometric properties of X. For instance, if A is a special R-algebra, |Spf A| can be irreducible even when Spec A is reducible (e.g. for
A = R[[x, y]]/(xy), where (Spf A) 0 = Spec k). Conversely, |Spf A| can be reducible even when A is integral (e.g. for A = R{x, y}/(π − xy)).
Therefore, a more subtle definition of the irreducible components of X is needed. We will use the normalization map X → X constructed in [15] (there normalization was already used to define the irreducible components of a rigid variety). The normalization map is a finite morphism of special formal R-schemes.
Definition 2.26. Let X be a special formal R-scheme, and let X → X be a normalization map. We say that X is irreducible if | X| is connected.
Definition 2.27. Let X be a special formal R-scheme, and let h : X → X be a normalization map. We denote by X i , i = 1, . . . , r the connected components of X (defined topologically), and by h i the restriction of h to X i .
For each i, we denote by X i the reduced closed subscheme of X defined by the kernel of the natural map ψ i :
O X → (h i ) * O e Xi . We call X i , i = 1, . . . , r the irreducible components of X. Note that (h i ) * O e
Xi is coherent, by finiteness of h, so the kernel of ψ i is a coherent ideal sheaf on X and X i is well-defined.
In the affine case, the irreducible components of X correspond to the minimal prime ideals of the ring of global sections, as one would expect: Lemma 2.28. Let X = Spf A be an affine special formal R-scheme, and denote by P i , i = 1, . . . , r the minimal prime ideals of A. Then the irreducible components of X are given by Spf A/P i , i = 1, . . . , r.
Proof. If A → A is a normalization map, then A = r i=1 A/P i . Hence, X i = Spf ( A/P i ) are the connected components of X, and X i = Spf A/P i for each i.2.3], X i ∩ h −1 (U) is a union of connected components of U, i.e.
it is of the form Spf j∈J A/P j for some subset J of {1, . . . , q}. Then by definition, X i ∩ U is the closed formal subscheme defined by the ideal P = ∩ j∈J P j , and the minimal prime ideals of A/P are precisely the images of the ideals P j , j ∈ J. This means that
h −1 (U) ∩ X i → U ∩ X i is a normalization map.
There is an important pathology in comparison to the scheme case: a non-empty open formal subscheme of an irreducible special formal R-scheme is not necessarily irreducible, as is shown by the following example. Put A = R{x, y}/(xy − π), and
X = Spf A. Then X is irreducible since A is a domain. However, if we denote by O the point of X defined by the open ideal (π, x, y), then X \ {O} is disconnected.
Proposition 2.30. Let Y be either an excellent Noetherian scheme over R or a special formal R-scheme. In the first case, let I be a coherent ideal sheaf on Y such that the I-adic completion Y of Y is a special formal R-scheme; in the second, let I be any open coherent ideal sheaf on Y , and denote again by Y the I-adic completion of Y .
If h : Y → Y is a normalization map, then its I-adic completion h : Y → Y is also a normalization map.
Proof. We may assume that Y is affine, say Y = Spec A (resp. Spf A), so that I is defined by an ideal I of A. Moreover, we may assume that A is reduced. Denote by A the I-adic completion of A. Since A is finite over A, B := A ⊗ A A is the I-adic completion of A. Hence, by [15, 1.2.2], it suffices to show that B is normal, and that B/P is reduced for all minimal prime ideals P of A.
Normality of B follows from excellence of A. Let P be any minimal prime ideal of A. Then A/P is reduced, and hence, so is B/P , again by excellence of A (see [24, 7.8
.3(v)]).
Corollary 2.31. We keep the notation of Lemma 2.30, and we denote by Z 1 , . . . , Z q the connected components of the closed subscheme Z of Y defined by I. If Y 1 , . . . , Y r are the irreducible components of Y , then the irreducible components of Y are given by the irreducible components of Y i /Z j for i = 1, . . . , r and j = 1, . . . , q (where Y i /Z j may be empty for some i, j).
Proof. Denote by h i : Y i → Y the restriction of h to Y i , for each i, and by h i : U i → Y /Z its I-adic completion. By exactness of the completion functor, the kernel of O d Y /Z → ( h i ) * O Ui
is the defining ideal sheaf of the completion Y i /Z, for each i.
Strict normal crossings and resolution of singularities.
Definition 2.32. A special formal R-scheme X is regular if, for any point x of X, the local ring O X,x is regular (it suffices to check this at closed points).
Let X be a regular special formal R-scheme. We say that a coherent ideal sheaf I on X is a strict normal crossings ideal, if the following conditions hold:
(1) there exists at each point x of X a regular system of local parameters
(x 0 , . . . , x m ) in O X,x such that I x is generated by m j=0 (x j ) Mj for some tuple M in N {0,...,m} .
(2) if E is the closed formal subscheme of X defined by I, then the irreducible components of E are regular.
Lemma 2.33. If (1) holds, then condition (2) is equivalent to the condition that O Ei,x is a domain for each irreducible component E i of E and each point x of E i . Proof. If E i is regular, then O Ei,U = Spf A of x in X such that x 0 , . . . , x m are defined on U and such that I is generated by m j=0 (x j ) Mj on U. Denote by M the maximal ideal of A defining x. Since O X,x /(x j ) is regular for each j, so is (A/(x j )) M (these2.3], E i ∩ U is a union of such components. However, since O Ei,x is a domain, we see that it is of the form O X,x /(x j ) for some j. In particular, O Ei,x is regular.
Hence, if E is a scheme, condition (2) follows from condition (1) (since any local ring of an irreducible scheme is a domain).
We say that a closed formal subscheme E of a regular special formal R-scheme X is a strict normal crossings divisor if its defining ideal sheaf is a strict normal crossings ideal. We say that a special formal R-scheme Y has strict normal crossings if Y is regular and the special fiber Y s is a strict normal crossings divisor, i.e. if the ideal sheaf πO Y is a strict normal crossings ideal. Now let X be a regular special formal R-scheme, and let I be a strict normal crossings ideal on X, defining a closed formal subscheme E of X. We can associate to each irreducible component E i of E a multiplicity m(E i ) as follows. Choose any point x on E i , denote by P i the defining ideal sheaf of E i , and by P i,x its stalk at x.
Lemma 2.34. The ring (O X,x ) Pi,x is a DVR. Proof. The ring (O X,x ) Pi,x is regular since X is regular, so it suffices to show that P i,x is principal. However, if (x 0 , . . . , x m ) is a regular system of local parameters in O X,x such that I x is generated by m i=0 (x i ) Mi
, then we've seen in the proof of Lemma 2.33 that P i,x is generated by x j at x, for some index j.
We define the multiplicity m(
E i , x) of E i at x as the length of the (O X,x ) Pi,x - module (O E,x ) Pi,x .
Lemma-Definition 2.35. The multiplicity m(E i , x) does not depend on the point x. Therefore, we denote it by m(E i ), and we call it the multiplicity of E i in E.
If x is any point of E i , and if I x is generated by m j=0 x Mj j
in O X,x , with (x 0 , . . . , x m ) a regular system of local parameters in O X,x , then P i,x is generated by x j for some index j, and m(Z i ) = M j .
Proof. We've seen in the proof of Lemma 2.34 that P i,x is generated by x j for some index j, and that (O X,x ) Pi,x is a DVR with uniformizing parameter x j , so clearly
m(E i , x) = M j .
Moreover, there exists an open neighbourhood U of x in X such that x 0 , . . . , x m are defined on U, and such that P i is generated by x j on U and I by m j=0
x Mj j . This shows that m(E i , y) = M j for each point y in a sufficiently small neighbourhood of x. Hence, m(E i , y) is locally constant on E i , and therefore constant since E i is connected.
If X is a regular special formal R-scheme and E is strict normal crossings divisor, then we write E = i∈I N i E i to indicate that E i , i ∈ I, are the irreducible components of E, and that N i = m(E i ) for each i. We say that E is a tame strict normal crossings divisor if the multiplicities N i are prime to the characteristic exponent of the residue field k of R. We say that X has tame strict normal crossings if X is regular and X s is a tame strict normal crossings divisor.
If Z is a separated R-scheme of finite type, Z is regular, and its special fiber Z s is a (tame) strict normal crossings divisor (in the classical sense), then we say that Z has (tame) strict normal crossings.
For any non-empty subset J of I, we define
E J := ∩ i∈J E i (i.e. E J is defined by the sum of the defining ideal sheaves of E i , i ∈ J), E J := ∩ i∈J (E i ) 0 and E o J := E J \ (∪ i / ∈J (E i ) 0 ). Moreover, we put m J := gcd{N i | i ∈ J} If i ∈ I, we write E i instead of E {i} = (E i ) 0 . Note that E J is regular and E J = (E J ) 0
for each non-empty subset J of I.
Example 2.36. Consider the special formal R-scheme
X = Spf R[[x, y]]/(π − x N1 y N2 ) Then X s = Spf k[[x, y]]/(x N1 y N2 ), and we get X s = N 1 E 1 + N 2 E 2 with E 1 = Spf k[[y]] and E 2 = Spf k[[x]]. Note that E o 1 = E o 2 = ∅, while E {1,2} is a point (the maximal ideal (π, x, y)).
The varieties E i are not necessarily irreducible. Consider, for instance, the smooth special formal R-scheme
X = Spf R[[x]]{y, z}/(x − yz) Then X s is the formal k-scheme Spf k[[x]][y, z]/(x − yz) which is irreducible, since k[[x]][y, z]/(x − yz) has no zero-divisors. However, X 0 = Spec k[y, z]/(yz) is re- ducible.
Definition 2.37. Let X be a regular special formal R-scheme with strict normal crossings, with X s = i∈I N i E i , and let J be a non-empty subset of I. We say that an integer 38. Let X be regular special formal R-scheme with strict normal crossings. There exists a sequence of admissible blow-ups
d > 0 is J-linear if there exist integers α j > 0, j ∈ J, with d = j∈J α j N j . We say that d is X s -linear if d is J-linear for some non-empty subset J of I with |J| > 1 and E o J = ∅. Lemma 2.π (j) : X (j+1) → X (j) , j = 0, . . . , r − 1 such that • X (0) = X,
• the special fiber of X (j) is a strict normal crossings divisor
X (j) s = i∈I (j) N (j) i E (j) i , • π (j) is the formal blow-up with center E (j) J (j) , for some subset J (j) of I (j) , with |J (j) | > 1, • d is not X (r) s -linear.
Proof. The proof of [32, 5.17] carries over verbatim to this setting. Definition 2.39. A resolution of singularities of a generically smooth flat R-variety X (resp. a generically smooth, flat special formal R-scheme), is a proper morphism of flat R-varieties (resp. a morphism of flat special formal R-schemes) h : X ′ → X, such that h induces an isomorphism on the generic fibers, and such that X ′ is regular, with as special fiber a strict normal crossings divisor X ′ s . We say that the resolution h is tame if X ′ s is a tame strict normal crossings divisor. Lemma 2.40. Let A be a special R-algebra, let X be a stf t formal R-scheme, and let Z be a closed subscheme of X s . Finally, let U be a Noetherian scheme, and let V be a closed subscheme of U .
If M is an open prime ideal of A, defining a point x of Spf A, then the local morphism A M → O Spf A,x induces an isomorphism on the completions (w.r.t. the respective maximal ideals)
A M ∼ = O Spf A,x If x is a point of Z, then the local morphism O X,x → O d X/Z,x induces an isomor- phism on the completions O X,x ∼ = O d X/Z,x If x is a point of V , then the local morphism O U,x → O d U/V ,x induces a canonical isomorphism on the completions O U,x ∼ = O d U/V ,x
Proof. The first point is shown in the proof of [15, 1.2.1]. As for the second, if J is the defining ideal sheaf of Z on X, then
(O X,x )/(J n ) ∼ = O X/J n ,x ∼ = (O X/Z,x )/(J n )
for each n ≥ 1. The proof of the third point is analogous. Lemma 2.41. Let A be a special R-algebra, let X be a separated scheme of finite type over R, or a stf t formal R-scheme, and let Z be a closed subscheme of X s . Let U be a Noetherian R-scheme, and let V be a closed subscheme of U .
(1) Spec A is regular iff Spf A is regular. Moreover, X is regular at the points of Z iff X/Z is regular, and U is regular at the points of V iff U/V is regular.
(2) If (Spec A) s is a strict normal crossings divisor, then (Spf A) s is a strict normal crossings divisor. Moreover, if X s is a strict normal crossings divisor at the points of Z, then ( X/Z) s is a strict normal crossings divisor 2 .
The same holds if we replace X by U and Z by V , and if we assume that U/V is a special formal R-scheme and that U is excellent.
(3) Spf A is generically smooth, iff (A ⊗ R K) M is geometrically regular over K, for each maximal ideal M of A ⊗ R K. Moreover, if X is generically smooth, then X/Z is generically smooth. (4) If K is perfect, any regular special formal R-scheme X is generically smooth.
Proof. Regularity of a local Noetherian ring is equivalent to regularity of its completion [23, 17.1.5], so (1) follows from Lemma 2.40, the fact that regularity can be checked at maximal ideals, and the fact that any maximal ideal of an adic topological ring is open. Point (2) follows from the fact that, for any local Noetherian ring S, a tuple (x 0 , . . . , x m ) in S is a regular system of local parameters for S iff it is a regular system of local parameters for S. The only delicate point is that we have to check if condition (2) in Definition 2.32 holds for ( X/Z) s and ( U/V ) s . This, however, follows from Corollary 2.31. Now we prove (3). By [12, 2.8], smoothness of X η is equivalent to geometric regularity of O Xη,x over K, for each point x of X η . By [16, 7.1.9], x corresponds canonically to a maximal ideal M of A ⊗ R K, and the completions of O Xη,x and (A ⊗ R K) M are isomorphic. We can conclude by using the same arguments as in [31, 2.4(3)]. To prove the second part of (3), we may assume that X is a stf t formal R-scheme: if X is a generically smooth separated scheme of finite type over R, then its π-adic completion X is generically smooth by [31, 2.4(3)], and we have (3) follows from the fact that ( X/Z) η is canonically isomorphic to the tube ]Z[, which is an open rigid subvariety of X η .
X/Z = X/Z. If X is a stf t formal R-scheme,
For point (4), we may assume that X = Spf A is affine. It suffices to show that (A ⊗ R K) M is geometrically regular over K for each maximal ideal M, by point (3). But A is regular by (1), and since K is perfect, (A ⊗ R K) M is geometrically regular over K.
Proposition 2.42. If k has characteristic zero, any affine generically smooth flat special formal R-scheme X = Spf A admits a resolution of singularities by means of admissible blow-ups.
Proof. By Temkin's resolution of singularities for quasi-excellent schemes of characteristic zero [36], Spec A admits a resolution of singularities Y → Spec A by means of blow-ups whose centers contain a power of π. Completing w.r.t. an ideal of definition I of A yields a resolution h : Y → Spf A by means of admissible blow-ups, by Lemma 2.41(2). In (2), the completions can be taken either w.r.t. the adic topologies on X and Y, or w.r.t. the topologies defined by the respective maximal ideals.
Lemma 2.44. Let X be a regular special formal R-scheme (or a regular R-variety), such that X s is a strict normal crossings divisor. Let h : Y → X be anétale morphism of adic formal schemes. Then Y is regular, and Y s is a strict normal crossings divisor. It is tame if X s is tame.
Proof. Let y be a closed point on Y s , and put x = h(y). Take a regular system of local parameters (x 0 , . . . , x m ) in O X,x , such that π = u m i=0 x Ni i , with u a unit. By Lemma 2.43, h * : O X,x → O Y,y is flat and unramified. In particular, (h * x 0 , . . . , h * x m ) is a regular system of local parameters in O Y,y . It satisfies
π = h * u · m i=0 (h * x i ) Ni
Finally, it is clear that the irreducible components of Y s are the connected components of the regular closed formal subschemes Y s × Xs E j , with E j , j ∈ J the irreducible components of X s .
Computation of nearby cycles on formal schemes
3.1. Algebraic covers. Definition 3.1. Let X be a flat special formal R-scheme. A nice algebraizable cover for X at a closed point x of X 0 is a surjective finite adicétale morphism of special formal R-schemes Z → U, with U a Zariski-open neighbourhood of X, such that Z 0 /U 0 is a tameétale covering, and such that each point of Z has a Zariski-open neighbourhood which is isomorphic to the formal completion of a regular R-variety Z with tame strict normal crossings, along a closed subscheme of Z s . Proposition 3.2. We assume that k is perfect. Let X be a regular special formal R-scheme with tame strict normal crossings. Then X admits a nice algebraizable cover at any closed point x of X 0 .
Proof. We may assume that X is affine, say X = Spf A, and that there exist elements x 0 , . . . , x m in A with π = u m i=0 x Ni i , with u a unit and N i ∈ N, and such that (x 0 , . . . , x m ) is a regular system of local parameters on X at x. Put d : = gcd(N 0 , . . . , N m ), and consider the finiteétale morphism
g : Z := Spf A[T ]/(uT d − 1) → X
Since X s is tame, d is prime to the characteristic exponent p of k, and Z 0 is a tamé etale covering of X 0 .
By Bezout's Lemma, we can find integers a j , j = 0, . . . , m, such that d = given by h * (y i ) = z i , mapping the fiber over x to the origin. Moreover, (z 0 , . . . , z m ) is a regular system of local parameters at any point z of Z lying over x, and by Lemma 2.43, h isétale at z. Hence, shrinking X, we may assume that h isétale on Z.
By [3, 7.12], if z is any point of Z lying over x, there exists a Zariski-open neighbourhood of z in Z which is a formal completion of an adicétale Y -scheme Z along a closed subscheme of Z s ; Z is automatically a regular R-variety with tame strict normal crossings, by Lemma 2.44.
In particular, every regular special formal R-scheme with tame strict normal crossings is algebraizable locally w.r.t. theétale topology. Combining this with Proposition 2.42, we see that, if k has characteristic zero and X is a generically smooth affine special formal R-scheme, then there exist a morphism of special formal R-schemes h : X ′ → X such that h η is an isomorphism, and anétale cover {U i } of X ′ by algebraizable special formal R-schemes U i .
3.2.
Computation of the nearby cycles on tame strict normal crossings. We can use the constructions in the preceding section to generalize Grothendieck's computation of the tame nearby cycles on a tame strict normal crossings divisor [1, Exp. I], to the case of special formal R-schemes [7]. Let X be a regular special formal R-scheme, such that the special fiber X s is a tame strict normal crossings divisor i∈I N i E i . For any non-empty subset J of I, we denote by M J the kernel of the linear map
Z J → Z : (z j ) → j∈J N j z j .
Proposition 3.3. Suppose that k is algebraically closed. Let X be a regular special formal R-scheme, such that X s is a tame strict normal crossings divisor i∈I N i E i . Let M be a torsion ring, with torsion orders prime to the characteristic exponent of k. For each non-empty subset J ⊂ I, and each i ≥ 0, the i-th cohomology sheaf of tame nearby cycles R i ψ t η (M ) associated to X, is tamely lisse on E o J . Moreover, for each i > 0, and each point x on E o J , there are canonical isomorphisms
R i ψ t η (M ) x = R 0 ψ t η (M ) x ⊗ i M ∨ J and R 0 ψ t η (M ) x ∼ = M FJ ,
where F J is a set of cardinality m J , on which G(K t /K) acts transitively.
Proof. By [7], Cor. 2.3, whenever h : Y → X is an adicétale morphism of special formal R-schemes, we have
R i ψ t η (M | Yη ) ∼ = h * 0 R i ψ t η (M | Xη )
Hence, by Proposition 3.2, we may suppose that X is the formal completion of a regular R-variety X with tame strict normal crossings, along a closed subscheme of X s . By Berkovich' Comparison Theorem [6, 5.1], it suffices to prove the corresponding statements for X instead of X. The computation of the fibers was done in [1, Exp. I]. To see that R i ψ t η (M ) is tamely lisse on E o J , one can argue as follows: by the arguments in the proof of Proposition 3.2, one can reduce to the case
X = Spec R[x 0 , . . . , x m ]/(π − x N0 0 · . .
. · x Nq q ) with q ≤ m, N j > 0 for each j, and with E o J defined by x 0 = . . . = x q = 0. By smooth base change, one reduces to the case where m = q, and E o J is the origin. Now the statement is trivial.
Corollary 3.4. Suppose that k is algebraically closed. Let X be a regular special formal scheme over R, such that the special fiber X s is a tame strict normal crossings divisor i∈I N i E i . Let x be a closed point on E o J , for some non-empty subset J ⊂ I, and let e > 0 be an integer. Let ϕ be a topological generator of the tame geometric monodromy group G(K t /K).
• T r(ϕ e | Rψ t η (Q ℓ ) x ) = 0, if |J| > 1, or if J is a singleton {i} and N i ∤ e, • T r(ϕ e | Rψ t η (Q ℓ ) x ) = N i , if J = {i} and N i |e.
Corollary 3.5. If k is algebraically closed, of characteristic zero, then Rψ t η (Q ℓ ) is constructible on X 0 , for any generically smooth special formal R-scheme X, and the action of G(K t /K) is continuous.
Proof. This follows from Proposition 2.42, Proposition 3.3, and Corollary [7, 2.3].
Motivic integration on special formal schemes
Throughout this section, we assume that k is perfect. A possible approach to define motivic integration on special formal R-schemes X, would be to introduce the Greenberg scheme Gr R (X) of X (making use of the fact that V (J) is of finite type over R if J is an ideal of definition on X) and to generalize the constructions in [35] and [29] to this setting. We will take a shortcut, instead, making use of appropriate stf t models for special formal R-schemes. The theory of motivic integration on stf t formal R-schemes was developed in [35], and this theory was used to define motivic integrals of differential forms of maximal degree on smooth quasi-compact rigid varieties in [29]. The constructions were refined to a relative setting in [32], and extended to so-called "bounded" rigid varieties in [34].
Definition 4.1. Let X be a special formal R-scheme. A Néron smoothening for X is a morphism of special formal R-schemes Y → X such that Y is adic smooth over R, and such that Y η → X η is an open embedding satisfying Y η (K sh ) = X η (K sh ).
Remark. If X is stf t over R, we called this a weak Néron smoothening in [32], to make a distinction with a stronger variant of the definition. Since this distinction is irrelevant for our purposes, we omit the adjective "weak" from our terminology.
Any generically smooth stf t formal R-scheme X admits a Néron smoothening Y → X, by [13, 3.1]. Moreover, by [32, 6.1]
, the class [Y s ] of Y s in M X0 /(L − [X 0 ])
does not depend on Y. Following [29], we called this class the motivic Serre invariant of X, and denoted it by S(X). We will generalize this result to special formal Rschemes.
Lemma 4.2. If X is a flat special formal R-scheme, and h : Y → X is the dilatation with center X 0 , then Y is stf t over R, and h induces an open embedding Y η → X η with Y η (K sh ) = X η (K sh ).
Proof. By the universal property in Proposition 2.22, any R sh -section of X lifts uniquely to Y. Proof. We may assume that X is flat over R. Take Y as in Lemma 4 Likewise, if ω is a differential form of maximal degree (resp. a gauge form) on X η , then the image of Y |ω| coincides with the image of X |ω| in M U , resp. M U .
.2; it is generically smooth, since Y η is open in X η . If Y ′ → Y is a Néron smoothening of Y, then the composed morphism Y ′ → X is a Néron smoothening for X.
Proof. If we denote by h : Y ′ → X be the blow-up of X with center U , then Y is, by definition, an open formal subscheme of Y ′ . If g : Z ′ → Y ′ is a Néron smoothening, and if we put Z = g −1 (Y), then by the universal property of the dilatation in Proposition 2.22, the induced open embedding of Greenberg schemes
Gr R (Z) → Gr R (Z ′ ) is an isomorphism onto the cylinder (h s • g s • θ 0 ) −1 (U ) (here θ 0 denotes the truncation morphism Gr R (Z ′ ) → Z ′ s )
. We recall that the motivic integral X |ω|, with X generically smooth and stf t over R, was defined in [32, §6], refining the construction in [29, 4.1.2] Definition 4.5. Let X be a generically smooth, flat special formal R-scheme, and let h : X ′ → X be the dilatation with center X 0 . We define the motivic Serre invariant S(X) of X by
S(X) := S(X ′ ) in M X0 /(L − [X 0 ])
If ω is a differential form of maximal degree (resp. a gauge form) on X η , then we put
X |ω| := X ′ |ω| in M X0 , resp. M X0 .
If X is a generically smooth special formal R-scheme, we denote by X f lat its maximal flat closed subscheme (obtained by killing π-torsion) and we put S(X) = S(X f lat ) and X |ω| := X f lat |ω| It follows from Lemma 4.4 that this definition coincides with the usual one if X is stf t over R. Note that even in this case, the dilatation X ′ → X is not necessarily an isomorphism, since X s might not be reduced. In fact, X ′ might be empty, for instance if X has strict normal crossings with all multiplicities > 1.
Proposition 4.6. Let X be a generically smooth special formal R-scheme. If Y → X is a Néron smoothening for X, then
S(X) = [Y 0 ] ∈ M X0 /(L − [X 0 ])
For any differential form of maximal degree (resp. gauge form) ω on X η , we have
X |ω| = Y |ω| in M X0 , resp. M X0 .
Proof. We may assume that X is flat over R; let X ′ → X be the dilatation with center X 0 . By the universal property of the dilatation in Proposition 2.22 and the fact that Y s is reduced, any Néron smoothening Y → X factors through a morphism of stf t formal R-schemes Y → X ′ , and by Lemma 4.2, this is again a Néron smoothening. So the result follows from (the proof of) [32, 6.11] (see also [34] for an addendum on the mixed dimension case).
We showed in [34] that the generic fiber X η of a generically smooth special formal R-scheme X is a so-called "bounded" rigid variety over K (this also follows immediately from Lemma 4.2), and we defined the motivic Serre invariant S(X η ) of X η , as well as motivic integrals Xη |ω| of differential forms ω on X η of maximal degree.
Proposition 4.7. If X is a generically smooth special formal R-scheme, then S(X η ) is the image of S(X) under the forgetful morphism
M X0 /(L − [X 0 ]) → M k /(L − 1)
If ω is a gauge form on X η , then S(X) is the image of X |ω| in M X0 /(L−[X 0 ]). If ω is a differential form of maximal degree (resp. a gauge form) on X η , then Xη |ω| is the image of X |ω| under the forgetful morphism M X0 → M k , resp. M X0 → M k .
Proof. This is clear from the definitions, and the corresponding properties for stf t formal R-schemes [32, 6.4].
Definition 4.8. Suppose that k has characteristic zero. Let X be a generically smooth special formal R-scheme. Let ω be a gauge form on X η . We define the volume Poincaré series of (X, ω) by
S(X, ω; T ) := d>0 X(d) |ω(d)| T d in M X0 [[T ]]
Remark. The motivic Serre invariant S(X) and the motivic integral X |ω ′ | (for any differential form ω ′ of maximal degree on X η ) are independent of the choice of uniformizer π. The volume Poincaré series, however, depends on the choice of π, or more precisely, on the K-fields K(d). If k is algebraically closed, then K(d) is the unique extension of degree d of K, up to K-isomorphism, and S(X, ω; T ) is independent of the choice of π. See also the remark following Definition 7.37.
Proposition 4.9. Let X be a generically smooth special formal R-scheme, and let U be a locally closed subscheme of X 0 . Denote by U the formal completion of X along U . Then S(U) is the image of S(X) under the base change morphism
M X0 /(L − [X 0 ]) → M U /(L − [U ])
If ω is a gauge form on X η , then U |ω| is the image of X |ω| under the base change morphism M X0 → M U (the analogous statement holds if ω is merely a differential form of maximal degree).
If, moreover, k has characteristic zero, then S(U, ω; T ) is the image of S(X, ω; T ) under the base-change morphism
M X0 [[T ]] → M U [[T ]]
In particular, if X is stf t over R, then S(U) and S(U, ω; T ) coincide with the invariants with support S U (X) and S U (X, ω; T ) defined in [31].
Proof. We may assume that X is flat. If U is open in X 0 , then these results are clear from the definitions, since dilatations commute with flat base change; so we may suppose that U is a reduced closed subscheme of X 0 . Let h : X ′ → X be the dilatation with center X 0 , and let U ′ → U be the dilatation with center U 0 = U . By Proposition 2.23, there exists a unique morphism of stf t formal R-schemes U ′ → X ′ such that the square 4.11. Let X be a special formal R-scheme. A special Néron smoothening for X is a morphism of special formal R-schemes Y → X such that Y is smooth over R, and such that Y η → X η is an open embedding satisfying Y η (K sh ) = X η (K sh ).
U ′ − −−− → U X ′ − −−− → X commutes, and U ′ → X ′ is the dilatation with center X ′ s × X0 U .
In particular, any Néron smoothening is a special Néron smoothening (which makes the terminology somewhat paradoxical).
Proposition 4.12. If X is a generically smooth flat special formal R-scheme, then there exists a composition of admissible blow-ups Y → X such that Sm(Y) → X is a special Néron smoothening for X.
Proof. Let I be the largest ideal of definition for X, and let h : X ′ → X be the admissible blow-up with center I. We denote by U the open formal subscheme of X ′ where IO X ′ is generated by π, i.e. U → X is the dilatation with center X 0 , and U is stf t over R.
By [13, 3.1] and [10, 2.5], there exists an admissible blow-up g : Y ′ → U such that Sm(Y ′ ) → U is a Néron smoothening. We will show that this blow-up extends to an admissible blow-up Y → X ′ . Choose an integer j > 0 such that π j is contained in the center J of g. By [22, 9.4.7], the pull-back of J to V (I j O U ) extends to a coherent ideal sheaf J ′ on V (I j O X ′ ); we'll denote again by J ′ the corresponding coherent ideal sheaf on X ′ . Since formal blow-ups commute with open embeddings, the admissible blow-up Y → X ′ with center J ′ extends g.
Finally, let us show that the composed morphism Sm(Y) → X is a special Néron smoothening. It suffices to show that the natural map Sm(Y) η (K sh ) → X η (K sh ) is surjective. By Lemma 4.2, U η (K sh ) → X η (K sh ) is surjective, and since Sm(Y ′ ) → U is a Néron smoothening, Sm(Y ′ ) η (K sh ) → U η (K sh ) is surjective; but Y ′ is an open formal subscheme of Y, so the result follows. Proof. Stratifying X 0 in regular pieces, we might as well assume that X 0 is regular from the start, by Corollary 4.10. Moreover, we may suppose that X = Spf A is affine, and that X 0 is defined by a regular sequence (π, x 1 , . . . , x q ) in A. The dilatation of X with center X 0 is given by
Y := Spf A{T 1 , . . . , T q }/(x i − πT i ) i=1,...,q → X Now Y is flat, and Y s = Y 0 = Spec (A/(π, x 1 , . . . , x q )) [T 1 , . . . , T q ] ∼ = X 0 × k A q k
Since k is perfect and X 0 is regular, Y 0 is smooth over k, and hence Y is smooth over R, and
S(X) = S(Y) = [Y 0 ] = [X 0 ] in M X0 /(L − [X 0 ]).
Corollary 4.14. If h : Y → X is a special Néron smoothening, then
S(X) = [Y 0 ] ∈ M X0 /(L − [X 0 ])
Computation of Serre invariants and motivic integrals
Throughout this section, we assume that k is perfect.
5.1.
Serre invariants of the ramifications. If X is a special formal R-scheme, we denote by X → X the normalization morphism [15].
Theorem 5.1. Let X be a regular special formal R-scheme, such that X s is a tame strict normal crossings divisor i∈I N i E i , and let d > 0 be an integer, prime to the characteristic exponent of k. If d is not X s -linear, then
h : Sm( X(d)) → X(d)
is a special Néron smoothening. Moreover, if we put
E(d) o i = ( X(d)) 0 × X0 E o i for each i in I, then Sm( X(d)) 0 = Ni|d E(d) o i and the E o i -variety E(d) o i is canonically isomorphic to E o i (defined in Section 2.5).
Proof. The fact that
Sm( X(d)) 0 = Ni|d E(dSm( X(d)) η (K(d) sh ) = X(d) η (K(d) sh )
i.e. h is a special Néron smoothening.
Corollary 5.2. Let X be a regular special formal R-scheme, such that X s is a strict normal crossings divisor i∈I N i E i , and let d > 0 be an integer, prime to the characteristic exponent of k.
If d is not X s -linear, then
S(X(d)) = i∈I,Ni|d [ E o i ] in M X0 /(L − [X 0 ]).
Proof. If X ′ is the completion of X along ⊔ Ni|d E o i , then S(X(d)) = S(X ′ (d)), since d is not X s -linear, by a straightforward generalization of [32, 5.15]. Hence, we may as well assume that X ′ = X. In this case, since d is prime to the characteristic exponent of k, X s is tame, so we can use Theorem 5.1 and Corollary 4.14 to conclude.
Order of a top form at a section.
This subsection is a straightforward generalization of [31, §6.2]. Let X be a generically smooth special formal R-scheme, of pure relative dimension m. Let R ′ be a finite extension of R, of ramification index e, and denote by K ′ its quotient field. Definition 5.3. For any element ψ of X(R ′ ), and any ideal sheaf I on X, we define ord(I)(ψ) as the length of the R ′ -module R ′ /ψ * I.
We recall that the length of the zero module is 0, and the length of R ′ is ∞. For any element ψ of X(R ′ ), the R ′ -module M := (ψ * Ω m X/R )/(torsion) is a free rank 1 sublattice of the rank 1 K ′ -vector space (ψ η ) * (Ω m Xη /K ).
Definition 5.4. For any global section ω of Ω m X/R and any section ψ in X(R ′ ), we define the order ord(ω)(ψ) of ω at ψ as follows: we choose an integer a ≥ 0 such that ω ′ := π a ψ * η (ω) belongs to the sublattice M of (ψ η ) * (Ω m Xη/K ), and we put
ord(ω)(ψ) = length R ′ (M/R ′ ω ′ ) − e.a
This definition does not depend on a.
If e = 1, this definition coincides with the one given in [29, 4.1]. It only depends on the completion of X at ψ(0) ∈ X 0 . If ω is a gauge form on X η , ord(ω)(ψ) is finite. Now let h : Z → X be a morphism of generically smooth special formal Rschemes, both of pure relative dimension m. Let R ′ be a finite extension of R, and fix a section ψ in Z(R ′ ). The canonical morphism
h * Ω m X/R → Ω m Z/R
induces a morphism of free rank 1 R ′ -modules (ψ * h * Ω m X/R )/(torsion) → (ψ * Ω m Z/R )/(torsion) We define ord(Jac h )(ψ) as the length of its cokernel.
If Ω m Z/R /(torsion) is a locally free rank 1 module over O Z , we define the Jacobian ideal sheaf J ac h of h as the annihilator of the cokernel of the morphism h * Ω m X/R → Ω m Z/R /(torsion) and we have ord(Jac h )(ψ) = ord(J ac h )(ψ). The following lemmas are proved as their counterparts [31], Lemma 6.4-5.
Lemma 5.5. Let h : Z → X be a morphism of generically smooth special formal R-schemes, both of pure relative dimension m. Let R ′ be a finite extension of R. For any global section ω of Ω m X/K , and any ψ ′ ∈ Z(R ′ ), ord(h * ω)(ψ) = ord(ω)(h(ψ)) + ord(Jac h )(ψ) Lemma 5.6. Let e ∈ N * be prime to the characteristic exponent of k. Let X be a regular special formal R-scheme with tame strict normal crossings. Let ω be a global section of Ω m Xη/K . We denote by ω(e) the pullback of ω to the generic fiber of X(e). Let R ′ be a finite extension of R(e), and let ψ(e) be a section in Sm( X(e))(R ′ ). If we denote by ψ its image in X(R ′ ), then ord(ω)(ψ) = ord( ω(e))(ψ(e))
5.3.
Order of a gauge form on a smooth formal R-scheme. Let X be a smooth special formal R-scheme of pure relative dimension m, and let ω be a Xbounded gauge form on X η (see Definition 2.11). We denote by Irr(X 0 ) the set of irreducible components of X 0 (note that X 0 is not always smooth over k; see Example 2.36). Let C be an irreducible component of X 0 , and let ξ be its generic point. The local ring O X,ξ is a UFD, since X is regular. Since X is smooth over R, (Ω m X/R ) ξ is a free rank 1 module over O X,ξ by [4, 4.8] and [3, 5.10], and π is irreducible in O X,ξ .
Definition 5.7. Let A be a UFD, and let a be an irreducible element of A. Let N be a free A-module of rank one, and let n be an element of N . We choose an isomorphism of A-modules A ∼ = N , and we define ord a n as follows: if n = 0, ord a n is the largest q ∈ N such that a q |n in A. If n = 0, we put ord a n = ∞. This definition does not depend on the choice of isomorphism A ∼ = N .
Definition 5.8. Let X be a smooth special formal R-scheme of pure relative dimension m, let C be an irreducible component of X 0 , and denote by ξ its generic point.
If ω is a X-bounded m-form on X η , we can choose b ∈ N such that π b ω extends to a section ω ′ of (Ω m X/R ) ξ , and we put
ord C ω := ord π ω ′ − b
This definition does not depend on b.
Lemma 5.9. Let X be a smooth connected special formal R-scheme, and let f be an element of O X (X). If f is a unit on X η , and f is not identically zero on X 0 , then f is a unit on X.
Proof. We may assume that X = Spf A is affine. By the correspondence between maximal ideals of A ⊗ R K and points of X η explained in [16, 7.1.9], we see that f is a unit in A ⊗ R K, so there exists an element q in A and an integer i ≥ 0 such that f q = π i . We may assume that either i = 0 (in which case f is a unit), or q is not divisible by π in A. Since X is smooth, π is a prime in A, so π divides f if f is not a unit; this contradicts the hypothesis that f does not vanish identically on X 0 .
Lemma 5.10. Let X be a smooth special formal R-scheme, and let ω be a X-bounded gauge form on X η . Let R ′ be a finite unramified extension of R, and consider a section ψ ∈ X(R ′ ). If C is an irreducible component of X 0 containing ψ(0), then
ord C (ω) = ord(ω)(ψ)
Proof. We denote by ξ the generic point of C. Multiplying with powers of π, we may assume that ω is defined on X. Moreover, we may assume that there exists a section ω 0 in Ω m R (X/R) which generates Ω m X/R at each point of X, and we write ω = f ω 0 with f ∈ O X (X). Dividing by an appropriate power of π, we may assume that π ∤ f in O X,ξ and ord C (ω) = 0. Since ω is gauge on X η , f is a unit on X η , so by Lemma 5.9, f is a unit on X. Hence, ord(ω)(ψ) = ord π ψ * (f ) = 0 Corollary 5.11. If C 1 and C 2 are irreducible components of the same connected component C of X 0 , then ord C1 (ω) = ord C2 (ω), and we denote this value by ord C (ω).
Proof. We can always find a section ψ in X(R ′ ) with R ′ /R finite and unramified, and with ψ(0) ∈ C 1 ∩ C 2 .
Corollary 5.12. If X 0 is connected, and Z is a locally closed subset of X 0 , then
ord C (ω) = ord X0 (ω)
for any connected component C of Z, where the left hand side is computed on the completion X/Z. Lemma 5.13. Let X be a smooth connected special formal R-scheme, and let Z be a regular closed subscheme of X 0 . If we denote by h : Y → X the dilatation with center Z, and by c the codimension of Z in X, then for any X-bounded gauge form ω on X η , ord X0 (ω) = ord Y0 (ω) + c − 1
Proof. By Corollary 5.12 and Proposition 2.21, we may assume that X 0 = Z. Now the result follows from Lemma 5.10 and Lemma 5.5, since J ac h = (π c−1 ).
Proposition 5.14. Let X be a smooth special formal R-scheme, of pure relative dimension m, and denote by C(X 0 ) the set of connected components of X 0 . For any X-bounded gauge form ω on X η ,
X |ω| = L −m C∈C(X0) [C]L −ordC (ω) in M X0 .
Proof. By Corollaries 4.10 and 5.12, we may assume that X 0 is regular and connected. By definition,
X |ω| = Y |ω|
where Y → X is the dilatation with center X 0 . In the proof of Proposition 4.13, we saw that Y is smooth, and Y 0 = [X 0 ]L c−1 , with c the codimension of X 0 in X. Hence, we can conclude by Lemma 5.13.
Corollary 5.15. Let X be a generically smooth special formal R-scheme ofpure relative dimension m, and let ω be a gauge form on X η . If Y → X is a special Néron smoothening, then
X |ω| = L −m C∈C(Y0) [C]L −ordC (ω)
in M X0 .
The trace formula
Let F be any field, let Z be a variety over F , and let A be an abelian group. where S is a finite stratification of Z into constructible subsets, and a S ∈ A for S ∈ S, and where 1 S denotes the characteristic function of S. Note that a constructible function on Z is completely determined by its values on the set of closed points Z o of Z. If A is a ring, C(Z, A) carries a natural ring structure. If A = Z, we call C(Z, A) the ring of constructible functions on Z, and we denote it by C(Z).
For any constructible A-function ϕ = S∈S a S .1 S on Z as above, we can define its integral w.r.t. the Euler characteristic as follows:
Z ϕdχ := S∈S a S χ top (S)
If the group operation on A is written multiplicatively, we write × Z instead of Z . The calculus of integration with respect to the Euler characteristic was, to our knowledge, first introduced in [39]. Lemma 6.1. Suppose that F is algebraically closed, and let Z be a variety over F . Let ℓ be a prime number, invertible in F , and let L be a tamely constructible Q s ℓ -adic sheaf on Z. Suppose that a finite cyclic group G with generator g acts on L. We denote by Z o the set of closed points on Z.
(1) the mapping
T r(g | L * ) : Z o → Q s ℓ : x → T r(g | L x ) defines a constructible Q s ℓ -function on Z, and T r(g | ⊕ i≥0 H i c (Z, L)) = Z T r(g | L * )dχ
(2) the mapping
ζ(g | L * ; T ) : Z o → Q s ℓ [[T ]] × : x → ζ(g | L x ; T ) defines a constructible Q s ℓ [[T ]] × -function on Z, and ζ(g | ⊕ i≥0 H i c (Z, L); T ) = × Z ζ(g | L * ; T )dχ
Proof. First, we prove (1). By additivity of H c (. ), we may suppose that Z is normal and L is tamely lisse on Z. In this case, T r(g | L * ) is constant on Z, and the result follows from [31, 5.1]. Now (2) follows from the identity [17, 1.5.3]
det(Id − T.M | V ) −1 = exp( d>0 T r(M d | V ) T d d )
for any endomorphism M on a finite dimensional vector space V over a field of characteristic zero.
Lemma 6.2. Let G be a finite group, let F be an algebraically closed field, and fix a prime ℓ, invertible in F . If f : Y → X is a morphism of separated F -schemes of finite type, and L is a constructible E[G]-sheaf on Y , for some finite extension E of Q ℓ , then
f * [L] = f ! [L] in K 0 (X; E[G]) In particular, if X = Spec F , then T r(g | ⊕ i≥0 H i (Y, L)) = T r(g | ⊕ i≥0 H i c (Y, L)) for each element g of G.
Proof. If G is the trivial group, then this is a well-known theorem of Laumon's [28]. His proof carries over verbatim to the case where G is any finite group. Corollary 6.3. Let G =< g > be a finite cyclic group. Let F be an algebraically closed field, let U be a variety over F , and let L be a tamely constructible Q s ℓ [G]sheaf on U , for any prime ℓ invertible in F . Then
T r(g | ⊕ i≥0 H i (U, L)) = T r(g | ⊕ i≥0 H i c (U, L)) = U T r(g | L * ) dχ
Proof. The first equality follows from Lemma 6.2, while the equality between the second and the third expression follows from Lemma 6.1.
The following theorem is a broad generalization of [31, 5.4]. It implies, in particular, that the assumptions that X is algebraic and Z is proper, are superfluous in the statement of [31, 5.4].
Theorem 6.4 (Trace formula). Assume that k is perfect. Let ϕ be a topological generator of the tame geometric monodromy group G(K t /K sh ). Let X be a generically smooth special formal R-scheme, and suppose that X admits a tame resolution of singularities h : Y → X, with Y s = i∈I N i E i . For any integer d > 0, prime to the characteristic exponent of k, we have
χ top (S(X η (d))) = T r(ϕ d | H( X η )) = Ni|d N i χ top (E o i )
Proof. We may assume that Y = X, and that k is algebraically closed (since the motivic Serre invariant is clearly compatible with unramified extensions of the base R). The equality
χ top (S(X η (d))) = Ni|d N i χ top (E o i )
can be proven as in [31, 5.4]: the expression holds if d is not X s -linear, by Corollary 5.2 and the fact that E o i is a degree N i finiteétale cover of E i . Moreover, the right hand side of the equality does not change under blow-ups with center E J with ∅ = J ⊂ I, by the obvious generalization of [31, 5.2], so the expression holds in general by Lemma 2.38.
So it suffices to prove that
T r(ϕ d | H( X η )) = Ni|d N i χ top (E o i ) By [7, 2.3(ii)], there is for each i ≥ 0 a canonical isomorphism H i (X 0 , Rψ t η (Q ℓ | Xη )) ∼ = H i ( X η , Q ℓ ) and hence, (6.1) T r(ϕ d | H( X η )) = T r(ϕ d | H(X 0 , Rψ t η (Q ℓ | Xη )
)) By our local computation in Proposition 3.3 we can filter R j ψ t η (Q ℓ | Xη ) by constructible subsheaves which are stable under the monodromy action and such that the action of ϕ on successive quotients has finite order. Hence, we can apply Lemma 6.3. Combined with the computation in Corollary 3.4, we obtain the required equality. Corollary 6.5. If k has characteristic zero, and X is a generically smooth special formal R-scheme, then
χ top (S(X η (d))) = T r(ϕ d | H( X η ))
for any integer d > 0. In particular,
χ top (S(X(d))) = T r(ϕ d | H( X ))
for any smooth quasi-compact rigid variety X over K.
Proof. Since generically smooth affine special formal R-schemes admit a resolution of singularities if k has characteristic zero, by Proposition 2.42, we can cover X by a finite family of open affine formal subschemes, such that
χ top (S(V(d))) = T r(ϕ d | H( V η ))
whenever V is an intersection of members of this cover. But both sides of this equality are additive w.r.t. V (for the right hand side, use equation (6.1) and Lemma 6.3), which yields the result for V = X. Corollary 6.6. If k is an algebraically closed field of characteristic zero, and X an K is the analytification of a proper smooth variety X K over K, then
χ top (S(X an K (d))) = T r(ϕ d | ⊕ i≥0 H i (X K × K K s , Q ℓ )) Proof.
If X is any flat proper R-model for X K , then X an K is canonically isomorphic to the generic fiber X η of the π-adic completion X, by [8, 0.3.5]. Moreover, by [5, 7.5.4], there is a canonical isomorphism
H i (X K × K K s , Q ℓ ) ∼ = H i (X η )
for each i ≥ 0.
As we observed in [31, §5], some tameness condition is necessary in the statement of the trace formula: if R is the ring of Witt vectors W (F s p ), and X is Spf R[x]/(x p − π), then S(X) = 0 while the trace of ϕ on the cohomology of X η is 1. It would be interesting to find an intrinsic tameness condition on X η under which the trace formula holds.
It would also be interesting to find a proof of the trace formula which does not rely on explicit computation, and which does not use resolution of singularities. One could use the following strategy. There's no harm in assuming that k is algebraically closed. After admissible blow-up, we may suppose that the R-smooth part Sm(X) of X is a weak Néron model for X, by Proposition 4.12. On Sm(X) 0 , the tame nearby cycles are trivial, so the trace of ϕ on the cohomology of Sm(X) η yields χ top (S(X)). Hence, it suffices to prove that the trace of ϕ on the cohomology of Rψ t η (Q ℓ | Xη )| Y vanishes, where Y denotes the complement of Sm(X) 0 in X 0 . We'll assume the following result: for each i ≥ 0, there is a canonical G(K t /K)equivariant isomorphism [7, 3.1]. If K has characteristic zero and X is stf t over R, then it can be proven in the same way as Huber's result [26, 3.15] (which is the corresponding result for Berkovich' functor RΘ K from [7, §2]).
(6.2) H i (Y, Rψ t η (Q ℓ | Xη )| Y ) ∼ = H i (]Y [, Q ℓ ) This is known if X is algebraizable, by Berkovich' comparison theorem
Assuming (6.2), everything reduces to the following assertion: if X η is the smooth generic fiber of a special formal R-scheme and satisfies X η (K) = ∅ and an appropriate tameness condition (in particular if k has characteristic zero), then T r(ϕ | H(X η )) = 0 At this point, I don't know how to prove this without making an explicit computation on a resolution of singularities.
The volume Poincaré series, and the motivic volume
Throughout this section, we assume that k has characteristic zero. 7.1. Order of a gauge form along strict normal crossings. Throughout this subsection, X denotes a regular special formal R-scheme of pure relative dimension m, whose special fiber is a strict normal crossings divisor X s = i∈I N i E i . If X is stf t over R, then E i = E i , and if we denote by ξ i the generic point of E i , then P i,ξi is the maximal ideal of O X,ξi . So O X,Ei,ξi = O X,ξi , and Ω X,Ei,ξi is the module Ω i considered in [31, 6.7]. Note that in the general case, E i is not necessarily irreducible (see Example 2.36).
h * Ω m X/R → Ω m Y/R
induces an isomorphism Ω X,Ei,x ⊗ O X,E i ,x O Y,C,y ∼ = Ω Y,C,y for each point y on C = C 0 and with x = h(y).
Proof. Since h isétale,
h * Ω m X/R → Ω m Y/R
is an isomorphism by [4, 4.10]. Let P ′ y be the prime ideal in O Y,y defining C. Since h isétale, the local morphism h * : O X,x → O Y,y is a flat, unramified monomorphism by Lemma 2.43, and by localization, so is Proof.
O X,Ei,x → O Y,C,y The isomorphism Ω m X/R,x ⊗ O X,x O Y,y ∼ = Ω m Y/R,y localizes to an isomorphism (Ω m X/R,x ) Pi,x ⊗ O X,E i ,x O Y,C,y ∼ = (Ω m Y/R,y ) P ′
Since Ω X,Ei,x is finite over O X,Ei,x and torsion-free, and O X,Ei,x is a PID, the module Ω X,Ei,x is free over O X,Ei,x . Let us show that its rank equals 1. By Lemma 7.2, we may pass to anétale cover and assume that there exists a regular system of local parameters (x 0 , . . . , x n ) in O X,x with P i,x = (x i ) and π = n i=0 x Ni i . Deriving this expression, we see that Ω X,Ei,x is generated by
dx 0 ∧ . . . ∧ dx i ∧ . . . ∧ dx n .
Note that the natural map Ω m X/R (X) → Ω X,Ei,x factors through Ω m X/R (X)/(π − torsion) since Ω X,Ei,x has no torsion.
Ω X,Ei,x /(O X,Ei,x · ω)
and we denote it by ord Ei ,x ω.
If ω is a X-bounded m-form on X η , there exists an integer a ≥ 0 and an affine open formal subscheme U of X containing x, such that π a ω belongs to
Ω m X/R (U)/(π − torsion) ⊂ Ω m X/R (U) ⊗ R K We define the order of ω along E i at x as ord Ei ,x ω := ord Ei,x (π a ω) − aN i
This definition does not depend on a. If X is smooth, it coincides with the one given in Section 5.3 in the following sense: if X is connected and x is any point of X 0 , then ord X0 ω = ord Xs,x ω. Lemma 7.5. Fix i in I, and let x and y be points of E i such that y belongs to the Zariski-closure of {x}. For any X-bounded m-form ω on X η , ord Ei,x ω = ord Ei ,y ω Proof. We may suppose that ω ∈ Ω m X/R (X). The natural localization map O X,y → O X,x induces a flat, unramified local homomorphism O X,Ei,y → O X,Ei,x , and Corollary 7.6. Fix i in I. If ω is a X-bounded m-form on X η , then the function For each i ∈ I, we denote by I Ei the defining ideal sheaf of E i in X. For any finite extension R ′ of R and any ψ ∈ X(R ′ ), we denote ord(I Ei )(ψ) by ord Ei (ψ) (see Definition 5.3). If R ′ has ramification degree e over R, and the closed point ψ(0) of the section ψ is contained in E o i , then the equality π = x Ni i · (unit) in O X,ψ(0) implies that ord Ei (ψ) = e/N i .
Ω X,Ei,x ∼ = Ω X,Ei,y ⊗ O X,E i ,y O X,Ei,xE i → Z : x → ord Ei ,x ω is constant on E i .
The following results are proven exactly as their counterparts [31, 6.11-13]. where the left hand side is computed on the smooth special formal R-scheme Sm( X(e)).
Volume Poincaré series.
Theorem 7.12. Let X be a regular special formal R-scheme of pure relative dimension m, whose special fiber is a strict normal crossings divisor X s = i∈I N i E i .
Let ω be a X-bounded gauge form on X η , and put µ i = ord Ei ω for each i ∈ I. Then for any integer d > 0,
X(d) |ω(d)| = L −m ∅ =J⊂I (L − 1) |J|−1 [ E o J ]( k i ≥1,i∈J P i∈J kiNi=d L − P i kiµi ) in M X0
Proof. We'll show how the proof of the corresponding statement in [31, 7.6] can be generalized. First, suppose that d is not X 0 -linear. Then it follows from Theorem 5.1, Corollary 5.15 and Proposition 7.11 that
X |ω(d)| = L −m Ni|d [ E o i ]L −dµi/Ni = L −m ∅ =J⊂I (L − 1) |J|−1 [ E o J ]( k i ≥1,i∈J P i∈J kiNi=d L − P i kiµi ) ,( * )
By Lemma 2.38, it suffices to show that the expression ( * ) is invariant under formal blow-ups with center E J , |J| > 1. This can be done as in [31, 7.6], using an immediate generalization of the local computation in [31, 7.5].
Corollary 7.13. Let X be a generically smooth special formal R-scheme, of pure relative dimension m. Suppose that X admits a resolution of singularities X ′ → X, with special fiber X ′ s = i∈I N i E i . Let ω be a X-bounded gauge form on X η . The volume Poincaré series S(X, ω; T ) is rational over M X0 . In fact, if we put µ i := ord Ei ω, then the series is given explicitly by
S(X, ω; T ) = L −m ∅ =J⊂I (L − 1) |J|−1 [ E o J ] i∈J L −µi T Ni 1 − L −µi T Ni in M X0 [[T ]]
By Proposition 2.42, any affine generically smooth special formal R-scheme admits a resolution of singularities. By the additivity of the motivic integral, we obtain an expression for the volume Poincaré series in terms of a finite atlas of local resolutions. In particular, we obtain the following result.
Corollary 7.14. Let X be a generically smooth special formal R-scheme, of pure relative dimension m. Let ω be a X-bounded gauge form on X η . The volume Poincaré series S(X, ω; T ) is rational over M X0 . More precisely, there exists a finite subset S of Z × N * such that S(X, ω; T ) belongs to the subring
M X0 L a T b 1 − L a T b (a,b)∈S of M X0 [[T ]].
7.3. The Gelfand-Leray form and the local singular series. Let X be a special formal R-scheme, of pure relative dimension m. Then X is also a formal scheme of pseudo-finite type over Spec k, in the terminology of [4], and the sheaves of continuous differential forms Ω i X/k are coherent, by [4, 3.3]. Consider the morphism of coherent O X -modules
i : Ω m X/k dπ∧ − −−− → Ω m+1 X/k : ω → dπ ∧ ω Since dπ ∧ Ω m−1
X/k is contained in its kernel, and
Ω m X/k / dπ ∧ Ω m−1 X/k ∼ = Ω m X/R
by [4, 3.10], i descends to a morphism of coherent O X -modules
i : Ω m X/R dπ∧ − −−− → Ω m+1 X/k
We've seen in Section 2.1 that any coherent module F on X induces a coherent module F rig on X η , and this correspondence is functorial. Hence, i induces a morphism of coherent O Xη -modules [16, 7.1.12].
i : Ω m Xη/K dπ∧ − −−− → (Ω m+1 X/k ) rig by
Definition 7.15. If X is a special formal R-scheme, the Koszul complex of X is by definition the complex of coherent O Xη -modules
0 − −−− → O Xη dπ∧ − −−− → (Ω 1 X/k ) rig dπ∧ − −−− → (Ω 2 X/k ) rig dπ∧ − −−− → . . .
Lemma 7.16. If X is a special formal R-scheme, and i > 0 is an integer, then there exists a canonical exact sequence of O Xη -modules
(Ω i−1 X/k ) rig dπ∧ − −−− → (Ω i X/k ) rig − −−− → Ω i Xη/K − −−− → 0 Proof.
Since the functor (.) rig is exact by Proposition 2.6, we get a canonical exact sequence of O Xη -modules Lemma 7.17. Let X be a variety over k, and consider a morphism f : X → A 1 k = Spec k[π] which is smooth of pure relative dimension m over the torus G m = Spec k[π, π −1 ]. If we denote by X the π-adic completion of f , then the Koszul complex of X is exact, and
(Ω i−1 X/k ) rig dπ∧ − −−− → (Ω i X/k ) rig − −−− → (Ω i X/k /dπ ∧ Ω i−1 X/k ) rig − −−− → 0 But Ω i X/k /dπ ∧ Ω i−1 X/k ∼ = Ω i X/R ,i : Ω m Xη/K dπ∧ − −−− → (Ω m+1 X/k ) rig is an isomorphism. Proof. Put X ′ = X × A 1 k G m . Since f is smooth over G m , 0 − −−− → O X ′ dπ∧ − −−− → Ω 1 X ′ /k dπ∧ − −−− → Ω 2 X ′ /k dπ∧ − −−− → . . .
is exact, and hence the cokernels of the inclusion maps dπ ∧ Ω i−1 X/k → ker (dπ∧ : Ω i X/k → Ω i+1 X/k ) are π-torsion modules. Taking π-adic completions and using [4, 1.9], we see that the cokernels of the maps dπ ∧ Ω i−1 X/k → ker (dπ∧ : Ω i X/k → Ω i+1 X/k ) are π-torsion modules, so they vanish by passing to the generic fiber. We can conclude by exactness of (.) rig (Proposition 2.6) that the Koszul complex of X is exact. Moreover, Ω m+2 X/k is the π-adic completion of Ω m+2 X/k ; hence, it is π-torsion, and (Ω m+2 X/k ) rig = 0. By Lemma 7.16, this implies that
i : Ω m Xη/K dπ∧ − −−− → (Ω m+1 X/k ) rig is an isomorphism.
Lemma 7.18. Let h : X → Y be a morphism of special formal R-schemes, such that h η : X η → Y η isétale. If the Koszul complex of X is exact, then the natural map
ϕ : h * η ((Ω i Y/k ) rig ) = (h * Ω i Y/k ) rig → (Ω i X/k ) rig is an isomorphism of coherent O Xη -modules, for each i ≥ 0. If, moreover, h η is surjective, then the Koszul complex of Y is exact.
Proof. We proceed by induction on i. For i = 0, the statement is clear, so assume i > 0. We put Ω −1 X/k = 0 and Ω −1 Y/k = 0. Now consider the commutative diagram
(h * Ω i−2 Y/k ) rig dπ∧ − −−− → (h * Ω i−1 Y/k ) rig dπ∧ − −−− → (h * Ω i Y/k ) rig − −−− → h * η (Ω i Y/K ) − −−− → 0 (Ω i−2 X/k ) rig dπ∧ − −−− → (Ω i−1 X/k ) rig dπ∧ − −−− → (Ω i X/k ) rig − −−− → Ω i Xη/K − −−− → 0
The bottom row is exact by exactness of the Koszul complex on X and by Lemma 7.16. The upper row is exact except maybe at (h * Ω i−1 Y/k ) rig , by Lemma 7.16 (applied to Y) and flatness of h η . The first and second vertical arrows are isomorphisms by the induction hypothesis, and the fourth one is an isomorphism since h η isétale [12, 2.6]. Now a diagram chase shows that the third vertical arrow is an isomorphism as well. If h η is also surjective, then by faithful flatness the Koszul complex of Y is exact since the complex
0 − −−− → h * η O Yη dπ∧ − −−− → h * η (Ω 1 Y/k ) rig dπ∧ − −−− → h * η (Ω 2 Y/k ) rig dπ∧ − −−− → . . .
is isomorphic to the Koszul complex of X and hence exact.
Proposition 7.19. If X is a generically smooth special formal R-scheme of pure relative dimension m, then the Koszul complex of X is exact, and
i : Ω m Xη/K dπ∧ − −−− → (Ω m+1 X/k ) rig is an isomorphism.
Proof. We may assume that X is affine, say X = Spf A. We use the notation of Section 2.1. The morphism of special formal R-schemes
Y := Spf B n → X
induces an open embedding on the generic fibers. By Lemma 7.16 and Lemma 7.18, it suffices to show that the Koszul complex of Y is exact and that (Ω m+2 Y/k ) rig = 0. Hence, we may as well assume that A is topologically of finite type over R. By resolution of singularities (Proposition 2.42) and the proof of Proposition 3.2, and again applying Lemma 7.18, we may assume that X = Spf A is endowed with ań etale morphism of formal R-schemes Corollary 7.20. If, moreover, X is affine, then the natural map
f : Z → Spf R{x 0 , . . . , x m }/(π − m i=0 x Ni i ) with N i ∈ N. Since f isétale,Ω m X/R (X) ⊗ R K dπ∧ − −−− → Ω m+1 X/k (X) ⊗ R K is an isomorphism, and it fits in a commutative diagram Ω m X/R (X) ⊗ R K dπ∧ − −−− → Ω m+1 X/k (X) ⊗ R K Ω m Xη/K (X η ) dπ∧ − −−− → (Ω m+1 X/k ) rig (X η )
where the vertical arrows are injections and the horizontal arrows are isomorphisms.
Proof. This follows immediately from Lemma 2.10. Xη/K (X η ) induced by ω/dπ. It is obvious from the constructions that this form coincides with our Gelfand-Leray form associated to the element of Ω m+1 X/k (X) obtained from ω by completion.
Corollary 7.22. If X is a regular flat special formal R-scheme of pure relative dimension m, and if ω is an element of Ω m+1 X/k (X), then ω/dπ is X-bounded. If, moreover, ω is a gauge form on X (i.e. a nowhere vanishing section of Ω m+1 X/k (X)), then ω/dπ is a bounded gauge form on X η .
Proof. Boundedness follows from Corollary 7.20. Now suppose that ω is a gauge form on X. We may assume that X = Spf A is affine. The fact that ω is gauge means that ω / ∈ MΩ m+1 X/k (X) for each prime ideal M of A; using [16, 7.1.9], we see that this implies that de image of ω in (Ω m+1 X/k ) rig (X η ) does not vanish at any point x of X η . Hence, since the map i of Proposition 7.19 is an isomorphism of coherent O Xη -modules, ω/dπ is gauge.
Lemma 7.23. If h : Y → X is a morphism of generically smooth special formal R-schemes, both of pure relative dimension m, and if ω is a global section of Ω m+1 X/k , then
(h * ω)/dπ = (h η ) * (ω/dπ) in Ω m Yη /K (Y η ).
Proof. It suffices to show that
dπ ∧ ((h η ) * α) = (h η ) * (dπ ∧ α)
for any m-form α on X η ; substituting α by ω/dπ yields the result. For any i ≥ 0, the square
h * Ω i X/k − −−− → Ω i Y/k dπ∧ dπ∧ h * Ω i+1 X/k − −−− → Ω i+1 Y/k commutes, and therefore h * Ω m X/R − −−− → Ω m Y/R dπ∧ dπ∧ h * Ω m+1 X/k − −−− → Ω m+1 Y/k
commutes. We can conclude by passing to the generic fiber.
Lemma 7.24. If X is a separated formal scheme of pseudo-finite type over Spec F , with F a perfect field, and X is regular, then X is smooth over F .
Proof. Let x be a closed point of X 0 , and let (x 0 , . . . , x m ) be a regular system of local parameters on X at x. These define a morphism of formal schemes of pseudo-finite type over Spec F
h : U → A m+1 F on some open neighbourhood U of x in X. Since A m+1 F
is smooth over F , it suffices to show that h isétale at x. This follows immediately from Lemma 2.43 and the fact that F is perfect.
Lemma 7.25. Let X be a flat regular special formal R-scheme, of pure relative dimension m. Then X is smooth over k, of pure dimension m + 1, and Ω m+1 X/k is a locally free sheaf of rank 1 on X.
Proof. The fact that X is smooth over k follows from Lemma 7.24, since k has characteristic zero. The fact that it has pure dimension m + 1 follows from the flatness of X over R.
By [4, 4.8], the sheaf of continuous differential forms Ω 1 X/k is locally free; by [3, 5.10], it has rank m + 1.
Corollary 7.26. If X is a regular special formal R-scheme, then we can cover X by open formal subschemes U such that U η admits a U-bounded gauge form.
Proof. By Lemma 7.25, we can cover X by open formal subschemes U such that Ω m+1 X/k ∼ = O U . By Corollary 7.22, each U η admits a U-bounded gauge form.
If h : X ′ → X is a morphism of smooth formal k-schemes of pseudo-finite type of pure dimension m + 1, we define the Jacobian ideal sheaf of h as the annihilator of the cokernel of the natural map of locally free rank one O X ′ -modules
ψ : h * Ω m+1 X/k → Ω m+1 X ′ /k
and we denote this ideal sheaf by J ac h/k to distinguish it from the Jacobian ideal sheaf J ac h from Section 5.2.
Lemma 7.27. Let h : X ′ → X be a morphism of regular flat special formal Rschemes, both of pure relative dimension m, such that h η isétale. Then the Jacobian ideal sheaf J ac h/k is invertible, and contains a power of π.
Proof. By Lemma 7.18 and exactness of the functor (.) rig , we see that coker(ψ) rig = 0. This means that J ac h/k contains a power of π, by Corollary 2.9.
Both h * Ω m+1 X/k and Ω m+1 X ′ /k are line bundles on X ′ , by Lemma 7.25. Covering X ′ by sufficiently small affine open formal subschemes U = Spf A, we may assume that they are trivial; let ω and ω ′ be generators for h * Ω m+1 X/k resp. Ω m+1 X ′ /k . Then we can write ψ(ω) = f ω ′ on U with f in A, and f generates the ideal sheaf J ac h/k on U.
Let X be a regular special formal R-scheme, whose special fiber is a strict normal crossings divisor i∈I N i E i . Let J be an invertible ideal sheaf on X, and fix i ∈ I. One can show as in Lemma 7.5 that the length of the O X,Ei,x -module O X,Ei,x /J O X,Ei,x is independent of the point x of E i . We call this value the multiplicity of J along E i . Definition 7.28. Let h : X ′ → X be a morphism of regular flat special formal Rschemes, both of pure relative dimension m, such that h η isétale. If X ′ s is a strict normal crossings divisor i∈I N i E i , then we denote by ν i − 1 the multiplicity of J ac h/k along E i , and we write K X ′ /X = i∈I (ν i − 1)E i Lemma 7.29. Let h : X ′ → X be a morphism of regular flat special formal Rschemes, both of pure relative dimension m, such that h η isétale. If X ′ s is a strict normal crossings divisor i∈I N i E i , and if K X ′ /X = i∈I (ν i − 1)E i then for any gauge form ω on X and any i ∈ I,
ord Ei (h * η (ω/dπ)) = ν i − N i
Proof. First of all, note that h * η (ω/dπ) = (h * ω)/dπ by Lemma 7.23. Choose an index i in I and a point x ′ on E i , and put x = h(x ′ ). Shrinking X to an open formal neighbourhood of x, we may suppose that there exists an integer a such that φ := π a (ω/dπ) belongs to
Ω m X/R /(π − torsion) ⊂ Ω m Xη/K (X η )
since ω/dπ is X-bounded by Lemma 7.22. Consider the commutative diagram
h * Ω m X/R dπ∧ − −−− → h * Ω m+1 X/k Ω m X ′ /R dπ∧ − −−− → Ω m+1 X ′ /k
Definition 7.31. If X is a regular special formal R-scheme, and X admits a gauge form ω, then we define the local singular series of X by
F (X; * ) : N * → M X0 : d → X(d) | ω dπ (d)|
This definition only depends on X, and not on ω, by Proposition 7.30. If X is any regular special formal R-scheme, we choose a finite cover {U i } i∈I of X by open formal subschemes such that each U i admits a gauge form ω i , and we define the local singular series of X by
F (X; * ) : N * → M X0 : d → ∅ =J⊂I (−1) |J|+1 F (∩ j∈J U j ; d)
This definition does not depend on the chosen cover, by additivity of the motivic integral. We define the Weil generating series of X by
S(X; T ) = d>0 F (X; d)T d ∈ M X0 [[T ]]
In the terminology of [14, 4.4], the Weil generating series is the Mellin transform of the local singular series. If X admits a gauge form ω, then by definition, S(X; T ) = S(X, ω/dπ; T ).
The term "Weil generating series" is justified by the fact that F (X; d) can be seen as a measure for the set ∪ K ′ X η (K ′ ) where K ′ varies over the unramified extensions of K(d). Moreover, we have the following immediate corollary of the trace formula in Theorem 6.4. Proposition 7.32. Let ϕ be a topological generator of G(K s /K sh ). If X is a regular special formal R-scheme, then for any integer d > 0, χ top (F (X; d)) = T r(ϕ d | H(X η )) 7.4. The motivic volume. Let X be a generically smooth, special formal Rscheme, of pure relative dimension m, and let ω be a X-bounded gauge form on X η . It is not possible to associate a motivic Serre invariant to X × R R s in a direct way, since the normalization R s of R in K s is not a discrete valuation ring. We will define a motivic object by taking a limit of motivic integrals over finite ramifications of X, instead.
: M X0 L a T b 1 − L a T b (a,b)∈Z×N * −→ M X0 mapping (a,b)∈I L a T b 1 − L a T b
to (−1) |I| = (−1) |I| [X 0 ], for each finite subset I of Z × N * . We call the image of an element its limit for T → ∞.
Proposition 7.34. Let X be a generically smooth, special formal R-scheme, of pure relative dimension m, and let ω be a X-bounded gauge form on X η . The limit of −S(X, ω; T ) for T → ∞ is well-defined, and does not depend on ω. If X ′ → X is any resolution of singularities, with X ′ s = i∈I N i E i , then this limit is given explicitly by
L −m ∅ =J⊂I (1 − L) |J|−1 [ E o J ] in M X0 .
Proof. This follows immediately from the computation in Corollary 7.13, and the observation preceding Corollary 7.14.
Definition 7.35. Let X be a generically smooth special formal R-scheme of pure relative dimension, and assume that X η admits a X-bounded gauge form. The motivic volume S(X; K s ) ∈ M X0 is by definition the limit of −S(X, ω; T ) for T → ∞, where ω is any X-bounded gauge form on X η .
If h : Y → X is a morphism of generically smooth special formal R-schemes such that h η is an isomorphism, and if X η admits a X-bounded gauge form, then it is clear from the definition that S(X; K s ) = S(Y; K s ) in M X0 .
In definition 7.35, the condition that X η admits a gauge form can be avoided as follows.
Proposition-Definition 7.36. If X is a generically smooth special formal Rscheme which admits a resolution of singularities, then there exists a morphism of special formal R-schemes h : Y → X such that h η is an isomorphism, and such that Y has a finite open cover {U i } i∈I such that U i has pure relative dimension and (U i ) η admits a U i -bounded gauge form for each i. Moreover, the value
S(X; K s ) = ∅ =J⊂I (−1) |J|+1 S(∩ i∈J U i ; K s ) ∈ M X0
only depends on X.
Proof. Since X admits a resolution of singularities, we may assume that X is regular and flat; now it suffices to put Y = X and to apply Lemma 7.26.
The fact that the expression for S(X; K s ) only depends on X, follows from the additivity of the motivic integral, and the fact that we can dominate any two such morphisms h by a third by taking the fibered product.
Definition 7.37. Let X be a generically smooth special formal R-scheme, and take a finite cover {U i } i∈I of X by affine open formal subschemes. Then we can define the motivic volume S(X; K s ) by
S(X; K s ) = ∅ =J⊂I (−1) |J|+1 S(∩ i∈J U i ; K s )
This definition only depends on X.
Note that the terms S(∩ i∈J U i ; K s ) are well-defined, since each U i admits a resolution of singularities by Proposition 2.42 and hence, Proposition-Definition 7.36 applies.
Remark. Beware that the motivic volume S(X; K s ) depends on the choice of uniformizer π in R (more precisely, on the fields K(d)), if k is not algebraically
closed. For instance, if k = Q and X = Spf R[x]/(x 2 − 2π) then S(X; K s ) = [Spec Q( √ 2)] in M k , while for X = Spf R[x]/(x 2 − π)
we find S(X; K s ) = 2 (to see that these are distinct elements of M k , look at theirétale realizations in the Grothendieck ring of ℓ-adic representations of G(Q/Q)).
If k is algebraically closed, the motivic volume is independent of the choice of uniformizer, since K(d) is the unique extension of degree d of K in K s ; see the remark following Definition 4.8.
It is not hard to see that for any unramified extension R ′ of R, and any generically smooth special formal R-scheme X, the motivic volume of X ′ = X × R R ′ is the image of the motivic volume of X under the natural base change morphism M X0 → M X ′ 0 . Now we define the motivic volume of a smooth rigid K-variety X η that can be realized as the generic fiber of a special formal R-scheme X. If X η is quasi-compact, this definition was given in [31, 8.3]: the image of S(X; K s ) under the forgetful morphism M X0 → M k only depends on X η , and it was called the motivic volume S(X η ; K s ) of X η . I do not know if this still holds if X η is not quasi-compact; the problem is that it is not clear if any two formal R-models of X η can be dominated by a third. Therefore, we need an additional technical condition (which might be superfluous).
Definition 7.38. Let X be a special formal R-scheme, and suppose that X η is reduced. For any i ≥ 0, a section of
Ω i Xη (X η ) is called a universally bounded i-form on X η , if it is Y-bounded for each formal R-model Y of X η .
I don't know an example of a bounded i-form which is not universally bounded. If X η is reduced, then an analytic function on X η is bounded iff it is universally bounded, by Lemma 2.14. If X η is quasi-compact, then any differential form on X η is universally bounded.
Lemma 7.39. If X is an affine special formal R-scheme, and X η is reduced, then any X-bounded i-form ω on X η is universally bounded.
Proof. Since it suffices to prove that π a ω is universally bounded, for some integer a, we may suppose that ω belongs to the image of the natural map Ω i X/R (X) → Ω i Xη/K (X η ) by Lemma 2.12. This means that we can write ω as a sum of terms of the form a 0 (da 1 ∧ · · · ∧ da i ) with a 0 , . . . , a i regular functions on X, and hence a 0 , . . . , a i are bounded by 1 on X η . If Y is any formal R-model for X η , then by Lemma 2.14, the functions a 0 , . . . , a i on X η are Y-bounded; so ω is Y-bounded.
Proposition-Definition 7.40. Let X be any generically smooth special formal R-scheme, and assume that X η admits a universally bounded gauge form ω. The image of S(X; K s ) under the forgetful morphism M X0 → M k only depends on X η ; we call it the motivic volume of X η , and denote it by S(X η ; K s ).
Proof. If X is any formal R-model for X η , then ω is X-bounded, and the image of S(X; K s ) in M k coincides with
− lim T →∞ d>0 X(d)η |ω(d)| T d
by Proposition 4.7. Hence, it does not depend on X (and neither on ω).
In particular, this definition applies to the generic fiber of an affine regular special formal R-scheme X that admits a gauge form ω ∈ Ω max X/k (X), by Corollary 7.22 and Lemma 7.39. Hence, for any generically smooth special formal R-scheme X, we can cover X η by a finite number of open rigid subvarieties U i , i ∈ I such that S(∩ i∈J U i ; K s ) is defined for each non-empty subset J of I, by Proposition 2.42. However, it is not clear if the value ∅ =J⊂I (−1) |J|+1 S(∩ i∈J U i ; K s ) is independent of the chosen cover: if V ℓ , ℓ ∈ L is another such cover, I do not know if V ℓ ∩ U i admits a universally bounded gauge form for all i and ℓ.
If X is stf t, we recover the definitions from [31].
Proposition 7.41. Let X be a generically smooth special formal R-scheme, and V a locally closed subset of X 0 . If we denote by V the formal completion of X along V , then S(V; K s ) coincides with the image of S(X; K s )under the base-change morphism M X0 → M V .
Proof. This follows immediately from Proposition 4.9.
If X is stf t over R, then we called in [31] the image of S(X; K s ) in M V the motivic volume of X with support in V , and we denoted it by S V (X; K s ). The above proposition shows that
S V (X; K s ) = S(V; K s ) in M V .
In particular, it only depends on V, and not on the embedding in X.
Proposition 7.42. If X is a generically smooth special formal R-scheme, then
χ top (S(X; K s )) = χé t (X η )
where χé t is the Euler characteristic associated to Berkovich'étale ℓ-adic cohomology for non-archimedean analytic spaces.
In particular, if X η admits a universally bounded gauge form, then
χ top (S(X η ; K s )) = χé t (X η )
Proof. Let ϕ be a topological generator of the geometric monodromy group G(K s /K sh ). By definition,
χ top (S(X; K s )) = − lim T →∞ d>0 χ top (S(X(d) η ))T d
Hence, by our Trace Formula in Theorem 6.4,
χ top (S(X; K s )) = − lim T →∞ d>0 T r(ϕ d | H(X η ) )T d Recall the identity [17, 1.5.3] d>0 T r(F d , V )T d = T d dT log(det(1 − T F, V ) −1 ) = − T d dT (det(1 − T F, V )) det(1 − T F, V )
for any endomorphism F on a finite dimensional vector space V over a field of characteristic zero. Taking limits, we get
− lim T →∞ d>0 T r(F d , V )T d = dim(V )
Applying this to F = ϕ and V = H(X η ) yields the result.
The analytic Milnor fiber
In this section, we prove that the analytic Milnor fiber introduced in [31] determines a singularity up to formal equivalence. We do not impose any restriction on the residue field k. Remark. This notion should not be confused with the branches of the special fiber X s at x. For instance, if X = Spf R{x, y}/(xy − π), then X s = Spec k[x, y]/(xy) has two branches at the origin, while X is normal.
Proposition 8.2. Let X be a flat special formal R-scheme, and let x be a closed point on X 0 . Suppose that ]x[ is normal, and let x 1 , . . . , x m be the points lying over x in the normalization h : X → X. There is a canonical isomorphism . Let X and Y be flat special formal R-schemes, and let x and y be closed points of X 0 , resp. Y 0 . Then ]x[ and ]y[ are isomorphic over K, iff the disjoint unions of the branches of (X, x), resp. (Y, y) are isomorphic over R. In particular, if X and Y are normal at x, resp. y, then the R-algebras O X,x and O Y,y are isomorphic iff ]x[ and ]y[ are isomorphic over K. Lemma 8.4. Let X be a smooth special formal R-scheme, let R ′ be any finite unramified extension of R, and denote by k ′ its residue field. The natural map
O ]x[ ( ]x[ ) = m i=1 O ]xi[ ( ]x i [ ) Moreover, m i=1 O e X,X(R ′ ) → X 0 (k ′ )
is surjective.
Proof. By formal smoothness, the map X(R ′ /π n+1 ) → X(R ′ /π n ) is surjective for each n ≥ 0, so since R ′ is complete, we see that X(R ′ ) → X 0 (k ′ ) is surjective.
Corollary 8.5. Consider a flat special formal R-scheme X and a point x of X 0 (k). Then X is smooth at x, iff ]x[ is isomorphic to an open unit polydisc
B m K = (Spf R[[x 1 , . . . , x m ]]) η for some m ≥ 0.
Proof. Replacing X by its formal completion at x, we may as well assume that X 0 = {x}. If X is smooth at x, then X(R) is non-empty by Lemma π)). Let f : X → Spec k[π] be a morphism from a k-variety X to the affine line, let x be a closed point on the special fiber X s of f , and assume that f is flat at x. Denote by X the π-adic completion of f ; it is a stf t formal R-scheme. The tube F x := ]x[ of x in X is canonically isomorphic to the generic fiber of the flat special formal R-scheme Spf O X,x (the R-structure being given by f ), by [8, 0.2.7]. In [31], we called F x the analytic Milnor fiber of f at x, based on a topological intuition explained in [33, 4.1] and a cohomological comparison result: if k = C and X is smooth at x, then theétale ℓ-adic cohomology of F x corresponds to the singular cohomology of the classical topological Milnor fiber of f at x, by [31, 9.2]. If f has smooth generic fiber (e.g. when X − X s is smooth and k has characteristic zero), then the analytic Milnor fiber F x of f at x is a smooth rigid variety over K.
The arithmetic and geometric properties of F x reflect the nature of the singularity of f at x (see for instance Proposition 8.9). We will see in Proposition 8.7 that F x is even a complete invariant of the formal germ of the singularity (f, x), if X is normal at x. Definition 8.6. Let X and Y be k-varieties, endowed with k-morphisms f : X → Spec k[π] and g : Y → Spec k[π]. We say that (f, x) and (g, y) are formally equivalent if O X,x and O Y,y are isomorphic as R-algebras (the R-algebra structures being given by f , resp. g).
Proposition 8.7. Let X and Y be irreducible k-varieties, and let f : X → Spec k[π] and g : Y → Spec k[π] be dominant morphisms. Let x and y be closed points on the special fibers X s , resp. Y s , and assume that X and Y are normal at x, resp. y. The analytic Milnor fibers F x and F y of f at x, resp. g at y, are isomorphic over K, iff (f, x) and (g, y) are formally equivalent.
More precisely, the completed local ring O X,x is recovered, as a R-algebra, as the algebra of analytic functions f on F x with |f | sup ≤ 1.
Proof. By Proposition 8.2, it suffices to show that O X,x and O Y,y are normal and flat. Normality follows from normality of O X,x and O Y,y , by excellence, and flatness follows from the fact that f and g are flat.
Proposition 8.8. Let X be any k-variety, let f : X → Spec k[π] be a morphism of k-varieties, let x be a k-rational point on the special fiber X s of f , and assume that f is flat at x. Then f is smooth at x iff F x is isomorphic to an open unit polydisc B m K for some m ≥ 0. Proof. This follows from Corollary 8.5 (smoothness of f at x is equivalent to smoothness of Spf O X,x over R. Proposition 8.9. Let X be a smooth irreducible k-variety, let f : X → Spec k[π] be a dominant morphism, and let x be a closed point of X s whose residue field k x is separable over k. The following are equivalent:
(1) the morphism f is smooth at x, (2) the analytic Milnor fiber F x of f at x contains a K ′ -rational point for some finite unramified extension K ′ of K. If k is perfect, then each of the above statements is also equivalent to (3) the analytic Milnor fiber F x of f at x is smooth over K and satisfies S(F x ) = 0.
If k has characteristic zero, and if we denote by ϕ a topological generator of the geometric monodromy group G(K s /K sh ), then each of the above statements is also equivalent to (4) the analytic Milnor fiber F x of f at x satisfies T r(ϕ | H(F x × K K s , Q ℓ )) = 0
If k = C, and if we denote by F x the classical topological Milnor fiber of f at x and by M the monodromy transformation on the graded singular cohomology H sing (F x , C) = ⊕ i≥0 H i sing (F x , C) then each of the above statements is also equivalent to (5) the topological Milnor fiber F x of f at x satisfies T r(M | H sing (F x , C)) = 0
Proof. The implication (1) ⇒ (2) (with K ′ = k x ((π))) follows from Lemma 8.4. The implication (2) ⇒ (1) follows from [11, 3.1/2]: denote by R ′ the normalization of R in K ′ . Since X R := X × k[π] R is regular and R-flat, the existence of a R ′section through x on the R-scheme X R implies smoothness of X R at x. But the set F x (K ′ ) is canonically bijective to the set of R ′ -sections on X R through x. The implication (4) ⇒ (3) follows from the trace formula (Theorem 6.4), (3) ⇒ (2) is obvious, and (1) ⇒ (4) follows from [7, 3.5] and the triviality of the ℓadic nearby cycles of f at x. Finally, the equivalence (4) ⇔ (5) follows from the comparison theorem [31, 9.2].
The equivalence of (1) and (5) (for k = C) is a classical result by A'Campo [2].
Comparison to the motivic zeta function
We suppose that k has characteristic zero, and we put R = k[[π]]. Let X be a smooth, irreducible k-variety, of dimension m, and consider a dominant morphism f : X → Spec k[π]. The formal π-adic completion of f is a generically smooth, flat stf t formal R-scheme X. We denote by X η its generic fiber.
Definition 9.1. We call X η the rigid nearby fiber of the morphism f . It is a separated, smooth, quasi-compact rigid variety over K = k((π)). 9.1. The monodromy zeta function. Definition 9.2. Suppose that k is algebraically closed. For any locally closed subset V of X s , we define the monodromy zeta function of f at V by
ζ f,V (T ) = i≥0 det(1 − T ϕ | H i ( ]V [ × K K s , Q ℓ )) (−1) i+1 ∈ Q ℓ [[T ]]
where ϕ is a topological generator of the Galois group G(K s /K). For k = C, the function ζ f,x (T ) is known as the monodromy zeta function of f at x. Let h : X ′ → X be an embedded resolution for f , with X ′ s = i∈I N i E i , and with Jacobian divisor K X ′ |X = i∈I (ν i − 1)E i .
By [21], Theorem 3. Remark. In fact, the local zeta function Z x (f ; T ) carries additional structure, coming from a µ(k)-action on the spaces X d,1 (see [21, 3. In Corollary 9.6 and Theorem 9.7, we'll realize Z x (f ; T ) and S f,x explicitly in terms of the analytic Milnor fiber F x . 9.3. Comparison to the motivic zeta function. We define the local singular series associated to f by If we denote by U the formal completion of X along U , S U (f ; T ) coincides with S(U; T ) by Proposition-Definition 4.9; in particular, it depends only on U.
We recall the following result [31, 9.10]. where ω is any gauge form on Spf O X,x and where we view F x as a rigid variety over k x ((π)), with k x the residue field of x.
Hence, modulo a normalization by powers of L, we recover the motivic zeta function and the local motivic zeta function at x as the Weil generating series of X, resp. Spf O X,x . Proof. The result is obtained by taking a limit T → ∞ of the equality in Theorem 9.5, and applying Proposition-Definition 4.9. Note that S(F x ; K s ) is well-defined, since F x admits a universally bounded gauge form by Corollary 7.22 and Lemma 7.39.
Proposition 9.8. Assume that k is algebraically closed, and let ϕ be a topological generator of the absolute Galois group G(K s /K). For any integer d > 0,
χ top S(F x (d)) = T r(ϕ d | H(F x × K K s , Q ℓ ))
Proof. This is a special case of the trace formula in Theorem 6.4. Corollary 9.9. Suppose k = C, denote by F x the topological Milnor fiber of f at x, and by M the monodromy transformation on the graded singular cohomology space H(F x , C). For any integer d > 0,
χ top S(F x (d)) = T r(M d | H(F x , C))
Proof. This follows from the cohomological comparison in [31, 9.2].
Lemma 2 . 10 .
210Let X = Spf A be an affine special formal R-scheme, and let f : M → N be a morphism of coherent O X -modules such that the induced morphism of coherent O Xη -modules f rig : M rig → N rig is an isomorphism. Then the natural map f : M (X) ⊗ R K → N (X) ⊗ R K
( 1 )
1If U = Spf A is an affine open formal subscheme of X, then the restriction of h over U is the J (U)-adic completion of the scheme-theoretic blow-up of Spec A at the ideal I(U) of A.
Lemma 2 . 18 .
218Let A be a special R-algebra, with ideal of definition J, and let I = (f 1 , . . . , f q ) be a π-open ideal in A. Put X = Spf A. Let h : X ′ → X be the admissible blow-up of X at I. The scheme-theoretic blow-up of Spec A at I is covered by open charts Spec A i , i = 1, . . . , p, where
Lemma 2 . 29 .
229With the notations of Definition 2.27, the morphism h i : X i → X i induced by h is a normalization map for each i. Hence, X i is irreducible for each i.Proof. Fix an index i in {1, . . . , r}, let U = Spf A be an open affine formal subscheme of X, and denote by P j , j = 1, . . . , q the minimal prime ideals in A. Since normalization commutes with open immersions[15, 1.
Noetherian local rings have isomorphic M-adic completions by the proof of [15, 1.2.1]). Hence, shrinking U, we may assume that A/(x j ) is a domain for each j. Then the irreducible components of E ∩ U are defined by x j = 0 for j = 0, . . . , m, M j = 0, and since normalization commutes with open immersions [15, 1.
2. 6 .
6Etale morphisms of special formal schemes. We defineétale and adić etale morphisms of formal R-schemes as in [4, 2.6]. A local homomorphism of local rings (A, M) → (B, N) is called unramified if N = MB and B/N is separable over A/M. We recall the following criterion.
Lemma 2 . 43 .
243Let h : Y → X be a morphism of pseudo-finite type of Noetherian adic formal schemes, and let y be a point of Y. Then the following properties are equivalent: (1) The local homomorphism h * : O X,h(y) → O Y,y is flat and unramified. (2) The local homomorphism h * : O X,h(y) → O Y,y is flat and unramified. (3) h isétale at y. Proof. Use [3], (3.1), (4.5) ,(6.5).
j N j . On Z, we have π = m j=0 (T −aj x j ) Nj . The sections z j := T −aj x j on Z define a morphism of formal schemes over Spec R h : Z → Y := Spec R[y 0 , . . . , y m ]/
Proposition 4. 3 .
3Any generically smooth special formal R-scheme X admits a Néron smoothening.
Lemma 4. 4 .
4Let X be a flat, generically smooth stf t formal R-scheme, and let U be a closed subscheme of X s . If we denote by Y → X the dilatation with center U , then the image of S(Y) under the forgetful morphism M Y0 /(L − [Y 0 ]) → M U /(L − [U ]) coincides with the image of S(X) under the base change morphism M X0 /(L − [X 0 ]) → M U /(L − [U ])
Now we can conclude by Lemma 4.4.
Corollary 4. 10 .
10If {U i , i ∈ I} is a finite stratification of X 0 into locally closed subsets, and U i is the formal completion of X along U i , then i , ω; T ) (we applied the forgetful morphisms M Ui → M X0 to the right-hand sides).
Definition
Proposition 4 . 13 .
413If X is a smooth special formal R-scheme, thenS(X) = [X 0 ] in M X0 /(L − [X 0 ]).
The abelian group C(Z, A) of constructible A-functions on Z is the subgroup of the abelian group of functions of sets Z → A, consisting of mappings of the form ϕ = S∈S a S .1 S
Definition 7 . 1 .
71For each i ∈ I, and each point x of E i , we denote by P i,x the (not necessarily open) prime ideal in O X,x corresponding to E i , and we define O X,Ei,x as the localization O X,Ei,x = (O X,x ) Pi,x Moreover, we introduce the O X,Ei,x -module Ω X,Ei,x := (Ω m X/R,x ) Pi,x /(O X,Ei,x − torsion) By Lemma 2.34, O X,Ei,x is a discrete valuation ring. Note that the valuation of π in O X,Ei,x equals N i .
Lemma 7 . 2 .
72If h : Y → X is anétale morphism, and C is a connected component of h −1 (E i ), then the natural map
y which, by flatness of O X,Ei,x → O Y,C,y , induces an isomorphism Ω X,Ei,x ⊗ O X,E i ,x O Y,C,y ∼ = Ω Y,C,y Lemma 7.3. For each i ∈ I and each point x of E i , the O X,Ei,x -module Ω X,Ei,x is free of rank one.
Definition 7 . 4 .
74Fix i ∈ I and let x be a point of E i . For any ω ∈ Ω m X/R (X)/(π − torsion) we define the order of ω along E i at x as the length of the O X,Ei,x -module
Hence, we can conclude by the following algebraic property: if g : A → A ′ is a flat,unramified morphism of discrete valuation rings, if M is a free A-module of rank 1, and m is an element of M , then the length of the A-module M/(Am) equals the length of the A ′ -module (M ⊗ A ′ )/(A ′ m). Indeed: fixing an isomorphism of A-modules A ∼ = M , the length of A/(Am) is equal to the valuation of m in A.
Definition 7 . 7 .
77For any i ∈ I and any X-bounded m-form on X η , we define the order of ω along E i by ord Ei ω := ord Ei ,x ω where x is any point on E i . By Corollary 7.6, this definition does not depend on the choice of x. Lemma 7.8. Let h : Y → X be anétale morphism of special formal R-schemes. Let C be a connected component of h −1 (E i ). For any X-bounded m-form ω on X η , ord Ei ω = ord C h * η ω Proof. This follows immediately from Lemma 7.2 and the algebraic argument in the proof of Lemma 7.5.
Lemma 7. 9 .
9Fix a non-empty subset J of I.Let R ′ be a finite extension of R, and let ψ be an element of X(R ′ ), such that its closed point ψ(0) lies on E o J . For any X-bounded gauge form ω on X η ,ord(ω)(ψ) = i∈J ord Ei (ψ)(ord Ei ω − 1) + max i∈J {ord Ei (ψ)} Proposition 7.10.Let ω be a X-bounded gauge form on X η . Take a subset J of I, with |J| > 1, and E o J = ∅. Let h : X ′ → X be the formal blow-up with center E J , and denote by E ′ 0 its exceptional component. We have ord E ′ 0 ω = i∈J ord Ei ω Proposition 7.11. Let ω be a X-bounded gauge form on X η . Fix an integer e > 0. Denote by ω(e) the pull-back of ω to the generic fiber of X(e). For each i ∈ I, with N i |e, and each connected component C of Sm( X(e)) 0 × X0 E i , we have ord C ( ω(e)) = (e/N i ).ord Ei ω
it is enough to prove the result forX = Spf R{x 0 , . . . , x m }/(π − m i=0 x Ni i )Now we can conclude by Lemma 7.17.
Definition 7 .
721 (Gelfand-Leray form). If X is a generically smooth special formal R-scheme of pure relative dimension m, and if ω is an element of Ω m+1 X/k (X), then we denote by ω/dπ the inverse image of ω under the isomorphismi : Ω m Xη/K dπ∧ − −−− → (Ω m+1X/k ) rig and we call it the Gelfand-Leray form associated to ω.Remark. Let us compare this definition with the construction made in[31, §9.2]. Let X be a smooth irreducible variety over k, of dimension m + 1, and let f :X → A 1 k = Spec k[t]be a morphism of k-varieties, smooth over the torus Spec k[t, t −1 ]. Let ω be a gauge form on X, and denote by V the complement in X of the special fiber X s of f . In [31, §9.2], we constructed a relative form ω/dπ in Ω defined the Gelfand-Leray form as the element of Ω m
Definition 7.33 ([25], (2.8)). There is a unique M X0 -linear morphism lim T →∞
8. 1 .
1Branches of formal schemes. Definition 8.1. Let X be a flat special formal R-scheme, and let x be a closed point of X 0 . Consider the normalization X → X, and let x 1 , . . . , x m be the points on X 0 lying over x. We call the special formal R-schemes Spf O e X,x1 , . . . , Spf O e X,xm the branches of X at x.
Lemma 9 . 3 .
93If k = C, and x is a closed point of X s , then ζ f,x (T ) coincides withζ(M | ⊕ i≥0 H i sing (F x , Q); T )where F x denotes the topological Milnor fiber of f at x, and M is the monodromy transformation on the graded singular cohomology space ⊕ i≥0 H i sing (F x , Q). Proof. This follows from the comparison result in[31, 9.2].
9. 2 .
2Denef and Loeser's motivic zeta functions. As in[20, p.1], we denote, for any integer d > 0, by L d (X) the k-scheme representing the functor(k − algebras) → (Sets) : A → X(A[t]/(t d+1 ))Following[21, 3.2], we denote by X d and X d,1 the X s -varietiesX d := {ψ ∈ L d (X) | ord t f (ψ(t)) = d} X d,1 := {ψ ∈ L d (X) | f (ψ(t)) = t d mod t d+1 }where the structural morphisms to X s are given by reduction modulo t.In [21, 3.2.1], the motivic zeta function Z(f ; T ) of f is defined asZ(f ; T ) = ∞ d=1 [X d,1 ]L −md T d ∈ M Xs [[T ]]and the naïve motivic zeta function Z naïve (T ) is defined asZ naïve (f ; T ) = ∞ d=1 [X d ]L −md T d ∈ M Xs [[T ]]If U is a locally closed subscheme of X s , the local (naïve) motivic zeta functionZ U (f ; T ) (resp. Z naïve U (f ; T )) withsupport in U is obtained by applying the base change morphism M Xs [[T ]] → M U [[T ]].
L
−νi T Ni 1 − L −νi T Ni in M Xs [[T ]] Z naïve (f ; T ) = ∅ =J⊂I (L − 1) |J| [E o J ] i∈J L −νi T Ni 1 − L −νi T Ni in M Xs [[T ]]Inspired by the p-adic case[18], Denef and Loeser defined the motivic nearby cycles S f by taking formally the limit of −Z(f ; T ) for T → ∞, i.e.S f = ∅ =J⊂I (1 − L) |J|−1 [ E o J ] ∈ M XsFor each closed point x of X s , they denote by S f,x the image of S f under the base change morphism M Xs → M x , and they called S f,x the motivic Milnor fiber of f at x. This terminology is justified by the fact that, when k = C, the mixed Hodge structure of S f,x ∈ M C coincides with the mixed Hodge structure on the cohomology of the topological Milnor fiber of f at x (in an appropriate Grothendieck group of mixed Hodge structures); see [19, 4.2]. Theorem 9.4. Let x be a closed point of X s . The local zeta functions Z x (f ; T ) and Z naïve x (f ; T ), and the motivic Milnor fiber S f,x , depend only on the rigid K-variety F x , the analytic Milnor fiber of f at x. Proof. This follows from Proposition 8.7, since all these invariants can be computed on the R-algebra O X,x , i.e. they are invariant under formal equivalence. To see this, note that any arc ψ : Spec k ′ [[t]] → X with origin x factors through a morphism of k-algebras O X,x → k ′ [[t]], and that f (ψ) ∈ k ′ [[t]] is simply the image of π under
2.1]); the resulting µ(k)action on the motivic Milnor fiber S f,x captures the semi-simple part of the monodromy action on the cohomology of the topological Milnor fiber, by [21, 3.5.5].The same argument as above shows that F x completely determines the zeta function Z x (f ; T ) localized Grothendieck ring of varieties over x with good µ(k)-action[21, 2.4].
F d>0 F
d>0(f ; d) = F ( X; d) ∈ M Xs (see Definition 7.31). We define the motivic Weil generating series associated to f by S(f ; T ) := S( X; T ) = (f ; d)T d ∈ M Xs [[T ]] For any locally closed subscheme U of X s , we define the motivic Weil generating series S U (f ; T ) with support in U as the image of S(f ; T ) under the base-change morphism M Xs [[T ]] → M U [[T ]]
Theorem 9. 5 .
5We haveS(f ; T ) = L −(m−1) Z(f ; LT ) ∈ M Xs [[T ]]Corollary 9.6. For any closed point x on X s ,L −(m−1) Z x (f ; LT ) = S(Spf O X,x ; T ) = d>0 Fx(d) |ω/dπ(d)| T d ∈ M x [[T ]]
9. 4 .
4Comparison to the motivic nearby cycles.Theorem 9.7. We have S( X; K s ) = L −(m−1) S f ∈ M XsFor any closed point x on X s , we haveS(F x ; K s ) = L −(m−1) S f,x ∈ M x
x is regular and hence a domain. Conversely, assume that O Ei,x is a domain. Using the notations in condition (1) of Definition 2.32, we choose an open affine neighbourhood
], h η is an open embedding. Since d is not X s -linear, the obvious generalization of[32, 5.15] implies that) o
i
and that the E o
i -variety E(d) o
i is canonically isomorphic to E o
i , can be proven
exactly as in [31, 4.4].
Since X(d) η is smooth and, a fortiori, normal, and normalization commutes with
taking generic fibers [15, 2.1.3
so we can conclude by[16, 7.1.12].
xi is canonically isomorphic to the subring of O ]x[ ( ]x[ ) consisting of the analytic functions f on ]x[ with supremum norm |f | sup ≤ 1. Proof. The map h η : X η → X η is a normalization map by [15, 2.1.3], and so is its restriction over ]x[, by [15, 1.2.3]. Hence, h η is an isomorphism over ]x[, so it is clear that ]x[ ∼ = ⊔ m i=1 ]x i [. Therefore, we may as well assume that X is normal. In this case, the result follows from [16, 7.4.1].Corollary 8.3
8.4, and hence X ∼ = Spf R[[x 1 , . . . , x m ]] by [11, 3.1/2]. Hence, ]x[ is isomorphic to the open unit polydisc B m K . For the converse implication, assume that X η ∼ = B m K . Then X is normal since B m K is normal and connected; so we can apply Corollary 8.3 to (X, x) and (Spf R[[x 1 , . . . , x m ]], 0). 8.2. The analytic Milnor fiber. In this section, we put R = k[[π]] and K = k((
The converse implication is false, as is seen by taking a regular flat formal curve X over R whose special fiber Xs is an irreducible curve with a node x, and putting Z = {x}.
Since Ω m+1 X/k and Ω m+1 X ′ /k are locally free, we get a commutative diagram (using the notation in Section 7.1)Since dπ ∧ φ = π a ω and ω generates Ω m+1 X/k , we see that the O X ′ ,Ei,x ′ -module, so it suffices to show that the cokernel of the lower horizontal mapSince this value does not change if we pass to anétale cover of X ′ whose image contains x ′ (by the algebraic argument used in the proof of Lemma 7.5), we may assume that π = m j=0 x Nj j on X ′ , with (x 0 , . . . , x m ) a regular sequence. Hence, taking differentials, we see thatgenerates Ω X ′ ,Ei,x ′ , and it is clear that ord xi (dπ ∧ ω 0 ) = N i − 1 in Ω m+1 X ′ /k ⊗ O X ′ ,Ei,x ′ (see Definition 5.7 for the notation ord xi (.) ).Proposition 7.30. Let X be a regular flat special formal R-scheme of pure relative dimension m, and let ω be a gauge form in Ω m+1 X/k (X). The volume Poincaré series S(X, ω/dπ; T ) only depends on X, and not on ω. In fact, if X ′ → X is any resolution of singularities, with X ′ s = i∈I N i E i and K X ′ /X = i∈I (ν i − 1)E i , then S(X, ω/dπ; T ) is given explicitly byProof. By additivity of the motivic integral, we may assume that X is affine. Then X admits a resolution of singularities by Proposition 2.42, and the expression for S(X, ω/dπ; T ) follows from Corollary 7.13 and Lemma 7.29. This expression is clearly independent of ω.Remark. The fact that S(X, ω/dπ; T ) does not depend on ω follows already from the fact that ω/dπ is independent of ω up to multiplication with a unit on X. Hence, for each d > 0, X(d) |(ω/dπ)(d)| does not depend on ω. Beware that S(X, ω/dπ; T ) depends on the choice of π if k is not algebraically closed; see the remark following Definition 4.8.
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. CNRS -UMR. 852459655Cité ScientifiqueUniversité Lille 1, Laboratoire PainlevéUniversité Lille 1, Laboratoire Painlevé, CNRS -UMR 8524, Cité Scientifique, 59655
France E-mail address: [email protected]. Villeneuve D'ascq, Cédex, frVilleneuve d'Ascq Cédex, France E-mail address: [email protected]
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[] |
[
"CHAIN OF THOUGHT PROMPTING UNDER STREAMING BATCH: A CASE STUDY",
"CHAIN OF THOUGHT PROMPTING UNDER STREAMING BATCH: A CASE STUDY"
] |
[
"Yuxin Tang [email protected] \nDepartment of Computer Science\nRice University Houston\n77005TXUSA\n"
] |
[
"Department of Computer Science\nRice University Houston\n77005TXUSA"
] |
[] |
Recently, Large Language Models (LLMs) have demonstrated remarkable capabilities. Chain-of-Thought (CoT) has been proposed as a way of assisting LLMs in performing complex reasoning. However, developing effective prompts can be a challenging and labor-intensive task. Many studies come out of some way to automatically construct CoT from test data. Most of them assume that all test data is visible before testing and only select a small subset to generate rationales, which is an unrealistic assumption. In this paper, we present a case study on how to construct and optimize chain-of-thought prompting using batch data in streaming settings.
| null |
[
"https://export.arxiv.org/pdf/2306.00550v1.pdf"
] | 258,999,174 |
2306.00550
|
811115d36e1eabe2cef03b38a0809514e40b658e
|
CHAIN OF THOUGHT PROMPTING UNDER STREAMING BATCH: A CASE STUDY
Yuxin Tang [email protected]
Department of Computer Science
Rice University Houston
77005TXUSA
CHAIN OF THOUGHT PROMPTING UNDER STREAMING BATCH: A CASE STUDY
10.18653/v1/D15-1202Published as a Tiny Paper at ICLR 2023
Recently, Large Language Models (LLMs) have demonstrated remarkable capabilities. Chain-of-Thought (CoT) has been proposed as a way of assisting LLMs in performing complex reasoning. However, developing effective prompts can be a challenging and labor-intensive task. Many studies come out of some way to automatically construct CoT from test data. Most of them assume that all test data is visible before testing and only select a small subset to generate rationales, which is an unrealistic assumption. In this paper, we present a case study on how to construct and optimize chain-of-thought prompting using batch data in streaming settings.
PROBLEM STATEMENT
The problem of chain-of-thought under streaming is first proposed by Zhang et al. (2023a). It assumes that the test dataset, denoted as D, will be evenly partitioned into m batches, each containing N samples. These batches are fed into a large language model M in a continuous stream-like manner. At each time step t k for k = 1, 2, . . . , m, M processes one batch of questions q
(k) 1 , q (k) 2 , . . . , q (k) N
using the same prompt P . After processing a batch, rationales generated by M are denoted as c
(k) 1 , c (k) 2 , . . . , c (k) N . A prompting optimization function f (P |(q (k) 1 ||c (k) 1 ), (q (k) 2 ||c (k) 2 ), . . . , (q (k) N ||c (k)
N )) 1 is applied to update prompt P before processing the next batch. This allows the model to maintain a coherent chain-of-thought within each batch in the stream, known as the intra-batch chain-of-thought.
Function f adopted in Zhang et al. (2023a) is a simple concatenation function by appending all the newly generated question-rationale pairs (q
(k) 1 ||c (k) 1 ), (q (k) 2 ||c (k) 2 ), . . . , (q (k) N ||c (k)
N ) to the previous prompting P . However, this approach can quickly reach the maximum input sequence length of the Language Model (e.g. 2048 tokens) and is not particularly scalable or efficient, potentially leading to high levels of redundancy in the prompting and high costs for querying LLM. Therefore, an alternative approach may be needed to address these limitations.
PROMPTING OPTIMIZATION FUNCTION
Prompt optimization is a black-box optimization problem, as it can only be evaluated based on the quality of the generated rationales or the correctness of the answers provided by LLM. In this regard, we present our empirical findings on how to design f to update prompting constrained by the limit input sequence length based on two attributes of CoT: correctness and depth. Correctness is a crucial criterion for prompting engineers to update the prompting. Here, we want to ask the question: Can a valid rationale be replaced with an invalid one without performance drop? Depth refers to the number of reasoning steps which can be reflected by the length of CoT (different steps are separated by comma or \n). A deeper CoT is longer and typically contains more complex rationales, while a shallower CoT is more straightforward and contains fewer reasoning steps. Here, the question is: Given the similar question, can a deep CoT always be replaced by a shallow CoT?
EXPERIMENTS
Our evaluation is conducted on model text-davinci-002 from OpenAI. We choose multiple datasets across different tasks including arithmetic reasoning (GSM8K Cobbe et al. (2021), Multi-Arith Roy & Roth (2015)), commonsense reasoning (StrategyQA Geva et al. (2021)), and symbolic reasoning (Letter ). We compare our methods with Zero-Shot-CoT Kojima et al. (2022), and Bootstraping Auto-CoT Zhang et al. (2023a). We manually partition each dataset into 10 batches and generate rationales for each batch under the streaming setting. We report the test accuracy for ten batches. The number of samples in each batch is in Appendix A.
RIGHT OR WRONG COT
To address this question, we substitute ground-truth rationales that produce the correct answer with Zero-Shot-CoT that produce the wrong answer. During the substitution, we ensure that more than 50% of the rationales in the prompt are incorrect. This substitution method is referred to as Wrong-CoT. Conversely, we have Correct-CoT, which includes only the correct rationales. Figure 1(a) shows the results. Wrong-CoT is manually selected through human evaluation by running the same query multiple times.
DEEP OR SHALLOW COT
In order to address the aforementioned question, we exclusively utilize CoTs that are deemed correct in the prompt. To distinguish between deep CoT and shallow CoT, we have established a simple heuristic parameter denoted as ξ, which is based on the number of \n in the CoT. If the number of \n surpasses ξ, we classify it as a deep CoT. Conversely, if a CoT has fewer \n than ξ, we classify it as a shallow CoT. Shallow-CoT are selected by replacing lengthier rationales by shorter rationales in a batch. Figure 1(b) shows the results.
CONCLUSION
This paper presents a straightforward case study on how to update and replace prompts used for large language model in a streaming batch setting. Our findings indicate that incorrect chain-ofthought promptings can be valuable and do not significantly diminish performance. Additionally, promptings that consist of shorter rationales exhibit superior performance compared to those with lengthier rationales.
URM STATEMENT
The authors acknowledge that at least one key author of this work meets the URM criteria of ICLR 2023 Tiny Papers Track.
Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Olivier Bousquet, Quoc Le, and Ed Chi. Least-to-most
Models Tang et al. (2023) have shown emergent abilities. Wei et al. (2022a) Wei et al. (2022b) discover Chain-of-Thought (CoT) prompting as a simple and broadly applicable method for enhancing reasoning in language models. Many work Zhou et al. (2022a); Wang et al. (2022b); Shi et al. (2022); Zhang et al. (2023a;b); Wang et al. (2022a); Zhou et al. (2022b); Fei et al. (2023); Yang et al. (2023); Shi et al. (2023); Diao et al. (2023) have tried to make further improvements based on CoT.
Figure 1 :Figure 2 :
12Left: accuracy for MultiArith dataset under Correct-CoT and Wrong-CoT. Right: accuracy for MultiArith dataset under Deep-CoT and Shallow-CoT with ξ = 3. Left: accuracy for GSM8K dataset under Correct-CoT and Wrong-CoT. Right: accuracy for GSM8K dataset under Deep-CoT and Shallow-CoT with ξ = 3.
Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers. arXiv preprint arXiv:2211.01910, 2022b.prompting enables complex
reasoning in large language models. arXiv preprint arXiv:2205.10625, 2022a.
A APPENDIX
Table 1: Batch size of four different datasets
MultiArith
GSM8K
StrategyQA
Letter
Batch Size
60
64
32
81
1 2 3 4 5 6 7 8 9 10
The concatenation of question and rationale is denoted as a question-rationale pair (q||c)
Training verifiers to solve math word problems. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, John Schulman, Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems, 2021. URL https://arxiv. org/abs/2110.14168.
Active prompting with chain-of-thought for large language models. Shizhe Diao, Pengcheng Wang, Yong Lin, Tong Zhang, arXiv:2302.12246arXiv preprintShizhe Diao, Pengcheng Wang, Yong Lin, and Tong Zhang. Active prompting with chain-of-thought for large language models. arXiv preprint arXiv:2302.12246, 2023.
Reasoning implicit sentiment with chain-of-thought prompting. Hao Fei, Bobo Li, Qian Liu, Lidong Bing, Fei Li, Tat-Seng Chua, arXiv:2305.11255arXiv preprintHao Fei, Bobo Li, Qian Liu, Lidong Bing, Fei Li, and Tat-Seng Chua. Reasoning implicit sentiment with chain-of-thought prompting. arXiv preprint arXiv:2305.11255, 2023.
Did aristotle use a laptop? a question answering benchmark with implicit reasoning strategies. Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, Jonathan Berant, Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, and Jonathan Berant. Did aristotle use a laptop? a question answering benchmark with implicit reasoning strategies, 2021. URL https://arxiv.org/abs/2101.02235.
Large language models are zero-shot reasoners. Takeshi Kojima, Shane Shixiang, Machel Gu, Yutaka Reid, Yusuke Matsuo, Iwasawa, arXiv:2205.11916arXiv preprintTakeshi Kojima, Shixiang Shane Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. Large language models are zero-shot reasoners. arXiv preprint arXiv:2205.11916, 2022.
Solving general arithmetic word problems. Subhro Roy, Dan Roth, 10.18653/v1/D15-1202Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsSubhro Roy and Dan Roth. Solving general arithmetic word problems. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pp. 1743-1752, Lisbon, Por- tugal, September 2015. Association for Computational Linguistics. doi: 10.18653/v1/D15-1202. URL https://aclanthology.org/D15-1202.
Language models are multilingual chain-of-thought reasoners. Freda Shi, Mirac Suzgun, Markus Freitag, Xuezhi Wang, Suraj Srivats, Soroush Vosoughi, Hyung Won, Yi Chung, Sebastian Tay, Denny Ruder, Zhou, arXiv:2210.03057arXiv preprintFreda Shi, Mirac Suzgun, Markus Freitag, Xuezhi Wang, Suraj Srivats, Soroush Vosoughi, Hyung Won Chung, Yi Tay, Sebastian Ruder, Denny Zhou, et al. Language models are multi- lingual chain-of-thought reasoners. arXiv preprint arXiv:2210.03057, 2022.
Large language models can be easily distracted by irrelevant context. Freda Shi, Xinyun Chen, Kanishka Misra, Nathan Scales, David Dohan, Ed Chi, Nathanael Schärli, Denny Zhou, arXiv:2302.00093arXiv preprintFreda Shi, Xinyun Chen, Kanishka Misra, Nathan Scales, David Dohan, Ed Chi, Nathanael Schärli, and Denny Zhou. Large language models can be easily distracted by irrelevant context. arXiv preprint arXiv:2302.00093, 2023.
Does synthetic data generation of llms help clinical text mining?. Ruixiang Tang, Xiaotian Han, Xiaoqian Jiang, Xia Hu, arXiv:2303.04360arXiv preprintRuixiang Tang, Xiaotian Han, Xiaoqian Jiang, and Xia Hu. Does synthetic data generation of llms help clinical text mining? arXiv preprint arXiv:2303.04360, 2023.
Towards understanding chain-of-thought prompting: An empirical study of what matters. Boshi Wang, Sewon Min, Xiang Deng, Jiaming Shen, You Wu, Luke Zettlemoyer, Huan Sun, arXiv:2212.10001arXiv preprintBoshi Wang, Sewon Min, Xiang Deng, Jiaming Shen, You Wu, Luke Zettlemoyer, and Huan Sun. Towards understanding chain-of-thought prompting: An empirical study of what matters. arXiv preprint arXiv:2212.10001, 2022a.
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Chain-of-thought prompting elicits reasoning in large language models. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le, Denny Zhou, Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le, and Denny Zhou. Chain-of-thought prompting elicits reasoning in large language models, 2022b. URL https://arxiv.org/abs/2201.11903.
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|
[] |
[
"Diffusive and Ballistic Transport in Ultra-thin InSb Nanowire Devices Using a Few-layer-Graphene-AlOx Gate",
"Diffusive and Ballistic Transport in Ultra-thin InSb Nanowire Devices Using a Few-layer-Graphene-AlOx Gate"
] |
[
"Lior Shani \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Pim Lueb \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Gavin Menning \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Mohit Gupta \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Colin Riggert \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Tyler Littman \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Frey Hackbarth \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Marco Rossi \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Jason Jung \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Ghada Badawy \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Marcel A Verheijen \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Paul Crowell \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n",
"Erik P A M Bakkers \nDepartment of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands\n",
"Vlad S Pribiag [email protected] \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n"
] |
[
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA",
"Department of Applied Physics\nEindhoven University of Technology\nEindhovenThe Netherlands",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA"
] |
[] |
Quantum devices based on InSb nanowires (NWs) are a prime candidate system for realizing and exploring topologically-protected quantum states and for electrically-controlled spin-based qubits. The influence of disorder on achieving reliable topological regimes has been studied theoretically, highlighting the importance of optimizing both growth and nanofabrication. In this work we investigate both aspects. We developed InSb nanowires with ultra-thin diameters, as well as a new gating approach, involving few-layer graphene (FLG) and Atomic Layer Deposition (ALD)-grown AlOx. Low-temperature electronic transport measurements of these devices reveal conductance plateaus and Fabry-Pérot interference, evidencing phase-coherent transport in the regime of few quantum modes. The approaches developed in this work could help mitigate the role of material and fabricationinduced disorder in semiconductor-based quantum devices.
| null |
[
"https://export.arxiv.org/pdf/2306.00117v1.pdf"
] | 258,999,310 |
2306.00117
|
0a391575492e05bbb2bd925c806274a825d61cf0
|
Diffusive and Ballistic Transport in Ultra-thin InSb Nanowire Devices Using a Few-layer-Graphene-AlOx Gate
Lior Shani
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Pim Lueb
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Gavin Menning
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Mohit Gupta
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Colin Riggert
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Tyler Littman
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Frey Hackbarth
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Marco Rossi
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Jason Jung
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Ghada Badawy
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Marcel A Verheijen
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Paul Crowell
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Erik P A M Bakkers
Department of Applied Physics
Eindhoven University of Technology
EindhovenThe Netherlands
Vlad S Pribiag [email protected]
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
Diffusive and Ballistic Transport in Ultra-thin InSb Nanowire Devices Using a Few-layer-Graphene-AlOx Gate
1 * Equal Contribution & Corresponding Author:
Quantum devices based on InSb nanowires (NWs) are a prime candidate system for realizing and exploring topologically-protected quantum states and for electrically-controlled spin-based qubits. The influence of disorder on achieving reliable topological regimes has been studied theoretically, highlighting the importance of optimizing both growth and nanofabrication. In this work we investigate both aspects. We developed InSb nanowires with ultra-thin diameters, as well as a new gating approach, involving few-layer graphene (FLG) and Atomic Layer Deposition (ALD)-grown AlOx. Low-temperature electronic transport measurements of these devices reveal conductance plateaus and Fabry-Pérot interference, evidencing phase-coherent transport in the regime of few quantum modes. The approaches developed in this work could help mitigate the role of material and fabricationinduced disorder in semiconductor-based quantum devices.
Introduction
Improving the electronic cleanliness of quantum devices based on low-dimensional semiconductors, such as nanowires or quantum wells, is of paramount importance for creating robust and tunable quantum states and for enabling quantum technologies. InSb and InAs NWs have a wide range of desirable properties, including strong spin-orbit 1 coupling (SOC) and large g-factors 2,3 , that make their use appealing for realizing qubits based on Majorana Zero Modes (MZM) 4-7 and electrically-controlled spin-based qubits [8][9][10] .
MZMs, which could be used to develop topologically-protected qubits, require proximitizing of nanowires with a conventional superconductor, such as Al. A key precursor of MZMs is the spin-helical state, in which momentum and spin become correlated. This arises in the normal state, without superconductors, and requires spin-orbit coupling and robust ballistic transport. However, establishing clean ballistic transport and the presence of the helical state have been sidestepped in most recent experiments on MZMs, which have prioritized more complex devices involving superconductors even when these underlying requirements were not reliably confirmed.
Recent theoretical analysis 11,12 suggests that disorder, due to inhomogeneous dielectric environments, surface charges on the nanowire and impurities in the crystal lattice that occur during the growth process, is a key factor in determining whether the topological regime of MZMs can be reliably achieved. The presence of quantized conductance is a clear indication of whether a nanowire device is sufficiently clean to host helical states and hence, potentially, MZMs, making investigations of quantized transport a critical first step towards ascertaining that MZMs can be realized using a given type of nanowire and nanofabrication process. Therefore, optimizing the materials and nanofabrication is an important task that is necessary in order to align the quality of quantum devices with the requirements of MZMs.
Past progress in InSb NW growth has produced high quality wires with a typical diameter of 90 − 120 13 . However, the diameter of the wires was still 3-4 times larger than the Fermi wavelength 14 of InSb is (~ 30 ). Therefore, reducing the diameter of the wire may unravel new phenomena by further enhancing quantum confinement, leading to larger inter-subband energy splitting and potentially reduced scattering in the relevant regime of few occupied transverse quantum modes.
Recent advances in the development of 2D materials offer new ways to electrostatically gate quantum devices by taking advantage of the low surface roughness and flexibility in stacking 2D materials. Owing to its high electrical conductivity, graphene has emerged as a widely studied material for the development of new, high-performance nanoelectronics devices such as sensors and transistors 15,16 . However, because monolayer graphene is a Dirac semimetal 17 , external gating is required to achieve a 'metallic' electrical conductivity 18 . In addition, the surface of graphene is chemically inactive 19 , which inhibits the growth of thin dielectrics using conventional techniques such as Physical Vapor Deposition (PVD) and Atomic Layer Deposition (ALD), thus restricting the usefulness of graphene as an electrostatic gate for tuning quantum devices. A possible solution for the issues above is using a graphene multilayer, since it is intrinsically metallic and has a more chemically active surface.
In this work we demonstrate the structural and electrical characterization of ultra-thin where under high temperature the organo-indium and organo-antimony bonds are broken, allowing the indium and antimony to interact with the gold droplet. The gold acts as the catalyst particle for the vapor-liquid-solid (VLS) method 21 .
The fabrication of the FLG-AlOx back gate begins with depositing FLG to a prepatterned chip with gold rectangles. FLG was exfoliated from Highly Oriented Pyrolytic Graphite (HOPG, HQ Graphene) and positioned deterministically using a transfer station.
The FLG flake is placed on SiO2 substrate and partially overlapping with a gold rectangle to connect the FLG to the voltage source. The typical areas of the FLG flakes were selected to be larger than 50 2 to have a sufficient overlap with the gold rectangle and to contain the entire length of the wire. A ~55 of AlOx was deposited on the entire chip using Plasma Enhanced Atomic Layer Deposition (PE-ALD, Fiji 2, Ultratech) at 150 . While the FLG surface is more reactive compared to monolayer graphene, nonetheless, it still lacks dangling bonds to react with the precursor using only thermal treatment, which causes the dielectric to grow with pinholes. Therefore, additional surface treatment is necessary to create nucleation sites for the precursor 22,23 . The PE-ALD allows deposition of dielectric layer on the FLG without pretreatment because it creates local defects using plasma that serves as nucleation site for the precursor.
To investigate the electrical properties of the NWs we used a mechanical transfer station with a micro manipulator and an optical microscope to position the NWs on the FLG-AlOx back gates, then patterned leads using conventional electron-beam lithography (see SI). The NW is coated with a layer of native oxide, therefore, to make contact to the wire it is necessary to remove the oxide before putting on the metallic leads. The native oxide was removed using Ar ion milling, followed by in-situ evaporation of 10 of Ti and 140 of Au. The devices were pumped overnight and measured in a dilution refrigerator at base indicated temperature of ~10 (unless indicated otherwise).
After the device has been electrically characterized, a transmission electron microscopy (TEM) lamella is prepared using a FEI Nova Nanolab 600i. Protective layers of carbon and platinum are deposited with an electron-beam in vacuum. A lamella is cut using a Gallium Focused Ion Beam (FIB) and transferred to a half-moon TEM grid. The lamella is thinned by FIB milling in steps at 30 , 16 and finally 5 , which creates a window thinner than 100 . This window is subsequently studied with TEM, a probe-corrected JEOL ARM 200F.
Results and discussion
The first goal is to realize thin and long InSb wires and, as reported before 20 , the size of the mask opening influences the resulting wire diameter. The challenge is to create small, but well-defined holes. Here, we found that RIE provides finer control of the dimensions of the opening in the mask than wet chemical etching (see SI). By carefully tuning the growth temperature and incoming material flux (SI), the dimensions of the wire are optimized while retaining a high yield. At sufficiently low overall flux the radial growth rate is 6x lower than the axial growth rate as a function of time (SI), resulting in high aspect ratio nanowires. This can be understood by the fact that radial growth is driven by a vapor-solid (VS) mechanism, and axial growth by the VLS mechanism in which growth is catalyzed by the Au particle. By optimizing growth parameters, a yield of 90 + % has been obtained, widths in the range 50 − 60 and length of 14 + (Figure 1a and 1b). Because the NWs are very long, we are able to fabricate multiple devices on the same NW. In particular, we fabricated on the same NW a 'long' device with source-drain spacing ≈ 6 much larger than the mean-free-path ( ≈ 300 ) and a 'short' device with To extract the field-effect mobility, , from this pinch-off data we treat the wire as a variable-conductance channel coupled to the gate electrode with a capacitance .
Additionally, we consider this channel in series with a fixed resistance , which includes the contact resistance of the normal metal/wire interfaces, as well as the resistance of the fridge lines, filters, and measurement apparatuses. Together, this gives the gate-dependent conductance of the device as, . c) SEM image of a NW device placed on FLG-AlOx back gate, scale bar is 1 .The inset to the figure shows the transport data for the device with the 6 spacing, the black line is the measured data, and the red line is the fitting to eq.1. d) Cross-sectional TEM overview of the measured NW device, showing defect free hexagonal crystal structuring having a width of ~52 and native oxide layer of ~3 .
where is the channel length and ℎ is the threshold gate voltage at which the Fermi level enters the conduction band and the device becomes conductive 13 .
In order to fit this equation to our data, we calculate the capacitance between our gate and wire, using a self-consistent, finite-element Schrödinger-Poisson solver to simulate the exact geometry of our device, as described in Ref 24 . For the ultra-thin 6 µ long channel, we calculate this capacitance to be 588 . Fitting the data in Fig 1c using this capacitance, we find our mobility to be µ = 12,700 ± 300 2 / . The appearance of the diamond shape pattern indicates that the device acts as a Fabry−Pérot (FP) interferometer, a partially transmitting cavity for electron waves. The reflection may occur in part as a result of band bending near the contacts, as expected for a metal-semiconductor interface, and was observed before in InSb and InAs nanowires [25][26][27] . In addition, metallic contacts screen the electric field, such that the back-gate tunability on the channel close to the contacts is reduced, resulting in different electron densities there vs. in-between the contacts. As a result of these barriers, electrons experience multiple partial reflections near the contacts while propagating phase-coherently, which gives rise to FP interference, setting a lower bound of 200 for the length over which phase-coherent and quasi-ballistic transport can be achieved in these devices. The FP oscillations are very well defined, with a large relative conductance modulation suggestive of transport that is close to the 1D limit. Though revealing phase-coherent transport, FP oscillations may obscure other quantum phenomena, in particular quantized conductance plateaus associated with individual transverse quantum modes.
Applying an external magnetic field to the ultra-thin NW lifts the spin degeneracy and may cause suppression of FB oscillation by inducing dephasing between multiple trajectories. To investigate the presence of conductance plateaus, we apply an out-of-plane magnetic field ( = 2.8 ) to suppress the FP oscillations (Figure 3). Figure 3b shows a series of line cuts taken at −2 < < 2 from the differential conductance map, confirming that the FP oscillations are substantially reduced with respect to the = 0 data.
The line cuts reveal a conductance plateau that remains close to 0 over a wide range of source-drain bias values, compatible with quantized transport superimposed with some residual oscillations due to FP interference.
In addition, a further plateau-like feature appears near 1. To further investigate the robustness of the observed 1 0 plateau, we measure conductance at a higher temperature, which is expected to reduce the phase coherence and thus further suppress resonances such as those due to FP interference. Figure 4a shows a differential conductance map as function of and taken at 400 and = 0. Note that the pinch-off voltage increased to ~3 after the temperature change, possibility due to a showing a suppression of FB interference. b) A series of line cuts from the 2D differential conductance map (Fig 4a) between = −1.5 and = 1.5 as a function of , purple and brown curve respectively. The curves show the emergence of a plateau at 1 0 thus highlight the fact the FB interference can suppress potential conductance plateaus. spontaneous charge switch in the dielectric environment, either in the native oxide shell surrounding the wire or the ALD deposited gate dielectric. The finer oscillatory features seen at 10
in Figure 2a are suppressed at 400 , however clear FP interference is still present, as evidenced by the diamonds in the differential conductance map. Figure 4b shows a series of line cuts for −1.5 < < 1.5 . The oscillations appear to contain fewer harmonic components than in the 10 data, consistent with suppression of phase coherence and hence interference between fewer coherent trajectories. For 3.2 < < 3.8
the conductance values again coalesce near the value of 1 0 , providing further evidence that this plateau feature is likely due to the lowest transverse subband in the nanowire.
In order to gain better understanding of the origin of the FB oscillations and their interplay with subband quantization, we performed tight-binding numerical simulations using the KWANT 28 Python software package. We show that one possible origin of FP oscillations in our system is the difference in the chemical potential between the metallic leads and the semi-conducting nanowire. The Hamiltonian of our nanowire is defined as:
(2) = ( 2 2 * − ) − − 2 ⃗ • ⃗⃗
The first term of the Hamiltonian is the kinetic energy, the second term of the Hamiltonian describes the Rashba SOC, where is the Rashba spin-orbit coefficient and is the zcomponent of the Pauli matrix, and the third term is the Zeeman coupling due to an applied magnetic field ⃗⃗ .
We simulate this by discretizing over a square mesh with a lattice constant =
5
. The width of nanowire is set to 60 with a contact spacing of 200 29 . The
Hamiltonian of the leads includes only the kinetic energy term from the above defined .
Additional details regarding the simulation parameters are provided in the SI. Figure 5a shows the subbands for a one-dimensional semiconductor with spin-orbit coupling and Zeeman coupling. The subbands are shifted relative to each-other along the k-axis due to the SOC. The red curve in Figure 5b shows conductance as a function of energy through this system. This is valid under the assumption that there is no chemical potential difference between leads and the wire and we obtain clean quantized conductance plateaus. However, this does not represent a typical NW experiment because the chemical potential of the NW is tuned by applying electric field using a gate, and the field in the leads area is screened by the metallic contacts. This gives rise to a non-uniform potential profile along the length of the wire. To account for this we implement the gate potential as a hyperbolic tangent function as described in Ref. 30 . This can result in FP oscillations in pristine wires without any defects. The blue curve in Figure 5b shows the conductance, , in units of 0 as a function of gate voltage. The curve shows conductance oscillations, making the previously visible plateaus obscure. The results obtained from our simulations agree qualitatively with the data in Figure 2b.
In conclusion, we fabricated ~10 long and ~55 wide InSb nanowires. We used
Few-Layer-Graphene as the conducting layer for local back gate, yielding reduced surface roughness relative to Ti/Au backgates. The electrical properties of the ultra-thin InSb nanowires with FLG-AlOx local back gate were characterized at temperatures. By leveraging the large length of the nanowires (~10 ) we explored electronic transport in the diffusive transport regime on the same wire on which the ballistic regime was also accessed, the first with contact spacing of ~6 and the second with contact spacing of ~200 . In the ballistic regime, the devices reveal Fabry-Pérot interference due to multiple partial reflections of the phase-coherently propagating electron waves. The visibility of the Fabry-Pérot oscillations competes with that of conductance plateaus, yet the plateaus remain detectable, suggesting that for the channel length of ~200 transport occurs phase-coherently and ballistically through a few discrete quantum modes. Quantized conductance at zero magnetic field has been very difficult to observe in an InSb nanowire devices 31 and remains a key test for a device's suitability for MZM experiments. Our results suggest that reducing the nanowire diameter, along with optimizing gate materials could be a viable path towards realizing MZMs.
~10longInSb nanowires with a diameter of ~50 . To optimize the dielectric environment surrounding the wire we developed a facile fabrication technique based on Few-Layer-Graphene (FLG) and ALD-grown AlOx to electrostatically gate the nanowires. Experimental The fabrication of stemless InSb NWs closely matches earlier techniques 20 , where an InSb(111)B substrate with SiNx mask is used. However, differing from previously reported InSb NWs, the holes in the mask are created with reactive ion etching (RIE) (Oxford Plasmalab System 100) as opposed to wet chemical etching methods. The nanoholes in the mask are filled with gold droplets, allowing for growth using a metalorganic vapor-phase epitaxy (MOVPE) technique. Precursors of trimethylindium and trimethylantimony are used,
Figure
1c shows a false color SEM image of a typical NW-on-FLG-AlOx-back-gate device. The blue area is the location of the deposited FLG. The entire chip is coated with conformal AlOx. We also fabricated NW devices, on a different chip, with a Ti/Au-AlOx back gate (see SI). The electrodes were placed on top of the ultra-thin InSb NW, shown in gold and green respectively. The surface underneath the NW, FLG coated with AlOx, appears to be much smoother than the Ti/Au leads. AFM measurements confirm that the surface of the FLG-AlOx is significantly smoother compared to the Ti/Au-AlOx with typical roughness of ~ 170 ± 70and ~ 860 ± 70 respectively.The cross-sectional TEM image (Figure 1d) shows a cut of the measured nanowire device, reported in all of the measured data. The wire has a hexagonal shape and width of~52. The wire is capped with ~3 oxide, and no crystal defects have been observed in the wire. In addition, the TEM cross-section of the FLG-AlOx back gate revealed that the thickness of the FLG is ~4.5 , see SI.
~200, on the order of . The long device provides insight into diffusive transport, while the short device gives insight into quasi-ballistic transport. The inset toFigure 1cshows a conductance plot as a function of gate voltage (Vg) of the ultra-thin NW device with contact spacing of ~6 . The conductance increases as a function of gate voltage up to the point of saturation at ~0.3 . Measuring the pinch-off and conductance of the ultra-thin NW with contact spacing of ~6 allows us to extract the field-effect mobility value, which is a property of the diffusive transport regime.
Figure 1 .
1a) SEM image of a NW field taken under a 30°, where the scalebar is 30 . b) Close up of the NW field, showing high aspect ratio NWs. The larger objects appearing behind the NWs are parasitic growth on the mask. Scale bar is 1
Figure 2a
2ashows the differential conductance ( / ) map as a function of gate voltage ( ) of the short device. The conductance map as function of shows quasiperiodic diamond shape patterns. A series of line cuts taken from the conductance map (Figure 2a) for −1.95 < < 1.95 as a function of )shown in Figure 2b ( demonstrate that the oscillations persist over a substantial range of gate voltages.
Figure 2 .
2a) Differential conductance ( / ) vs and of the device with contact spacing of 200 at base temperature showing diamond shape pattern i.e., an indication of the FB interference. b) A series of line cuts from the 2D differential conductance map(Fig. 2a)between = −1.95 and = 1.95 as a function of , purple and brown curve respectively.
Figure 3 .
3a) Differential conductance ( / ) vs and of the device with contact spacing of 200 under 2.8 out-of-plane magnetic field showing a suppression of FB interference. b) A series of line cuts from the 2D differential conductance map at = 2.8(Fig 3a)between = −2 and = 2 as a function of , purple and brown curve respectively. The curves show the emergence of a plateau at 1 0 , and a second plateau near 1.5 0 compatible with Zeeman splitting due to the applied B-field.
5 0
0, consistent with Zeemansplit subbands. At more positive gate voltages conductance oscillations, possibly due to FP interference, prevent the observation of any other clear plateaus. However, for MZM experiments, identification of the lowest one or two subbands is generally sufficient.
Figure 4
4. a) Differential conductance ( / ) vs and of the device with contact spacing of 200 at 400
Figure 5 .
5a) subband in a semi-conducting nanowire with spin-orbit interaction and Zeeman coupling. b) The red curve shows an energy sweep along the subbands of the wire. The y-axis is in the units of 0 and the xaxis (upper) is a normalized energy scale in arbitrary units. The blue curve shows the conductance as a function of gate voltage under non-uniform gate potential. The Y-axis is in the units of 0 as a function and the X-axis (lower) is the gate voltage in arbitrary units.
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|
[] |
[
"Complete Formalism of Cross Sections and Asymmetries for Longitudinally and Transversely Polarized Leptons and Hadrons in Deep Inelastic Scattering",
"Complete Formalism of Cross Sections and Asymmetries for Longitudinally and Transversely Polarized Leptons and Hadrons in Deep Inelastic Scattering"
] |
[
"Paul Anderson \nWilliam and Mary\n23185WilliamsburgVAUSA\n",
"Douglas Higinbotham \nThomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA\n",
"Sonny Mantry \nUniversity of North Georgia\n30597DahlonegaGAUSA\n",
"Xiaochao Zheng \nUniversity of Virginia\n22904CharlottesvilleVAUSA\n"
] |
[
"William and Mary\n23185WilliamsburgVAUSA",
"Thomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA",
"University of North Georgia\n30597DahlonegaGAUSA",
"University of Virginia\n22904CharlottesvilleVAUSA"
] |
[] |
Studies of the Deep Inelastic Scattering (DIS) have provided fundamental information of the nucleon structure for decades. The electron-ion collider (EIC) will be the first collider capable of DIS study with both polarized lepton and polarized hadron beams, providing the possibility of accessing new electroweak structure functions of the nucleon. In this work, we completed the DIS cross section derivations for both longitudinally and transversely polarized leptons and hadrons, with no approximations made, and with all three contributions -γγ, γZ, ZZ -included. These results were derived using primarily tensor algebra and Feynman calculus, starting from previously established leptonic and hadronic tensors and carry out their contraction. Our results are presented in terms of both spin-averaged and spin-dependent cross sections, allowing direct comparison with experimentally measured cross sections and their asymmetries. We include also in our discussion comparisons of different conventions that exist in the literature.
| null |
[
"https://export.arxiv.org/pdf/2306.00097v1.pdf"
] | 258,999,638 |
2306.00097
|
590c1c0257ab6e94c311408415786cf87fde708c
|
Complete Formalism of Cross Sections and Asymmetries for Longitudinally and Transversely Polarized Leptons and Hadrons in Deep Inelastic Scattering
Paul Anderson
William and Mary
23185WilliamsburgVAUSA
Douglas Higinbotham
Thomas Jefferson National Accelerator Facility
23606Newport NewsVAUSA
Sonny Mantry
University of North Georgia
30597DahlonegaGAUSA
Xiaochao Zheng
University of Virginia
22904CharlottesvilleVAUSA
Complete Formalism of Cross Sections and Asymmetries for Longitudinally and Transversely Polarized Leptons and Hadrons in Deep Inelastic Scattering
(Dated: June 2, 2023)Presented at DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related Subjects, Michigan State University, USA, 27-31 March 2023
Studies of the Deep Inelastic Scattering (DIS) have provided fundamental information of the nucleon structure for decades. The electron-ion collider (EIC) will be the first collider capable of DIS study with both polarized lepton and polarized hadron beams, providing the possibility of accessing new electroweak structure functions of the nucleon. In this work, we completed the DIS cross section derivations for both longitudinally and transversely polarized leptons and hadrons, with no approximations made, and with all three contributions -γγ, γZ, ZZ -included. These results were derived using primarily tensor algebra and Feynman calculus, starting from previously established leptonic and hadronic tensors and carry out their contraction. Our results are presented in terms of both spin-averaged and spin-dependent cross sections, allowing direct comparison with experimentally measured cross sections and their asymmetries. We include also in our discussion comparisons of different conventions that exist in the literature.
I. INTRODUCTION
With the construction of the Electron-Ion Collider (EIC) underway, it will soon be possible to perform Deep Inelastic Scattering (DIS) studies at a collider setting where both lepton and hadron beams have a spin polarization. Previous collider facilities have only had the capability to polarize at most one of the particles involved in the collision, while the kinematic reach of fixed-target facilities was limited. As such, there was previously no need for a complete and explicit cross section formalism with full spin degree of freedom and that includes all electroweak terms. With the EIC, this has changed. In order to derive the cross sections and asymmetries for lepton-hadron collisions in the DIS regime where both particles have a spin polarization, we start with defining all 4-momenta and spin vectors as shown in Fig. 1. Here, m(M ) is the mass of the lepton (hadron), E(E ′ ) and k(k ′ ) are the initial (final) energy and momentum of the lepton, respectively, s(s ′ ) is the initial (final) spin of the lepton, S is the initial spin and P the initial momentum of the hadron. We will use the commonly used notations: q := k − k ′ is the momentum transfer; Q 2 := −q 2 is the negative square momentum transfer; x := Q 2 2(P ·q) is the Bjorken scaling variable; and y := P ·q P ·k is the inelasticity. We also have the normalization s 2 = S 2 = −1 for the lepton and the hadron spins, respectively.
We focus on the following three specific spin polarization cases and provide the results. Among these spin cases, DIS between longitudinally polarized lepton and longitudinally or transversely polarized hadrons is the most relevant for JLab and EIC. DIS between transversely polarized leptons and unpolarized hadron is in- cluded because this is one of the primary background contribution for precision parity-violating electron scattering (PVES) experiments. Additionally, we have also derived the general case with arbitrary spin polarization direction of both the lepton and hadron, but will not include the results here due to space limitation.
A. Longitudinally Polarized Lepton
Longitudinally Polarized Hadron
For longitudinally polarized leptons, we start with the leptonic tensor given in Eq. (
L (γ) µν = 2 k µ k ′ ν + k ′ µ k ν − (k · k ′ )g µν − iλ ℓ ε µναβ k α k ′β ,(1)L (j) µν = C j L γ µν ,(2)
where j = γ, γZ, and Z. The relevant conventions include: (1, −1, −1, −1) for the metric tensor g µν , ε 0123 = 1 = −ε 0123 for the Levi-Civita symbol, and λ ℓ = ±1 for the left-and right-handed lepton helicities, which differs from the convention λ ℓ = ± 1 2 of [1]. It should be noted that the tensor of Eq. (1) is based on the assumption of a fast-moving lepton, s µ ≈ λ ℓ m k µ and a negligible lepton mass. For the C coefficients, C γ = 1,
C γZ = −(g V − λ ℓ g A ), and C Z = (g V − λ ℓ g A ) 2 ,(3)
where g V = − 1 2 + 2 sin 2 θ W with θ W the weak mixing angle and g A = − 1 2 are the vector and axial-vector couplings of the lepton to the Z boson, respectively. Note that C γZ differs by a minus sign from [1], arising from the charge of the electron being left out in [1].
For the hadronic tensor, we start from Eq. (2.2.7) of [1], but with the sign of g 4 term flipped [2] to be consistent with the parton model interpretation, g j 4 = 2xg j 5 , and Eq.(18.18) of Particle Data Group [3].
1 2M W µν(j) = − g µν M F j 1 + P µ P ν M (P · q) F j 2 + i ε µναβ 2(P · q) P α q β M F j 3 + 2q α S β g j 1 − 4xP α S β g j 2 − P µ S ν + S µ P ν 2(P · q) g j 3 −
S · q (P · q) 2 P µ P ν g j 4 +
S · q P · q g µν g j 5 .
Note that the hadronic tensor of Eq. (18.6) in [3] is 1/2 of Eq. (4), and the sign for both g 4,5 are the opposite. The F 1,2,3 and g 1,4,5 can be interpreted in the parton model [3]. One could in addition set F γ 3 = g γ 3 = g γ 4 = g γ 5 = 0 based on the current understanding that parity is conserved in electromagnetic interactions.
The neutral-current (NC) cross sections are formed from leptonic and hadronic tensors as:
d 2 σ NC dxdy = πyα 2 Q 4 j=γ,γZ,Z η j L (j) µν W µν(j)(5)
where η γ = 1,
η γZ = GM 2 Z 2 √ 2πα Q 2 Q 2 + M 2 Z ; and η Z = η γZ 2 . (6)
Here, α is the fine structure constant, M Z is the mass of the Z 0 boson, and G is the Fermi coupling constant. The factor πyα 2 Q 4 of Eq. (5) differs from PDG by factor two due to the different definitions of the hadronic tensor, as explained above. Our next step was to carry out tensor contraction for longitudinally polarized hadrons. The results involve a number of 4-vector dot products, most of which commonly used in DIS. However, a few complicated terms appear that are worth listing: (
1,−P )·k E+| ⃗ P | ≈ 0, k · S = λ H (P ·k) M , k ′ · S = λ H (P ·k) M 1 − y − 2M 2 x 2 y 2 Q 2 , q · S = λ H (P ·k) M 1 + 2M 2 x 2 y Q 2
, where λ H = ±1 is the helicity of the hadron and the lepton mass has been neglected. Then tensor contraction is:
L (γ) µν W µν(j) = 4Q 2 F j 1 + 4Q 2 xy 2 1 − y − M 2 x 2 y 2 Q 2 F j 2 − 2λ ℓ Q 2 (2 − y) y F j 3 + 4λ ℓ λ H Q 2 y 2 − y − 2M 2 x 2 y 2 Q 2 g j 1 − 32M 2 x 2 λ ℓ λ H g j 2 − 4Q 2 λ H xy 2 1 − y − M 2 x 2 y 2 Q 2 g j 3 − 4Q 2 λ H xy 2 1 + 2M 2 x 2 y Q 2 1 − y − M 2 x 2 y 2 Q 2 g j 4 −4λ H Q 2 1 + 2M 2 x 2 y Q 2 g j 5 .(7)
For L γZ,Z W j , we multiply the RHS of Eq. (7) by the C j . Finally, we note that coefficients for the g 3 term differ from [3].
A cross section decomposition based on its spin dependence can be performed in a similar manner to that of [4]. Abbreviating d 2 σ dx dy as dσ:
dσ 0 = 1 4 dσ| ++ + dσ| +− + dσ| −+ + dσ| −− , dσ ℓ = 1 4 dσ| ++ + dσ| +− − dσ| −+ − dσ| −− , dσ H = 1 4 dσ| ++ − dσ| +− + dσ| −+ − dσ| −− , dσ ℓH = 1 4 dσ| ++ − dσ| +− − dσ| −+ + dσ| −− .(8)
where the first ± refer to the lepton helicity λ = ±1 and the second to the hadron helicity λ H = ±1. We present in the following results from explicit calcu-
lation, with Υ = 1 − y − M 2 x 2 y 2 Q 2 , ζ = 2 − y − 2M 2 x 2 y 2 Q 2 , ξ = 1 − y − M 2 x 2 y Q 2 , and Ξ = 1 + 2M 2 x 2 y Q 2 : dσ 0 = 4πyα 2 Q 2 F γ 1 − η γZ g V F γZ 1 + η Z (g 2 V + g 2 A )F Z 1 + 4πα 2 xyQ 2 Υ F γ 2 − η γZ g V F γZ 2 + η Z (g 2 V + g 2 A )F Z 2 − 2πα 2 (2 − y) Q 2 η γZ g A F γZ 3 − 2η Z g V g A F Z 3 ,(9)dσ ℓ = 4πyα 2 Q 2 η γZ g A F γZ 1 − 2η Z g V g A F Z 1 + 4πα 2 xyQ 2 Υ η γZ g A F γZ 2 − 2η Z g V g A F Z 2 (10) − 2πα 2 (2 − y) Q 2 −η γZ g V F γZ 3 + η Z (g 2 V + g 2 A )F Z 3 , dσ H = 4πα 2 Q 2 ζ η γZ g A g γZ 1 − 2η Z g V g A g Z 1 − 16πM 2 α 2 x 2 y Q 4 η γZ g A g γZ 2 − 2η Z g V g A g Z 2 − 4πα 2 xyQ 2 Υ −η γZ g V g γZ 3 + η Z (g 2 V + g 2 A )g Z 3 (11) − 4πα 2 xyQ 2 ΞΥ −η γZ g V g γZ 4 + η Z (g 2 V + g 2 A )g Z 4 − 4πyα 2 Q 2 Ξ −η γZ g V g γZ 5 + η Z (g 2 V + g 2 A )g Z 5 , dσ ℓH = 4πα 2 Q 2 ζ g γ 1 − η γZ g V g γZ 1 + η Z (g 2 V + g 2 A )g Z 1 − 16πM 2 α 2 x 2 y Q 4 g γ 2 − η γZ g V g γZ 2 + η Z (g 2 V + g 2 A )g Z 2 − 4πα 2 xyQ 2 Υ η γZ g A g γZ 3 − 2η Z g V g A g Z 3 (12) − 4πα 2 xyQ 2 ΞΥ η γZ g A g γZ 4 − 2η Z g V g A g Z 4 − 4πyα 2 Q 2 Ξ η γZ g A g γZ 5 − 2η Z g V g A g Z 5 .
Transversely Polarized Hadron
The case of longitudinally polarized lepton and transversely polarized hadron was considered next. The calculation is almost the same as the previous section, except the hadron spin S changed, along with all dot products involving S. The tensor contraction was found to be:
L (γ) µν W µν(j) = 4Q 2 F j 1 + 4Q 2 xy 2 ΥF j 2 − 4λ ℓ Q 2 (2 − y) y F j 3 + 16M λ ℓ x(k ′ · S)g j 1 + 32M xλ ℓ y (k ′ · S)g j 2 − 4M y (k ′ · S)g j 3 + 8M y 2 Υ(k ′ · S)g j 4 + 8M x(k ′ · S)g j 5 . (13)
Note that for g 4 and g 5 , the dot product q · S = (k · S) − (k ′ · S) was needed. We first note that S = (0, cos(φ), sin(φ), 0) -in the convention where the first number is the time component -in the rest frame of the hadron, and is invariant under Lorentz boosts in the zdirection. One straightforward way to calculate the q · S product is thus to use the rest frame of the lepton where k = (E, ⃗ 0). We found k · S = 0 and q · S = −k ′ · S.
When considering the k ′ · S term, two possible bases could be considered. The first is to let S along an arbitrary direction in the x, y-plane, while k ′ only possesses components in the x, z plane. The second is to let S be fixed along the x-axis, while k ′ is entirely arbitrary. In short, the decision is whether to have an azimuthal angle φ on S or k ′ . The vectors for the first case are: k ′ = E ′ , ⃗ k ′ sin(θ), 0, ⃗ k ′ cos(θ) and S = (0, cos(φ), sin(φ), 0), while for the second case:
k ′ = E ′ , ⃗ k ′ cos(φ) sin(θ), ⃗ k ′ sin(φ) sin(θ), ⃗ k ′ cos(θ)
and S = (0, 1, 0, 0) . The results are:
dσ 0 = 4πyα 2 Q 2 F γ 1 − η γZ g V F γZ 1 + η Z (g 2 V + g 2 A )F Z 1 + 4πα 2 xyQ 2 Υ F γ 2 − η γZ g V F γZ 2 + η Z (g 2 V + g 2 A )F Z 2 − 2πα 2 (2 − y) Q 2 η γZ g A F γZ 3 − 2η Z g V g A F Z 3 ,(14)dσ ℓ = 4πyα 2 Q 2 η γZ g A F γZ 1 − 2η Z g V g A F Z 1 + 4πα 2 xyQ 2 Υ η γZ g A F γZ 2 − 2η Z g V g A F Z 2 (15) − 2πα 2 (2 − y) Q 2 −η γZ g V F γZ 3 + η Z (g 2 V + g 2 A )F Z 3 , dσ H = 8πα 2 M xyk ′ 1 Q 4 η γZ g A g γZ 1 − 2η Z g V g A g Z 1 + 16πα 2 M xk ′ 1 Q 4 η γZ g A g γZ 2 − 2η Z g V g A g Z 2 − 4πα 2 M k ′ 1 Q 4 −η γZ g V g γZ 3 + η Z (g 2 V + g 2 A )g Z 3 (16) + 8πα 2 M k ′ 1 yQ 4 Υ −η γZ g V g γZ 4 + η Z (g 2 V + g 2 A )g Z 4 + 8πM xyα 2 k ′ 1 Q 4 −η γZ g V g γZ 5 + η Z (g 2 V + g 2 A )g Z 5 ,
and dσ ℓH =
8πα 2 M xyk ′ 1 Q 4 g γ 1 − η γZ g V g γZ 1 + η Z (g 2 V + g 2 A )g Z 1 + 16πα 2 M xk ′ 1 Q 4 g γ 2 − η γZ g V g γZ 2 + η Z (g 2 V + g 2 A )g Z 2 − 4πα 2 M k ′ 1 Q 4 η γZ g A g γZ 3 − 2η Z g V g A g Z 3 (17) + 8πα 2 M k ′ 1 yQ 4 ξ η γZ g A g γZ 4 − 2η Z g V g A g Z 4 + 8πM xyα 2 k ′ 1 Q 4 η γZ g A g γZ 5 − 2η Z g V g A g Z 5 ,
where k ′ 1 := k ′ · S = | ⃗ k ′ | cos(φ) sin(θ).
B. Transversely Polarize Lepton
The case of transversely polarized lepton was discussed briefly in [1] and it was stated that the lepton spindependent cross section difference is proportional to the lepton mass and thus is negligible. We derive below explicitly the result for DIS of transversely polarized lepton and unpolarized hadron case because this could potentially be a background for parity-violating electron scattering (PVES) experiments, which are now carried out to higher and higher precision at a number of fixed-target facilities.
Unpolarized Hadron Case
For the transversely polarized lepton case, the assumption of the fast-moving electron was dropped, and the lepton mass was kept. The leptonic tensor for transversely polarized leptons is:
L (γ) µν = 2(k µ k ′ ν + k ν k ′ µ ) +2g µν (m 2 − k · k ′ ) +2ms α (k − k ′ ) β ε µναβ i (18) L (γZ) µν = −2g V L (γ) µν −4g A m(k ′ · s)g µν − m(k ′ µ s ν + k ′ ν s µ ) +k α k ′β ε µναβ i (19) L (Z) µν = (g 2 V + g 2 A )L (γ) µν − 4g 2 A m 2 g νν −4g 2 A ms α k β ε µναβ i + 4g V g A m(k ′ · s)g µν −m(k ′ µ s ν + k ′ ν s µ ) + k α k ′β ε µναβ i .(20)
Note that unlike longitudinally polarized lepton case, the γZ and Z components of the leptonic tensor can no longer be written as proportional to the γ component.
To obtain the case for unpolarized hadrons, the average was taken over the hadron spin S and −S, and we form only two cross section combinations:
dσ 0 = 1 2 dσ| s + dσ| −s ,(21)dσ ℓ = 1 2 dσ| s − dσ| −s ,(22)
where s is the lepton spin. Similar to the longitudinally polarized lepton and transversely polarized hadron case, there were two potential options for the basis of k ′ and s.
If we choose to fix the lepton spin along x and let ⃗ k ′ direction be arbitrary, we can write s = (0, 1, 0, 0) and frame P = (M, ⃗ 0) and conclude P · s = 0. This specific result simplies the cross section results, which are found to be:
k ′ = E ′ , | ⃗ k ′ | cos(φ) sin(θ), | ⃗ k ′ | sin(φ) sin(θ), ⃗ k ′ cos(θ) , seedσ 0 = − 4πyα 2 Q 4 2m 2 − Q 2 F γ 1 −2 2m 2 − Q 2 g V η γZ F γZ 1 + g 2 V + g 2 A 2m 2 − Q 2 − 8g 2 A m 2 η Z F Z 1 + 4πα 2 xyQ 2 ΥF γ 2 − 2g V η γZ ΥF γZ 2 +η Z g 2 V + g 2 A Υ − 4g 2 A m 2 M 2 x 2 y 2 Q 4 F Z 2 + 4πα 2 g A (y − 2) Q 2 η γZ F γZ 3 − g V η Z F Z 3 (23) dσ ℓ = 16πyα 2 mg A (k ′ · s) Q 4 η γZ F γZ 1 − g V η Z F Z 1 − 16πmM 2 xyα 2 g A (k ′ · s) Q 6 η γZ F γZ 2 − g V η Z F Z 2 + 4πα 2 m(k ′ · s) Q 4 − 2yg V η γZ F γZ 3 + y g 2 V + g 2 A − 2g 2 A η Z F Z 3 .(24)
Note that the lepton-spin dependent cross section difference dσ ℓ is proportional to k ′ · s and thus would have a cos ϕ modulation. In typical PVES experiments, the main parityviolating observable of interest is the cross section asymmetry between right-and left-handed incident electrons on an unpolarized target. In reality, the spin of the incident electron beam may have a transverse component, causing background signal due to two-photon exchange effects which are of pure electromagnetic nature. All PVES experiments thus had dedicated measurements of the beam "normal" single-spin asymmetry, A n , to directly access this background using transversely polarized electron beam. In recent experiments with full azimuthal acceptance of the scattered electron, the A n is determined by fitting the sin ϕ component of the ob-served cross section asymmetry associated with the electron beam spin flip, see e.g. [5,6]. However, as can be seen above, there can be a cos ϕ component caused by parity-violation. Our results show that the parityviolating asymmetry of transversely polarized lepton is of the order of η γZ m 2 /Q 2 ≈ m 2 /M 2 Z ≈ 10 −11 , and is likely to be too small to be observed in such A n measurements.
II. DISCUSSION AND SUMMARY
We have completed the DIS cross section formalism in the most complete form known to date, that includes lepton and hadron spin in arbitrary directions, all neutral current terms, and both lepton and hadron mass terms. These results should serve as a foundation for future asymmetry experiments at Jefferson Lab and the Electron Ion Collider. Results on the spin-dependent cross section asymmetry for transversely polarized electron case are also relevant for beam-normal single spin asymmetry measurements. From here, projections of the size of various asymmetries for possible experimental measurements at Jefferson Lab and EIC can be made, and feasibility of new measurements studied.
Code Availability
We made use of the FeynCalc package of Mathematica [7][8][9] for cross-checking our calculations. The Mathematica code used can be downloaded from https://www.dropbox.com/s/gwlkhibxerpf0jd/ DIS%20Cross%20Sections.nb?dl=0.
FIG. 1. Feynman diagram for lepton-hadron collision.
FIG. 2 .
2The convention chosen for longitudinally polarized lepton and transversely polarized hadron. The x axis is chosen to be along the hadron spin ⃗ S, while ⃗ k ′ is attributed a polar angle θ and an azimuthal angle ϕ.The second basis was chosen for this work, see Fig. 2 for the definition of spin orientations and angles. The cross sections of Eq. (8) were then generalized, with λ H = ±1 replaced with ±S: dσ 0 = 1 4 dσ| +,S + dσ| +,−S + dσ| −,S + dσ| −,−S dσ ℓ = 1 4 dσ| +,S + dσ| +,−S − dσ| −,S − dσ| −,−S dσ H = 1 4 dσ| +,S − dσ| +,−S + dσ| −,S − dσ| −,−S dσ ℓH = 1 4 dσ| +,S − dσ| +,−S − dσ| −,S + dσ| −,−S
Fig. 3 .
3Since s is invariant under Lorentz boosts in the z-direction, we can utilize that in the hadron rest FIG. 3. The convention chosen for transversely polarized lepton scattering. The x axis is chosen to be along the lepton spin ⃗ s, while ⃗ k ′ is attributed a polar angle θ and an azimuthal angle ϕ. The hadron spin ⃗ S is averaged in the calculation and is not shown.
ACKNOWLEDGMENTS
This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract number DE-SC0014434 (U. of Virginia); and Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships Program. Travel support to the DIS2023 conference was provided by the organizers and the College of William & Mary.
TABLE I .
IList of spin cases presented in this work.
with Eqs. (2.2.2-4) of [1] re-written as:
The Theory and phenomenology of polarized deep inelastic scattering. M Anselmino, A Efremov, E Leader, 10.1016/0370-1573(95)00011-5hep-ph/9501369Phys. Rept. 261M. Anselmino, A. Efremov and E. Leader, The Theory and phenomenology of polarized deep inelastic scattering, Phys. Rept. 261 (1995) 1-124, [hep-ph/9501369].
T Martini, Deep Inelastic Lepton Scattering with Polarized Nucleons. Johannes Gutenberg-Universit at MainzPh.D. thesisT. Martini, Deep Inelastic Lepton Scattering with Polarized Nucleons, Ph.D. thesis, Johannes Gutenberg-Universit at Mainz, 2012.
. R L Workman, Particle Data Group collaboration10.1093/ptep/ptac097Review of Particle Physics. 2022PTEPParticle Data Group collaboration, R. L. Workman et al., Review of Particle Physics, PTEP 2022 (2022) 083C01.
Neutral-current electroweak physics and SMEFT studies at the EIC. R Boughezal, A Emmert, T Kutz, S Mantry, M Nycz, F Petriello, 10.1103/PhysRevD.106.0160062204.07557Phys. Rev. D. 10616006R. Boughezal, A. Emmert, T. Kutz, S. Mantry, M. Nycz, F. Petriello et al., Neutral-current electroweak physics and SMEFT studies at the EIC, Phys. Rev. D 106 (2022) 016006, [2204.07557].
Measurement of the Beam-Normal Single-Spin Asymmetry for Elastic Electron Scattering from 12 C and 27 Al. D Androić, QWeak collaboration10.1103/PhysRevC.104.0146062103.09758Phys. Rev. C. 10414606QWeak collaboration, D. Androić et al., Measurement of the Beam-Normal Single-Spin Asymmetry for Elastic Electron Scattering from 12 C and 27 Al, Phys. Rev. C 104 (2021) 014606, [2103.09758].
New Measurements of the Beam-Normal Single Spin Asymmetry in Elastic Electron Scattering over a Range of Spin-0 Nuclei. Prex, D Crex Collaboration, Adhikari, 10.1103/PhysRevLett.128.1425012111.04250Phys. Rev. Lett. 128142501PREX, CREX collaboration, D. Adhikari et al., New Measurements of the Beam-Normal Single Spin Asymmetry in Elastic Electron Scattering over a Range of Spin-0 Nuclei, Phys. Rev. Lett. 128 (2022) 142501, [2111.04250].
Feyn calccomputer-algebraic calculation of feynman amplitudes. R Mertig, M Böhm, A Denner, 10.1016/0010-4655(91)90130-DComputer Physics Communications. 64R. Mertig, M. Böhm and A. Denner, Feyn calc - computer-algebraic calculation of feynman amplitudes, Computer Physics Communications 64 (1991) 345-359.
V Shtabovenko, R Mertig, F Orellana, 10.1016/j.cpc.2016.06.008New developments in FeynCalc 9.0. 207V. Shtabovenko, R. Mertig and F. Orellana, New developments in FeynCalc 9.0, Computer Physics Communications 207 (oct, 2016) 432-444.
V Shtabovenko, R Mertig, F Orellana, 10.1016/j.cpc.2020.107478FeynCalc 9.3: New features and improvements. 256107478V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: New features and improvements, Computer Physics Communications 256 (nov, 2020) 107478.
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"Nekoo \nGRVC Robotics Lab\nGRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nJosé Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nUniversidad de Sevilla\n\n",
"Anibal Ollero [email protected] \nGRVC Robotics Lab\nGRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nJosé Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nUniversidad de Sevilla\n\n"
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"GRVC Robotics Lab\nGRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nJosé Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla\nGRVC Robotics Lab. Universidad de Sevilla\nUniversidad de Sevilla\n"
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One of the motivations for exploring flapping-wing aerial robotic systems is to seek energy reduction, by maintaining manoeuvrability, compared to conventional unmanned aerial systems. A Flapping Wing Flying Robot (FWFR) can glide in favourable wind conditions, decreasing energy consumption significantly. In addition, it is also necessary to investigate the power consumption of the components in the flapping-wing robot. In this work, two sets of the FWFR components are analyzed in terms of power consumption: a) motor/electronics components and b) a vision system for monitoring the environment during the flight. A measurement device is used to record the power utilization of the motors in the launching and ascending phases of the flight and also in cruising flight around the desired height. Additionally, an analysis of event cameras and stereo vision systems in terms of energy consumption has been performed. The results provide a first step towards decreasing battery usage and, consequently, providing additional flight time.
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[
"https://export.arxiv.org/pdf/2306.00848v1.pdf"
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2306.00848
|
2f4119fcf088a066b60b3ebf31d29b9933f1092f
|
EXPERIMENTAL ENERGY CONSUMPTION ANALYSIS OF A FLAPPING-WING ROBOT
Raul Tapia [email protected]
GRVC Robotics Lab
GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
José Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
Universidad de Sevilla
Alvaro Cesar Satue
GRVC Robotics Lab
GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
José Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
Universidad de Sevilla
Saeed Rafee [email protected]
GRVC Robotics Lab
GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
José Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
Universidad de Sevilla
Nekoo
GRVC Robotics Lab
GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
José Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
Universidad de Sevilla
Anibal Ollero [email protected]
GRVC Robotics Lab
GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
José Ramiro Martínez-de Dios GRVC Robotics Lab. Universidad de Sevilla
GRVC Robotics Lab. Universidad de Sevilla
Universidad de Sevilla
EXPERIMENTAL ENERGY CONSUMPTION ANALYSIS OF A FLAPPING-WING ROBOT
This paper has been accepted for publication at
One of the motivations for exploring flapping-wing aerial robotic systems is to seek energy reduction, by maintaining manoeuvrability, compared to conventional unmanned aerial systems. A Flapping Wing Flying Robot (FWFR) can glide in favourable wind conditions, decreasing energy consumption significantly. In addition, it is also necessary to investigate the power consumption of the components in the flapping-wing robot. In this work, two sets of the FWFR components are analyzed in terms of power consumption: a) motor/electronics components and b) a vision system for monitoring the environment during the flight. A measurement device is used to record the power utilization of the motors in the launching and ascending phases of the flight and also in cruising flight around the desired height. Additionally, an analysis of event cameras and stereo vision systems in terms of energy consumption has been performed. The results provide a first step towards decreasing battery usage and, consequently, providing additional flight time.
Introduction
Flapping-wing flying robots (FWFR) are bioinspired platforms that use flapping wings to generate lift and thrust. They have high maneuverability without fast rotating propellers, which makes them less dangerous and more robust against collisions than multirotors. Their potentialities in a wide range of applications have motivated significant R&D interest in the last years [1]. The design of the flapping-wing robots needs a reduction of weight in all possible components of the robot to save payload for other equipment, such as [2], vision camera for monitoring [3,4], bio-inspired claws [5], or even manipulators. The current E-Flap robot weighs 500g with almost 500g payload capacity [6]. One important element (relatively heavy ≈ 13.4% of the weight) of the robot is a single battery that provides all the power of the components from the motors to the cameras. Then, all of the components should be optimized in terms of power consumption.
Online onboard perception of flapping-wing robots involves strong challenges. In fact, FWFR control and guidance methods have traditionally relied on external sensors such as motion capture systems [7,8]. In the last years, the advances in miniaturization have facilitated the integration of online perception systems on board flapping-wing robots. One of the first was the obstacle avoidance method proposed in [9], which used a lightweight stereo camera to detect static obstacles using disparity maps. Also, an event-based dynamic obstacle avoidance method for flapping-wing robots was presented in [3]. An FWFR guidance method based on event cameras was investigated using line tracking algorithms, and a visual-servoing controller [10]. However, onboard online perception systems for the autonomous navigation of flapping-wing robots is still an under-researched topic. The paper [11] presents a flapping-wing perception dataset, including measurements recorded from an event camera and a traditional camera.
Another relevant advantage of FWFR over multirotor drones is their energy efficiency. Flapping-wing platforms can frequently exchange flight phase between flapping-wing and gliding, and this drastically reduces the consumed energy [12]. Energy consumption is critical and depends on the type of flight and also on the onboard sensors and electronics. This has implications not only in flight endurance but also -due to the need for lighter batteries-in payload and weight distribution. We are interested in developing methods that enable us to predict the consumed energy by autonomous flapping-wing robots in different types of flights using different onboard sensors. This paper presents an experimental energy consumption analysis of an autonomous flapping-wing robot as a way of predicting the consumed energy in a given type of flight with pre-installed onboard sensors. The analysis includes the energy consumed both for flying and by the onboard perception system. The study includes experiments in test benches and also flight tests conducted with the E-Flap robot developed by the GRVC Robotics Lab. at the University of Sevilla, shown in Figure 1.
This paper is structured as follows. The measurement method and description of the consumption of energy during the FWFR flight are presented in Section 2. Section 3 analyses the consumed energy of the onboard perception system. Section 4 closes the paper with the conclusions.
Flapping-Wing Flight Energy Consumption
In order to measure the flying power consumption of a bio-inspired aerial robot, it is important to define different components of the platform in the analysis. A detailed description of the platform could be found in [6]. Here we briefly revisit the work with more focus on power management. The source and consumers of energy in the E-Elap robot are shown in Fig. 2. It includes a four-cell-in-series Lithium Polymer (LiPo) battery. The main consumers are: a flapping permanent magnet synchronous motor (PMSM), tail servomotors for rudder and elevator, and the electronics, mainly composed of regulators, sensors, main computer with WiFi connectivity (Cortex-A7 CPU), and a digital signal processor (DSP) board (Cortex-M4 CPU).
The power consumption of each of the above elements depends mainly on wind conditions, flight trajectory, and speed. In the indoor test bed there is no external wind as a disturbance; nevertheless, the counteracting flow of the air on the tail and surface of the wings (drag force) affects the consumption of the power in the corresponding actuators. Ambient temperature also augments power consumption during flights. While temperature affects negatively the overall efficiency of electronics, motors, and drivers, wind speed can be favorable or unfavorable with respect to power management depending on the flight trajectory. The conducted experiments were performed in the GRVC Robotics Lab. indoor test bed; hence, the influence of external wind disturbance was not considered in this analysis.
The flight trajectory of an aerial robot has a great dependence on energy consumption [13]. Two different types of trajectories have been analyzed: ascending (i.e. increasing altitude) and cruising (keeping altitude around steady-state condition). A set of experiments has been performed and one of the results is shown in Fig. 3. Notice that flappingwing robots can perform descending flights in gliding mode with negligible energy consumption as they could be stabilized without flapping and through the tail, as an actual bird would do. In this analysis, the energy consumption is measured assuming an approximate constant flapping rate (with changes less than 5%). The goal of this work is to provide a first approach to flying consumption for our platform at a constant flapping frequency, proposing for further research a more extensive comparison of different trajectories and control strategies. The consumption of the onboard computer and electronics (without the camera, which will be analyzed separately in Section 3) was measured using an INA219-based Gravity I2C digital wattmeter, resulting in less than 2W. The required power from the tail servomotors working around the steady-state position in cruising flight is around 1W in average. However, the peaks depending on counteracting airflow and aggressive maneuvers could increase up to 10W. The flapping motor is the main actuator of the altitude control, resulting in the highest power consumption, taking an average of 160W for a cruising flight. The recorded data revealed that the flapping wing, servomotors, and rest of the electronics consume 93.02%, 5.81%, and 1.16%, respectively. Even when the tail is working in aggressive maneuvers, the main consumer is the flapping motor, which limits the flying time in ascending and cruising. A control strategy that optimizes flying range by combining flapping and gliding is under development for further energy analysis. To compute the consumed energy of the flight in each zone of Fig. 3, the following aspects should be regarded. The FWFR starts the flight at an initial speed of 4m/s, provided by an external launcher. This energy of the launching system is not visible in the measured graph of the results. The second aspect is that the flapping exists throughout the whole flight in this preliminary experiment for the indoor test. Since the flight zone is limited to 15m diagonal space in the testbed, the possibility of hybrid flight (gliding and flapping) has not been tried in this work. During the ascending flight, the tail servomotors, particularly the elevator, consume a lot of its current, close to saturation to keep the elevator steady against the airflow. This situation does not exist in the cruise flight when the bird is regulated slightly around the desired point. Regarding the mentioned points, and omitting the time of launching from the graph, the flight time is found 5.5s. The consumed energy of the launching phase and ascending part is found 52.9% and the cruise flight 47.1%.
An effect that has been confirmed experimentally corresponds to the positive mean lift force during a flapping cycle. As shown in Fig. 4, maximum consumption is achieved during the down-stroke, while a lower peak consumption is measured during the upstroke. That results in an irregular cyclic discharge of the battery, whose effect on battery lifespan and usable capacity for LiPo chemistry needs to be analyzed in further work.
Perception System Energy Consumption
Although the energy consumption of the perception system is much smaller than that of the power components, in some phases of the flight, such as gliding for surveillance applications, it may be relevant. The design of perception systems for FWFRs has strong constraints. First, flapping strokes generate mechanical vibrations and abrupt pitch changes, that can perturb the onboard sensor measurements [14]. Besides, they can glide and flap at different flapping frequencies. Also, flapping-wing robots have strict payload and weight distribution restrictions, which impact the weight and shape of the sensors, electronics, batteries, and other onboard components [15]. These constraints prevent the use of sensors that are common in multirotor drones, such as LiDARS, which their weight usually exceeds the FWFR's available payload.
Vision sensors have low size and weight. They are suitable for online onboard flapping-wing robot perception, and in fact, all existing related works selected different types of cameras as the main sensor. Flapping-wing robots have employed stereo vision as they mimic the perception of large birds [9]. In addition, stereo vision enables the use of standard perception algorithms and provides depth estimation, which is crucial for applications such as navigation and obstacle avoidance.
Event cameras have gained more and more attention in recent years. These bioinspired sensors provide high responsiveness, dynamic range, and robustness against motion blur and offer significantly lower power consumption compared to standard cameras [16]. This is particularly interesting for flapping-wing robots, where energy efficiency is critical. Although event-based vision requires specialized algorithms that differ from those used in standard vision, there have been significant advancements in event-based vision in recent years, offering better solutions for visual perception for flapping-wing robots [10,3].
In this section, we present an experimental analysis of the energy consumption of three widely-used stereo vision systems and one event camera. The cameras considered are 1) a RealSense D435 stereo-pair widely used in many robotics applications, 2) a large stereo pair with high resolution Stereolabs ZED, 3) a small lightweight stereo camera eCapture G53, and 4) a DAVIS346 event camera which contains a Dynamic Vision Sensor (DAVIS346 DVS). The camera specifications are presented in Table 1.
The experiment consisted in placing each of the cameras in front of a scenario containing various objects and patterns. As the power consumption of the event cameras depends on the number of events triggered, we tested the DAVIS346 DVS under three different conditions: Test1 involved moving the objects in the scene rapidly; Test2 involved moving the objects slowly, and Test3 involved keeping the objects stationary. To emulate the flapping strokes of the FWFR, all cameras were mounted on a benchmark that oscillated -varying the pitch angle in 60 degrees-at different frequencies. To measure the pitch rate, we used a VectorNav VN-200 sensor, and for electrical consumption measurements, we used an INA219 wattmeter. This allowed us to analyze how different scenarios and flapping frequencies affect the camera's power consumption.
The experimental results, see Fig. 5-top, clearly show that DAVIS346 DVS offers significantly lower power consumption (< 200 mW) compared to the rest of the cameras (in a range from 300 to 700 mW). The events generated in the experiments are shown in Fig. 5-bottom. Although the energy consumed by the perception system is much lower than by other FWFR components (e.g., PMS motor), its analysis is relevant as it becomes significant in certain applications such as monitoring during long-distance gliding. In addition, it is interesting to consider that the dependency between power consumption and event generation rate is beneficial for FWFR applications since only certain parts of the scene (those with relevant information to process) trigger events and hence entail power consumption. In contrast, scenes with no relevant information (e.g., the open sky during long flights) trigger a very low number of events, involving negligible consumption.
Conclusions
This work investigated the power consumption of flight and monitoring components of a flapping-wing aerial robot. The power consumption was measured during the three different phases of the flight: launching, ascending, and cruising. The consumed energy of the launching phase and ascending part is found 52.9% and the cruising flight 47.1%. The vision monitoring section also analyzed event cameras and stereo cameras in terms of energy consumption. The experimental results confirmed event cameras as the most energy efficient perception sensor (< 200 mW).
Figure 1 :
1The E-Flap robot developed at the GRVC Robotics Lab. of the University of Seville, placed on an artificial branch for manipulation.
Figure 2 :
2The diagram of relevant consumers and source of energy in E-Flap aerial robot prototype.
Figure 3 :
3The typical flying profile: it starts with launching, continues with ascending flight, and finishes with cruising flight.
Figure 4 :
4Mean power cycle of a flap during the flight shown inFig. 3: down-stroke (red) and upstroke (green).
Figure 5 :
5Energy consumption of the evaluated cameras as a function of pitch rates. The event rate is measured in millions of events per second (Meps).
Table 1 :
1Specifications of the evaluated cameras.Name
FPS Resolution Ideal Depth Range
RealSense D435
30
1920×1080
0.3m -3m
Stereolabs ZED
60
1280×720
1.5m -20m
eCapture G53
30
640×400
0.15m -2m
DAVIS346 DVS
-
346×260
-
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
89.1
107
128
154
185
222
266
319
383
460
552
662
Pitch rate [Hz]
Power [mW]
RealSense D435
Stereolabs ZED
eCapture G53
DAVIS346 DVS on Test 1
DAVIS346 DVS on Test 2
DAVIS346 DVS on Test 3
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0.1
1
10
Pitch rate [Hz]
Event rate [Meps]
AcknowledgementsThis work was supported by the GRIFFIN ERC Advanced Grant 2017, Action 788247. Partial funding was obtained from the Spanish Projects ROBMIND (Ref. PDC2021-121524-I00), HAERA (PID2020-119027RB-I00), and the Plan Estatal de Investigación e Innovación of the Spanish Ministry (FPU19/04692).
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[] |
[
"Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories",
"Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories"
] |
[
"Xie Chen \nDepartment of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA\n",
"Ho Tat Lam \nCenter for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n",
"Xiuqi Ma \nDepartment of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA\n"
] |
[
"Department of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA",
"Center for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Department of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA"
] |
[] |
Infinite-component Chern-Simons-Maxwell theories with a periodic K matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -the size of the K matrix. It can grow exponentially or polynomially, cycle over finitely many values, or fluctuate erratically inside an exponential envelope. We relate these different patterns of the ground state degeneracy with the roots of the "determinant polynomial", a Laurent polynomial, associated to the periodic K matrix. These roots also determine whether the theory is gapped or gapless. Based on the ground state degeneracy, we formulate a necessary condition for a gapped theory to be a foliated fracton order. CONTENTS arXiv:2306.00291v1 [cond-mat.str-el] 1 Jun 2023
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[
"https://export.arxiv.org/pdf/2306.00291v1.pdf"
] | 258,999,929 |
2306.00291
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f1d98d26fe6d49817db2d280061ee31cf0d7fda1
|
Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories
Xie Chen
Department of Physics
Institute for Quantum Information and Matter
California Institute of Technology
91125PasadenaCaliforniaUSA
Ho Tat Lam
Center for Theoretical Physics
Massachusetts Institute of Technology
02139CambridgeMAUSA
Xiuqi Ma
Department of Physics
Institute for Quantum Information and Matter
California Institute of Technology
91125PasadenaCaliforniaUSA
Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories
(Dated: June 2, 2023)
Infinite-component Chern-Simons-Maxwell theories with a periodic K matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -the size of the K matrix. It can grow exponentially or polynomially, cycle over finitely many values, or fluctuate erratically inside an exponential envelope. We relate these different patterns of the ground state degeneracy with the roots of the "determinant polynomial", a Laurent polynomial, associated to the periodic K matrix. These roots also determine whether the theory is gapped or gapless. Based on the ground state degeneracy, we formulate a necessary condition for a gapped theory to be a foliated fracton order. CONTENTS arXiv:2306.00291v1 [cond-mat.str-el] 1 Jun 2023
I. INTRODUCTION
Fracton orders [1][2][3][4] are exotic phases of matter characterized by particles with restricted mobility. For example, fractons are completely immobile, lineons can move only along a one-dimensional line, and planons can move only within a two-dimensional plane. Another perculiar feature of fracton orders is that their ground state degeneracy (GSD) depends not only on the topology of the underlying manifold but also on the size and the geometry of the underlining lattice [5]. The GSD of different fracton orders can have drastically different behaviors. For example, it can grow exponentially or polynomially with the linear system size or fluctuates erratically as the system size varies. See the examples in Table I. This sensitive dependence of the GSD on the lattice details is a manifestation of the peculiar UV/IR mixing which is the reason why fracton orders defy a conventional continuum limit [6,7]. Many of these unconventional properties are closely related to the exotic symmetries associated to the fracton orders [6,[8][9][10][11]. See [12,13] for reviews on fracton orders.
A novel class of three-dimensional fracton orders described by an infinite-component Chern-Simons-Maxwell (iCS) theory was proposed in [21]. The theory is made of an infinite number of stacked two-dimensional layers where each layer supports an equal number of twodimensional U (1) gauge fields A I,a µ=0,1,2 . Here, the index I serves as a discrete coordinate of the transverse x 3 direction while the other index a = 1, ..., L labels different gauge fields on the same layer. The gauge fields have components only in the x 0 , x 1 , x 2 directions. Therefore, charged particles (if localized) have currents only along these three directions, and are hence planons. The gauge fields are coupled to each other through an integer-valued symmetric K matrix that gives rise to a Euclidean Lagrangian L = 1 4g 2 F I,a µν F µν I,a + i 4π K IL+a,JL+b ϵ µνρ A I,a µ ∂ ν A J,b ρ , (1) where F I,a µν is the field strength of A I,a µ . Imposing Fracton Model GSD Pattern of GSD X-cube model [4] 2 6N −3 [4] Grow exponentially
Ising cage-net model [14] 1 8 9 3N + 3 × 9 2N + 15 × 9 N + 45 [15] Grow as a sum of exponentials
Haah code [2] 2 r+1 (1 + 6q 7 + 6q 21 + 30q 31 + 60q 63 + · · · ) [2] Fluctuate erratically 2d U (1) vector charge tensor gauge theory [16][17][18] N 2 [19] Grow polynomially Tridiagonal iCS Theory GSD Pattern of GSD (2). In both cases, we impose periodic boundary conditions. In the former cases, N is the linear size of the lattice while in the latter ones, N is the number of layers. The formula for the Haah code is valid only for 2 ≤ N ≤ 100. Here, r = r(N ) is the largest integer such that 2 r divides N , and qn = qn(N ) equals to 1 if n divides N and equals to 0 otherwise. Another fracton model with erratic pattern of GSD was discussed recently in [20].
translation invariance in the x 3 direction constrains the K matrix to be a periodic matrix with a periodicity of L. Such a K matrix looks like
K = M 2 M T 2 . . . . . . M 0 M 0 M 0 M 1 M 1 M T 1 M T 1 L ,(2)
where M k are L × L matrices defined in (3). In order to avoid long-range interactions, K IL+a,JL+b should vanish for sufficiently large |I − J|. Mathematically, such a matrix is known as an integer symmetric quasi-diagonal block-circulant matrix. iCS theories exhibit many unconventional properties such as irrational braiding statistics [22], gapless surfaces states [23,24], exotic global symmetries [25,26] and etc. They generate a plethora of gapped fracton orders as well as new types of robust deconfined gapless fracton orders [25,26]. Some of them can emerge at low energy from three-dimensional fractional quantum Hall systems consisting of stacked two-dimensional electron gases in a perpendicular magnetic field [22,27].
Given this vast zoo of iCS theories, it is natural to wonder whether there is a systematic way to organize them. This question is a simpler version of the much more difficult task of classifying all fracton orders. Most crudely, we can divide the iCS theories into gapped and gapless theories. Among the gapped theories, we can further classify them into foliated and non-foliated fracton orders as defined in [5]. A foliated fracton order has the property that changing the system size is equivalent to adding or removing decoupled twodimensional topological orders. The simplest example of foliated iCS theory is the one with a diagonal K matrix. On the other hand, non-foliated fracton orders are more exotic and our understanding of their renormalization is yet preliminary. See [28] for a recent progress. In general, it is hard to determine whether or not an iCS theory is foliated, let alone study its renormalization. In this paper, we make progress towards answering these questions by studying the GSD of iCS theories and analyzing their behaviors as functions of the linear system size. This then leads us to a necessary condition, stated in Proposition 3, for the theory to be foliated.
More specifically, we compactify the x 1 , x 2 direction of the iCS theories on a torus, restrict the number of layers to a finite number N and impose periodic boundary condition in the x 3 direction. The index I then runs from 1 to N and the K matrix becomes an N L-dimensional matrix. We find that the GSD of these iCS theories exhibits a very rich pattern as N varies.
See the examples in Table I and Figure 1. In order to analyze these patterns, we derive analytic formulae (21) and (22) for the GSD of iCS theories and show that these In (a), the GSD is not exactly an exponential of N but grows exponentially with N for large N . In (b), the GSD fluctuates erractically with an exponentially growing upper bound. Numerics suggests that there is also a lower bound with the same base. In (c), the GSD oscillates as a function of N . In (d), the GSD remains a constant for odd N and grows linearly with N for even N different behaviors of the GSD are simply governed by the roots of the "determinant polynomial" associated to the periodic K matrix, defined in (6). Mathematically, the GSD is given by the product of the non-zero invariant factors of the K matrix. When the K matrix is nondegenerate, it is simply the determinant of the K matrix. For the degenerate case, we compute the GSD using a perturbative method.
The rest of the paper is organized as follows. In Section II, we collect some useful mathematical results related to periodic symmetric integer matrices. In Section III, we review the plane wave spectrum and the braiding statistics in iCS theories. Section IV contains the main results of this paper which relates the behaviors of the GSD of iCS theories with the roots of the determinant polynomial of the periodic K matrix. In Section V, we formulate a necessary condition for the iCS theory to have foliated fracton order. Appendix A provides more mathematical details regarding polynomial rings and cyclotomic polynomials and proves the two propositions stated in Section III. Appendix B discusses the structure of the fusion group of an iCS theory when its period is one.
II. PERIODIC K MATRIX
In this section, we collect some mathematical results regarding periodic integer symmetric matrices, which we generally refer to as periodic K matrices.
A periodic K matrix of period L can be parametrized as
K IL+a,(I+k)L+b = M k,ab ,(3)
where M −k = (M k ) T because of the symmetric property of the K matrix. We can map the K matrix to an Ldimensional matrix
P (u) = k M k u k ,(4)
whose entries are integer-coefficient Laurent polynomials of a complex variable u. The symmetric property of the K matrix implies that P (u −1 ) = P (u) T . On the unit circle u = exp(iq), P [exp(iq)] is essentially the Fourier transform of the periodic K matrix. The eigenvectors of the K matrix take the form of
v IL+a (q) = v a (q)e iqI ,(5)
where v a (q) is an eigenvector of P [exp(iq)] and q ∼ q+2π. For a given q there are L independent eigenvectors. We denote their eigenvalues by λ a (q) where a = 1, ..., L.
When the size of the K matrix -N L is finite, q are quantized to be 2πk/N with integer k and in particular can only be a rational multiple of 2π. The quantization condition is removed when N → ∞.
The determinant of P (u) is also a Laurent polynomial
D(u) = det[P (u)] ,(6)
which will play a central role in our calculations of the GSD. We refer to D(u) as the determinant polynomial of the K matrix. We can factorize D(u) as
D(u) = Cu −ξ α (u − u α ) Γα(7)
where u α are the distinct roots of D(u), Γ α are their multiplicities and
ξ = 1 2 α Γ α(8)
is the power of the leading order term in D(u). Because of the symmetric property of the K matrix, D(u) = D(u −1 ) and the roots of D(u) are distributed such that they are invariant under the map u → u −1 .
The roots on the unit circle are particularly special because they are related to the zero eigenvalues of K. On the unit circle, we have
D(e iq ) = L a=1 λ a (q) .(9)
Thus, when D(u) has a root u α = exp(iq α ) on the unit circle, there is at least one eigenvalue vanishing at q = q α and vice versa. In principle, there could be more than one zero eigenvalue. We will refer to these roots on the unit circle as unit roots and those away from the unit circle as non-unit roots.
Among the unit roots, those whose phases are rational multiples of 2π are even more special, and we refer to them as rational roots; unit roots that are not rational are referred to as irrational roots. There are two useful mathematical results concerning rational roots, which we list here and prove in Appendix A: Proposition 1. An mth root of unity exp(2πik/m) is called primitive if k is coprime with m. If D(u) has a primitive mth root of unity as one of its roots, then it has all the other primitive mth roots of unity as its roots with the same multiplicity.
Proposition 2. If D(u) has leading coefficient C = 1 and has only unit roots, then all of its roots are rational.
III. REVIEW OF ICS THEORIES
In this section, we review the plane wave spectrum [21,25,26] and the braiding statistics [21,26] in iCS theories. In particular, we highlight how different types of roots of the determinant polynomial affect these physical observables. In Section IV, we will again see that different types of roots have drastically different effects on the behavior of the GSD as a function of the number of layers N .
A. Plane Wave Spectrum
An iCS theory has propagating plane waves
A I,a µ = C µ v a (q)e iωt+ikx+iky+iqI ,(10)
with dispersion relations
ω 2 = k 2 x + k 2 y + g 4 4π 2 λ(q) 2 .(11)
Here q ∼ q + 2π is the momentum in the x 3 direction and v(q) is an eigenvector of P [exp(iq)] with eigenvalue λ(q). The theory is gapless at momentum q if λ a (q) = 0 for any a = 1, ..., L, or equivalently if D(u) has a root on the unit circle at u = exp(iq). Below, we summarize the effect of the three types of roots on the plane wave spectrum:
• A non-unit root does not lead to any gapless mode.
• An irrational root leads to gapless modes in the N → ∞ limit but not when N is finite. It is because when N is finite, the momenta q are quantized and can only be rational multiples of 2π. As N → ∞, q can take finer and finer values and eventually the gap closes at the irrational momentum associated to the irrational root.
• A rational root at a primitive mth root of unity u = exp(2πik/m) implies that the spectrum closes at q = 2πk/m when N is a multiple of m. Because of Proposition 1, when the spectrum closes at q = 2πk/m, it necessarily closes at q = 2πk ′ /m for all other k ′ coprime to m.
B. Braiding Statistics
We can take a charge at layer I and braid it with another charge at layer I + z. This induces a braiding phase exp[iΦ(z)]. In the N → ∞ limit, it was shown that as the size of the braiding trajectory approaches infinity, Φ(z) converges and depends only topologically on the braiding trajectory if the theory is gapped or gapless with linear dispersion around the gapless points [26]. In contrast, if there exists a gapless point with a higher order dispersion, Φ(z) diverges as the size of the braiding trajectory approaches infinity. Below, we focus on the former situation so that Φ(z) has a convergent asymptotic values and summarize the effect of the three types of roots on the asymptotic braiding phase:
• A non-unit root u α contributes to the asymptotic braiding phase an exponentially decaying term proportional to |u α | ±|z| . The sign in the exponent is chosen such that this term decays with |z|.
• A unit root u α = exp(iq α ), including both rational and irrational root, contributes to the asymptotic braiding phase an oscillatory term proportional to sin(q α z).
IV. GROUND STATE DEGENERACY
The GSD of an iCS theory can be inferred from its oneform global symmetry [29]. Suppose the Smith normal form of the K matrix is
V KW = R = r 1 r 2 . . . r N ,(12)
where V and W are GL(N L, Z) matrices, r i are integers that divide the succeeding ones r i+1 . Let N 0 and N 1 be the number of zero and non-zero eigenvalues of the K matrix respectively. Then the first N 1 of the r i 's are nonzero, whereas the last N 0 of them are zero. The one-form global symmetry is generated by the transformation
A I,a µ → A I,a µ + Λ I,a µ , Λ I,a µ = N L i=1 W IL+a,i λ i µ ,(13)
where λ i µ are Z ri gauge field for 1 ≤ i ≤ N 1 and flat U (1) gauge fields for N 1 < i ≤ N L [26]. This transformation shifts the Lagrangian (1) by
L → L + i 2π ϵ µνρ K IL+a,JL+b Λ I,a µ ∂ ν A J,b ρ + i 4π ϵ µνρ K IL+a,JL+b Λ I,a µ ∂ ν Λ J,b ρ .(14)
Although the shift of the Lagrangian is non-trivial, it integrates to an integer multiple of 2πi as
I K IL+a,JL+b Λ I,a µ(15)
is a flat gauge field with 2πZ holonomy. The partition function is therefore invariant under the transformation. In summary, the theory has a
U (1) N0 × N1 i=1 Z ri(16)
one-form global symmetry, or relatedly a N i=1 Z ri fusion group with Z 0 identified as a trivial group.
In Appendix B, we show that when the K matrix has period one and gcd{M k } = 1, all of the r i are equal to one except for the last 2ξ of them, where ξ is defined in (8). In this case, the fusion group of the fractional excitations is
G = Z r N −2ξ+1 × · · · × Z r N ,(17)
which has a finite number of large cyclic components for large N . Here, a group such as Z 6 = Z 2 × Z 3 is viewed as having one cyclic component instead of two. Now, consider compactifying the x 1 , x 2 direction of the iCS theory to a two-dimensional compact manifold. This leads to a discrete spectrum that is equally degenerate at every energy level. This degeneracy in particular is the same as the GSD. It is worth emphasizing that the theory may or may not be gapless when the volume of the compact manifold approaches infinity, but this does not affect the well-definedness and the value of the GSD.
The non-trivial GSD is a consequence of the spontaneous breaking of the discrete one-form symmetry. 1 It depends only on the topology of the two-dimensional manifold. When the manifold is a torus, the GSD is given by the order of the discrete one-form symmetry group. On a more general genus g manifold,
GSD = N1 i=1 r i g .(18)
Although the GSD depends only topologically on the manifold in the x 1 , x 2 direction, it can have a nontrivial dependence on the size of the x 3 direction, i.e. the number of layers N , as in other fracton orders. Our main focus in the next two subsections is to compute the GSD as a function of N . Without loss of generality, we will restrict the manifold in the x 1 , x 2 direction to be a torus in the rest of the paper. (7) on the GSD.
A. Non-degenerate K Matrix
Let us first compute the GSD when the K matrix is non-degenerate i.e. has no zero eigenvalues. In this case, the GSD reduces to the absolute value of the determinant of the K matrix (see for example the review [31])
GSD = N L i=1 r i = det(V −1 KW −1 ) = |det(K)| ,(19)
where we used the fact that every GL(N L, Z) matrix V has | det V | = 1. Determinant of the K matrix is given by the product of the eigenvalues. Using the relation (9) between the eigenvalues and the determinant polynomial, we obtain
GSD = N k=1 L a=1 λ a 2πk N = N k=1 D(e 2πik/N ) . (20)
Substituting (7) into the above formula, the GSD can then be expressed in terms of u α , the roots of D(u)
GSD = |C| N α N k=1 (e 2πik/N − u α ) Γα = |C| N α u N α − 1 Γα .(21)
In the last equality, we used the fact that u N − 1 has N distinct roots at exp(2πik/N ) with k = 1, ..., N and thus can be factorized into
N k=1 [u − exp(2πik/N )]
. The overall coefficient of the factorization is fixed by comparing the coefficient of u N with the expansion of the factorization. By expanding (21), we learn that the GSD for non-degenerate K matrices can be expressed as a sum of exponentials of N .
Mathematically, (21) can be thought of as the absolute value of the resultant of the polynomials u ξ D(u) and u N − 1. We refer to [32] for more details on resultant.
B. Degenerate K Matrix
Next, we move on to consider degenerate K matrices. Because there are zero eigenvalues, the GSD is no longer simply given by | det(K)|. Nevertheless, we can still determine how it depends on N . We will first state the result and then derive it.
The main result is as follows: Let u α be the distinct roots of the determinant polynomial D(u), Γ α be the multiplicity of u α , and ∆ α be the kernel dimension of P (u α ) in (4). ∆ α is always upper bounded by Γ α . 2 Define also an index set I = α : u N α = 1 , the set of roots of D(u) which correspond to gapless modes when the number of layer is N . The set I generally depends on N , and when discussing the dependence of the GSD on N , we only consider those values of N that yield the same I. For example, in the context of D(u) = u−1+u −1 , the roots are u = exp(±2πi/6) and thus we consider those N that are divisible by 6 separately from the rest. For the values of N that share the same index set I, the GSD takes the form
GSD = A|C| N N α∈I (Γα−∆α) β / ∈I |u N β − 1| Γ β ,(22)
where A is an N -independent constant. 3 The dependence of the GSD on N has an additional polynomial factor besides the usual terms in (21). This is controlled by the roots in I.
We now derive the formula (22) starting from (18). In order to evaluate (18), we add a small perturbation ϵ (times the identity matrix) to K and turn it into K +ϵI. Since K +ϵI is not integral, it does not physically correspond to a gauge invariant iCS theory. Instead, this perturbation is merely a mathematical technique. Now take V, W as in (12) and consider
V (K + ϵI)W = R + O(ϵ) N1×N1 O(ϵ) N1×N0 O(ϵ) N0×N1
ϵ H N0×N0 HereR = diag(r 1 , r 2 , ..., r N1 ) and H is an N 0 ×N 0 matrix with order 1 coefficients. To leading order in ϵ,
det(K + ϵI) = det(V (K + ϵ)W ) = ϵ N0 det(H) det(R) ,
and hence
GSD = | det(R)| = det(K + ϵ) ϵ N0 det(H) .(23)
This is the key equation for calculating the GSD.
Because the numerator is the determinant of the nondegenerate periodic matrix K + ϵI, we can calculate it using (21)
+ ϵI] = det[P (u α ) + ϵI] + η α (u − u α ) Γα + · · · ,(24)
where η α ̸ = 0 and "· · · " stands for higher order terms in u − u α . Let λ a be the eigenvalues of P (u α ) with the first ∆ α of them vanishing. Then (26) and the new roots are u α,m (ϵ) = u α + B α ϵ ∆α 1/Γα e 2πim/Γα .
det[P (u α ) + ϵI] = L a=1 (λ a + ϵ) = b α ϵ ∆α + · · · . (25) where b α = L i=∆α+1 λ a ̸ = 0. Thus (24) becomes det[P (u) + ϵI] = η α (u − u α ) Γα + b α ϵ ∆α + · · · ,
Here m = 1, . . . , Γ α labels the new roots and B α = −b α /η α . To avoid ambiguity, the Γ α th root of B α ϵ ∆α is chosen to be a specific one for all m. Here we see that the Γ α degenerate roots of D(u) at u = u α are split into Γ α distinct roots by the perturbation.
We now substitute the new roots into (21) to calculate det[P (u) + ϵI]. Each new root contributes a factor of u α,m (ϵ) N − 1. If α / ∈ I, the correction due to the perturbation is insignificant and
u α,m (ϵ) N − 1 = u N α − 1 + · · · .(28)
On the other hand, if α ∈ I, then u N α = 1 and
u α,m (ϵ) N −1 = N u −1 α B α ϵ ∆α 1/Γα e 2πim/Γα +· · · .(29)
Multiplying all the contributions together, the numerator of (23) is found to be
| det(K+ϵ)| = |C| N α∈I N Γα |B α ϵ ∆α | β / ∈I |u N β −1| Γ β +· · · .(30)
Next, we discuss the denominator of (23). LetŴ ,V T be the N L × N 0 matrix consisting of the last N 0 columns of W , V T of the Smith normal form (12) respectively. Then
H = ϵ −1V T (K + ϵ)Ŵ =V TŴ .(31)
The columns ofŴ ,V T are the null vectors of K with integral coefficients. They can be decomposed into linear combinations of null vectors of the form (5) with q α = −i log u α for α ∈ I. All the null vectors are thus periodic vectors and each can be constructed by repeating some vector O(N ) times. It then implies that every entry of V TŴ is proportional to N and thus
det(H) ∝ N N0 .(32)
Finally, using N 0 = α∈I ∆ α to cancel all the ϵ's and absorbing various constant factors into an overall constant A, (30) and (32) are combined to give (22). This concludes our derivation of (22).
C. Effects of Three Types of Roots
Equipped with the formula (21), (22) for GSD, we are now ready to examine how different types of roots affect the GSD.
Non-unit Roots
First, let us discuss the effect of the non-unit roots. Recall that the list of roots are invariant under the map u → u −1 . Thus every non-unit root u α is paired up with another non-unit root u α * = u −1 α and within a pair, one of the roots has magnitude greater than 1 while the other has magnitude less than one. According to (21), a pair of non-unit roots contribute to the GSD as
|(u N α − 1)(u N α * − 1)| Γα N ≫ 1 − −−− → |u α | N Γα , |u α | > 1 |u α | −N Γα , |u α | < 1 ,(33)
which grows exponentially with N for large N .
If an iCS theory has only non-unit roots, then half of the roots has magnitude greater than one while the other half has magnitude less than one. When N is large, the asymptotic GSD grows exponentially as
GSD = |C| |uα|>1 |u α | Γα N .(34)
As an example, consider a theory with a tridiagonal K matrix
K = . . . M 0 M 1 M 1 M 0 M 1 M 1 M 0 M 1 M 1 M 0 . . . .(35)
The theory has D(u) = M 1 u + M 0 + M 1 u −1 with roots at
u ± = M ± M 1 , M ± = −M 0 ± M 2 0 − 4M 2 1 2 .(36)
The roots are both non-unit roots if |M 0 | > 2|M 1 | and the GSD is
GSD = M N + + M N − − 2M N 1(37)
In Figure 1a, we plot the GSD as a function of N when (M 0 , M 1 ) = (3, 1). It grows exponentially for large N .
Irrational Roots
Next, we move on to discuss the effect of the irrational roots. An irrational root can be expressed as u α = exp(iq α ). Again as non-unit roots, it is paired up with another irrational root u α * = exp(−iq α ). According to (21), a pair of them contributes to the GSD as
|(u N α − 1)(u N α * − 1)| Γα = 2 sin q α N 2 2Γα ,(38)
which oscillates as a function of N . Since q α is an irrational multiple of 2π, the oscillation is erratic as N increases by integer steps.
If an iCS theory has irrational roots, its asymptotic GSD no longer grows exponentially with N but rather fluctuates erratically. Suppose there are only non-unit roots and irrational roots, then the asympotic GSD is
GSD = |C| |uα|>1 |u α | Γα N |u β |=1 u β =e iq β 2 sin q β N 2 Γ β ,(39)
with an exponentially growing envelop
GSD ≤ |C| |uα|>1 |u α | Γα N |u β |=1 2 Γ β .(40)
Note that the base of the exponentially growing envelop |C| |uα|>1 |u α | Γα is always greater than one if there exists an irrational root. If |C| > 1, the base is necessarily greater than one. An example of this kind is a period one K matrix with D(u) = 2u + 1 + 2u −1 , which has two irrational roots. If |C| = 1, there must be nonunit roots and thus the base is still greater than one. It is because otherwise all the roots are unit roots and according to Proposition 2 they must all be rational, which contradicts with the assumption that there exists at least one irrational root. An example of this kind is a period one K matrix with D(u) = u 2 +2u+1+2u −1 +u −2 , which has two irrational roots and two non-unit roots.
As an example, consider a theory with a tridiagonal K matrix (35). The roots are unit roots when |M 0 /M 1 | ≤ 2. They are generally irrational unless |M 0 /M 1 | = 0, ±1, ±2 according to the Niven's theorem [33]. The GSD is
GSD = 4|M 1 | N sin 2 qN 2 .(41)
where q = arccos(−M 0 /2M 1 ). In Figure 1b, we plot the GSD as a function of N when (M 0 , M 1 ) = (1, 2). It exhibits an erratic pattern with an exponentially growing upper bound. Numerically, it seems that the GSD also has an exponentially growing lower bound with the same base. However, we do not know how to prove the lower bound.
Rational Roots
Finally, we discuss the effect of the rational roots. A rational root can be expressed as u α = exp(2πik/m) where k and m are coprime integers. Such a rational root is a primitive mth root of unity. According to proposition 1, all the other primitive mth roots of unity are also roots of D(u) with the same multiplicity.
The contribution of these primitive mth roots of unity to the GSD depends on whether N is divisible by m or not. If N is not divisible by m, according to (21) the contribution to the GSD is an oscillatory function of N
If N is divisible by m, the K matrix becomes degenerate with zero eigenvalues generated by the primitive m roots of unity and according to (22) the contribution to the GSD is a mononomial in N
AN α (Γα−∆α) ,(43)
where A is an N -indepedent constant and the sum in exponent is over roots that are primitive mth roots of unity. Combining (42) and (43), we conclude that the pattern of the contribution to the GSD from the primitive mth roots of unity depends only on the value of N modulo m.
If an iCS theory has only roots at the primitive mth roots of unity, then the determinant polynomial must be a power of the mth cyclotomic polynomial Φ m (u) with an integer coefficient C (see Proposition 10 in Appendix A 4 for more details)
D(u) = Cu −ξ Φ m (u) Γ .(44)
The mth cyclotomic polynomial is defined as
Φ m (u) = 1≤k≤m gcd(k,m)=1 (u − e 2πik/m ) .(45)
It is in fact an integer polynomial although not manifest in the definition. The GSD of such an iCS theory is
GSD = A|C| N N γ , m | N |C| N 1≤k≤m gcd(k,m)=1 2 sin kπN m Γ , m ∤ N ,(46)
where γ = α (Γ − ∆ α ). Restricting to a subsequence of N with a fixed (N mod m) ̸ = 0, the GSD • is a constant when |C| = 1;
• grows exactly exponentially when |C| ̸ = 1.
On the other hand, on the subsequence of N that are divisible by m, the GSD
• is a constant when |C| = 1, γ = 0;
• grows polynomially when |C| = 1, γ ̸ = 0;
• grows exponentially when |C| ̸ = 1, γ = 0;
• grows exponentially with a polynomial overhead when |C| ̸ = 1, γ ̸ = 0.
As an example, consider a theory with a tridiagonal K matrix (35) with (M 0 , M 1 ) = (−1, 1). The theory has roots at u = exp(±2πi/6) with Γ = ∆ = 1. Its GSD is plotted in Figure 1c and is given by
GSD = 1 , 6 | N 4 sin 2 (πN/6) , 6 ∤ N ,(47)
which oscillates between 1, 3, 4, 3, 1, 1 periodically.
For a more sophisticated example, consider another theory with a tridiagonal K matrix (35) with (M 0 , M 1 ) = (2, 1). The theory has roots at u = −1 with Γ = 2 and ∆ = 1. Its GSD is plotted in Figure 1d and is given by
GSD = N , 2 | N 4 , 2 ∤ N ,(48)
which grows linearly for even N but retains a constant for odd N .
V. CONDITION FOR FOLIATION
Foliated fracton orders [5] provide a useful organizing principle for gapped fracton phases including the X cube model [4], the checkerboard model [4,34], the twisted foliated fracton order [35] and etc. As defined in [5], a foliated fracton order uses decoupled layers of twodimensional gapped topological orders as resources to increase its system size. As a result, the GSD of a foliated fracton order grows exponentially with the linear system size N
GSD = AM N ,(49)
where A is an N -independent constant and M ∈ Z is the GSD of the resource topological order. The simplest foliated iCS theory is the one with a diagonal K matrix
K = M M . . . M .(50)
It is a foliated fracton order as increasing the system size amounts to inserting a decoupled layer of U (1) M Chern-Simons theory. As a result, its GSD = M N grows exponentially with N . A more non-trivial example of foliated iCS theory was suggested in [35], which has a K matrix of period two
K = . . . 0 2 −1 2 0 −1 0 2 −1 2 0 −1 0 2 −1 2 0 −1 0 . . . .(51)
The theory has an exponentially growing GSD = 4 N . One can decouple the resource topological order from the iCS theory by performing a GL(N,
Z) transformation K → W T KW with the GL(N, Z) matrix W = . . . 1 1 1 1 −1 1 −1 1 1 1 −1 1 1 . . . .(52)
W is the identity matrix outside the region displayed above. This transformation yields a new K matrix
W T KW = . . . 0 2 −1 2 0 0 2 −1 0 2 0 0 0 −1 0 0 2 0 0 2 0 −1 0 2 2 0 . . . .
(53) There is a 4 × 4 block decoupled from the rest of the matrix which takes the same form as the original K matrix. The decoupled block is the resource layer, in this case a Z 2 × Z 2 twisted gauge theory.
Despite the examples above, generically a gapped iCS theory is expected to be non-foliated, such as the one considered in Figure 1a whose GSD (37) does not take the form as (49).
An interesting and important question is to determine which gapped iCS theories are foliated and which ones are not. Here we propose a necessary condition for a gapped iCS theory to be foliated:
Proposition 3. A gapped iCS theory is foliated only if its determinant polynomial D(u) is a constant.
We do not know whether the converse of the proposition is true i.e. whether the condition is sufficient.
Proposition 3 can be proven by comparing the GSD of the gapped iCS theory (21) with the GSD of a foliated fracton order (49). If D(u) were not a constant, then it has some roots u α which renders the GSD formula (21) a sum over exponentials of N . The sum includes at least two terms: an exponential with the largest base |C| |uα|>1 |u α | Γα and an exponential with the smallest base |C| |uα|<1 |u α | Γα . Such a sum can never be expressed as a single exponential of N . Hence the theory cannot be foliated.
This condition is also consistent with the expectation from the braiding statistics. Charges in a foliated iCS theory are expected to have non-trivial braiding only if they are separated by a finite range in the x 3 direction. It is such that charges with non-trivial braiding can be grouped into anyons of the resource topological order. However, a non-unit root of the determinant polynomial leads to a long-ranged interaction in the braiding phase that decays exponentially with the separation in the x 3 direction. Thus, a foliated iCS theory is expected to have a constant determinant polynomial.
We now check the condition in Proposition 3 against the two examples of foliated iCS theories. The diagonal K matrix (50) has a constant D(u) = M , which is consistent with the condition. The period two K matrix (51) has
P (u) = u + u −1 2 2 0 , D(u) = 4 .(54)
The constant D(u) is again consistent with the condition. A consequence of Proposition 3 is that any gapped iCS theory with a non-diagonal K matrix of period one cannot be a foliated fracton order as its D(u) = k M k u k is not a constant.
Most of the statements can be found in standard mathematics textbooks such as [32,36]. Appendix A 1, A 2, A 3 discusses the background materials, Appendix A 4 proves Proposition 1 using the properties of cyclotomic polynomials, and Appendix A 5 proves Proposition 2.
Definitions
Let Z[u] (resp. Q[u]) be the ring of polynomials in u with integer (resp. rational) coefficients. Although in the main text we have used Laurent polynomials, none of the statements in this appendix depends on whether terms with negative powers are allowed. Therefore, we will only consider ordinary polynomials here. Take
α(u) = a n u n + · · · + a 0 ∈ Z[u] ,(A1)
where a n ̸ = 0. We say that α(u) is monic if a n = 1. The content of α(u) is
c(α) = gcd{a k } .(A2)
We say that α(u) is primitive if c(α) = 1. 4 We say that α(u) is reducible if it can be written as
α(u) = β(u)γ(u) ,(A3)
where neither of β(u) and
γ(u) is invertible in Z[u]. Otherwise, α(u) is irreducible. In Z[u]
, the only invertible elements are ±1. We have the following lemma:
Lemma 4. Let α(u), β(u) ∈ Z[u] with α(u) monic. If γ(u) = β(u) α(u) ∈ Q[u] ,(A4)then actually γ(u) ∈ Z[u].
This lemma is clear from the procedure of long division. We will use the lemma several times in this appendix. Due to its simplicity, we will not always quote the lemma explicitly when it is used. Content and reducibility can also be defined in Q[u], where the convention for gcd is e.g.
gcd 1 2 , 1 3 = 1 6 ,(A5)
and the invertible elements are q ∈ Q, q ̸ = 0. We have the following lemmas:
Lemma 5. If α(u), β(u) ∈ Q[u]
, then the content of the product α(u)β(u) is c(αβ) = c(α)c(β). In particular, the product of two primitive polynomials is also primitive. These lemmas are known collectively as Gauss's lemma.
Algebraic Number and Minimal Polynomial
Now take a number z ∈ C. We say that z is an algebraic number if it satisfies some polynomial α(u) ∈ Q[u], i.e. α(z) = 0. An equivalent definition is to replace Q[u] by Z[u], since if α(z) = 0 then bα(z) = 0 for any integer b ̸ = 0. Define the annihilator of an algebraic number z as the set
Ann(z) = {α(u) ∈ Q[u] | α(z) = 0} ,(A6)
and the minimal polynomial α z (u) of z as the element of Ann(z) of the minimal degree. For example, if z = 1/2 then we can take α z (u) = 2u − 1. Note that the minimal polynomial is always defined up to multiplication by q ∈ Q, q ̸ = 0. Aside from this ambiguity, α z (u) is unique, which follows from the fact that
α z (u) = gcd (Ann(z)) ,(A7)
where we are taking the gcd of polynomials in Q[u]. To give an example, gcd{β(u), γ(u)} is the polynomial of the maximal degree that divides both β(u) and γ(u). We can prove (A7) by contradiction: Suppose that α z (u) does not divide some β(u) ∈ Ann(z). A property of the gcd is that there exist ϕ(u), χ(u) ∈ Q[u] such that
γ(u) = gcd{α z (u), β(u)} = ϕ(u)α z (u) + χ(u)β(u) .(A8)
This then implies that γ(u) ∈ Ann(z). Since α z (u) does not divide β(u), their gcd γ(u) has a smaller degree than α z (u), contradicting the definition of α z (u). Similarly, α z (u) is irreducible, since any non-trivial factor of α z (u) would have a lower degree than α z (u). Theses facts imply the following corollary:
Corollary 7.
The minimal polynomial of an algebraic number z is (up to multiplication by q ∈ Q, q ̸ = 0) the unique irreducible element of Ann(z).
This lemma will be used later to prove Proposition 10 and hence Proposition 1.
Algebraic Integer
We say that z ∈ C is an algebraic integer if it satisfies some monic polynomial α(u) ∈ Z [u]. Note that we are restricting the polynomials from Q[u] to Z [u] here. An algebraic integer is always an algebraic number, but not vice versa. We have the following lemma: Lemma 8. A number z ∈ C is an algebraic integer if and only if its minimal polynomial in Z[u] is monic.
In Z[u], the minimal polynomial is also required to be primitive besides having the minimal degree. The "if" part of the lemma is obvious. For the "only if" part, take a monic polynomial β(u) ∈ Ann(z) and let
γ(u) = β(u) α z (u) .(A9)
By Lemma 5,
c(γ) = c(β) c(α z ) = 1 ,(A10)
so γ(u) ∈ Z[u]. When multiplying two polynomials α z (u) and γ(u), the leading coefficient of the product is the product of the leading coefficients. Since β(u) is monic by assumption, so are α z (u) and γ(u). We have the following corollary:
Corollary 9. A rational number q ∈ Q is an algebraic integer if and only if it is an ordinary integer.
To see this, write q = r/s where r, s ∈ Z are coprime.
Then the minimal polynomial of q is
α q (u) = su − r ,(A11)
which is monic if and only if s = 1. This corollary will also be used to prove Proposition 10. Furthermore, it can be shown that algebraic integers form a ring.
Cyclotomic Polynomial and Proof of Proposition 1
Having prepared the basics, we now begin a discussion of roots of unity which eventually leads to Propositions 1 and 2. Recall that an mth root of unity e 2πik/m is called primitive if k is coprime with m. 5 We write
C m = {k ∈ Z m | gcd(k, m) = 1} ,(A12)
and define the mth cyclotomic polynomial Φ m (u) as
Φ m (u) = k∈Cm (u − ω k ) ,(A13)
where ω = e 2πi/m . Equivalently,
Φ m (u) = u m − 1 k / ∈Cm (u − ω k ) .
(A14) 5 Note that this is difference from "primitive polynomial".
The denominator of (A14) is a product of cyclotomic polynomials Φ m ′ (u) where m ′ < m. Thus by induction on m and Lemma 4, it is straightforward to show that Φ m (u) ∈ Z [u]. Clearly, Φ m (u) is monic and is satisfied by all primitive mth roots of unity. In fact:
Proposition 10. All primitive mth roots of unity share the same minimal polynomial, which is Φ m (u).
By Corollary 7, it is enough to show that Φ m (u) is irreducible. We give a proof found in [37]. Suppose that β(u) ∈ Z[u] divides u m − 1. Let u 0 be a root of β(u) (and hence an mth root of unity), and p a prime that does not divide m. We claim that u p 0 is also a root of β(u). If this claim is true, then we can choose u 0 to be a primitive mth root of unity and β(u) its minimal polynomial α u0 (u). By basic number theory, every primitive mth root of unity is of the form u s 0 for some s ∈ Z coprime with m. We can then apply the claim several times using the factorization s = p k1 1 · · · p kn n , where none of the prime factors p i divides m. Therefore, α u0 (u) has all primitive mth roots of unity as its roots, so Φ m (u) divides α u0 (u). However, the minimal polynomial α u0 (u) of u 0 is irreducible, so actually α u0 (u) = Φ m (u). Since this argument works for all primitive mth roots of unity, Proposition 10 is proved.
To prove the above claim, we start by defining an auxiliary quantity
∆ = j<k ω j − ω k 2 = ± j̸ =k ω j − ω k = ± j ω j m−1 l=1 1 − ω l = ± j mω j = ±m m ,(A15)
where all labels take values in Z m . When going from the fourth to the fifth line, we set x = 1 in the polynomial
f (x) = m−1 l=1 x − ω l = x m − 1 x − 1 = m−1 l=0 x l ,(A16)
which gives f (1) = m. The quantity ∆ is known as the discriminant of the polynomial u m − 1, but we do not need a formal definition of this concept here. Suppose that the claim is false, i.e. that u p 0 is not a root of β(u). Without loss of generality, β(u) is monic. We can then use the roots u i of β(u) to write
β(u p 0 ) = i (u p 0 − u i ) ̸ = 0 .(A17)
Thus β(u p 0 ) is a product of differences of distinct mth roots of unity. Furthermore, β(u p 0 ) is also an algebraic integer since algebraic integers form a ring. Therefore, β(u p 0 ) divides ∆ as algebraic integers, i.e. ∆/β(u p 0 ) is an algebraic integer. Now we can apply the multinomial theorem to β(u) p to show that β(u p 0 ) = β(u 0 ) p = 0 (mod p) .
Here, "equality mod p" means that the difference of the two sides is p times an algebraic integer. Thus p divides β(u p 0 ) as algebraic integers. To explain this step in more detail, suppose that β(u) = k b k u k , where b k are ordinary integers. In the multinomial expansion of β(u) p , every cross term has a multinomial coefficient that is divisible by p, so
β(u) p = k b p k u kp (mod p) .(A19)
By applying Fermat's little theorem to the ordinary integer b k , we have b p k = b k (mod p). Therefore,
β(u) p = β(u p ) (mod p) .(A20)
Consequently, p divides ∆ as algebraic integers, i.e.
∆ p = ∆ β(u p 0 ) β(u p 0 ) p (A21)
is an algebraic integer. However, Corollary 9 says that the rational number ∆/p is an algebraic integer if and only if it is an ordinary integer. The conclusion is that p divides ∆ = ±m m as ordinary integers, which contradicts the assumption that p does not divide m. This proves the claim and hence Proposition 10. Proposition 1 can be derived easily from Proposition 10. Suppose that a primitive mth root of unity u i is a root of D(u) with multiplicity Γ i . Since D(u) ∈ Ann(u i ), the cyclotomic polynomial Φ m (u) must divide D(u) (up to some trivial technicality we can pretend D(u) ∈ Z[u]). If Γ i > 1, then the same argument can be applied to D(u)/Φ m (u). By induction, Φ m (u) Γi divides D(u). Therefore, all primitive mth roots of unity are roots of D(u) with the same multiplicity Γ i .
Proof of Proposition 2
Finally, we prove Proposition 2 following [38]. Let u 1 , . . . , u n be the roots of D(u). Here, a repeated root of multiplicity Γ is listed Γ times as different roots. For k = 1, . . . , n, let e k (x 1 , . . . , x n ) be the kth elementary symmetric polynomial in n variables. For example,
e 1 (x 1 , . . . , x n ) = i x i , e 2 (x 1 , . . . , x n ) = i<j x i x j , e 3 (x 1 , . . . , x n ) = i<j<k x i x j x k . (A22)
By Vieta's theorem, e k (u 1 , . . . , u n ) ∈ Z for all k. Now for each integer s > 0, define a polynomial
β s (u) = n k=1 (u − u s k ) = n k=0 b sk u k .(A23)
Each coefficient b sk is a symmetric polynomial in the n variables u 1 , . . . , u n with integer coefficients. By the fundamental theorem of symmetric polynomials, b sk can be written uniquely as b sk = γ sk (e 1 (u 1 , . . . , u n ), . . . , e n (u 1 , . . . , u n )) , (A24)
where γ sk (x 1 , . . . , x n ) is a polynomial in n variables with integer coefficients. The conclusion is that b sk ∈ Z. Also, since |u k | = 1, we have the bound
|b sk | ≤ n k . (A25)
As a result, the list (b s1 , . . . , b sn ) can only take finitely many values, so there exist integers 0 ≤ g < h such that β 2 g (u) = β 2 h (u). Therefore, the list (u 2 h 1 , . . . , u 2 h n ) is a permutation of the list (u 2 g 1 , . . . , u 2 g n ). Let c = 2 g and d = 2 h−g . Then this permutation σ ∈ S n of the list (u c 1 , . . . , u c n ) is achieved by raising each element to the power d. Let r be the order of σ in the group S n . Since the action of raising to the power d repeated r times is trivial, we see that u rcd k = 1 for all k. This shows that all roots of D(u) are rational roots.
Appendix B: Fusion Group of iCS Theories with Period One
In this appendix, we study the fusion group or relatedly the one-form symmetry group of iCS theories with period L = 1. Consider a period one K matrix with
D(u) = ξ k=−ξ M k u k ,(B1)
where M ξ ̸ = 0. We will prove the following bound on the fusion group:
Proposition 11. If the K matrix has period one and gcd{M k } = 1, then the fusion group G has at most 2ξ cyclic components.
As defined in Appendix A 1, gcd{M k } = 1 means that D(u) is a primitive polynomial. In the proposition, we always merge the cyclic components in the fusion group G as much as possible. For example, Z 6 = Z 2 × Z 3 is viewed as having one cyclic component instead of two. To demonstrate the proposition, consider a period one K matrix with D(u) = u + 3 + u −1 . The fusion group is G = Z F N × Z 5F N which has two cyclic components, where F N is the N th Fibonacci number [21]. As explained in Section IV, the fusion group is
G = N i=1 Z ri ,(B2)
where r i are the diagonal entries of the Smith normal form (12) of K, and we use the convention that Z 0 is the trivial group. Therefore, it suffices to show that r 1 = · · · = r N −2ξ = 1 .
Since r i are integers, this is equivalent to showing that
N −2ξ i=1 r i = 1 . (B4)
This product is suitable for the current problem due to the following fact:
gcd {minor of K of order m} = m i=1 r i ,(B5)
where a minor of K of order m is the determinant of an m × m submatrix of K. By (B5), the problem becomes showing that gcd {minor of K of order N − 2ξ} = 1 .
Therefore, we will pick some submatrices of K obtained by deleting 2ξ rows and 2ξ columns, and show that their determinants are coprime as a whole. First, delete the first 2ξ columns and the last 2ξ rows of the K matrix and consider the submatrix
T ξ = M ξ M ξ−1 M ξ . . . . . . . . . 0 · · · M ξ−1 M ξ .(B7)
Clearly det(T ξ ) = M N −2ξ ξ since T ξ is lower-triangular. Next, instead of deleting the first 2ξ columns of the K matrix, delete the first 2ξ−1 columns and the last column and consider the submatrix
T ξ−1 = M ξ−1 M ξ M ξ−2 M ξ−1 M ξ . . . . . . . . . . . . 0 · · · M ξ−2 M ξ−1 M ξ 0 0 · · · M ξ−2 M ξ−1 . (B8)
When expanding det(T ξ−1 ) into a sum of terms using the definition of determinant, a term either is M N −2ξ ξ−1 , or contains a factor of M ξ . Therefore,
det(T ξ−1 ) = M N −2ξ ξ−1 (mod M ξ ) .(B9)
Suppose there exists a prime p that divides both det(T ξ ) and det(T ξ−1 ). It then also divides M ξ since det(T ξ ) = M N −2ξ ξ . Using the relation (B9), we can further deduce that M N −2ξ ξ−1 is divisible by p and so does M ξ−1 . Likewise, we can define submatrices T ξ−2 , . . . , T 0 , and show inductively that if p divides all det(T i ) it must also divide all M i . This however contradicts our assumption that gcd{M k } = 1, so such a prime p cannot exist and we must have gcd{det(T k )} = 1. Since a subset of minors is already coprime, the set (B6) of all minors of order N −2ξ is also coprime, and hence Proposition 11 is proved.
More generally, if b = gcd{M k } > 1, we can apply the proposition to the integer matrix K/b. After multiplying b back, we see that the fusion group G has at least N − 2ξ cyclic components that are Z b . In general, the other 2ξ cyclic components still grow (asymptotically) exponentially with N , affirming the non-foliated nature of the theory unless K is diagonal.
FIG. 1 :
1GSD of various tridiagonal iCS theories, labeled by their (M0, M1), as a function of the number of layers N . Note that the y axis is logarithmic in subfigures (a) and (b) but not in (c) or (d).
, which relates the determinant with the roots of det[P (u) + ϵI]. For small ϵ, the roots of det[P (u) + ϵI] are close to the roots of det[P (u)]. To find the new roots, we expanding det[P (u) + ϵI] in the vicinity of u α det[P (u)
Lemma 6 .
6If α(u) ∈ Z[u] is primitive and irreducible in Z[u], then it is also irreducible when viewed as an element of Q[u].
TABLE I :
IThe GSD of various fracton models on square/cubic lattice and various tridiagonal iCS theories. The iCS theories are labeled by their (M0, M1) as defined in
TABLE II :
IIThe effect of the leading coefficient C and the roots uα of the determiant polynomial D(u) in
In contrast, a U (1) one-form symmetry cannot be spontaneously broken in two spatial dimensions[29,30].
To understand the difference between Γα and ∆α better, it is helpful to draw an analogy with the algebraic multiplicity Γ and the geometric multiplicity ∆ of an eigenvalue λ in linear algebra. In linear algebra, the polynomial of interest is the characteristic polynomial of a matrix K, the algebraic multiplicity is the number of roots of the characteristic polynomial at the given eigenvalue λ and the geometric multiplicity is the kernel dimension of K − λI. The quantity Γ − ∆ intuitively measures the obstacle to diagonalize the matrix.3 The precise value of A is a complicated expression involving basic algebraic number theory, and we will not determine it in this paper. Practically, A is most conveniently fixed by fitting(22) for a small N .
Note that this is different from "primitive mth root of unity".
ACKNOWLEDGMENTSWe are grateful to Daren Chen, Meng Cheng, Roman Geiko, Yi Ni and Yiyue Zhu for discussions. X.M. and X.C. are supported by the National Science Foundation under award number DMR-1654340, the Simons collaboration on "Ultra-Quantum Matter" (grant number 651440), the Simons Investigator Award (award ID 828078) and the Institute for Quantum Information and Matter at Caltech. X.C. is also supported by the Walter Burke Institute for Theoretical Physics at Caltech. H.T.L. is supported in part by a Croucher fellowship from the Croucher Foundation, the Packard Foundation and the Center for Theoretical Physics at MIT. The authors of this paper were ordered alphabetically.Appendix A: Supplementary mathematicsIn this appendix, we discuss the mathematics necessary for proving the two propositions at the end of Section II.
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|
[] |
[
"Pursuing Equilibrium of Medical Resources via Data Empowerment in Parallel Healthcare System",
"Pursuing Equilibrium of Medical Resources via Data Empowerment in Parallel Healthcare System"
] |
[
"Yi Yu [email protected] ",
"Shengyue Yao [email protected] ",
"Kexin Wang ",
"Yan Chen [email protected] ",
"Fei-Yue Wang [email protected] ",
"Yilun Lin [email protected] ",
"\nSchool of Health Policy and Management\nUrban Computing Lab Shanghai AI Laboratory Shanghai Urban Computing Lab Shanghai AI Laboratory Shanghai\nNanjing Medical University\nNanjingChina, China, China\n",
"\nSchool of Health Policy and Management\nNanjing Medical University Affiliated Suzhou Hospital\nSuzhouChina\n",
"\nSKL-MCCS\nInstitute of Automation Chinese Academy of Sciences\nThe Macau Institute of Systems Engineering, Macau University of Science and Technology Macau\nUrban Computing Lab Shanghai AI Laboratory Shanghai\nNanjing Medical University\nNanjing, BeijingChina, China, China, China\n"
] |
[
"School of Health Policy and Management\nUrban Computing Lab Shanghai AI Laboratory Shanghai Urban Computing Lab Shanghai AI Laboratory Shanghai\nNanjing Medical University\nNanjingChina, China, China",
"School of Health Policy and Management\nNanjing Medical University Affiliated Suzhou Hospital\nSuzhouChina",
"SKL-MCCS\nInstitute of Automation Chinese Academy of Sciences\nThe Macau Institute of Systems Engineering, Macau University of Science and Technology Macau\nUrban Computing Lab Shanghai AI Laboratory Shanghai\nNanjing Medical University\nNanjing, BeijingChina, China, China, China"
] |
[] |
The imbalance between the supply and demand of healthcare resources is a global challenge, which is particularly severe in developing countries. Governments and academic communities have made various efforts to increase healthcare supply and improve resource allocation. However, these efforts often remain passive and inflexible. Alongside these issues, the emergence of the parallel healthcare system has the potential to solve these problems by unlocking the data value. The parallel healthcare system comprises Medicine-Oriented Operating Systems (MOOS), Medicine-Oriented Scenario Engineering (MOSE), and Medicine-Oriented Large Models (MOLMs), which could collect, circulate, and empower data. In this paper, we propose that achieving equilibrium in medical resource allocation is possible through parallel healthcare systems via data empowerment. The supply-demand relationship can be balanced in parallel healthcare systems by (1) increasing the supply provided by digital and robotic doctors in MOOS, (2) identifying individual and potential demands by proactive diagnosis and treatment in MOSE, and (3) improving supply-demand matching using large models in MOLMs. To illustrate the effectiveness of this approach, we present a case study optimizing resource allocation from the perspective of facility accessibility. Results demonstrate that the parallel healthcare system could result in up to 300% improvement in accessibility.Index Terms-parallel healthcare system, parallel intelligence, equilibrium, medical resource allocation,
| null |
[
"https://export.arxiv.org/pdf/2306.00408v1.pdf"
] | 259,000,140 |
2306.00408
|
d267f802477e320bf6dc2e4f8cd5304f470ae48a
|
Pursuing Equilibrium of Medical Resources via Data Empowerment in Parallel Healthcare System
Yi Yu [email protected]
Shengyue Yao [email protected]
Kexin Wang
Yan Chen [email protected]
Fei-Yue Wang [email protected]
Yilun Lin [email protected]
School of Health Policy and Management
Urban Computing Lab Shanghai AI Laboratory Shanghai Urban Computing Lab Shanghai AI Laboratory Shanghai
Nanjing Medical University
NanjingChina, China, China
School of Health Policy and Management
Nanjing Medical University Affiliated Suzhou Hospital
SuzhouChina
SKL-MCCS
Institute of Automation Chinese Academy of Sciences
The Macau Institute of Systems Engineering, Macau University of Science and Technology Macau
Urban Computing Lab Shanghai AI Laboratory Shanghai
Nanjing Medical University
Nanjing, BeijingChina, China, China, China
Pursuing Equilibrium of Medical Resources via Data Empowerment in Parallel Healthcare System
Index Terms-parallel healthcare systemparallel intelligenceequilibriummedical resource allocation
The imbalance between the supply and demand of healthcare resources is a global challenge, which is particularly severe in developing countries. Governments and academic communities have made various efforts to increase healthcare supply and improve resource allocation. However, these efforts often remain passive and inflexible. Alongside these issues, the emergence of the parallel healthcare system has the potential to solve these problems by unlocking the data value. The parallel healthcare system comprises Medicine-Oriented Operating Systems (MOOS), Medicine-Oriented Scenario Engineering (MOSE), and Medicine-Oriented Large Models (MOLMs), which could collect, circulate, and empower data. In this paper, we propose that achieving equilibrium in medical resource allocation is possible through parallel healthcare systems via data empowerment. The supply-demand relationship can be balanced in parallel healthcare systems by (1) increasing the supply provided by digital and robotic doctors in MOOS, (2) identifying individual and potential demands by proactive diagnosis and treatment in MOSE, and (3) improving supply-demand matching using large models in MOLMs. To illustrate the effectiveness of this approach, we present a case study optimizing resource allocation from the perspective of facility accessibility. Results demonstrate that the parallel healthcare system could result in up to 300% improvement in accessibility.Index Terms-parallel healthcare system, parallel intelligence, equilibrium, medical resource allocation,
I. INTRODUCTION
The impact of urbanization and population aging on healthcare services is becoming increasingly apparent, marked by the great gap between the supply and demand of medical resources, particularly severe in developing countries [1], [2]. Firstly, the gap is caused by limited medical resource supply due to insufficient medical staff, equipment, and facilities [1], [2]. For instance, over 20% of countries lack physicians to provide hypertension patients with at least one annual medical consultation [1]. Additionally, the gap is exacerbated † for co-first author, * for corresponding author by mismatches between supply and demand, such as redundant diagnostic procedures and imbalanced medical facility layouts. Consequently, healthcare quality has declined, and doctorpatient relationships have become strained, highlighting the need for a healthcare system reform [3].
To alleviate the medical resource gap, governments and academic communities conduct substantial efforts from different perspectives. Governments increase healthcare expenditure to increase the supply, including the expansion of healthcare institutions and procurement of medical equipment. For example, China issued the "Health China 2030" plan to establish a comprehensive medical service system that covers urban and rural areas [4]. Academic communities conducted numerous studies focusing on optimizing the supply and allocation of medical resources, such as healthcare facility layout optimization, outpatient service procedures design, and medical diagnostic model construction [5]- [9]. However, certain challenges still remain unresolved. One issue is that most optimization methods tend to be passive in nature, which are primarily designed to respond and take action only when a problem arises [10]. These passive approaches often fail to address potential issues proactively or utilize systematic and well-planned strategies [10]. Furthermore, the efforts often concentrate on optimizing specific aspects without considering the need for achieving a global equilibrium by simultaneously adjusting the supply, demand, and matching processes. These challenges underscore the need for healthcare system reforms that can enhance the fairness and efficiency of healthcare services.
Alongside these problems and needs, there are great opportunities with a surge of new intelligent technology facilities, exemplified by Artificial Intelligence(AI). Conversational AI, such as ChatGPT, offers opportunities to enhance the supply of medical resources via preprocessing information and transforming communication forms [11]. New foundational models, such as Graph Transformers, make it possible to improve resource scheduling via automatically identifying areas for [11], [12]. Scenario Engineering ensures the trustworthiness of AI techniques, and Cyber-Physical-Social Systems (CPSS) provide insight into transforming traditional hospital information systems into Medicine-Oriented Operating Systems [13]- [19]. These burgeoning technologies support the parallel healthcare systems, which consist of 5% biological doctors, 15% robotic doctors, and a substantial 80% digital doctors [18]- [24]. The parallel healthcare system will have capabilities for describe (personalized demand identification), predict (potential demand prediction), and prescribe (intelligent healthcare supply). It can offer a pathway to the medical resource equilibrium by boosting the medical resource supply, as shown in Fig. 4. In this paper, we suggest that the equilibrium of medical resource allocation can be attained via the parallel healthcare system, and this optimization is essentially achieved via data empowerment. By collecting, circulating, and empowering large-scale medical data within parallel healthcare systems, we can unlock the value of data. The parallel healthcare systems aim to balance the supply-demand relationship through three key aspects: (1) understanding demands via describing and predicting in proactive diagnosis; (2) boosting supply via conducting treatments by digital and robotic doctors; (3) enhancing supply-demand matching using novel foundational models. By implementing parallel healthcare systems, medical resource allocation could shift from passive and fixed processes to active and asynchronous support, from symptom alleviation to health improvement. This transformation ultimately leads to a fairer and more efficient allocation of resources [18].
II. FRAMEWORK, COMPOSITION, AND SCENARIOS OF PARALLEL HEALTHCARE SYSTEM
The essence of achieving medical resource equilibrium through parallel healthcare systems is the empowerment of data value. The parallel healthcare system encompasses Medicine-Oriented Large Models (MOLMs), Medicine-Oriented Scenario Engineering (MOSE), and Medicine-Oriented Operating Systems (MOOS). In this chapter, we will use AI technologies to incentive data circulation, utilization, and analysis in healthcare systems, thereby unlocking the value of data in optimizing medical resources.
A. Framework of Parallel Healthcare System
The framework of the parallel healthcare system, as shown in Fig. 2, reveals significant transformations in demand acquisitions and supply of medical resources, occurring in a parallel, concurrent, and asynchronous way. Serving as the system's gateway, MOOS acts as the backbone for efficient coordination and seamless interactions, optimizing resource allocation and decision-making processes. MOSE plays a crucial role in decomposing complex tasks into achievable scenarios and processes, establishing robust workflows for all medical tasks. MOLMs contribute a lot by supporting the MOOS and MOSE meanwhile continuously being trained using aggregated data records generated by MOOS and MOSE. These components of the parallel healthcare system will be elaborated upon.
B. Medicine-oriented operating system
The efficient coordination of Medical resource supply and demand requires a sophisticated infrastructure that can orchestrate various tasks and processes. At the highest level, the Medicine-Oriented Operating System (MOOS) serves as the backbone of the parallel healthcare system. MOOS organizes and optimizes the dynamic exchange of resources, facilitating seamless interactions between patients and doctors. By leveraging AI technologies, MOOS facilitates the doctor and patient virtualization, concurrent communication, and the persistence of knowledge, thus optimizing resource allocation, decisionmaking, and overall system efficiency.
1) Virtualization: The medical process in MOOS can be transformed into a virtualized form. MOOS integrates and analyzes large-scale medical data, establishing digital patient and doctor portraits. In addition, MOOS offers automatic and asynchronous approaches based on digital portraits.
2) Concurrency: Healthcare experts and patients can interact across time and space. On one hand, MOOS concurrently employs AI technologies to acquire demands. On the other hand, MOOS automatically, asynchronously, and concurrently provides support based on these demands.
3) Persistence: MOOS digitally and persistently stores a vast amount of experts' knowledge and experience, including medical literature and case databases. Through intelligent organization and management, it supports healthcare experts in their decision-making and diagnostic processes.
The characteristics of MOOS enhance the efficiency and flexibility of healthcare, achieving the functionalities of demand acquisition and supply provision through three distinct modes: Autonomous Mode (AM), Parallel Mode (PM), and Expert Mode (EM). In Autonomous Mode, MOOS facilitates efficient and autonomous resource allocation by autonomously acquiring medical demands and providing healthcare supplies based on predefined goals and constraints. In Parallel Mode, MOOS has the capability to enhance decision-making and improve the quality of diagnosis and treatment by employing a collaborative approach where artificial intelligence collaborates with human doctors. Moreover, Expert Mode empowers experts to analyze complex medical demands and refine resource allocation through the integration of innovative algorithms. As a vital component of parallel healthcare systems, the development of MOOS by combining AM, PM, and EM modes lays the foundation for a more efficient and sustainable allocation of medical resources.
C. Medicine-oriented scenario engineering
To effectively address the diverse demands and supply of medical resources, Medicine-Oriented Scenario Engineering (MOSE) plays a crucial role [25]. MOSE decomposes complex tasks into achievable scenarios and processes, ensuring compatibility with predefined criteria. By carefully designing and validating these scenarios, MOSE establishes a robust framework for resource allocation and optimization [16], [25], [26].
1) Identification and Interpretation (I&I): MOOS could identify and interpret scenarios from patients' preferences, historical, and real-time data that align with individual demands and system requirements.
2) Calibration and Certification (C&C): MOSE could design and develop algorithms and models that can calibrate and certify according to abstract scenarios, catering to diverse medical settings. It enables efficient allocations of healthcare resources.
3) Verification and Validation (V&V): To validate the capabilities and feasibility, MOSE employs real or simulated medical data and scenarios for experiments. The outcomes derived from Verification and Validation can be utilized for correction, guaranteeing the system's accuracy of healthcare resource allocation challenges in practical applications.
D. Medicine-Oriented Large Models
Underlying the MOOS and MOSE framework are Medicine-Oriented Large Models (MOLMs). MOLMs encompass comprehensive representations of the medical system, incorporating vast amounts of data, behavioral patterns, and domain knowledge. Using language models, diagnostic models, and decision models as examples, these large-scale models are trained using aggregated data records generated by MOOS and MOSE. Through continuous learning and refinement, MOLMs contribute to adaptability and responsiveness.
1) Large language models: Language Large Models (LLMs) can capture the intricate dynamics of resource supply and demand through natural language processing and machine learning techniques [11], [27]. By doing so, it transforms the patient-doctor communication from a 'black box (Human)-black box' interaction to a 'white box (LLMs)-black box' interaction [28]. LLMs can also integrate multiple data sources, characterized by a high initial slope of the cumulative value function. In the new communication form, LLMs could leverage history data, provide real-time responsiveness, and integrate multiple data sources from other MOLMs. It results in highly efficient communication from the very beginning, ultimately enhancing the efficiency of healthcare resource allocation, as shown in Fig. 3.
2) Large diagnosis models: Large diagnosis models facilitate the supplementation of supply and the clarification of demand, thereby breaking down the barriers between the field of healthcare and other domains through the integration of diverse knowledge. Large diagnosis models can analyze and identify patterns and correlations within the data and offer diagnostic support.
3) Large optimization models: Large optimization models have the potential to enhance resource allocation by leveraging machine learning, deep learning, and related techniques for handling and analyzing the supply and demand of healthcare resources [29]. For instance, models such as Graph Transformers and pricing mechanisms can automatically optimize resource scheduling patterns [30].
E. Resource allocation scenarios
In practical applications, resource allocation can be categorized into three types: macro, meso, and micro, based on the corresponding relationship between supply and demand, as well as optimization objectives. Constructing optimization models according to these categories enables effective resolution of specific resource allocation optimization problems.
Micro resource allocation problems often involve situations where demand exceeds supply, such as hospital bed allocation, management of medication, and medical equipment allocation.
Meso resource allocation problems typically involve situations where there are multiple demands and multiple supplies. Examples include operating room scheduling, doctor's outpatient scheduling, as well as optimization of outpatient processes.
Macro resource allocation problems usually involve multiobjective optimization with multiple demands and multiple supplies, such as city-level allocation of medical resources and optimization of hospital locations.
By integrating MOOS, MOSE, and MOLMs, parallel healthcare systems can collect, process, and analyze largescale medical data. Through the application of AI technologies, the value of data in medical resources can be unlocked. The supply-demand curve of medical resources will undergo changes as shown in Fig. 4. The shifted demand-supply curves are caused by (1) the increasing supply in parallel healthcare systems provided by robotic doctors and digital doctors, and (2) changed demand with accurate description and prediction of potential demand, which could increase or decrease the demand. Apart from supply and demand, there have been significant transformations in the supply-demand matching of medical resources. These transformations reflect in several aspects: (1) a shift from passive to proactive medical support, (2) a departure from fixed linear processes to asynchronous concurrent support, and (3) a transition of medical knowledge from transient to persistent, which pave ways to achieve equilibrium and balance in the supply-demand relationship of medical resources.
III. APPLICATION OF PARALLEL HEALTHCARE SYSTEM FOR MEDICAL RESOURCE ALLOCATION
In this paper, we present a case study about accessibilityoriented optimization of healthcare facilities in the parallel healthcare system, as shown in Fig. 2. The parallel healthcare system could ensure the expansion and optimization of healthcare facilities, supported by digital doctors and robotic doctors.
A. Accessibility definition and objectives
The accessibility of healthcare resources can be obtained by computing the attributes of healthcare resource demand and supply in spatial dimensions, using the Two-step Floating Catchment Area (2SFCA) method. Fig. 4: Parallel healthcare systems could support high-quantity and low-price services due to the shifted demand-supply curves of medical resources. (1) The supply is increased in parallel healthcare systems due to the additional treatment provided by robotic doctors and digital doctors. (2) The demand is changed because parallel healthcare systems could describe and predict potential demand, which could increase or decrease the demand value The accessibility A i based on the 2SFCA method is formulated as:
A i = j∈t≤tσ γG(t ij )S j n∈{tnj ≤tσ} G(t ij )D n (1a) where G(t ij ) = e − 1 2 × t ij tσ 2 − e − 1 2 1 − e − 1 2 (1b)
where G(t ij ) is the time-decay function, quantifying the decay in accessibility as the travel time t ij between locations i and j increases; S j is the number of healthcare facilities at location j; γ is the parameter that adjusts the weight of the supply-demand ratio; t σ is a travel time threshold to determine whether locations j should be considered; n is the index representing the district is in the area where its travel time t nj ≤ t σ ; D n is the volume of demand represented by population within the catchment area of location. To achieve a balance between accessibility and fairness in healthcare facilities, the objective function is set as2:
min α · k + β · m i=1 (∆A i ) 2 s.t. ∀i ∈ m A i ≥ A σ(2)
where α and β are weights in the objective function; m is the number of demand locations; ∆A i = |A i − A σ | in which A σ is the target accessibility and A i is the accessibility in location i.
B. Experiment Setting
We conduct the case study in the Gulou District, Nanjing. The district spans an area of 54.18 km 2 and accommodates a population exceeding 900,000 individuals. We take the population as a representative of the medical resource demand, and the data on population distribution were obtained on April 4, 2023. Previous COVID test huts are regarded as available medical resource supplies. According to a survey conducted on March 9, 2023, there are currently 16 healthcare facilities in place. These huts can be repurposed as healthcare facilities to enhance the supply of medical resources.
In this study, we set target accessibility A σ = 0.135, weights in objective function α, and β as 1. The capability of the healthcare facility of supplying is set as 1500 individuals per day. In addition, we determine that the average walking speed for the general public is 80m per minute, with a maximum walking distance of 700m [31]. For the elderly, the average walking speed is 70m per minute, with a maximum walking distance also limited to 700m [31]. By utilizing the road network data from OpenStreetMap the accessibility can be computed.
C. Evaluation results
Based on the aforementioned method and objective function, in the parallel healthcare system, we could achieve both accessibility and fairness of healthcare facilities layout by only The parallel healthcare system further gives out an optimized facility layout. Comparisons of accessibility results are demonstrated in Fig. 6, revealing an overall optimization of accessibility. Before optimizations, only 3.65% of residents have a very high level of accessibility, indicating a deficiency in fairness, as indicated by Fig. 6. Following the establishment of 15 healthcare facilities using parallel healthcare systems, the overall coverage for the general public notably increased to over 95%. The range of highly accessible areas expanded significantly, with up to a 300% increase in coverage for populations above medium accessibility.
Regarding the elderly population, their overall accessibility level was considerably lower before optimization. The coverage was limited, with only approximately 7.27% of elderly individuals able to reach healthcare facilities within 10 minutes, as illustrated in Fig. 6. The prevalence of low accessibility was notably 70.47%, revealing imbalanced resource allocation. After optimization, there was a significant improvement with overall coverage increased to 95% or higher.
In summary, significant improvements have been achieved in the overall accessibility for both the general public and the elderly. These improvements address parallel healthcare systems' potential of approaching medical resources equilibrium.
IV. CONCLUSION AND FUTURE WORKS
In this paper, we discuss the parallel healthcare system, which presents a promising path forward equilibrium in medical resource allocation via data empowerment. By analyzing the supply and demand of medical resources and leveraging AI technologies, we proposed a parallel healthcare system framework to balance the supply-demand relationship.
Through the implementation of parallel healthcare systems, which include Medicine-Oriented Operating Systems (MOOS), Medicine-Oriented Scenario Engineering (MOSE), and Medicine-Oriented Large Models (MOLMs), we can boost resource supply, optimize allocation, predict demand, and improve medical resource allocation, leading to a more efficient and social-centric healthcare system. An application of medical facilities optimization demonstrated a threefold improvement in accessibility, showcasing the potential of parallel healthcare systems to optimize medical resource allocation.
While parallel healthcare systems offer a promising approach to optimizing medical resource allocation, there is still much work left for the future. (1) Constructing MOOS that aligns with practical application requirements (2) Integrating MOSE with domain knowledge in medicine to facilitate smoother workflows. (3) Exploring solutions to privacypreserving problems when using large models in MOLMs. We expect the community to delve deeper into this field, extending and refining its framework and applications. Through continuous exploration and innovation in parallel healthcare systems, we can unlock its full potential and establish a more equitable and efficient healthcare system worldwide.
Fig. 1 :
1Differences of volume and proportion in the medical resource supply.
Fig. 2 :
2Parallel healthcare systems composed of Medicine-Oriented Large Models (MOLMs), Medicine-Oriented Scenario Engineering (MOSE), and Medicine-Oriented Operating Systems (MOOS).
Fig. 3 :
3Cumulative value functions in conversations: In humanhuman interactions, communication involves incremental information disclosure, which requires a substantial amount of low-value information. In human-machine (white box -black box) interactions, communication can directly address value information based on records.
Fig. 5 :
5Application workflow of parallel healthcare systems for optimizing medical resource allocation
Fig. 6 :
6Population proportions of different accessibility coverage levels for healthcare facilities constructing 15 additional healthcare facilities, which means a total of 31 healthcare facilities.
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[] |
[] |
[
"Fangjian Lin \nShanghai AI Laboratory\nShanghai China\n\nUniversity of Chinese Academy of Sciences\n\n",
"Yizhe Ma \nShanghai AI Laboratory\nShanghai China\n",
"Sitong Wu ",
"Long Yu \nShanghai AI Laboratory\nShanghai China\n",
"Shengwei Tian \nShanghai AI Laboratory\nShanghai China\n",
"\nSchool of Software\nXinjiang University\nUrumqiChina\n"
] |
[
"Shanghai AI Laboratory\nShanghai China",
"University of Chinese Academy of Sciences\n",
"Shanghai AI Laboratory\nShanghai China",
"Shanghai AI Laboratory\nShanghai China",
"Shanghai AI Laboratory\nShanghai China",
"School of Software\nXinjiang University\nUrumqiChina"
] |
[] |
Recently Transformer has shown good performance in several vision tasks due to its powerful modeling capabilities. To reduce the quadratic complexity caused by the attention, some outstanding work restricts attention to local regions or extends axial interactions. However, these methos often lack the interaction of local and global information, balancing coarse and fine-grained information. To address this problem, we propose AxWin Attention, which models context information in both local windows and axial views. Based on the AxWin Attention, we develop a context-aware vision transformer backbone, named AxWin Transformer, which outperforming the state-of-the-art methods in both classification and downstream segmentation and detection tasks.
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10.48550/arxiv.2305.01280
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[
"https://export.arxiv.org/pdf/2305.01280v1.pdf"
] | 258,436,739 |
2305.01280
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84499860fe6a8d1be31c535be69a3ec38ce9afe9
|
Fangjian Lin
Shanghai AI Laboratory
Shanghai China
University of Chinese Academy of Sciences
Yizhe Ma
Shanghai AI Laboratory
Shanghai China
Sitong Wu
Long Yu
Shanghai AI Laboratory
Shanghai China
Shengwei Tian
Shanghai AI Laboratory
Shanghai China
School of Software
Xinjiang University
UrumqiChina
AXWIN TRANSFORMER: A CONTEXT-AWARE VISION TRANSFORMER BACKBONE WITH AXIAL WINDOWSIndex Terms-Transformer, Backbone
Recently Transformer has shown good performance in several vision tasks due to its powerful modeling capabilities. To reduce the quadratic complexity caused by the attention, some outstanding work restricts attention to local regions or extends axial interactions. However, these methos often lack the interaction of local and global information, balancing coarse and fine-grained information. To address this problem, we propose AxWin Attention, which models context information in both local windows and axial views. Based on the AxWin Attention, we develop a context-aware vision transformer backbone, named AxWin Transformer, which outperforming the state-of-the-art methods in both classification and downstream segmentation and detection tasks.
INTRODUCTION
Recently, Transformer has shown remarkable potential in computer vision. Since Dosovitskiy et al. [1] proposed Vision Transformer (ViT), the design of the Attention module has become one of the main research hotspots. Several works [2,3,4,5,6,7,8] have achieved high accuracy on classification tasks, but the performance on downstream tasks, especially dense prediction tasks, has not been the same. For dense prediction tasks that include complex scene changes, there is required to have two properties for the backbone: 1. long-range global modeling capability. 2. excellent local information extraction capability. The former not only models rich context information but also obtains higher shape bias (i.e., Some work [9,10] has demonstrated that shape bias is critical for downstream tasks). the latter enables the model to focus on the key regions of the feature map. But, balancing these two aspects is a very challenging task.
For the global modeling capability of the attention, a classical representation is Vision Transformer (ViT) [1]. As shown in Figure 1 (a), it can model every pixel in an image. However, for downstream tasks where images have high † Corresponding author. resolution, the quadratic complexity of global self-attention is unbearable on the one hand, and it lacks local modeling capability on the other. One way to solve the quadratic complexity of global self-attention is to divide the global image into multiple local regions, as shown in Figure 1 (b), Swin Transformer [7] reduces the computational complexity to a tolerable level and enables the model to focus its attention on the regions inside each window. Although its shift transform can expand the receptive fields, this operation is not sufficient for global dependencies. Axial-based methods are a more friendly choice for downstream tasks, where axial attention can obtain higher shape bias and greatly reduce computational complexity. As shown in Figure 1 (c), such as CSWin Transformer [8] or Pale Transformer [11], global context interactions are constructed by alternating rows and columns. However, as the image resolution increases in the downstream task, the blank portion of the row and column alternation increases, which may lack some critical information in the image. In short, axial-based methods lack the local modeling capability to capture effective context information.
In this work, we propose an axial window of self-attention to solve the above problem, window region for focusing on local features and axial region for modeling global features to obtain higher shape bias and capture richer context dependencies. In addition, we devise a Multi-Scale Patch Embedding (MSPE) downsampling strategy to enrich the context information of high-resolution images. Finally, we slightly improve the classical MLP architecture with a built-in depthwise convolution operation to enhance the local bias. The architecture of the whole network and the details of each module are shown in Figure 2.
RELATED WORK
Recent Vision Transformer backbones focus on two main aspects: (1) Enhanced local modeling capabilities. (2) Efficient global attention implementation.
Windows-Based Attention. The classical ViT architecture uses global attention and lacks local inductive bias. Swin Transformer [7] enhances the ability to extract local information and greatly reduces the computation of self-attention by confining the attention within the window. Consequently, T2T-ViT [12], Shuffle Transformer [13] facilitates the development of Window-based attention by enhancing local connections across windows. Efficent Global Attention. CSWin [8] enhances global context awareness and image shape bias by using axial-based attention to establish global connections across rows and columns. Pale Transformer extends attention to multiple rows and columns, balancing performance and efficiency.
Different from the above two attention mechanisms, our attention module integrates both local and global attention to overcome the problem of insufficient local information extraction and limited global representation.
METHOD
In this section, we first show the Multi-Scale Patch Embedding (MSPE) module. Then we describe the efficient implementation of AxWin Attention. Finally the overall architecture of AxWin Transformer and various variant configurations are shown. Multi-Scale Patch Embedding. In order to capture multiscale context information, we propose the MSPE module. As shown in Figure 2(a), given an input feature map X ∈ R h×w×c , the output feature map Y ∈ R h 2 × w 2 ×2c . Corresponding to stages 1-4, the number of branches in MSPE is 4-1. For branch i, i light-weight 3×3 depth-wise convolutions with stride = 2 are performed, the output feature map X i ∈ R h 2 i × w 2 i ×2c . For branch i and i+1, A top-down connect operation is used to fuse multi-scale features (i.e., bilinear interpolation to upsample the low-scale feature map), followed by a 3×3 depth-wise convolution with stride = 1 and a 1×1 convolution. Axial Window Self-Attention. In order to capture both fine-grained local features and coarse-grained global information, we propose the Axial Window Self-Attention (AxWin Attention), which computes self-attention within an axial window region. As shown in the green shadow of Figure 2(d), given an input feature X ∈ R h×w×c , first the fully connected layer is used to perform the mapping of X to generate the Query (X q ), Key (X k )) and Value (X v ). Then {X q , X K , X v } is divided into window group {X wq , X wk , X wv } ∈ R h×w× c 2 and axial group {X aq , X ak , X av } ∈ R h×w× c 2 according to channel dimension. For the window group, the matrix {X wq , X wk , X wv } is split in a non-overlapping manner [7], then perform multihead self-attention. For the axial group, refer to previous work [11], We rearrange {X aq , X ak , X av } into two separate regions of rows {X r aq , X r ak , X r av } ∈ R sar×w× c 4 and columns {X c aq , X c ak , X c av } ∈ R h×sac× c 4 . Here s ar = s ac and indicates how many alternating rows and columns. We perform the connection operation along channel dimension after the division is complete, we get {X rc aq , X rc ak , X rc av } to perform multi-head self-attention (MHSA) [1].
X rc = M HSA(X rc aq , X rc ak , X rc av ),(1)X w = M HSA(X wq , X wk , X wv ),(2)X = Concat(X rc ,X w ).(3)
AxWin Transformer Block. As shown in Figure 2(c), there [14] is used to dynamically generate implicit position embedding, our AxWin Attention is used to capture local and global context information, the proposed ICFFN module is based on the MLP module (i.e., consists of two fully connected layers) with a 3x3 depth-wise convolution to add local information extraction capability for feature projection. The forward process is as follows:
X i = X i−1 + CPE(X i−1 ),(4)X i =X i + AxWin Attention LN(X i ) ,(5)X i =X i + ICFFN LN(X i ) .(6)
We use layer normalization (LN) for feature normalization.
Overall Architecture and Variants. Our AxWin Transformer consists of a stem layer, four hierarchical stages, and a classifier head. As shown in Figure 2 (a), the stem layer [15] (i.e., a 3×3 convolution layer with stride = 2 and two 3×3 convolution layers with stride = 1) makes the output features smoother. After the stem, each stage contains a MSPE module and multiple AxWin Transformer blocks. The final classifier head is a linear layer. There are three different variants, including AxWin-T (Tiny), AxWin-S (small), and AxWin-B (base), whose detailed configurations are shown in Table 1. The above variants differ primarily in the channel dimension and the number of heads.
EXPERIMENTS
We first compare our AxWin Transformer with the state-ofthe-art methods on ImageNet-1K [16] for image classification. To demonstrate the generalization of our method, we performed experiments on several downstream tasks, including ADE20k [17] for semantic segmentation, COCO [18] for object detection, and instance segmentation. Finally, we give the analysis of ablation studies for each module. Image Classification on ImageNet-1K. Table 2 compares the performance of our AxWin Transformer with the state-ofthe-art methods on ImageNet-1K validation set. Our method boosts the top-1 accuracy by an average of 1.5% for all variants compared to the most relevant sota methods. Object Detection and Instance Segmentation on COCO. As shown in Table 3, for object detection, our method average improves 3.2% box AP. For instance segmentation, AxWin Transformer average improves by 2.4% mask AP. Also as the input resolution increases, the average FLOPs of our method decrease by 68G. Semantic Segmentation on ADE20K. The results on ADE20K dataset are shown in Table 5. Compared to other methods, our AxWin Transformer params and FLOPs decrease more on average as the image resolution increases. Meanwhile, the performance of our single-scale mIoU and multi-scale mIoU is improved by 2.7% and 2.5% respectively. Ablation Study. Table 6 compares the different attention modes and shows that our Axwin attention achieves excellent results. Table 4 demonstrates the benefits of the MSPE module, bringing performance gains with only a small increase in computation. Table 7 shows the performance of different split size, for tiny, small, and base models, the split size is 7, 7 and 12 respectively.
CONCLUSION
This work proposes an efficient self-attention mechanism, called AxWin Attention, which models both local and global context information. Based on AxWin Attention, we develop a context-aware vision transformer backbone, called AxWin Transformer, which achieves the state-of-the-art performance in ImageNet-1k image classification and outperforms previous ones in ADE20k semantic segmentation and COCO object detection and instance segmentation methods.
Fig. 1 :
1Comparison with different self-attention mechanisms in Transformer backbones. The blue area indicates performing attention operation. (a) is the standard global selfattention. (b) is the window-based self-attention. It restricts the computation of attention to the inside of each window. (c) is the Axis-based self-attention. It expands the receptive fields by alternating rows and columns. (d) Ours, it expands the receptive fields by alternating single rows and columns, and adds windows to focus on local features.
Fig. 2 :
2(a) The overall architecture of our AxWin Transformer. (b) Illustration of Multi-Scale Patch Embedding downsampling strategy. (c) The composition of each block. (d) The specific implementation of AxWin Attention. (e) Structure of Inner Convolution Feed-Forward Network.
Table 1 :
1Detailed configurations of AxWin Transformer variants. P i means the spatial reduction factor. C i is the channel dimension. H i , S i and R i represent the number of heads, the split-size (i.e., window size and row-column size) for AxWin-Attention, and the expand ratio in ICFFN.Stage/Stride
Layer
AxWin-T
AxWin-S
AxWin-B
Stride=2
Stem
P0 = 2, C0 = 32
P0 = 2, C0 = 48
P0 = 2, C0 = 56
Stage 1
Stride=4
MSPE
P1 = 2, C1 = 64
P1 = 2, C1 = 96 P1 = 2, C1 = 112
AxWin
Block
H1=2
S1=7
R1=4
× 2
H1=2
S1=7
R1=4
× 2
H1=4
S1=12
R1=4
× 2
Stage 2
Stride=8
MSPE
P2 = 2, C2 = 128 P2 = 2, C2 = 192 P2 = 2, C2 = 224
AxWin
Block
H2=4
S2=7
R2=4
× 2
H2=4
S2=7
R2=4
× 2
H2=8
S2=12
R2=4
× 2
Stage 3
Stride=16
MSPE
P3 = 2, C3 = 256 P3 = 2, C3 = 384 P3 = 2, C3 = 448
AxWin
Block
H3=8
S3=7
R3=4
× 17
H3=8
S3=7
R3=4
× 17
H3=16
S3=12
R3=4
× 17
Stage 4
Stride=32
MSPE
P4 = 2, C4 = 512 P4 = 2, C4 = 768 P4 = 2, C4 = 896
AxWin
Block
H4=16
S4=7
R4=4
× 2
H4=16
S4=7
R4=4
× 2
H4=32
S4=12
R4=4
× 2
are three main modules in our AxWin Transformer block:
the conditional position encoding (CPE), AxWin Attention
and Inner Convolution Feed-Forward Network (ICFFN). The
CPE
Table 2 :
2Comparisonsof different backbones on Ima-
geNet1K validation set. Avg-improve represents the average
performance improvement per variant (T, S, B). Flops are cal-
culated with the resolution of 224×224.
Method
Params FLOPs Top-1 Acc. (%)
PVT-S [2]
25M
3.8G
79.8
Swin-T [7]
29M
4.5G
81.3
CSWin-T [8]
23M
4.3G
82.7
Pale-T [11]
22M
4.2G
83.4
AxWin-T(ours)
22M
3.5G
83.9
PVT-M [2]
44M
6.7G
81.2
Swin-S [7]
50M
8.7G
83.0
CSWin-S [8]
35M
6.9G
83.6
Pale-S [11]
48M
9.0G
84.3
AxWin-S(ours)
48M
7.6G
84.6
Swin-B [7]
88M
15.4G
83.3
CSWin-B [8]
78M
15.0G
84.2
Pale-B [11]
85M
15.6G
84.9
AxWin-B(ours)
84M
12.7G
85.1
Table 3 :
3Comparisonof different backbones on COCO
val2017 using Mask R-CNN framework, and 1x training
schedule for object detection and instance segmentation.
Flops are calculated with a resolution of 800×1280.
Table 4 :
4Ablation study for different downsampling manner. Flops are calculated with the resolution of 512×512, note that only the down-sampling modules are tested here.Down-sampling
ImageNet-1K ADE20K
Top-1 acc
SS mIoU
Params GFLOPs
Patch Merging[1]
83.8
51.0
0.7M
0.6
MSPE (ours)
83.9
51.3
0.7M
1.1
Table 5 :
5Comparisons of different backbones with UperNet as decoder on ADE20K for semantic segmentation. FLOPs are calculated with a resolution of 512 × 2048.Backbone
Params FLOPs mIoU(SS) mIoU(MS)
Swin-T [7]
60M
945G
44.5
45.8
CSWin-T [8]
60M
959G
49.3
50.4
Pale-T [11]
52M
996G
50.4
51.2
AxWin-T (ours)
52M
910G
51.3
52.2
Swin-S [7]
81M
1038G
47.6
49.5
CSWin-S [8]
65M
1027G
50.0
50.8
Pale-S[11]
80M
1135G
51.2
52.2
AxWin-S (ours)
80M
995G
52.0
52.9
Swin-B [7]
121M
1188G
48.1
49.7
CSWin-B [8]
109M
1222G
50.8
51.7
Pale-B[11]
119M
1311G
52.2
53.0
AxWin-B (ours)
97M
1050G
52.8
53.7
Table 6 :
6Ablation study for different attention modes.Attention
ImageNet-1K ADE20K
COCO
Top-1 acc
SS mIoU AP box AP mask
Axial
83.0
48.8
46.9
41.8
Window
82.6
47.6
45.6
40.9
AxWin (ours)
83.9
51.3
48.2
43.4
Table 7 :
7Ablation study for different choices of split size. The padding operation is performed when the image length and width cannot be divided.split-size
in four stages
ImageNet-1K
ADE20K
COCO
Top-1 (%)
SS mIoU (%) AP box AP mask
3 3 3 3
83.4
49.8
47.5
42.8
5 5 5 5
83.6
50.0
47.8
43.2
7 7 7 7
83.9
51.3
48.2
43.4
9 9 9 9
83.8
51.2
48.0
43.1
12 12 12 12
83.9
51.5
48.3
43.5
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|
[] |
[
"Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces",
"Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces"
] |
[
"Florian Fischer \nInstitute of Mathematics\nUniversity of Potsdam\nGermany\n",
"Norbert Peyerimhoff [email protected] \nDepartment of Mathematical Sciences\nDurham University\nGreat Britain, MSC 2020: 26D1031C12, 58J60\n",
"Berchio / Ganguly ",
"/ Grillo ",
"Berchio / Ganguly ",
"/ Grillo ",
"/ Pinchover "
] |
[
"Institute of Mathematics\nUniversity of Potsdam\nGermany",
"Department of Mathematical Sciences\nDurham University\nGreat Britain, MSC 2020: 26D1031C12, 58J60"
] |
[] |
We show various sharp Hardy-type inequalities for the linear and quasi-linear Laplacian on non-compact harmonic manifolds with a particular focus on the case of Damek-Ricci spaces. Our methods make use of the optimality theory developed by Devyver/Fraas/Pinchover and Devyver/Pinchover and are motivated by corresponding results for hyperbolic spaces by
| null |
[
"https://export.arxiv.org/pdf/2305.01288v1.pdf"
] | 258,436,911 |
2305.01288
|
2391f2c198584738f7b5e3d837d87c8acf36e621
|
Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces
2 May 2023 May 3, 2023
Florian Fischer
Institute of Mathematics
University of Potsdam
Germany
Norbert Peyerimhoff [email protected]
Department of Mathematical Sciences
Durham University
Great Britain, MSC 2020: 26D1031C12, 58J60
Berchio / Ganguly
/ Grillo
Berchio / Ganguly
/ Grillo
/ Pinchover
Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces
2 May 2023 May 3, 2023Harmonic manifoldsDamek-Ricci spacesoptimal Hardy inequalitiesPoincaré-Hardy inequalitiesp-Laplacians
We show various sharp Hardy-type inequalities for the linear and quasi-linear Laplacian on non-compact harmonic manifolds with a particular focus on the case of Damek-Ricci spaces. Our methods make use of the optimality theory developed by Devyver/Fraas/Pinchover and Devyver/Pinchover and are motivated by corresponding results for hyperbolic spaces by
Introduction
The aim of this article is to derive Hardy-type inequalities for the Laplacian on non-compact harmonic manifolds and, in particular, Damek-Ricci spaces, and to discuss applications in the same spirit as in Devyver/Fraas/Pinchover [18], Devyver/Pinchover [19], Berchio/Ganguly/Grillo [8] and Berchio/Ganguly/Grillo/ Pinchover [9]. The two latter paper focus mainly on the real hyperbolic space H n = H n (R) with n ≥ 3. The starting point is the classical Euclidean Hardy inequality
R n |∇φ(x)| 2 dx ≥ (n − 2) 2 4 R n φ 2 (x) |x| 2 dx
for all smooth and compactly supported functions φ ∈ C ∞ c (R n ) and n ≥ 3 (see, e.g., [6,Cor. 1.2.6] for p = 2). This inequality was generalised by G. Carron [13,Prop. 2.1] to arbitrary n-dimensional Cartan-Hadamard manifolds X (that is, complete and simply connected Riemannian manifolds (X, g) of non-positive curvature) with |x| 2 replaced by d(o, x) 2 , where o ∈ X is an arbitrary point (pole) and d is the Riemannian distance function on X.
One of the main results in [8] is the following Poincaré-Hardy inequality for the n-dimensional real hyperbolic spaces H n = H n (R), which can be seen as a variation of Carron's Hardy inequality for Cartan-Hadamard manifolds with an additional term in this special case (see [8,Theorem 2.1]): Given n ≥ 3 and any point o ∈ H n , we have
H n |∇φ(x)| 2 dx ≥ λ 0 (H n ) H n φ 2 (x)dx + 1 4 H n φ 2 (x) d(o, x) 2 dx + (n − 1)(n − 3) 4 H n φ 2 (x) sinh 2 d(o, x) dx (1.1)
for all functions φ ∈ C ∞ c (H n ). Here λ 0 (H n ) denotes the bottom of the spectrum of the (positive) Laplacian −∆ = − div •∇ on H n , which is known to be (n−1) 2 4 . If we disregard the extra term (n−1)(n−3) 4 H n φ 2 (x) sinh 2 d(o,x) dx, the variation can be understood to include an additional L 2 -norm φ 2 H n with factor λ 0 (H n ) at the expense of the constant in front of the "Hardy" term H n φ 2 (x) d(o,x) 2 dx. Moreover, it is shown in [8] that the constants in inequality (1.1) cannot be improved.
Real hyperbolic spaces H n = H n (R) have constant sectional curvature −1 and are examples of non-compact rank-one symmetric spaces. Besides them, there also exist the (real 2n-dimensional) complex hyperbolic spaces H n (C), the (real 4n-dimensional) quaternionic hyperbolic spaces H n (H) and the (real 16dimensional) Cayley plane H 2 (O) based on the octonians O. These spaces comprise (besides the "simple" flat space R), the class of all non-compact rank-one symmetric spaces, and they are all Cartan-Hadamard manifolds with sectional curvatures within the interval [−4, −1].
All these spaces are also non-compact harmonic manifolds. A spectral geometric characterization of harmonic manifolds is that they are complete Riemannian manifolds on which all harmonic functions φ (that is ∆φ = 0) satisy the Mean Value Property (that is φ(x 0 ) = 1 vol(Sr(x0)) Sr (x0) φ(x)dx for all x 0 ∈ X and spheres S r (x 0 ) of radius r > 0 around x 0 ). It was generally assumed (and referred to as the so-called "Lichnerowicz Conjecture") that all simply connected harmonic manifolds should be either Euclidean spaces or rank-one symmetric spaces. This was proved by Z. I. Szabó [45] in the compact case, and it came as a surprise when E. Damek and F. Ricci discovered in 1992 (see [15,16]) that a whole family of non-compact, non-Euclidean and generally non-symmetric homogeneous Riemannian manifolds were indeed also harmonic manifolds (thus disproving this conjecture in the non-compact case). There spaces where studied before (see [3,4,5,12,16,22,34,36,43,44] for a selection of papers investigating various of their harmonic analytic properties) and they are nowadays called Damek-Ricci spaces. They are solvable extensions N A of 2-step nilpotent groups N of Heisenberg-type (by a one-dimensional abelian group A) with leftinvariant metrics. They are associated with a pair of parameters (p, q) which are the dimensions of particular subspaces of the underlying nilpotent Lie algebra of N . While the parameters (p, q) do not always uniquely determine the Damek-Ricci space we will use for any Damek-Ricci space with these parameters the notation X p,q (by a slight abuse of notation due to this non-uniqueness).
Damek-Ricci spaces X p,q are homogeneous Cartan-Hadamard manifolds of dimension n = p + q + 1, with sectional curvatures in the interval [−1, 0]. The rank-one symmetric spaces H n (K) for the division algebras K = C, H, O areup to scaling of the metric by the constant factor 4 -Damek-Ricci spaces with special choices of the parameters (p, q). In fact, we can write (up to the metric scaling factor 4), H n (C) = X 2(n−1),1 , H n (H) = X 4(n−1),3 , and H 2 (O) = X 8,7 . While the real hyperbolic spaces H n can also be viewed as solvable Lie groups N A with left-invariant metrics, they are not Damek-Ricci spaces since the group N of H n is abelian and not 2-step nilpotent as required for Damek-Ricci spaces. The smallest dimension of a Damek-Ricci space is 4, and the only possible values (p, q) for Damek-Ricci spaces X p,q are given in the following table with a ≥ 0 and b ≥ 1 (see, e.g., [43, p. 64]). The derivation of these values goes back to [23, p. 150] and is based on the representation theory of Clifford algebras. q 8a + 1 8a + 2 8a + 3 8a + 4 8a + 5 8a + 6 8a + 7 8a + 8
p 2 4a+1 b 2 4a+2 b 2 4a+2 b 2 4a+3 b 2 4a+3 b 2 4a+3 b 2 4a+3 b 2 4a+4 b
The Damek-Ricci space of smallest dimension which is not-symmetric is the 7-dimensional space X 4,2 . It was shown by J. Heber [21] that any noncompact homogeneous harmonic manifold must either be Euclidean, a real hy-perbolic space or a (symmetric or non-symmetric) Damek-Ricci space, and it is not known and a challenging open problem whether there exist any further non-compact harmonic manifolds (which must then be necessarily nonhomogeneous).
Our main result is that the Poincaré-Hardy inequality (1.1) for H n has the following generalization to Damek-Ricci spaces with explicitely given constants:
Main Theorem A. Let X p,q be a Damek-Ricci space with a pole o ∈ X p,q and r = d(o, ·). Then we have for all φ ∈ C ∞ c (X p,q ),
X p,q |∇φ| 2 dx ≥ λ 0 (X p,q ) X p,q φ 2 dx + 1 4 X p,q φ 2 r 2 dx + p(p + 2q − 2) 16 X p,q φ 2 sinh 2 (r/2) dx + q(q − 2) 4 X p,q φ 2 sinh 2 (r) dx. (1.2)
Moreover, the constants on the right hand side of this inequality are optimal and can only be improved at the expense of the other constants.
In this paper, we will also discuss two applications of this inequality as well as various variations concerning the constants appearing in this inequality.
Let us briefly discuss the arguments behind this result. They are based on the fact that, in the case of a simply connected non-compact harmonic manifold (X, g) with a pole o ∈ X, the Laplacian ∆ of a radial function h(r) = h(d(o, x)) is again radial and given by
∆h(r) = h ′′ (r) + f ′ (r) f (r) h ′ (r),(1.3)
where f (r) is the volume density of the harmonic manifold. For a general Riemannian manifold (X, g), the volume density f o (x) = det g ij (p) in normal coordinates centered around o ∈ X is not radial and not independent of the center o ∈ X. In a harmonic manifold (X, g), the volume density is a radial function f (r) and all spheres of the same radius have the same volume. We have vol(S r (x)) = ω n f (r),
with ω n being the volume of the unit sphere in R n . Note however, that this does not mean that a harmonic manifold is a Riemannian model as described in [8,Section 4]. In fact, a non-compact harmonic manifold is such a Riemannian model only if it is the Euclidean space R n or the hyperbolic space H n . Inequality (1.2) is a consequence of the following general Hardy-type inequality for arbitrary non-compact harmonic manifolds:
Main Theorem B. Let (X, g) be a non-compact harmonic manifold with volume density f . Let o ∈ X be a pole and r = d(o, ·). Then we have, for all φ ∈ C ∞ c (X),
X |∇φ| 2 dx ≥ 1 4 X φ 2 r 2 dx + 1 4 X 2f (r)f ′′ (r) − (f ′ (r)) 2 f 2 (r) φ 2 dx. (1.4)
Inequality (1.2) is then a consequence of (1.4) by the following explicit expression of the volume density for a Damek-Ricci space X p,q (see, e.g., [43,Théorème 10(ii)]): f (r) = 2 p+q (sinh(r/2)) p+q (cosh(r/2)) q = 2 p (sinh(r/2)) p (sinh r) q .
(1.5)
This implies that
f ′ (r) f (r) = p 2 coth(r/2) + q coth r = p + 2q 2 coth(r/2) − q sinh(r) ,(1.6)
and we have for radial functions h(r) in X p,q ,
∆h(r) = h ′′ (r) + p 2 coth(r/2) + q coth r h ′ (r).
Since coth(x) ≥ 1/x and sinh x ≥ x, it follows from (1.6) for all r > 0 that
f ′ (r) f (r) ≥ p + q r . (1.7)
Moreover, the L 2 -spectrum of the operator −∆ on X p,q is given by σ
(−∆) = [ρ 2 , ∞) with ρ = p+2q 4
(see, e.g., [36, Remark 2.2(iii)]). Therefore, we have
λ 0 (X p,q ) = (p+2q) 2
16
. Since the Cheeger constant of X p,q is given by h(X p,q ) = 2ρ = p+2q 2 (see [36, Remark 2.2(i)]), this means that Cheeger's Inequality λ 0 (X p,q ) = h(X p,q ) 2 4 holds in this case with equality. In fact, this holds for arbitrary non-compact harmonic manifolds (X, g) (see [37,Corollary 5.2]), and the Cheeger constant agrees with other geometric quantities like the (constant) mean curvature of the horospheres or the exponential volume growth (see [37,Theorem 5.1]).
For readers interested in more details about harmonic manifolds and Damek-Ricci spaces, we provide this information and a description of the real hyperbolic space H n as a solvable Lie group with left-invariant metric in the Appendix.
Moreover, since the seminal works of Hardy and Landau on Hardy-type inequalities more than hundred years ago, these inequalities are studied for the more general quasi-linear case P > 1 as well. The inequalities discussed before correspond to the linear case P = 2. We show the related inequalities for P ≥ 2 in Subsection 3.3.
The structure of the paper is as follows: In the next section, we state and prove the above Main Theorems. Thereafter, we briefly show two famous applications: a version of Heisenberg-Pauli-Weyl's uncertainty principle and a Rellich-type inequality. In Section 3, we vary parts of the proof of the main result and get closely connected families of sharp Poincaré-Hardy-type inequalities. The first variation shows the effect of being slightly away from the bottom of the spectrum at the Poincaré-part of the inequality, the second focuses on the weighted version and the third on the P -Laplacian version. In Section 4 we show another Poincaré-Hardy-type inequality for the P -Laplacian using the P -Green function. Here the focus is on the asymptotic behaviour of the corresponding Hardy weight. This closes the main part of this paper. In the Appendix, we briefly introduce further information about harmonic manifolds and Damek-Ricci spaces.
A Poincaré-Hardy-type inequality and applications
This section is concerned with an analogue of [8, Theorem 2.1] for all Damek-Ricci spaces. Before we present this result, we first need to introduce some background. The main reference here is [18]. While the concepts hold in more general Lebesgue and Sobolev spaces, we restrict our considerations to the smooth setting.
Some background from Optimality Theory
Henceforth we always assume that (X, g) is a non-compact Riemannian manifold. We are also concerned with the associated Schrödinger operators −∆ + V with potentials V ∈ C ∞ (X) and their corresponding energy functionals E V on C ∞ c (X), which are quadratic forms defined via
E V (φ) := X |∇φ| 2 + V |φ| 2 dx.
We say that E V is non-negative in X and write
E V ≥ 0 in X, if E V (φ) ≥ 0 for all φ ∈ C ∞ c (X).
Let Ω ⊂ X be a domain (that is a non-empty connected open subset). A function u ∈ C ∞ (Ω) is called solution (with respect to the operator −∆ + V ) if (−∆+V )u = 0, subsolution if (−∆+V )u ≤ 0, and supersolution if (−∆+V )u ≥ 0 in Ω. Moreover, a function u ∈ C ∞ (Ω) is called (super-)solution near infinity if there exists a compact set K ⊂ X such that (−∆ + V )u = 0 in Ω \ K, (resp.,
(−∆ + V )u ≥ 0 in X \ K). Furthermore, u ∈ Ω is a solution near o ∈ X, if there is an open set O ⊆ X containing o such that (−∆ + V )u = 0 in (Ω ∩ O) \ {o}.
Let W ≥ 0 in X. A Hardy-type inequality then reads as
E V (φ) ≥ X W |φ| 2 dx for all φ ∈ C ∞ c (X).
One goal in Optimality Theory is -roughly speaking -to make W as large as possible with large support (confer [1, page 6] where this problem was proposed first). With this idea in mind, Devyver, Fraas and Pinchover came up with a definition of an optimal Hardy weight, see [18, Definitions 2.1, 4.8 and 4.10].
Definition 2.1. Let (X, g) be a non-compact Riemannian manifold and o ∈ X.
Let E V ≥ 0 in X and W ≥ 0 be a non-trivial function such that the following Hardy-type inequality holds:
E V (φ) ≥ X\{o} W |φ| 2 dx for all φ ∈ C ∞ c (X \ {o}).
Then W is called an optimal Hardy weight of the Schrödinger operator
−∆ + V in X \ {o} if 1. −∆+ (V − W ) is critical in X \ {o}, that is, for any W ≥ W with W = W , the Hardy-type inequality E V (φ) ≥ X\{o} W |φ| 2 dx for all φ ∈ C ∞ c (X \ {o})
does not hold. This is equivalent to the following (see e.g. [30,Lemma 2.11]): there exists a unique (up to a multiplicative constant) positive supersolution to −∆u + (V − W )u = 0 on X \ {o}. Such a function is also a solution and is called the (Agmon) ground state u Agmon .
2. −∆ + (V − W ) is null-critical with respect to W , that is, we have that u Agmon ∈ L 2 (X \ {o}, W dx).
The original definition of an optimal Hardy weight also requires a condition called optimality at infinity (which is part (b) in Definition 2.1 of [18], or see [30,Definition 2.14]). Recently, Kovařík and Pinchover showed in [30,Corollary 3.7] that in our setting, null-criticality implies optimality at infinity. For that reason, it is not necessary to give the definition of optimality at infinity, since it is covered by Condition 2. Now we are in a position to present the main result in this section for Damek-Ricci spaces. It is an analogue of [8, Theorem 2.1] which covers the case of real hyperbolic spaces. Recall that Damek-Ricci spaces do not include real hyperbolic spaces and that their smallest dimension is 4. Our proof is inspired by the proof given in Section 4 of [8].
Theorem 2.2 (Poincaré-Hardy-type inequality on Damek-Ricci spaces). Let X p,q be a Damek-Ricci space, o ∈ X p,q be a pole, and r = d(o, ·). We have for
all φ ∈ C ∞ c (X p,q ), X p,q |∇φ| 2 dx − λ 0 (X p,q ) X p,q |φ| 2 dx ≥ 1 4 X p,q |φ| 2 r 2 dx+ p(p + 2q − 2) 16 X p,q |φ| 2 sinh 2 (r/2) dx+ q(q − 2) 4 X p,q |φ| 2 sinh 2 (r) dx. Moreover, the operator −∆ + (V − W ) on X p,q with V (r) := −λ 0 (X p,q ) and W (r) := 1 4r 2 + p(p + 2q − 2) 16 sinh 2 (r/2) + q(q − 2) 4 sinh 2 (r) is critical in X p,q \ {o}. In particular, there is no W (r) ≥ 1 4r 2 + p(p+2q−2) 16 sinh 2 (r/2) + q(q−2) 4 sinh 2 (r) , W = W , such that X p,q |∇φ| 2 dx − λ 0 (X p,q ) X p,q |φ| 2 dx ≥ X p,q W |φ| 2 dx holds true for all φ ∈ C ∞ c (X p,q \ {o})
. Furthermore, if we choose as a weight W 1 (r) = 1 4r 2 , then this weight is optimal with respect to the operator
−∆ − λ 0 (X p,q ) − p(p + 2q − 2) 16 sinh 2 (r/2) − q(q − 2) 4 sinh 2 (r) .
The proof of Theorem 2.2 is given in the next subsection, and is based on a result known as Khas'minskiȋ-type criterion. During the proof, we formulate a Hardy-type inequality for the more general case of non-compact harmonic manifolds. Subsection 2.3 presents applications of this result: An uncertainty principle and a Poincaré-Rellich-type inequality.
Proof of the Poincaré-Hardy-type inequality
To prove Theorem 2.2, we will use the following result -known as Khas'minskiȋtype criterion. This is a variation of Proposition 6.1 in [18], confer also with [24, Lemma 9.2.6]. It is in fact an equivalence (see [2, Theorem 1]) but we only need one direction here.
We start with a non-compact Riemannian manifold X with a chosen point o ∈ X. We refer to o as a pole of X.
We need some more definitions, confer with [18,Definition 4.4] and also with [18, Definition 11.2]: Let K be a compact subset of X \ {o}, and let u be a positive function on X \ ({o} ∪ K) which is solution of (−∆ + V )u = 0 in X \ ({o} ∪ K). The function u has minimal growth at infinity if for every compact K ′ ⊆ X \ {o} with smooth boundary such that K ⊆ int(K ′ ) and for every positive
supersolution v ∈ C(X \ ({o} ∪ K ′ ∪ ∂K ′ )) in X \ ({o} ∪ K ′ ) with u ≤ v in ∂K ′ , we have u ≤ v in X \ ({o} ∪ K ′ ).
Let u be a positive function defined in a punctured neighbourhood Ω of the
pole o ∈ X which is a solution of (−∆ + V )u = 0 in Ω \ {o}. The function u has minimal growth at o if for every positive supersolution v in a punctured neighbourhood of o, there is a constant C > 0 such that u ≤ Cv in a punctured neighbourhood Ω ′ ⊆ Ω of o.
A global minimal solution is a positive function on X \ {o} which is solution of (−∆ + V )u = 0 in X \ {o}, and has both, minimal growth at infinity and o. Proof. The first part is Proposition 6.1 in [18], which is essentially an application of the maximum principle to an exhaustion of the space. The second part follows by a mild adaption of the same argument as in the proof of the cited proposition.
The proof of Theorem 2.2 uses some general explicit calculations for radial functions. These are extracted in the next lemmata. We mainly work in harmonic manifolds and use the explicit form of the volume density on Damek-Ricci spaces as late as possible.
Lemma 2.4. Let (X, g) be a non-compact harmonic manifold with volume den- sity f , o ∈ X p,q a pole, r = d(o, ·), and α, β ∈ R. Then Φ(r) = r α f β (r) satisfies on X \ {o} ∆Φ(r) = α(α − 1) r 2 + α(2β + 1) f ′ (r) rf (r) + β f ′′ (r) f (r) + β 2 (f ′ (r)) 2 f (r) 2 Φ(r).
In particular, if (α, β) = (1/2, −1/2), we have on X \ {o},
∆Φ(r) = (f ′ (r)) 2 − 2f (r)f ′′ (r) 4f (r) 2 − 1 4r 2 Φ(r).
Proof. For simplicity, we will drop the argument r in our computations. Differentiation yields
Φ ′ = αr α−1 f β + βr α f β−1 f ′ and Φ ′′ = α(α − 1)r α−2 f β + 2αβr α−1 f β−1 f ′ + β(β − 1)r α f β−2 (f ′ ) 2 + βr α f β−1 f ′′ .
Using (1.3) and substituting Φ back into the Laplace equation leads to
∆Φ = Φ ′′ + f ′ f Φ ′ = α(α − 1) r 2 + α(2β + 1)f ′ rf + βf ′′ f + β 2 (f ′ ) 2 f 2 Φ.
In the proof of the main results, we want to apply the Khas'minskiȋ-type criterion, Proposition 2.3. Therefore, we need to find a second positive solution. This is done next. Lemma 2.5. Let (X, g) be a non-compact harmonic manifold with volume density f , o ∈ X a pole, r = d(o, ·), and α, β ∈ R. Furthermore, set
Φ(r) = r α f β (r). Let h : X \ {o} → R be a smooth radial function, then on X \ {o} we have ∆(Φh)(r) − α(α − 1) r 2 + α(2β + 1)f ′ (r) rf (r) + βf (r)f ′′ (r) + β 2 (f ′ (r)) 2 f 2 (r) (Φ(r)h(r)) = Φ(r) h ′′ (r) + 2α r + (2β + 1)f ′ (r) f (r) h ′ (r) .
In particular, the right-hand side vanishes for (α, β, h(r)) = (1/2, −1/2, ln(r)).
Proof. In the following, we omit again the argument r. By the product rule, we have
∆(Φh) = Φ∆h + h∆Φ + 2Φ ′ h ′ .
Using this rule and Lemma 2.4, we obtain
∆(Φh) − α(α − 1) r 2 + α(2β + 1)f ′ rf + βf ′′ f + β 2 (f ′ ) 2 f 2 (Φh) = Φ∆h + 2Φ ′ h ′ .
Using once more that h is radial, i.e., ∆h = h ′′ + f ′ f h ′ , and that
Φ ′ = αr α−1 f β + βr α f β−1 f ′ = α r + βf ′ f Φ, we obtain Φ∆h + 2Φ ′ h ′ = Φ h ′′ + 2α r + (2β + 1)f ′ f h ′ .
This shows the first part of the statement. The other assertion follows by a simple computation.
Finally, we will need the following lemma, which is a special case of the Agmon-Allegretto-Piepenbrink theorem, see [39,Theorem 2.3] and see also Lemma 3.6. We remark that the implication in the lemma below is actually an equivalence, i.e., the non-negativity of the energy functional implies also the existence of a positive (super)solution.
φ ∈ C ∞ c (Ω), Ω |∇φ| 2 dx ≥ Ω (W − V )φ 2 dx.
Proof. Set φ = Φψ with ψ ∈ C ∞ c (Ω). Then we have, by Green's formula,
Ω |∇φ| 2 dx ≥ Ω ∇(ψ 2 Φ), ∇Φ dx = − Ω ψ 2 Φ(∆Φ)dx = Ω (W − V )φ 2 dx.
We link the previous lemmata and the Khas'minskiȋ-type criterion to obtain the following Hardy-type inequality on general harmonic manifolds, which is our Main Theorem B in the Introduction.
Theorem 2.7 (Hardy-type inequality on harmonic manifolds). Let (X, g) be a non-compact harmonic manifold with volume density f = f (r), o ∈ X a pole, and r = d(o, ·). Then, the Schrödinger operator
−∆ + (V − W ) on X with V (r) := (f ′ (r)) 2 − 2f (r)f ′′ (r) 4f 2 (r) and W (r) := 1 4r 2 is critical in X \ {o} with ground state r/f (r). Moreover, the weight W (r) = 1 4r 2 is optimal in X \ {o} with respect to the operator −∆ + V . Furthermore, for all φ ∈ C ∞ c (X), we have X |∇φ| 2 dx ≥ 1 4 X φ 2 r 2 dx + 1 4 X 2f (r)f ′′ (r) − (f ′ (r)) 2 f 2 (r) · φ 2 dx. (2.1)
Proof. The first part of the proof uses the same ideas as the proof of [18, Theorem 6.2] doing the necessary changes. The second part is motivated by [8].
Here are the details: Let us denote H := −∆ + V − W . By Lemma 2.4 and Lemma 2.5 applied with (α, β) = (1/2, −1/2), we see that Φ(r) = r 1/2 f −1/2 (r) and Φ(r) · ln(r), are solutions on X \ {o} with respect to H. Moreover, we obviously have
lim r→∞ Φ(r) Φ(r) ln(r) = 0.
By the Khas'minskiȋ-type criterion, Proposition 2.3, we get that Φ is a positive solution of minimal growth near infinity in X. By the same argument and using −Φ(r) ln(r) instead of Φ(r) ln(r), we see that Φ has minimal growth near o in X \ {o}.
Since Φ is a solution in X \ {o} and of minimal growth near infinity and o, it is a global minimal solution in X \ {o}. By [40, Theorems 4.2 and 5.8], it has to be the (Agmon)
ground state in X \ {o} which is the unique positive supersolution of H in X \ {o}. Thus, H is critical in X \ {o}. Let now W (r) = 1/(2r) 2 . Since Φ 2 L 2 (X\{o},W dx) = ω n ∞ 0 (r 1/2 f (r) −1/2 ) 2 f (r) dr (2r) 2 = ∞,
we conclude that −∆ + V is null-critical with respect to W , and thus W is an optimal Hardy weight of −∆ + V in X \ {o}.
Inequality (2.1) can be seen as follows: Since Φ is a strictly positive solution of HΦ = 0 on X \ {o}, we obtain the desired inequality on C ∞ c (X \ {o}) by Lemma 2.6. Next we briefly explain how to extend the inequality from C ∞ c (X \ {o}) to C ∞ c (X) (this is a standard argument but we could not find a good reference in the Schrödinger operator setting; for the p-Laplacian see [14, Appendix A]): By [40,Theorem 4.5], the capacity of {o} vanishes, i.e., there is
a sequence (φ j ) in C ∞ 0 (X) such that 0 ≤ φ j ≤ 1, φ j = 1 in a neighbourhood of {o} and φ j → 0 in D 0 (X) = C ∞ c (X) |·|0 where | · | 0 denotes the form norm.
The latter is defined via [14, Appendix A]): One can use dominated convergence for the potential part and then proceed with mild changes (note that the last step there is just another application of [40,Theorem 4.5]).
|φ| 2 0 := E V (φ) + | · | 2 . In particular, E V (φ j ) → 0. We set ψ j := (1 − φ j )ψ, where ψ is an arbitrary function in C ∞ c (X). Now it follows ψ j → ψ in D 0 (X \{o}) similarly as in
We now turn to Damek-Ricci spaces and explicitly calculate the term 2f
f ′′ −(f ′ ) 2 f 2 .
Later we will need similar calculations, and therefore, we state it here more generally.
Lemma 2.8. Let X p,q be a Damek-Ricci space, o ∈ X p,q a pole and r = d(o, ·). Let f (r) be the volume density of X p,q . Then, for all a, b ∈ R and r > 0, we have
a f ′ (r) f (r) 2 − b f ′′ (r) f (r) = 4(a − b)λ 0 (X p,q ) + q(q(a − b) + b) sinh 2 (r) + p((a − b)(p + 2q) + b) 4 sinh 2 (r/2) .
Proof. Recall that f (r) = 2 p+q (sinh(r/2)) p+q (cosh(r/2)) q .
Straightforward computations yield f ′ (r) = f (r) 2 (p + q) coth(r/2) + q tanh(r/2) .
and
f ′′ (r) = f (r) 4 (p + q) coth(r/2) + q tanh(r/2) 2 − p + q sinh 2 (r/2) + q cosh 2 (r/2) .
Note, furthermore, that for all α, β ∈ R α sinh 2 (r/2) − β cosh 2 (r/2) = α − β sinh 2 (r/2) + 4β sinh 2 (r) and α coth(r/2) + β tanh(r/2) 2 = (α + β) 2 + 4β 2 sinh 2 (r) + α 2 − β 2 sinh 2 (r/2) .
Hence,
a f ′ (r) f (r) 2 − b f ′′ (r) f (r) = (a − b)(p + 2q) 2 4 + q(q(a − b) + b) sinh 2 (r) + p((a − b)(p + 2q) + b) 4 sinh 2 (r/2) .
Using the fact that λ 0 (X p,q ) = (p + 2q) 2 /16, the statement follows.
Note that we obtain from this lemma for the special choice (a, b) = (1/4, 1/2):
2f (r)f ′′ (r) − (f ′ (r)) 2 4(f (r)) 2 = λ 0 (X p,q ) + q(q − 2) 4 sinh 2 (r) + p(p + 2q − 2) 16 sinh 2 (r/2) . (2.2)
Lemma 2.9. Let X p,q be a Damek-Ricci space, o ∈ X p,q a pole and r = d(o, ·). Let f (r) be the volume density of X p,q and Φ(r) = r 1/2 f −1/2 (r).
Then we have on
X p,q \ {o} −∆Φ(r) − λ 0 (X p,q )Φ(r) = p(p + 2q − 2) 16 sinh 2 (r/2) + q(q − 2) 4 sinh 2 (r) + 1 4r 2 Φ(r).
Proof. The statement of the lemma is a direct application of Lemma 2.4 by choosing (α, β) = (1/2, −1/2), and of (2.2).
Finally, we are in a position to prove Theorem 2.2.
Proof of Theorem 2.2. The result follows by applying Lemma 2.9 and Theorem 2.7.
Applications of the Poincaré-Hardy-type inequality
This subsection is devoted to two well-known applications of Hardy-type inequalities, which can be obtained easily in combination with the Cauchy-Schwarz inequality: an uncertainty-type principle and a Rellich-type inequality.
A Heisenberg-Pauli-Weyl's Uncertainty Principle
Here, we briefly show a shifted version of the famous Heisenberg-Pauli-Weyl uncertainty principle. It asserts in its classical form, that the position and momentum of a particle can not be determined simultaneously. For further information confer e.g. [6, Subsection 1.6] for a detailed discussion in the Euclidean space, [28,29,32] for Riemannian manifolds, or [9, Section 3] for a recent version in the hyperbolic space.
Recall that the dimension of a Damek-Ricci space is at least 4, i.e., p+ q ≥ 3.
Corollary 2.10. Let X p,q be a Damek-Ricci space with q ∈ {0, 2} and p = 0, o ∈ X p,q a pole, and r = d(o, ·). Then, we have for all φ ∈ C ∞ c (X p,q ),
X p,q |∇φ| 2 dx − λ 0 (X p,q ) X p,q φ 2 X p,q g(r)r 2 φ 2 dx ≥ 1 4 X p,q φ 2 dx 2 ,
with a function 0 < g < 1.
In fact, the function g in the corollary is explicitely given by
g(r) = 1 4r 2 W (r) (2.3) with W (r) := 1 4r 2 + p(p + 2q − 2) 16 1 sinh 2 (r/2) + q(q − 2) 4 1 sinh 2 (r)
.
Note that we have W (r) > 1/(2r) 2 , even for the smallest choice p ≥ 2 and q = 1.
Proof. By the Cauchy-Schwarz inequality and the Poincaré-Hardy-type inequality (Theorem 2.2), we obtain
X p,q φ 2 dx 2 ≤ X p,q W φ 2 dx X p,q φ 2 W dx ≤ X p,q |∇φ| 2 dx − λ 0 (X p,q ) X p,q φ 2 dx X p,q φ 2 W dx .
Since 1/W (r) = 4g(r)r 2 we obtain the desired inequality.
A Poincaré-Rellich-type inequality
Next, we show a shifted Rellich-type inequality. For more details on the history and generalisations of this inequality in different settings, we suggest the papers [9,25,28,29] and the monograph [6], and references therein.
Corollary 2.11. Let X p,q be a Damek-Ricci space with q ∈ {0, 2} and p = 0, o ∈ X p,q a pole, and r = d(o, ·). Then, we have for all φ ∈ C ∞ c (X p,q ),
X p,q φ 2 16g(r)r 2 dx ≤ X p,q g(r)r 2 (−∆φ − λ 0 (X p,q )φ) 2 dx,
with the function 0 < g < 1 from (2.3).
Proof. Let us denote as before,
W (r) := 1 4r 2 + p(p + 2q − 2) 16 1 sinh 2 (r/2) + q(q − 2) 4 1 sinh 2 (r)
.
Moreover, we set H := −∆ − λ 0 (X p,q ). Then, again by the Poincaré-Hardytype inequality (Theorem 2.2), the Green's formula and the Cauchy-Schwarz inequality, we obtain
X p,q W (r)φ 2 dx ≤ X p,q φHφdx ≤ X p,q W (r)φ 2 dx 1/2 X p,q W −1 (r)(Hφ) 2 dx 1/2 .
Since 1/W (r) = 4g(r)r 2 we obtain the desired inequality.
Variations of the main result
The aim of this section is to show the effect of perturbations of some parameters. In the first subsection, we will investigate the consequence of changing the function Φ from (r/f (r)) 1/2 to (r/f (r)) 1/2 · (r/f 1/(n−1) (r)) γ for some number γ > 0. This will then result in a new family of critical Schrödinger operators, where the effect of γ is only visible at the corresponding constants.
In the second subsection, we have a closer look at the weighted version of the Hardy inequality, i.e., we consider the following family of quadratic forms
X p,q |∇φ| 2 r 2α dx, α ≥ 0, φ ∈ C ∞ c (X p,q ),
and show the effect of α > 0. In a different perspective, this new weighted energy form is a specific ground state transform of the unweighted energy form. Finally, in the third subsection, we vary the operator and consider quasilinear P -Laplacians, P ≥ 2. Here the classical linear case corresponds to P = 2.
A family of Hardy-type inequalities
In the previous section, we investigated a Hardy-type theorem for the operator −∆ − 1 4r 2 on harmonic manifolds. Now, we will use this result to obtain similar inequalities for the operators −∆ − a 4r 2 for 0 ≤ a ≤ 1. The main result of this section reads as follows and is inspired by the corresponding result on hyperbolic spaces, see [9, Theorem 2.1]: Theorem 3.1 (Family of Hardy-type inequalities on harmonic manifolds). Let (X, g) be a non-compact harmonic manifold with volume density f , o ∈ X a pole, r = d(o, ·), and dimension n ≥ 3. Let h : X \ {o} → (0, ∞) be a smooth radial function satisfying h(r) ≥ C 1 r for all r ≥ 0 and h(r) ∼ C 2 r as r → 0 for some positive constants C 1 , C 2 > 0. Then, for all φ ∈ C ∞ c (X) and all γ ∈ [0, 1/2],
X |∇φ| 2 dx ≥ 1 − (2γ) 2 4 X φ 2 r 2 dx + X 1 2 f ′′ f − 1 4 f ′ f 2 + γ h ′′ h + (1 + 2γ) h ′ rh − (1 + γ) h ′ h 2 φ 2 dx.
(3.1)
Moreover, the Schrödinger operator
−∆ + V − W with V (r) := − 1 2 f ′′ f − 1 4 f ′ f 2 + γ h ′′ h − (1 + 2γ) h ′ rh − (1 + γ) h ′ h 2 and W (r) := 1 − (2γ) 2 4r 2 is critical in X \ {o} with ground state r 1/2+γ f −1/2 h −γ . Furthermore, if we choose as weight W (r) = 1−(2γ) 2 4r 2 , 0 ≤ γ < 1/2, then W is optimal in X \ {o} with respect to −∆ + V .
Note that Theorem 2.2 can be recovered from Theorem 3.1 if we set γ = 0. We derive this result with the help of the Liouville comparison theorem. For the readers' convenience, we state it here (see also [9, Theorem 6.3]).
H i = −∆ + V i , i = 0, 1,
with smooth potentials. Furthermore, assume that 1. H 0 is critical in Ω with ground state Φ;
H 1 is non-negative in Ω and there is a function
Ψ ∈ C ∞ (Ω) with H 1 Ψ ≤ 0 with Ψ + = 0, where Ψ + (x) = max{Ψ(x), 0}.
There exists a constant
C > 0 such that (Ψ + ) 2 ≤ C(Φ) 2 .
Then the operator H 1 is critical with ground state Ψ.
β) = (1/2 + γ, −1/2) yields − ∆(Φh −γ )(r) + γ 2 − 1/4 r 2 + 1 4 f ′ (r) f (r) 2 − 1 2 f ′′ (r) f (r) Φ (r)h −γ (r) = − γ(γ + 1) h ′ (r) h(r) 2 − γ h ′′ (r) h(r) − γ(1 + 2γ) h ′ (r) rh(r) Φ (r)h −γ (r).
Now we apply the Liouville comparison principle with Ω = X \ {o},
V 0 = − 1 4r 2 + 1 4 f ′ (r) f (r) 2 − 1 2 f ′′ (r) f (r)
and, for γ ∈ [0, 1/2],
V 1 = γ 2 − 1/4 r 2 + 1 4 f ′ (r) f (r) 2 − 1 2 f ′′ (r) f (r) + γ(γ + 1) h ′ (r) h(r) 2 − γ h ′′ (r) h(r) − γ(1 + 2γ) h ′ (r) rh(r)
.
Let H i = −∆ + V i for i = 0, 1.
We have shown in Theorem 2.7 that H 0 is critical with ground state Φ = (r/f ) 1/2 . Set Ψ = (r/f ) 1/2 · (r/h) γ > 0 on Ω, that is Ψ + = Ψ. By construction, we have H 1 Φ = 0, and non-negativity of H 1 on Ω follows from Lemma 2.6.
Since Ψ/Φ = (r/h) γ ≤ 1/C 1 , the Liouville comparison principle implies that H 1 is critical with ground state Ψ. We can then use the same approximation argument as in the proof of Theorem 2.2 to obtain inequality (3.1) for all φ ∈ C ∞ c (X p,q ) and not only φ ∈ C ∞ c (Ω). Let now W (r) = (1 − (2γ) 2 )/(2r) 2 and 0 < γ < 1/2. Since h(r) ∼ C 2 r n−1 as r → 0, we have Ψ ∈ L 2 (Ω, W dx), and H 1 is therefore null-critical with respect to W . This implies that W is an optimal Hardy weight of H 1 in X \ {o}.
The special choice h(r) = f 1/(n−1) (r) leads to the following result which, in the case of Damek-Ricci spaces X p,q , can be viewed as Hardy-type improvements of the operators −∆ − λ for λ ≤ λ 0 (X p,q ). Corollary 3.3. Let (X, g) be a non-compact harmonic manifold with volume density f , o ∈ X a pole, r = d(o, ·), and dimension n ≥ 3. Then, for all φ ∈ C ∞ c (X) and all γ ∈ [0, 1/2],
X |∇φ| 2 dx ≥ 1 − (2γ) 2 4 X φ 2 r 2 dx + γ(1 + 2γ) n − 1 X f ′ f · φ 2 r dx+ 1 2 + γ n − 1 X f ′′ f − 1 2 + γ n − 1 f ′ f 2 φ 2 dx. (3.2)
In the particular case of a Damek-Ricci space X p,q , we have
X p,q |∇φ| 2 dx − 1 − (2γ) 2 (p + q) 2 λ 0 (X p,q ) X p,q φ 2 dx ≥ 1 − (2γ) 2 4 X p,q φ 2 r 2 dx + γ(1 + 2γ) p + q X p,q coth(r/2) − 1 sinh(r) φ 2 r dx + q 1 2 + γ p + q + q γ 2 (p + q) 2 − 1 4 X p,q φ 2 sinh 2 (r) dx + p 1 2 + γ p + q + (p + 2q) γ 2 (p + q) 2 − 1 4 X p,q φ 2 4 sinh 2 (r/2) dx. (3.3)
Proof of Corollary 3.3.
Let h = f 1/(n−1) . Let us first verify that h satisfies the conditions required in Theorem 3.1. Let (X, g) be a non-compact harmonic manifold of dimension n ≥ 3 and H ≥ 0 the (constant) mean curvature of its horospheres. We know from a combination of [41] and [35] that non-compact harmonic manifolds with H = 0 are flat (see also [27,Section 6]). In this case we have f (r) = r n−1 and, therefore h(r) = r and the conditions of the theorem are satisfied with C 1 = C 2 = 1. On the other hand, we have for any n-dimensional Riemannian manifold that lim r→0 f (r) r n−1 = 1, that is, h(r) ∼ r as r → 0. Moreover, this implies that
C 1 = inf r>0 h(r) r > 0.
It is easy to see that
h ′ = h n − 1 · f ′ f and h ′′ = h n − 1 2 − n n − 1 · f ′ f 2 + f ′′ f .
Plugging this into (3.1) yields (3.2). Now we derive the inequality for a Damek-Ricci space with n − 1 = p + q.
Using Lemma 2.8 with a = 1 2 + γ (p+q) 2 and b = 1 2 + γ p+q , we obtain
1 2 + γ (p + q) 2 f ′ (r) f (r) 2 − 1 2 + γ p + q f ′′ (r) f (r) = (2γ) 2 (p + q) 2 − 1 λ 0 (X p,q ) + q γ 2 (p + q) 2 − 1 4 + 1 2 + γ p + q q sinh 2 (r) + (p + 2q) γ 2 (p + q) 2 − 1 4 + 1 2 + γ p + q p 4 sinh 2 (r/2) .
Plugging this into (3.2) yields (3.3), finishing the proof of the corollary.
We want to mention that also other choices of h lead to closely related inequalities. In [9] the choice of h is h(r) = sinh(r) which is a natural choice for hyperbolic spaces, and our aim was to generalize that result.
Weighted Hardy-type inequalities
Here, we use the method in [9, Theorem 5.1], which is concerned with the hyperbolic space, to obtain a similar result for arbitrary harmonic manifolds X. Note again that we obtain the main results in the Introduction by choosing α = 0 in the following theorem. Theorem 3.4 (Weighted Hardy-type inequality). Let (X, g) be a non-compact harmonic manifold with volume density f = f (r), o ∈ X a pole, r = d(o, ·), and dimension n ≥ 3. Assume that n ≥ 2(1 + α) for some α ≥ 0, then the following weighted Hardy-type inequality holds true for all φ ∈ C ∞ c (X),
X |∇φ| 2 r 2α dx ≥ X 2f (r)f ′′ (r) − (f ′ (r)) 2 4f (r) 2 − α f ′ (r) rf (r) + 4α + 1 4r 2 φ 2 r 2α dx.
Especially, for such Damek-Ricci spaces X p,q , we have for all φ ∈ C ∞ c (X) and
p + q − 1 ≥ 2α, X |∇φ| 2 − λ 0 (X p,q )φ 2 r 2α dx ≥ X p(p + 2q − 2) 16 sinh 2 (r/2) + q(q − 2) 4 sinh 2 (r) φ 2 r 2α dx + X α r q sinh(r) − p + 2q 2 coth(r/2) + 4α + 1 4r 2 φ 2 r 2α dx.
Proof. Given φ ∈ C ∞ c (X \ {o}) and Φ (which we will specify later), we choose ψ ∈ C ∞ c (X \ {o}) such that φ = r α Φψ. Then, we calculate
|∇φ| 2 r 2α = |∇Φ| 2 ψ 2 + |∇ψ| 2 Φ 2 + 2ψΦ ∇Φ, ∇ψ + 2α
Φψ r (ψΦ r + Φψ r ).
Note that Φψ
r (ψΦ r + Φψ r ) = φ r φ r 2α+1 − α φ 2 r 2α+2 .
Integrating, using that Φψ 2 ∈ C ∞ c (X \ {o}) and Green's formula, we obtain
X |∇φ| 2 r 2α dx = X (−∆Φ)Φψ 2 dx + X |∇ψ| 2 Φ 2 dx − 2α 2 X φ 2 r 2α+2 dx + 2α X φ r φ r 2α+1 dx.
Changing to polar coordinates in the latter integral, using integration by parts, f ∼ r n−1 as r → 0 and n − 1 ≥ 2α + 1, and then resubstituting, we get
2 X φ r φ r 2α+1 dx = ∞ 0 S n−1 (φ 2 ) r f (r) r 2α+1 dωdr = − ∞ 0 S n−1 f (r) r 2α+1 r φ 2 dωdr = (2α + 1) X φ 2 r 2α+2 dx − X f ′ (r) f (r)r 2α+1 φ 2 dx.
Thus, we obtain altogether
X |∇φ| 2 r 2α dx ≥ X (−∆Φ)Φψ 2 dx + α X φ 2 r 2α+2 dx − α X f ′ (r)φ 2 f (r)r 2α+1 dx.
Now, as usual, let Φ = (r/f (r)) 1/2 , and recall from Lemma 2.4 that
−∆Φ(r) = 2f (r)f ′′ (r) − (f ′ (r)) 2 4f (r) 2 + 1 4r 2 Φ(r).
Thus, we get
X |∇φ| 2 r 2α dx ≥ X 2f (r)f ′′ (r) − (f ′ (r)) 2 4f (r) 2 − α f ′ (r) rf (r) + 4α + 1 4r 2 φ 2 r 2α dx.
Moreover, we can use the same approximation argument as in the proof of Theorem 2.2 to obtain φ ∈ C ∞ c (X p,q ) and not only φ ∈ C ∞ c (X p,q \ {o}). This finishes the proof on harmonic manifolds.
On Damek-Ricci spaces X p,q , we know f explicitly, and by using (1.6) and (2.2) the result follows.
A Poincaré-Hardy-type inequality for the P -Laplacian
In this subsection we want to generalize our Poincaré-Hardy-type inequality to the quasi-linear P -Laplacian for P ∈ (1, ∞).
Let us first introduce the P -Laplacian on a Riemannian manifold (X, g). It is defined as ∆ P φ := div(|∇φ| P −2 ∇φ).
This quasi-linear second order operator reduces to the classical linear Laplace operator ∆ in the case P = 2. In the case of a non-compact harmonic manifold (X, g) and a smooth radial function φ ∈ C ∞ (X) around a pole o ∈ X, the P -Laplacian is again radial and given by
∆ P φ(r) = |φ ′ (r)| P −2 L P φ(r),(3.4)
where
L P φ(r) = (P − 1)φ ′′ (r) + f ′ (r) f (r) φ ′ (r). (3.5)
Moreover, we will need in Section 4 below that harmonicity of a radial function φ ∈ C ∞ (X\{o}) with nowhere vanishing derivative is equivalent to the condition that f |φ ′ | P −2 φ ′ is constant. This follows directly from the identity
(f |φ ′ | P −2 φ ′ ) ′ = f |φ ′ | P −2 L P φ = f ∆ P φ.
Our aim in this subsection is to prove a quasi-linear version of the Poincaré-Hardy inequality on Damek-Ricci spaces in the spirit of Theorem 2.2. This can be seen as an generalisation of Theorem 2.5 in [7] on hyperbolic spaces to Damek-Ricci spaces. The proof uses basically the quasi-linear version of the ideas from the linear case. Clearly, the terminology of the Optimality Theory in Subsection 2.1 needs to be generalized to the quasi-linear context. However, most of the generalizations are obvious and are not discussed here. We recommend [19] for further details and references. Moreover, we want to mention that the same restrictions on P and the dimension in the theorem below also appear in [7]. Theorem 3.5. Let P ≥ 2, and let X p,q be a Damek-Ricci space of dimension n = p + q + 1 ≥ 1 + P (P − 1). Let o ∈ X p,q be a pole, r = d(o, ·), h := h(X p,q ) = (p + 2q)/2 be its Cheeger constant, and
Λ P (X p,q ) := h P P Furthermore, we set g(r) := coth(r/2) − 2 p + 2q q sinh(r) + 1 r .
Then, g(r) ≥ p+q−1 h·r and g(r) → 1 as r → ∞, and we have for all φ ∈ C ∞ c (X p,q ),
X p,q |∇φ| P dx − Λ P (X p,q ) X p,q g(r) P −2 |φ| P dx ≥ h P −2 (P − 1) 2 P P X p,q g(r) P −2 r 2 |φ| P dx + h P −1 (P − 2) P P X p,q g(r) P −2 (g(r) + 1 hr ) r |φ| P dx + h P −2 P P X p,q g(r) P −2 q 2 − qP (P − 1) sinh 2 (r) + p(2h − P (P − 1)) 4 sinh 2 (r/2) |φ| P dx. (3.6)
Moreover, all integrands on the right hand side of this inequality are nonnegative functions.
Note again that the special choice P = 2 in the above theorem leads to our Main Theorem A in the Introduction.
The theorem will be obtained with the help of the Agmon-Allegretto-Piepenbrink Theorem. For convenience, we state it here in our smooth setting and for a strong solution. Note that it is a generalization of Lemma 2.6. Moreover, the statement is actually an equivalence but we only state the direction we need. Lemma 3.6 (see [39,Theorem 2.3]). Let Ω ⊂ X be a domain in a Riemannian manifold (X, g), P ∈ (1, ∞). Let Φ ∈ C ∞ (Ω) be a positive supersolution of
−∆ P Φ + (V − W )|Φ| P −2 Φ = 0 on Ω. Then we have for all φ ∈ C ∞ c (Ω), Ω |∇φ| P dx ≥ Ω (W − V )|φ| P dx.
Proof of Theorem 3.5. Set Φ(r) = (r/f (r)) 1/P and let L P be the operator introduced in (3.5). It is not difficult to see (omitting the argument r),
L P Φ = − P − 1 P 2 1 r 2 − P − 2 P 2 f ′ rf + P 2 − P − 1 P 2 (f ′ ) 2 f 2 − P − 1 P f ′′ f Φ = − 1 P 2 (P − 1) 2 r 2 + (P − 2)f ′ rf + (f ′ ) 2 f 2 + P (P − 1) f ′′ f − (f ′ ) 2 f 2 Φ. Substituting (3.7) into (−∆ P )Φ = h P −2 P P g P −2 · (P − 1) 2 r 2 + (P − 2)f ′ rf + (f ′ ) 2 f 2 + P (P − 1) f ′′ f − (f ′ ) 2 f 2 Φ P −1
and using f ′ f = h(g + 1 hr ), the inequality of the theorem follows from Lemma 3.6 and the usual approximation argument to extend it to all functions φ ∈ C ∞ c (X p,q ).
Remark. It follows from the proof that inequality (3.6) in the theorem can be simplified by replacing g(r) by the easier lower bound p+q−1 hr > 0, but we expect that the corresponding P -Schrödinger operator in the theorem is critical on X p,q \ {o}, which would be lost by this simplification.
Another Poincaré-Hardy-type inequality for the P -Laplacian
In this section we derive the Green function for the P -Lapacian and follow the general arguments given in [11, Section 5.1] (see also [7] for the case of the real hyperbolic space) to obtain a Hardy weight for the P -Laplacian on noncompact harmonic manifolds, which is optimal under a certain condition of the dimension. Similarly as in [11], we also study asymptotics of this Hard weight at the pole.
The Green function of the P-Laplacian on a non-compact harmonic manifold
The P -Green function G = G P : X × X \ {(x, x) | x ∈ X} → R is characterised by the following conditions (see, e.g., [33]):
(a) ∆ P,x G(x, y) = 0 for all x = y,
(b) G(x, y) ≥ 0 for all x = y,(c)
For all y ∈ X, we have lim x→∞ G(x, y) = 0, (d) If 1 < P ≤ n: For all y ∈ X, we have lim x→y G(x, y) = ∞, (e) For all φ ∈ C ∞ c (X) and all y ∈ X, we have
X |∇ x G(x, y)| p−2 ∇ x G(x, y), ∇φ(x) dx = φ(y),
that is, −∆ P,y G(x, ·) = δ y in the sense of distributions.
In the case of a harmonic manifold, we can fix a point o ∈ X and consider the Green function as a function of the radius, that is, we can write G(d(x, o)) = G(x, o). The Green function of the P -Laplacian of the Euclidean space R n for 1 < P < n is given by
G(r) = P − 1 (n − P ) ω −1/(P −1) n r − n−P P −1 ,
where ω n is the volume of the unit sphere S n−1 in R n . Let us now calculate the Green function of the P -Laplacian of a non-compact non-Euclidean harmonic manifold X of dimension n with density function f . Recall from Subsection 3.3 that P -harmonicity of a radial function G ∈ C ∞ (X \ {o}) with nowhere vanishing derivative is given when f |G ′ | P −2 G ′ is constant. Assuming G ′ < 0 and lim r→∞ G(r) = 0, and setting
f |G ′ | P −2 G ′ = −f |G ′ | P −1 ∼ = −β P −1 ,
for some constant β > 0, we find by integration
G(r) = β ∞ r dt (f (t)) 1/(P −1) . (4.1)
Since X is non-flat, the mean curvature h of its horospheres must be strictly positive (see, e.g., [27,Corollary 2.8]). This result is a consequence of the fact that X cannot have polynomial volume growth by [41,Theorem 4.2] and that the density function f of X is an exponential polynomial by [35]. Therefore, X must have exponential volume growth and the integral on the right hand side of (4.1) is finite for all r > 0. Let us study lim r→0 G(r) for G(r) defined in (4.1). It follows from [46, p. 82] for arbitrary Riemannian manifolds that
f (t) = t n−1 1 − s(o) 6n t 2 + O(t 4 ) for t → 0, (4.2)
where s(o) is the scalar curvature of X at o ∈ X. Consequently, we can find r ′ > 0 and 0 < c 1 < c 2 such that, for 0 < r < r ′ ,
c 1 r ′ r dt t (n−1)/(P −1) ≤ r ′ r dt (f (t)) 1/(P −1) ≤ c 2 r ′ r dt t (n−1)/(P −1) ,(4.3)
which implies lim r→0 G(r) = ∞ in the case 1 < P ≤ n. The Ansatz (4.1) implies
G ′ (r) = − β (f (r)) 1/(P −1) ,
which shows that the assumption G ′ < 0 was justified. The arguments so far and the explicit expression for G imply that G satisfies conditions (a), (b), (c) and (d).
It remains to verify condition (e). It is known that non-compact harmonic manifolds do not have conjugate points, since simply connected harmonic manifolds with conjugate points are Blaschke manifolds by Allamigeon's Theorem (see [10,Chapter 6F] or [31,Section 5.1]). We can therefore use global geodesic polar coordinates (r, θ) ∈ (0, ∞) × S n−1 → X \ {y} around the pole y ∈ X. For simplicity of notation, we switch between points of X and their representations in polar coordinates. We have ∇ x G(x, y) = G ′ (r)ξ(r, θ), where r = d(x, y) and ξ defined on X \{y} is the outward unit normal vector field of concentric spheres around y. Consequently, we have
|∇ x G(x, y)| P −2 ∇ x G(x, y), ∇φ(x) = |G ′ (r)| P −2 G ′ (r) ∂ ∂r φ(r, θ) = − β P −1 f (r) ∂ ∂r φ(r, θ).
This implies
X |∇ x G(x, y)| P −2 ∇ x G(x, y), ∇φ(x) dx = − S n−1 ∞ 0 β P −1 f (r) ∂ ∂r φ(r, θ)f (r)drdθ = β P −1 vol(S n−1 )φ(y).
This shows that (e) is satisfied if we choose β = 1/ω 1/(P −1) n . Therefore the Green function of the P -Laplacian of an n-dimensional non-compact harmonic manifold is given by
G(d(x, y)) = ∞ d(x,y) dt
(ω n f (t)) 1/(P −1) .
Moreover, we see from estimate (4.3) that, in the case n < P < ∞, the limit γ := lim r→0 G(r) is a finite positive real number. Note that this expression agrees with the boxed formula in [27, p. 51] for the standard Laplace operator, that is, the case P = 2.
A Poincaré-Hardy-type inequality for the P -Laplacian based on the P -Green function
The following theorem is an extension of [7, Proposition 1.1] from real hyperbolic spaces to arbitrary non-flat harmonic manifolds. The proof follows closely the one given in [7] which, in turn, is based on the arguments given in [11, Section 5.1].
Theorem 4.1. Let X be a non-compact non-Euclidean harmonic manifold of dimension n, o ∈ X a pole, and r = d(o, ·). Let 1 < P < ∞ and G(r) be the corresponding P -Green function. Then we have
W := P − 1 P P ∇G G P ≥ Λ P := h P P .
In the case 1 < P ≤ n, W is an optimal Hardy weight of −∆ P in X \ {o}.
For all 1 < P < ∞ and all φ ∈ C ∞ c (X), we have
X |∇φ| P dx − Λ P X |φ| P dx ≥ X W |φ| P dx (4.4)
with W = W − Λ P ≥ 0. Moreover, we have the following asymptotics,
W (r) ∼ n−P P P r −P for 1 < P < n P −1 P P |r log(r)| −P for P = n C P,f r − P (n−1) P −1 for n < P < ∞ as r → 0, (4.5)
where
C P,f = P −1 P P ∞ 0 dt (f (t)) 1/(P −1) −P .
We also have lim r→∞ W (r) = 0.
Proof. Let us first prove the inequality W (r) ≥ Λ P . Let f be the density function of X. We know from [42,Section 2] that the quotient
f ′ (r)/f (r) is monotone descreasing with h = lim r→∞ f ′ (r)/f (r). Since X is non-flat, we have h > 0. Using f ′ /f ≥ h, we obtain ∞ r dt (f (t)) 1/(P −1) ≤ 1 h ∞ r f ′ (t)dt (f (t)) 1+1/(P −1) = P − 1 h (f (r)) −1/(P −1) .
This implies that
G ′ (r) G(r) = (f (r)) 1/(P −1) ∞ r dt (f (t)) 1/(P −1) −1 ≥ h P − 1 ,
and therefore
W (r) = P − 1 P P G ′ (r) G(r) P ≥ P − 1 P P h P − 1 P = Λ P .
Applying [19, Theorem 1.5(1)], we conclude that W is an optimal Hardy weight of −∆ P in X\{o} in the case 1 < P ≤ n.
Choosing ρ = G in [14, Theorem 2.1], we conclude for all φ ∈ C ∞ c (X 0 ) that
X0 |∇φ| P dx ≥ X0 W |φ| P dx for X 0 = X \ {o} if 1 < P ≤ n and X 0 = X if P > n.
Since {o} ⊂ X is a compact set of zero p-capacity, we can apply [14, Corollary 2.3] to extend the inequality to the whole manifold X in the case 1 < P ≤ n. This shows inequality (4.4). It remains to prove the asymptotics. Let us begin with the asymptotics as r → 0. Note that W (r) ∼ a(r) is equivalent to W (r) ∼ a(r) in the case lim r→0 a(r) = ∞, and it suffices to prove the asymptotics in (4.5) for W instead of W .
Let us first consider the case P > n. Using the estimate (4.3), we conclude that r 0 dt (f (t)) 1/(P −1) = O(r P −n P −1 ) as r → 0, and therefore, employing (4.2),
G ′ (r) G(r) = ∞ 0 dt (f (t)) 1/(P −1) −1 + O(r (P −n)/(P −1) ) (f (r)) −1/(P −1) ∼ ∞ 0 dt (f (t)) 1/(P −1) −1 r −(n−1)/(P −1) as r → 0.
This shows, in the case P > n, that
W (r) = P − 1 P P G ′ (r) G(r) P ∼ C P,f r −P (n−1)/(P −1) as r → 0.
Now, we turn to the case 1 < P < n. We have lim r→0 G(r) = ∞ by property (d) of the P -Green function and, by using (4.2),
lim r→0 rG ′ (r) = − 1 ω 1/(P −1) lim r→0 r r (n−1)/(P −1) = −∞.
We can apply L'Hôpital and obtain lim r→0 rG ′ (r) G(r)
= 1 + lim r→0 rG ′′ (r) G ′ (r) = 1 − 1 P − 1 lim r→0 rf ′ (r) f (r)
.
Note that f ′ (r)/f (r) is the mean curvature of metric spheres of radius r, and it is well-known for arbitrary Riemannian manifolds that f ′ (r)/f (r) ∼ (n − 1)/r as r → 0 (in the case of a harmonic manifold, we have rf ′ (r)/f (r) = trace(C) with C in [10, 6.33] and the statement follows from [10, 6.36]). Consequently, we have
G ′ (r) G(r) ∼ P − n P − 1 · 1 r as r → 0.
From this we conclude that, in the case 1 < P < n,
W (r) = P − 1 P P G ′ (r) G(r) P ∼ n − P P P r −P as r → 0.
Let us now consider the remaining case P = n. Similarly as before we have lim r→0 r log rG ′ (r) = −∞ and L'Hôpital yields lim r→0 r log rG ′ (r) G(r)
= 1 + lim r→0 log r G ′ (r) + rG ′′ (r) G ′ (r) = 1 + lim r→0 log r 1 − 1 P − 1 rf ′ (r) f (r) = 1 + lim r→0 log r P − n P − 1 + O(r 2 ) .
Here we used the slightly stronger fact that rf ′ (r)/f (r) = (n − 1) + O(r 2 ) as r → 0 (this follows, e.g., from [20,Lemma 12.2], where an explicit Taylor expansion is given up to the sixth term). Since we assume P = n, we conclude that G ′ (r) G(r) ∼ 1 r log r as r → 0, and therefore, in the case P = n,
W (r) = P − 1 P P G ′ (r) G(r) P ∼ P − 1 P P |r log r| −P as r → 0.
Finally, the proof of lim r→∞ W (r) = 0 is a straightforward application of L'Hôpital. We have
lim r→∞ G ′ (r) G(r) = − lim r→∞ 1/(f (r)) 1/(P −1) ∞ r dt (f (t)) 1/(P −1) = − 1 P − 1 lim r→∞ f ′ (r) f (r) = − h P − 1 ,
and therefore
lim r→∞ W (r) = P − 1 P P lim r→∞ G ′ (r) G(r) P = P − 1 P P h P − 1 P = Λ P .
Remark. It follows from [19, Theorem 1.5 (2)] that, in the case n < P < ∞, an optimal Hardy weight of −∆ P in X \ {o} is given by
W := P − 1 P P ∇G G P · |γ − 2G| P −2 |γ − G| P γ 2 + 2(P − 2)G(γ − G) with γ := G(0) = ∞ 0
dt (ωnf (t)) 1/(P −1) . Remark. In the special case of a Damek-Ricci space X = X p,q , the asymptotics of W for r → ∞ can be evaluated more precisely by adapting the computations in [7, p.153]: The density function of X p,q is given by f (r) = 2 p+q (sinh(r/2)) p+q (cosh(r/2)) q , the mean curvature of horospheres is given by h = p+2q 2 , and we obtain with s = sinh(t/2) and α = q 2(P −1) + 1 2 ,
(2 p+q ω n ) 1 P −1 G(r) = ∞ r (sinh(t/2)) − p+q (P −1) (cosh(t/2)) − q 2(P −1) dt = 2 ∞ sinh r 2 s − p+q P −1 1 + s 2 −α ds = 2 ∞ sinh r 2 s − p+q P −1 −2α 1 − α s 2 + o 1 s 3 ds = P − 1 h sinh r 2 − 2h P −1 −2α 2h P − 1 + 2 −1 sinh r 2 − 2h P −1 −2 +o sinh r 2 − 2h P −1 −3
for r → ∞. This implies that
G ′ (r) G(r) P = coth r 2 − P P −1 ·q P − 1 h − 2α 2h P − 1 + 2 −1 sinh r 2 −2 + o sinh r 2 −3 −P ,
and therefore the following asymptotics of W at infinity:
W (r) = h P P coth r 2 − P P −1 ·q 1 + 2αP h 2h + 2P − 2 sinh r 2 −2 + o sinh r 2 −3 .
The corresponding P -Rellich inequality can be found in [19,Theorem 7.3], and a quasi-linear version of the uncertainty principle follows by using the Hölder inequality properly (confer with Subsection 2.3).
A Non-compact harmonic manifolds and Damek-Ricci spaces
Let us start with the original definition of a harmonic manifold: A complete Riemannian manifold (X, g) is harmonic if, for all o ∈ X, there exists a local non-constant radial harmonic function, that is, a function g(x) = g(d(o, x)), depending only on the distance to o ∈ X and satisfying ∆g = 0 on a small punctured neighbourhood of o. There are various other equivalent definitions of harmonicity which can be found, e.g., in [46, p. 224], [10,Prop. 6.21] or [43,Théorème 4]. A very natural way to characterize harmonic manifolds (going back to Willmore) is to require that all harmonic functions satisfy the Mean Value Property. This characterization was given in the introduction. It is known for any non-compact harmonic manifold that its volume density f (r), r = d(o, ·), is a strictly positive exponential polynomial, that is, a finite sum k i=1 (p i (r) sin(β i r) + q i (r) cos(β i r))e αi r (see [35,Theorem 2]). Moreover, if a non-compact harmonic manifold of dimension n has sub-exponential volume growth, then it must be isometric to the flat Euclidean space R n (this follows from [41]).
Let us now focus on an explicit family of non-compact harmonic manifolds, namely the Damek-Ricci spaces. They are particular Lie groups with left-invariant metrics. Their definition requires some preparation.
A 2-step nilpotent Lie algebra n with center z and an inner product ·, · : n × n → R is called a Lie algebra of Heisenberg-type if it has a orthogonal decomposition n = v ⊕ z and if the linear maps J Z : v → v for each Z ∈ z, defined by
J Z V, V ′ = Z, [V, V ′ ] satisfy J 2 Z (V ) = − Z 2 V. (A.1)
A Heisenberg-type Lie algebra n comes with the two parameters p, q ≥ 1 which are the dimensions p = dim(v) and q = dim(z). The above conditions restrict the possible pairs (p, q), for example, it follows from (A.1) that p needs to be an even number. Let a be a 1-dimensional Lie algebra generated by H ∈ a, that is, a = RH. We extend a Lie algebra n of Heisenberg-type to a solvable Lie algebra for all V ∈ v and Z ∈ z. We also extend the inner product on n to an inner product ·, · : s × s → R, by setting H = 1 and requiring that the decomposition (A.2) is orthogonal. We can now give the definition of a Damek-Ricci space.
Definition A.1. A Damek-Ricci space X p,q is the unique connected and simply connected Lie group S = N A associated to the solvable extension s of a Heisenberg-type Lie algebra n with parameters p, q by a = RH, as described above, and equipped with a left invariant metric such that the Riemannian metric of T e S agrees with the inner product on s under the canonical identification T e S ∼ = s, where e ∈ S denotes the neutral element of S. The space X p,q has therefore the dimension n = p + q + 1.
Finally, let us briefly explain how the real hyperbolic space H n can be described as a solvabe Lie group with left-invariant metric, even though this space is associated to a Lie algebra s = n ⊕ a where n is not 2-step nilpotent but abelian. For that reason, real hyperbolic spaces are not contained in the class of Damek-Ricci spaces.
Example A.2. The upper half space model of the real hyperbolic space H n is given by H n = {(x, y) : x ∈ R n−1 , y > 0} with Riemannian metric
g H n (v, w) = v, w 0 y 2 for v, w ∈ T (x,y) ∼ = R n ,
where ·, · 0 denotes the Euclidean inner product of R n . For this space we have λ 0 (H n ) = (n−1) In other words, S = N A can be identified with a semidirect product R n−1 ⋊ R + , where the group operation in R n−1 is addition and the group operation in R + is multiplication. We transfer the inner product of s to an inner product on T e S via the identification Z → d dt | t=0 exp(tZ) and extend it left-invariantly to a Riemannian metric g S on S. Then the map exp(n(x)) exp(sH) → (x, e s ) ∈ H n is an isometry between (S, g S ) and (H n , g H n ), which implies that (S, g S ) is a model of the n-dimensional real hyperbolic space.
Proposition 2 . 3 (
23Khas'minskiȋ-type criterion). Let K be a compact subset of X. If u and v ∞ are two positive functions on X \ K which are solutions of(−∆ + V )u = 0 in X \ K, and if lim x→∞ u(x) v ∞ (x) = 0,then u has minimal growth at infinity. If u and also v 0 are two positive functions in a punctured neighbourhood Ω of the pole o which are solutions of (−∆ + V )u = 0 in Ω \ {o}, u has minimal growth at o.
Lemma 2.6 (see[17, Theorem 1.5.12]). Let Ω ⊂ X be a domain in a Riemannian manifold (X, g), H := −∆ + V − W , and Φ ∈ C ∞ (Ω) be a positive solution of HΦ = 0 on Ω. Then we have, for all
Proposition 3 . 2 (
32Liouville comparison principle, Theorem 1.7 in [38]). Let Ω be a domain in a non-compact n-dimensional Riemannian manifold. Consider two Schrödinger operators on Ω defined via
Proof of Theorem 3. 1 .
1Let h : X \ {o} → (0, ∞) be a smooth radial function satisfying h(r) ≥ Cr for some positive constant C > 0. Applying Lemma 2.5 withΦ = r α f β and (α,
2 4 .
4For the description of H n as a solvable Lie group S = N A, we start with the Lie algebras n = n(x) : x ∈ R n−1 and a = {diag(0, 0, . . . , 0, t, −t) : t ∈ R} , It is easy to see that [H, X] = HX − XH = X and [X, Y ] = XY − Y X = 0 for all X, Y ∈ n. An inner product on the solvable Lie algebra s = n ⊕ a with Lie bracket [Z 1 , Z 2 ] = Z 1 Z 2 − Z 2 Z 1 is given by the requirements that n and a are perpendicular, H = 1, and n(x), n(y) = x, y 0 , where ·, · 0 denotes the Euclidean inner product on R n−1 . The maps = n ⊕ a → S = N A, (n(x), tH) → exp(n(x)) exp(bijection. If we identify S = N A with R n−1 × R + via exp(n(x)) exp(tH) → (x, e t ),the Lie group multiplication in S takes the form (y, e s ) · (x, e t ) = (y + e s x, e s+t ).
Some background from Optimality Theory . . . . . . . . . . . . . 6 2.2 Proof of the Poincaré-Hardy-type inequality . . . . . . . . . . . . 8 2.3 Applications of the Poincaré-Hardy-type inequality . . . . . . . A family of Hardy-type inequalities . . . . . . . . . . . . . . . . . 15 3.2 Weighted Hardy-type inequalities . . . . . . . . . . . . . . . . . . 18 3.3 A Poincaré-Hardy-type inequality for the P -Laplacian . . . . . . 201 Introduction
2
2 A Poincaré-Hardy-type inequality and applications
6
2.1 . 13
2.3.1 A Heisenberg-Pauli-Weyl's Uncertainty Principle . . . . . 13
2.3.2 A Poincaré-Rellich-type inequality . . . . . . . . . . . . . 14
3 Variations of the main result
15
3.1
Acknowledgements: F. F. wants to thank the University of Durham for their kind hospitality during a stay where the main idea of the paper was worked out. Moreover, F. F. wants to thank Noema Nicolussi for pointing out paper[9], and also the Heinrich-Böll-Stiftung for the support.The basic idea is to use Lemma 3.6, i.e., we need to show that Φ is a supersolution with respect to the corresponding P -Schrödinger operator, i.e., the value inside the latter parenthesis should be non-negative. Since f ′ /f > 0 (see[41,Prop. 2.2]Once this is shown, the lemma provides us with a particular Hardy inequality. However, it seems that we need more knowledge about f to prove non-negativity of this term. Since our Riemannian manifold is the Damek-Ricci space X p,q with the explicit expression (1.5) for the volume density f , we conclude from Lemma 2.8 by setting a = 1 − P (P − 1) and b = −P (P − 1) and the relation 4λUsing the assumption n − 1 = p + q ≥ P (P − 1), we estimatewhere we used sinh(r) = 2 sinh(r/2) cosh(r/2) and cosh(r/2) ≥ 1 for all r ∈ R in the last inequality. This shows on the one hand that the last integrand on the right hand side of (3.6) is non-negative, and also thatHence, for P (P − 1) ≤ n − 1 and P ≥ 2, Φ is a positive supersolution. By Lemma 3.6, we conclude the non-negativity of the corresponding energy functional.To finish the proof of the theorem, we only need to analyse ∆ P Φ(r) = |Φ ′ (r)| P −2 L P Φ(r). Note thatOne could use (1.7) and estimate | 1for a simplification. But we want to take the precise term instead and write with the help of (1.6),with g and h = h(X p,q ) defined as in the theorem. This yields the estimate g(r) ≥ p+q−1 h·r in the theorem as well as the asymptotics g(r) → 1 as r → ∞.
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[
"WHAT IS THE STANDARD MODEL HIGGS ?",
"WHAT IS THE STANDARD MODEL HIGGS ?"
] |
[
"Afsar Abbas :[email protected] \nInstitute of Physics\nBhubaneswar-751005India\n"
] |
[
"Institute of Physics\nBhubaneswar-751005India"
] |
[] |
It is shown that in the Standard Model, the property of charge quantization holds for a Higgs with arbitrary isospin and hypercharge. These defining quantum numbers of the Higgs remain unconstrained while the whole basic and fundamental structure of the Standard Model remains intact. Hence it is shown that the Higgs cannot be a physical particle. Higgs is the underlying 'vacuum' over which the whole edifice of the Standard Model stands.
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"https://export.arxiv.org/pdf/hep-ph/9912243v1.pdf"
] | 18,040,990 |
hep-ph/9912243
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68bf0c85e15fd4e2239e26fde726f9649f7eb590
|
WHAT IS THE STANDARD MODEL HIGGS ?
5 Dec 1999
Afsar Abbas :[email protected]
Institute of Physics
Bhubaneswar-751005India
WHAT IS THE STANDARD MODEL HIGGS ?
5 Dec 1999
It is shown that in the Standard Model, the property of charge quantization holds for a Higgs with arbitrary isospin and hypercharge. These defining quantum numbers of the Higgs remain unconstrained while the whole basic and fundamental structure of the Standard Model remains intact. Hence it is shown that the Higgs cannot be a physical particle. Higgs is the underlying 'vacuum' over which the whole edifice of the Standard Model stands.
The Standard Model (SM) of particle physics is a well-tested and established model. Higgs boson is the only missing link. Much effort, both theoretical and experimental (for a review see [ 1 ]) worldwide is being put to understand and detect it. It is a common belief that future colliders like LHC shall detect it in the near future. But do we understand Standard Model well enough to speak with confidence of the impending discovery of Higgs boson ?
The answer is No ! Until 1989/90 it was commonly believed (unfortunately many still believe it) that the SM does not have electric charge quantization built into it and that one has to go to the Grand Unified Theories (GUTs) to obtain charge quantization [ 2 ]. In fact one of the main motivations of invoking GUTs was precisely this. But it was shown that electric charge quantization is built into the SM [ 3,4,5,6 ]. However, some authors were making redundant assumptions of arbitrarily fixing up the Higgs hypercharge in these derivations [ 3,4 ]. This weakened their charge quantization arguments and was corrected by the author [ 5,6 ]. Pitfalls in such weakening assumptions were also pointed out by the author [ 6 ]. Until very recently, it was believed that the SM allows for the existence of the millicharged particles. This is not true [ 6 ] and the point was clarified in a recent letter by the author [ 7 ].
So has the last hurdle in understanding the SM been crossed ? No, because we have not yet found the Higgs. In this paper, the author would show that this is due to deeper reasons. We still have not understood the basic underlying structure of the SM and hence it is very relevant to ask, what is the standard model Higgs ?
The SM assumes a repetitive structure for each generation of quarks and leptons. The anomalies cancel generation by generation. Why this is so, is not answered by the SM. We accept this as a fact (any attempt to do so, would take one beyond the SM) and try to understand the deep and basic underlying structure of the SM.
Let us start by looking at the first generation of quarks and leptons (u, d, e, ν ) and assign them to
SU(N C ) ⊗ SU(2) L ⊗ U(1) Y (where N C = 3 )
representation as follows.
q L = u d L , (N C , 2, Y q ) u R ; (N C , 1, Y u ) d R ; (N C , 1, Y d ) l L = ν e ; (1, 2, Y l ) e R = (1, 1, Y e )(1)
To keep things as general as possible this brings in five unknown hypercharges.Let us now define the electric charge in the most general way in terms of the diagonal generators of SU(2) L ⊗ U(1) Y as
Q ′ = a ′ T 3 + b ′ Y(2)
We can always scale the electric charge once as
Q = Q ′ a ′ and hence (b = b ′ a ′ ) Q = T 3 + bY(3)
In the SM SU(N C ) ⊗ SU(2) L ⊗ U(1) Y is spontaneously broken through the Higgs mechanism to the group SU(3) c ⊗ U(1) em . In SM the Higgs is assumed to be a doublet. However we do not use this restriction either and assume the Higgs φ to have any isospin T and arbitrary hypercharge Y φ . The isospin T φ 3 component of the Higgs develops a nonzero vacuum expectation value < φ > o . Since we want the U(1) em generator Q to be unbroken we require Q < φ > o = 0. This right away fixes b in (3) and we get
Q = T 3 − ( T φ 3 Y φ )Y(4)
For the SM to be renormalizable we require that the triangular anomaly be canceled. This leads to three constraints.
T rY [SU(N C )] 2 = 0 (5) T rY [SU(2) L ] 2 = 0 (6) T r[Y 3 ] = 0(7)
The expression (6) yields
Y q = − Y l N C(8)
Next we use the fact that after the spontaneous breaking of
SU(N C ) ⊗ SU(2) L ⊗ U(1) Y to SU(N C ) ⊗ U(1) em , the L-and R-handed charges couple identically with photon. Using Q(u L ) = Q(u R ); Q(d L ) = Q(d R ); Q(e L ) = Q(e R ) yields respectively Y u = Y q − Y φ 2T φ 3 (9) Y d = Y q + Y φ 2T φ 3 (10) Y e = Y l + Y φ 2T φ 3(11)
When these are used in conjunction with the condition (7) one finds
Y l = Y φ 2T φ 3(12)
Hence
Y q = − Y φ 2T φ 3 N C(13)
When these hypercharges are put in eq. (4) for the electric charge one obtains :
Q(u) = 1 2 (1 + 1 N c ) Q(d) = 1 2 (−1 + 1 N c ) (14) Q(e) = −1 (15) Q(ν) = 0(16)
For N C = 3 these are the correct charges in the SM. Note that this charge quantization in the SM holds for Higgs for arbitrary T and arbitrary hypercharge. Hence as far as charge quantization is concerned, the values of T and Y φ remain unconstrained. This point for the special case of the Higgs doublet was already noted by the author earlier [ 5,6 ].
Let us continue with the rest of the structure of the SM and see how our general Higgs with unconstrained and unspecified isospin T and hypercharge Y φ fits into it. We can write the covariant derivative of the SM as
D µ = ∂ µ + ig 1 T φ 3 Y φ Y B µ − ig 2 T . W µ(17)
As the kinetic energy term for the gauge fields is
− 1 4 W µν a W aµν − 1 4 B µν B µν(18)
Only orthogonal combination of W 0 µ and B 0 µ would have independent kinetic energy terms. Hence the photon field A µ and the orthogonal Z µ are written as
A µ = g 2 B µ + g 1 ( 2T φ 3 Y φ Y l )W 0 µ g 2 2 + (g 1 2T φ 3 Y φ Y l ) 2 (19) Z µ = −g 1 ( 2T φ 3 Y φ Y L )B µ + g 2 W 0 µ g 2 2 + (g 1 2T φ 3 Y φ Y l ) 2(20)
with Y l from eq. (12) we get
A µ = g 2 B µ + g 1 W 0 µ g 2 1 + g 2 2(21)
and
Z µ = −g 1 B µ + g 2 W 0 µ g 2 1 + g 2 2(22)
With D µ given by eq.(17) we can write the lepton part of the SM Lagrangian as
L(lepton) =q L iγ µ (ig 1 T φ 3 Y φ Y l B µ )q L +ē R iγ µ (ig 1 T φ 3 Y φ Y e B µ )e R −q L iγ µ ig 2 T . W µ q L(23)
The electron term in the Lagrangian with B µ , W 0 µ replaced from eqs. (21) and (22) becomes
A µ ē L γ µ e L ( G 1 g 2 Y l X +ē R γ µ e R ( G 1 g 2 Y e 2X ) +Z µ ē L γ µ e L ( G 2 1 Y 2 l − g 2 2 2X ) +ē R γ µ e R ( G 2 1 Y e Y l 2X ) (24) where G 1 = − 2g 1 T φ 3 Y φ and X = g 2 2 + G 2 1 Y 2 l
As the electromagnetic interaction of particles with charge Q is
L em = QA µ [ē L γ µ e L +ē R γ µ e R ](25)
This form above is true provided Y e = 2Y l . This is exactly what we had found above in the context of the electric charge quantization.
As we require Q = −e for electron we find e = g 1 g 2
g 2 1 + g 2 2 (26) with g 2 = e sinθ W , g 1 = e cosθ W(27)
We also find that electron coupling to Z µ and neutron coupling to Z µ are as in the SM.
The point to be emphasized is that the whole structure of the SM stands and is independent of Higgs isospin and hypercharge, which all throughout remain unconstrained and undetermined. One should not fix any arbitrary values for them as nothing in the theory demands it.
We also find that
ρ = m 2 W m 2 Z Cos 2 (θ W ) = T (T + 1) − T φ 3 2 2T φ 3 2 (28) ρ = 1 when T (T + 1) − 3T φ 3 = 0 (29)
T = 1/2.T = ±1/2 does satisfy (29). Next solution is T = 3, T 3 ± 2 . Actually, as pointed out by Tsao [ 8 ] the complete set of solutions of (29) is infinite. One can generate all solutions from the lowest doublet solution by rewriting (29) as
X 2 − 3Y 2 = 1 (30) where T = X−1 2 , T 3 = ± Y 2 .
This is a special case of the Pell equation in number theory. All the solutions ( X n , Y n ), n = 1, 2, ... can be obtained from integer solutions of
X n + √ 3Y n = (2 + √ 3) n(31)
The solutions of this are infinite in number. Hence for ρ = 1, T, T 3 are infinite in number. Again, nothing in theory demands that one fix this to a particular value.
The point is that the full structure of the SM stands intact without constraining the quantum numbers isospin and/or the hypercharge of the Higgs to any specific value. All the particles that have been isolated in the laboratory or have been studied by any other means, besides having a specific mass, have definite quantum numbers which identify them. In the case of Higgs here, no one knows of its mass and more importantly its quantum numbers like isospin and hypercharge, as shown above, are not specified. The hypercharge of all the other particles are specified as being proportional to the Higgs hypercharge which itself remains unconstrained. That is, all the hypercharges of particles are rooted on to the Higgs hypercharge which itself remains free and unspecified. Hence Higgs is very different from all known particles. Because of the above reasons Higgs cannot be a physical particle which may be isolated and studied. It must be just the 'vacuum' which sets up the structure of the whole thing.
Higgs is a manifestation of the vacuum structure of the SM. Higgs shall never get pinned down as an isolated physical particle, but makes its presence felt through charge quantization and giving the SM its complete structure and consistency. Hence it is predicted that Higgs shall not be discovered as a particle.
No basic principle demands that the mass of the matter particle be given by Yukawa interaction, but since as we have no idea of where these masses come from, one just demands that they arise from such a coupling. If this be so then the Higgs isospin is necessarily T = 1/2. This just tells you that the 'vacuum' has this particular structure. But as Y φ is not constrained in any way, the Higgs cannot be a particle but just 'vacuum' which behaves in this fundamental and basic manner.
In conventional SM there is just one Higgs doublet and which gives e−µ− τ universality for the 3 generations. In our present calculations there is no reason that Higgs structure for each generation may be the same. There could be three different Higgs 'vacuum' structures and yet give e−µ−τ universality. Instead of being a static Higgs the Higgs in our view is a more dynamic 'vacuum'. In our model, as lack of Yukawa coupling implies unconstrained isospin of the Higgs, hence custodial symmetry question becomes doubtful. So should monopoles as well. The fact that Higgs is not a particle should also help in solving the cosmological constant problem.
In summary, we have shown that the basic and fundamental structure of the standard model stands intact without specifying and constraining the quantum numbers of the Higgs. As such Higgs is very different from any known physical particle. Hence Higgs cannot be a 'particle' but represents the omnipresent vacuum which provides the 'root' to support the Standard Model.
The Higgs Hunter's Guide. G F Gunion, H E Haber, G Kane, S Dawson, Addison-Wesley Pub. CoRedwood, California, USAG.F.Gunion, H.E.Haber, G.Kane and S. Dawson, "The Higgs Hunter's Guide", Addison-Wesley Pub. Co., Redwood, California, USA (1990).
Particle Physics and Cosmology. P D B Collins, A D Martin, E J Squires, John Wiley & SonsNew York, USAP.D.B.Collins, A.D.Martin and E.J.Squires, "Particle Physics and Cos- mology", John Wiley & Sons, New York, USA (1989)
. K S Babu, R N Mohapatra, Phys. Rev. Letts. 63938K.S.Babu and R.N.Mohapatra, Phys. Rev. Letts. 63 (1989) 938
. X.-G He, G C Joshi, H Lew, B H J Mckellar, R R Volkas, Phys. Rev. D. 403140X.-G.He, G.C.Joshi, H.Lew, B.H.J.McKellar and R.R.Volkas, Phys. Rev. D 40 (1989) 3140
. A Abbas, Phys. Lett. 238 B. 344A. Abbas, Phys. Lett. 238 B (1990) 344
. A Abbas, J. Phys. G. 16163A. Abbas, J. Phys. G. 16 (1990) L163
. A Abbas, Physics Today. A. Abbas, Physics Today , July 1999, p.81-82
H.-S Tsao, Proc. 1980 Guangzhou Conf. on Theo. Part. Phys. 1980 Guangzhou Conf. on Theo. Part. PhysEdH.-S.Tsao, Proc. 1980 Guangzhou Conf. on Theo. Part. Phys.; Ed.
. H Ning, T Hung-Yuan, Science Press1240BeijingH.Ning and T. Hung-Yuan, (Science Press, Beijing, 1980), p.1240
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[] |
[
"The Geodesic Diameter of Polygonal Domains *",
"The Geodesic Diameter of Polygonal Domains *"
] |
[
"Sang Won Bae [email protected] \nDepartment of Computer Science and Engineering\nPOSTECHPohangKorea\n",
"Matias Korman [email protected] \nComputer Science Department\nUniversité Libre de Bruxelles\nBelgium\n",
"Yoshio Okamoto [email protected] \nGraduate School of Information Science and Engineering\nTokyo Institute of Technology\nTokyoJapan\n"
] |
[
"Department of Computer Science and Engineering\nPOSTECHPohangKorea",
"Computer Science Department\nUniversité Libre de Bruxelles\nBelgium",
"Graduate School of Information Science and Engineering\nTokyo Institute of Technology\nTokyoJapan"
] |
[] |
This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), it is known that the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time. For general polygonal domains with h ≥ 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n 7.73 ) or O(n 7 (log n + h)). The algorithms are based on our new geometric observations, part of which states as follows: the geodesic diameter of a polygonal domain can be determined by two points in its interior, and in that case there are at least five shortest paths between the two points.
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10.1007/s00454-013-9527-8
|
[
"https://arxiv.org/pdf/1001.0695v2.pdf"
] | 9,306,114 |
1001.0695
|
1de397fbf1efcbb475219fb8817d90f358a79637
|
The Geodesic Diameter of Polygonal Domains *
5 Jan 2010
Sang Won Bae [email protected]
Department of Computer Science and Engineering
POSTECHPohangKorea
Matias Korman [email protected]
Computer Science Department
Université Libre de Bruxelles
Belgium
Yoshio Okamoto [email protected]
Graduate School of Information Science and Engineering
Tokyo Institute of Technology
TokyoJapan
The Geodesic Diameter of Polygonal Domains *
5 Jan 2010
This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), it is known that the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time. For general polygonal domains with h ≥ 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n 7.73 ) or O(n 7 (log n + h)). The algorithms are based on our new geometric observations, part of which states as follows: the geodesic diameter of a polygonal domain can be determined by two points in its interior, and in that case there are at least five shortest paths between the two points.
Introduction
A polygonal domain P with h holes and n corners V is a connected and closed subset of R 2 of genus h whose boundary ∂P consists of h + 1 simple closed polygonal chains of n total line segments. The holes and the outer boundary of P are regarded as obstacles so that any feasible path in P is not allowed to cross the boundary ∂P. The geodesic distance d(p, q) between any two points p, q in a polygonal domain P is defined as the (Euclidean) length of a shortest obstacle-avoiding path between p and q.
In this paper, we address the geodesic diameter problem in polygonal domains. The geodesic diameter diam(P) of a polygonal domain P is defined as diam(P) := max s,t∈P d(s, t). A pair (s, t) of points in P that realizes the geodesic diameter diam(P) is called a diametral pair. The geodesic diameter problem is to find the value of diam(P) and a diametral pair.
For simple polygons (i.e., h = 0), the geodesic diameter has been extensively studied and fully understood. Chazelle [6] provided the first O(n 2 )-time algorithm computing the geodesic diameter of a simple polygon, and Suri [18] presented an O(n log n)-time algorithm that solves the all-geodesicfarthest neighbors problem, computing the farthest neighbor of every corner and thus finding the geodesic diameter. At last, Hershberger and Suri [11] showed that the diameter can be computed in linear time using their fast matrix search technique.
On the other hand, to the best of our knowledge, no algorithm for computing diam(P) has yet been discovered when P is a polygonal domain having one or more holes (h ≥ 1). Mitchell [14] has posed an open problem asking an algorithm for computing the geodesic diameter diam(P). However, even for the corner-to-corner diameter max u,v∈V d(u, v), only known is a brute-force algorithm that takes O(n 2 log n) time, checking all the geodesic distances between every pair of corners. 1 This fairly wide gap between simple polygons and polygonal domains is seemingly due to the uniqueness of the shortest path between any two points; it is well known that there is a unique shortest path between any two points in a simple polygon [9]. Using this uniqueness, one can show that the diameter is indeed realized by a pair of corners in V ; that is, diam(P) = max u,v∈V d(u, v) if h = 0 [11,18]. For general polygonal domains with h ≥ 1, however, this is not the case. In this paper, we exhibit several examples where the diameter is realized by non-corner points on ∂P or even by interior points of P. (See Figure 1 and Appendix A.) This observation also shows an immediate difficulty in devising any exhaustive algorithm since the search space like ∂P or the whole domain P is not discrete.
The status of the geodesic center problem is also similar. The geodesic center is defined to be a point in P that minimizes the maximum geodesic distance from it to any other point of P. Asano and Toussaint [3] introduced the first O(n 4 log n)-time algorithm for computing the geodesic center of a simple polygon, and Pollack, Sharir and Rote [17] improved it to O(n log n) time. As with the diameter problem, there is no known algorithm for general polygonal domains. Note that computing the geodesic center involves computing the geodesic diameter because the geodesic center may be determined by the midpoint of a shortest path defining the geodesic diameter. See O'Rourke and Suri [16] and Mitchell [14] for more references on the geodesic diameter/center problem in simple polygons and polygonal domains.
Since the geodesic diameter/center of a simple polygon is determined by its corners, one can exploit the geodesic farthest-site Voronoi diagram of the corners V to compute the diameter/center, which can be built in O(n log n) time [2]. Recently, Bae and Chwa [4] presented an O(nk log 3 (n + k))-time algorithm for computing the geodesic farthest-site Voronoi diagram of k sites in a general polygonal domain. This can be used to compute the geodesic diameter max p,q∈S d(p, q) of a finite set S of points in P, but cannot be exploited for computing diam(P) without any characterization of the geodesic diameter of polygonal domains with h ≥ 1. Moreover, when S = V , this approach is no better than the brute-force O(n 2 log n)-time algorithm for computing the corner-to-corner diameter max u,v∈V d(u, v).
In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in O(n 7.73 ) or O(n 7 (log n + h)) time in the worst case. We also show that for small constant h Figure 1: Three polygonal domains where the geodesic diameter is determined by a pair (s * , t * ) of non-corner points; Gray-shaded regions depict the interior of the holes and dark gray segments depict the boundary ∂P. Recall that P, as a set, contains its boundary ∂P. (a) Both s * and t * lie on ∂P. There are three shortest paths between s * and t * . In this polygonal domain, there are two (symmetric) diametral pairs. (b) s * ∈ ∂P \ V and t * ∈ intP. Three triangular holes are placed in a symmetric way. There are four shortest paths between s * and t * . (c) Both s * and t * lie in the interior intP. Here, the five holes are packed like jigsaw puzzle pieces, forming narrow corridors (dark gray paths) and two empty, regular triangles. Observe that d(
1 Personal communication with Mitchell. t * s * (a) (b) (c) u 1 u 2 u 3 v 3 v 2 v 1 s * t * s * t *u 1 , v 1 ) = d(u 1 , v 2 ) = d(u 2 , v 2 ) = d(u 2 , v 3 ) = d(u 3 , v 3 ) = d(u 3 , v 1 )
. s * and t * lie at the centers of the triangles formed by the u i and the v i , respectively. There are six shortest paths between s * and t * . More details on this example can be found in Appendix A.2. the diameter can be computed much faster. Our new geometric results underlying the algorithms indeed show that the existence of any diametral pair consisting of non-corner points implies multiple shortest paths between the pair; as one of the cases, it is shown that if (s, t) is a diametral pair and both s and t lie in the interior of P, then there are at least five shortest paths between s and t.
Some analogies between polygonal domains and convex polytopes in R 3 can be seen. O'Rourke and Schevon [15] proved that if the geodesic diameter on a convex 3-polytope is realized by two non-corner points, at least five shortest paths exist between the two. Based on this observation, they presented an O(n 14 log n)-time algorithm for computing the geodesic diameter on a convex 3-polytope. Afterwards, the time bound has been improved to O(n 8 log n) by Agarwal et al. [1] and recently to O(n 7 log n) by Cook IV and Wenk [8]. This is also compared with the running time of our algorithms for polygonal domains, O(n 7.73 ) or O(n 7 (log n + h)).
Preliminaries
Throughout the paper, we frequently use several topological concepts such as open and closed subsets, neighborhoods, and the boundary ∂A and the interior intA of a set A; unless stated otherwise, all of them are supposed to be derived with respect to the standard topology on R d with the Euclidean norm · for fixed d ≥ 1. We denote the straight line segment joining two points a, b by ab. We are given as input a polygonal domain P with h holes and n corners. More precisely, P consists of an outer simple polygon in the plane R 2 and a set of h (≥ 0) disjoint simple polygons inside the outer polygon. As a subset of R 2 , P is the region contained in its outer polygon excluding the interior of the holes; thus P is a bounded, closed subset of R 2 . The boundary ∂P of P is regarded as a series of obstacles so that any feasible path inside P is not allowed to cross ∂P. Note that some portion or the whole of a feasible path may go along the boundary ∂P. The length of a path is the sum of the Euclidean lengths of its segments. It is well known from earlier work that there always exists a shortest (feasible) path between any two points p, q ∈ P [13]. The geodesic distance, denoted by d(p, q), is then defined to be the length of a shortest path between p ∈ P and q ∈ P.
Shortest path map. Let V be the set of all corners of P and π(s, t) be a shortest path between s ∈ P and t ∈ P. Then, it is represented as a sequence π(s, t) = (s, v 1 , . . . , v k , t) for some v 1 , . . . , v k ∈ V ; that is, a polygonal chain through a sequence of corners [13]. Note that possibly we may have k = 0 when d(s, t) = s − t . If two paths (with possibly different endpoints) induce the same sequence of corners, then they are said to have the same combinatorial structure.
The shortest path map SPM(s) for a fixed s ∈ P is a decomposition of P into cells such that every point in a common cell can be reached from s by shortest paths of the same combinatorial structure. Each cell σ s (v) of SPM(s) is associated with a corner v ∈ V or s itself, which is the last corner of π(s, t) for any t in the cell σ s (v). In particular, the cell σ s (s) is the set of points t such that π(s, t) passes through no corner in V and thus d(s, t) = s − t . Each edge of SPM(s) is an arc on the boundary of two incident cells σ s (v 1 ) and σ s (v 2 ) and thus determined by two corners v 1 , v 2 ∈ V ∪ {s}. Similarly, each vertex of SPM(s) is determined by at least three corners v 1 , v 2 , v 3 ∈ V ∪ {s}. Note that for fixed s ∈ P a point t that locally maximizes d s (t) := d(s, t) lies at either (1) a vertex of SPM(s), (2) an intersection between the boundary ∂P and an edge of SPM(s), or (3) a corner in V .
The shortest path map SPM(s) has O(n) complexity can be computed in O(n log n) time using O(n log n) working space [12]. For more details on shortest path maps, see [12][13][14].
Path-length function. If π(s, t) = st, then there are two corners u, v ∈ V such that π(s, t) is formed as the union of a shortest path from u to v and two segments su and vt. Note that u and v are not necessarily distinct. In order to realize such a path, we assert that s is visible from u and t is visible from v; thus, s ∈ VP(u) and t ∈ VP(v), where VP(p) for any p ∈ P is defined to be the set of all points q ∈ P such that pq ⊂ P. The set VP(p) is also called the visibility profile of p ∈ P [7].
We now define the path-length function len u,v : VP(u) × VP(v) → R for any fixed pair of corners
u, v ∈ V to be len u,v (s, t) := s − u + d(u, v) + v − t .
Then, len u,v (s, t) represents the length of the path from s to t that has the fixed combinatorial structure, entering u from s and exiting v to t. Also, unless d(s, t) = s − t (equivalently, s ∈ VP(t)), the geodesic distance d(s, t) can be expressed as the pointwise minimum of some path-length functions:
d(s, t) = min u∈VP(s), v∈VP(t)
len u,v (s, t).
Consequently, we have two possibilities for a diametral pair (s * , t * ); either we have d(s * , t * ) = s * − t * or the pair (s * , t * ) is a local maximum of the lower envelope of several path-length functions.
Local Maxima on the Lower Envelope of Convex Functions
In this section, we give some analysis on local maxima of the lower envelope of convex functions, which provides a key observation for our further discussions on the geodesic diameter and diametral pairs.
We start with a basic observation on the intersection of hemispheres on a unit sphere in the ddimensional space R d . For any fixed positive integer d, let S d−1 := {x ∈ R d | x = 1} be the unit sphere in R d centered at the origin. A closed (or open) hemisphere on S d−1 is defined to be the intersection of S d−1 and a closed (open, respectively) half-space of R d bounded by a hyperplane that contains the origin. We call a k-dimensional affine subspace of R d a k-flat. Note that a hyperplane in R d is a (d − 1)-flat and a line in R d is a 1-flat. Also, the intersection of S d−1 and a k-flat through the origin in R d is called a great (k − 1)-sphere on S d−1 . Note that a great 1-sphere is called a great circle and a great 0-sphere consists of two antipodal points. Then, we observe the following. Lemma 1 For any two positive integers d and m ≤ d, a set of any m closed hemispheres on S d−1 has a nonempty common intersection. Moreover, if the intersection has an empty interior relative to S d−1 , then it includes a great (d − m)-sphere on S d−1 .
Proof. Proof can be found in Appendix B.
Using Lemma 1 we prove the following theorem, which is the goal of this section. Theorem 1 Let F be a finite family of real-valued convex functions defined on an open and convex subset C ⊆ R d and g(x) := min f ∈F f (x) be their pointwise minimum. Suppose that g attains a local maximum at x * ∈ C and there are exactly m functions f 1 , . . . , f m ∈ F such that m ≤ d and f i (x * ) = g(x * ) for all i = 1, . . . , m. If none of the f i attains a local minimum at x * , then there exists a
(d + 1 − m)-flat ϕ ⊂ R d through x * such that g is constant on ϕ ∩ U for some neighborhood U ⊂ R d of x * with U ⊂ C.
Proof. Let x * ∈ C and m be as in the statement. For each i, consider the sublevel set
L i := {x ∈ C | f i (x) ≤ f i (x * )}.
Since each f i is convex and x * does not minimize f i , the set L i is convex and x * lies on the boundary ∂L i of L i . Therefore, there exists a supporting hyperplane h i to L i at x * . Denote by h ⊕ i the closed half-space that is bounded by h i and does not contain
L i . Note that f i (x * ) ≤ f i (x) for any x ∈ h ⊕ i ∩ C and f i (x * ) < f i (x) for any x ∈ (h ⊕ i \ h i ) ∩ C. Let H i := {x − x * | x ∈ h ⊕ i , x − x * = 1}
be a closed hemisphere on the unit sphere S d−1 centered at the origin.
Since g(x * ) = f i (x * ) for any i ∈ {1, . . . , m} and x * is a local maximum of g, the intersection H i has an empty interior relative to
S d−1 ; otherwise, there exists y ∈ S d−1 such that f i (x * + λy) > f i (x * )
for any i ∈ {1, . . . , m} and any λ > 0 with x * + λy ∈ C. Hence, by Lemma 1, H i has a nonempty intersection including a great
(d − m)-sphere G on S d−1 . Let ϕ be the corresponding (d − m + 1)-flat in R d through x * defined as ϕ := {x * + λy ∈ R d | y ∈ G and λ ∈ R}. Consider the restriction f i | ϕ∩C of f i on ϕ ∩ C.
Since f i is convex and ϕ is an affine subspace (thus convex), f i | ϕ∩C is also convex and their pointwise minimum g| ϕ∩C attains a local maximum at x * . Furthermore, each f i | ϕ∩C attains a local minimum at
x * ; since ϕ ⊆ h ⊕ i , we have f i (x * ) ≤ f (x) for any point x ∈ ϕ ∩ C. Hence, g| ϕ∩C also attains a local minimum at x * since g(x * ) = f i (x * ) for any i ∈ {1, . . . , m}. Consequently, g is locally constant at x * on ϕ; more precisely, there is a sufficiently small neighborhood U ⊂ R d of x * with U ⊂ C such that g is constant on U ∩ ϕ, completing the proof.
Remark that the theorem should have its own interest and find an application in problems of maximizing the pointwise minimum of several convex functions.
Properties of Geodesic-Maximal Pairs
We call a pair (s * , t * ) ∈ P × P maximal if (s * , t * ) is a local maximum of the geodesic distance function d. That is, (s * , t * ) is maximal if and only if there are two neighborhoods U s , U t ⊂ R 2 of s * and of t * , respectively, such that for any s ∈ U s ∩ P and any t ∈ U t ∩ P we have d(s * , t * ) ≥ d(s, t). For any pair (s, t), let Π(s, t) = {π 1 , . . . , π m } be the set of all distinct shortest paths from s to t, where m denotes the number of shortest paths. Let u i and v i be the first and the last corners in V along π i from s to t, and let V s := {u 1 , . . . , u m } and V t := {v 1 , . . . , v m }.
Let E be the set of all sides of P without their endpoints and B be their union. Note that B = ∂P \V , the boundary of P except the corners V . The goal of this section is to prove the following theorem, which is the main geometric result of this paper.
Theorem 2 Suppose that (s * , t * ) is a maximal pair in P and Π(s * , t * ), V s * , and V t * be defined as above. Then, we have the following implications.
(VV) s * ∈ V , t * ∈ V implies |Π(s * , t * )| ≥ 1, |V s * | ≥ 1, |V t * | ≥ 1; (VB) s * ∈ V , t * ∈ B implies |Π(s * , t * )| ≥ 2, |V s * | ≥ 1, |V t * | ≥ 2; (VI) s * ∈ V , t * ∈ intP implies |Π(s * , t * )| ≥ 3, |V s * | ≥ 1, |V t * | ≥ 3; (BB) s * ∈ B, t * ∈ B implies |Π(s * , t * )| ≥ 3, |V s * | ≥ 2, |V t * | ≥ 2; (BI) s * ∈ B, t * ∈ intP implies |Π(s * , t * )| ≥ 4, |V s * | ≥ 2, |V t * | ≥ 3; (II) s * ∈ intP, t * ∈ intP implies |Π(s * , t * )| ≥ 5, |V s * | ≥ 3, |V t * | ≥ 3.
Moreover, each of the above bounds is best possible by examples. To see the tightness of the bounds, we present examples with remarks in Figure 1 and Appendix A. In particular, one can easily see the tightness of the bounds on |V s * | and |V t * | from shortest path maps SPM(s * ) and SPM(t * ), when V ∪ {s * , t * } is in general position.
s * u i u ′ i π i s * u i = u ′ i π i s * u i u ′ i π i (a) u i u ′ i π i (b) s * s * u i u ′ i π i Figure 2: (a) How to determine u ′ i . (left to right) u i = u ′ i ; s * , u i ,
Throughout this section, for easy discussion, we assume that there is a unique shortest path between any two corners u, v ∈ V . This assumption does not affect Theorem 2 since multiple shortest paths between corners in V can only increase |Π(s * , t * )|. Note that this assumption implies that the pairs
(u i , v i ) are distinct, while the u i (also the v i ) are not necessarily distinct. We thus have |V s * | ≤ m, |V t * | ≤ m, and |{(u i , v i ) | 1 ≤ i ≤ m}| = m, where m = |Π(s * , t * )|.
The following lemma proves the bounds on |V s * | and |V t * | of Theorem 2. Proofs of the lemmas presented in this section can be found in Appendix B.
Lemma 2 Let (s * , t * ) be a maximal pair. Then, |V t * | ≥ 2 if t * ∈ B; |V t * | ≥ 3 if t * ∈ intP. Moreover, if t * ∈ e ∈ E, then there exists v ∈ V t * such that v is off the line supporting e; if t * ∈ intP, then t * lies in the interior of the convex hull of V t * . Lemma 2 immediately implies the lower bound on |Π(s * , t * )| when s * ∈ V or t * ∈ V since |Π(s * , t * )| ≥ max{|V s * |, |V t * |}.
This finishes the proof for Cases (V-). Note that Case (VV) is trivial.
From now on, we assume that both s * and t * are not corners in V . This assumption, together with Lemma 2, implies multiple shortest paths between s * and t * , and thus d(s * , t * ) > s * − t * . Hence, as discussed in Section 2, any maximal pair falling into one of Cases (BB), (BI), and (II) appears as a local maximum of the lower envelope of some path-length functions.
Case (II): When both s * and t * lie in intP. We will apply Theorem 1 to prove Theorem 2 for Case (II). For the purpose, we find m = |Π(s * , t * )| convex functions f i defined on a convex neighborhood C of (s * , t * ) such that the following requirements are satisfied: (i) the pointwise minimum g of the f i coincides with the geodesic distance d on C, (ii) f i (s * , t * ) = g(s * , t * ) = d(s * , t * ) for any i ∈ {1, . . . , m}, (iii) g attains a local maximum at (s * , t * ) ∈ C, and (iv) none of the f i attains a local minimum at (s * , t * ).
If there are exactly m pairs (u, v) of corners such that len u,v (s * , t * ) = d(s * , t * ), then we can apply Theorem 1 simply with the m path-length functions len u i ,v i . Unfortunately, this is not always the case; a single shortest path
π i ∈ Π(s * , t * ) may result in several pairs (u, v) of corners with u, v ∈ π i such that (u, v) = (u i , v i ) and len u,v (s * , t * ) = d(s * , t * )
. This happens only when either u, u i , s * or v, v i , t * are collinear. In this degenerate case, the path-length functions len u i ,v i violate the first requirement above. In the following, we thus define the merged path-length functions that satisfy all the requirements even under the degenerate case.
Recall that the combinatorial structure of each shortest path π i ∈ Π(s * , t * ) can be represented by a sequence (u i = u i,1 , . . . , u i,k = v i ) of corners in V . We define u ′ i to be one of the u i,j as follows: If s * does not lie on the line ℓ ⊂ R 2 through u i and u i,2 , then u ′ i := u i ; otherwise, if s * ∈ ℓ, then u ′ i := u i,j , where j is the largest index such that for any open neighborhood U ⊂ R 2 of s * there exists a point s ∈ (U ∩ VP(u i,j )) \ ℓ. Note that such u ′ i always exists, and if no three of V are collinear, then Figure 3: Illustration to Lemma 4; for any (s, t) ∈ D i and any sufficiently small δ if we pick (s ′ , t ′ ) such that s ′ is δ closer to u i than s and t ′ is δ farther from v i than t, then we have
s s ′ t ′ t u i π i v i δ δ s ′′ t ′′f i (s ′ , t ′ ) = f i (s, t). Symmetrically, f i (s ′′ , t ′′ ) = f i (s, t) with s ′′ − u i = s − u i + δ and t ′′ − v i = t − v i − δ.
we always have either u ′ i = u i or u ′ i = u i,2 ; Figure 2(a) illustrates how to determine u ′ i . Also, we define v ′ i in an analogous way. Let h s i and h t i be two functions defined as
h s i (s) := s − u ′ i if s ∈ VP(u ′ i ), s − u i + u i − u ′ i if s ∈ VP(u i ) \ VP(u ′ i ); h t i (t) := t − v ′ i if t ∈ VP(v ′ i ), t − v i + v i − v ′ i if t ∈ VP(v i ) \ VP(v ′ i ).
Then, the merged path-length function f i : D i → R is defined as
f i (s, t) := h s i (s) + d(u ′ i , v ′ i ) + h t i (t), where D i := (VP(u ′ i ) ∪ VP(u i )) × (VP(v ′ i ) ∪ VP(v i )) ⊆ P × P.
We consider P × P as a subset of R 4 and each pair (s, t) ∈ P × P as a point in R 4 . Also, we denote by (s x , s y ) the coordinates of a point s ∈ P and we write s = (s x , s y ) or (s, t) = (s x , s y , t x , t y ) by an abuse of notation. Observe that Figure 2(b).
f i (s, t) = min{len u i ,v i (s, t), len u ′ i ,v i (s, t), len u i ,v ′ i (s, t), len u ′ i ,v ′ i (s, t)} for any (s, t) ∈ D i if we define len u,v (s, t) = ∞ when s ∈ VP(u) or v ∈ VP(v); see
We first show the convexity of the functions f i .
Lemma 3
For any i ∈ {1, . . . , m} and any convex subset C ⊂ D i , f i is convex on C.
Observe that each of the f i is indeed not strictly convex. Figure 3 illustrates one such line in R 4 for a fixed (s, t) ∈ D i that f i stays constant when (s, t) moves locally along the line. We show that such a line in R 4 is unique for any fixed (s, t) ∈ D i . Lemma 4 For any i ∈ {1, . . . , m} and any (s, t) ∈ intD i , there exists a unique line ℓ i ⊂ R 4 through (s, t) such that f i is constant on ℓ i ∩ U for some neighborhood U of (s, t) with U ⊂ D i . Moreover, f i is constant on ℓ i ∩ C for any convex neighborhood C of (s, t) with C ⊂ D i . Now, we let g : D i → R be the pointwise minimum of the f i defined as g(s, t) = min i f i (s, t) for any (s, t) ∈ D i . Note that the intersection D i contains a nonempty interior and (s * , t * ) ∈ int D i by our construction. We show that the f i satisfy the aforementioned requirements to apply Theorem 1.
Lemma 5
The functions f i and their pointwise minimum g satisfy the following conditions.
(i) There exists a convex neighborhood C ⊂ R 4 of (s * , t * ) with C ⊆ D i such that d(s, t) = g(s, t)
for any (s, t) ∈ C. (ii) f i (s * , t * ) = g(s * , t * ) = d(s * , t * ) for any i ∈ {1, . . . , m}. (iii) g attains a local maximum at (s * , t * ). (iv) None of the f i attains a local minimum at (s * , t * ). Now, we take a convex neighborhood C ⊂ R 4 of (s * , t * ) that is as described in Lemma 5. We restrict Suppose that m < 5. Then, Theorem 1 implies that there exists at least one line ℓ ⊂ R 4 through (s * , t * ) such that g is constant on ℓ ∩ C. On the other hand, Lemma 4 implies that there is a unique line ℓ i through (s * , t * ) such that f i is constant on ℓ i ∩ C, and no other line along which f i is locally constant at (s * , t * ). Hence, there exists at least one index i such that ℓ = ℓ i . We also observe the following.
Lemma 6
There are at most two indices i ∈ {1, . . . , m} such that ℓ = ℓ i .
The last task is to check two possibilities; one or two of the f i are constant on ℓ ∩ C. Also, recall that m ≥ 3 by Lemma 2. Without loss of generality, we first assume that ℓ = ℓ 1 = ℓ i for any i ≥ 2. Along ℓ 1 , each f i with i ≥ 2 is not constant but convex. Since f 1 (s, t) = g(s, t) = min i f i (s, t) for any (s, t) ∈ ℓ ∩ C, by Lemma 4, f i with i ≥ 2 must strictly increase from (s * , t * ) in both directions along ℓ. Thus, for any (s, t) ∈ ℓ∩C with (s, t) = (s * , t * ), we have a strict inequality g(s, t) = f 1 (s, t) < f i (s, t). Then, by Lemmas 3 and 4 at any such (s, t) ∈ ℓ ∩ C there is a direction in which f 1 strictly increases: more precisely, for any arbitrarily small neighborhood U ⊂ C of (s * , t * ), g(s, t) = f 1 (s, t) < f i (s, t) for (s, t) ∈ ∂U ∩ ℓ and thus there exist a sufficiently small neighborhood U ′ ⊂ C of (s, t) and (s ′ , t ′ ) ∈ U ′ such that f 1 (s, t) < f 1 (s ′ , t ′ ) < f i (s ′ , t ′ ) for any i ≥ 2, which implies that g(s * , t * ) = g(s, t) < g(s ′ , t ′ ), a contradiction to that g attains a local maximum at (s * , t * ).
Thus, two of the f i must be constant on ℓ ∩ C. We assume that ℓ = ℓ 1 = ℓ 2 = ℓ i for i ≥ 3. In this case, for any (s, t) ∈ ℓ ∩ C with (s, t) = (s * , t * ), we have a strict inequality g(s, t) = f 1 (s, t) = f 2 (s, t) < f i (s, t). Then there exists a direction from (s, t) in which both of f 1 and f 2 strictly increase by Lemmas 3 and 4. Since g(s, t) = f 1 (s, t) = f 2 (s, t) < f i (s, t) for any i ≥ 3, we get a contradiction analogously to the above.
Hence, we achieve a bound m = |Π(s * , t * )| ≥ 5, as claimed in Case (II) of Theorem 2.
Case (BB): When both s * and t * lie on B.
In this case, we assume that s * ∈ e s ∈ E and t * ∈ e t ∈ E. Let p be an endpoint of e s and l s be the length of e s . We denote by s(ζ s ) the unique point on e s such that s(ζ s ) − p = ζ s for any 0 < ζ s < l s . Here, we consider s : (0, l s ) → e s as a bijective map between a real open interval (0, l s ) ⊂ R and a segment e s ⊂ R 2 except its endpoints. Analogously, we also define t(ζ t ). Let ζ * s and ζ * t be real numbers such that s * = s(ζ * s ) and t * = t(ζ * t ). The outline of proof is analogous to the above discussion for Case (II). We redefine f i : D i → R as
f i (ζ s , ζ t ) := h s i (s(ζ s )) + d(u ′ i , v ′ i ) + h t i (t(ζ t )), where D i := s −1 ((VP(u ′ i ) ∪ VP(u i )) ∩ e s ) × t −1 ((VP(v ′ i ) ∪ VP(v i )) ∩ e t )
. We consider D i as a subset of R 2 and each pair (ζ s , ζ t ) ∈ D i as a point in R 2 . Also, let g(ζ s , ζ t ) := min i f i (ζ s , ζ t ) for any (ζ s , ζ t ) ∈ D i .
The convexity of f i on any convex subset of D i is deduced from Lemma 3. Analogously to Lemma 5, one can show that (i) there exists a convex neighborhood C ⊂ R 2 of (ζ * s , ζ * t ) with C ⊂ D i such that g(ζ s , ζ t ) = min i f i (ζ s , ζ t ) = d(s(ζ s ), t(ζ t )) for any (ζ s , ζ t ) ∈ C, (ii) f i (ζ * s , ζ * t ) = g(ζ * s , ζ * t ) = d(s(ζ * s ), t(ζ * t )) for any i ∈ {1, . . . , m}, (iii) g attains a local maximum at (ζ * s , ζ * t ). Also, observe that (iv) none of the f i attains a local minimum at (ζ * s , ζ * t ) if m < 3: Assume that some f i attains a local minimum at (ζ * s , ζ * t ) and m < 3. This happens only when s * = s(ζ * s ) is the perpendicular foot of u i on e s and t * = t(ζ * t ) is the perpendicular foot of v i on e t . In this case, there always exists a direction along e s such that if we move s * in the direction, then h s i strictly increases for every 1 ≤ i ≤ m < 3, which contradicts to the assumption that (s * , t * ) is maximal.
In addition, we observe the following.
Lemma 7
If there exists a line ℓ ⊂ R 2 such that f i is constant on ℓ∩C, then u i lies on the line supporting e s and v i lies on the line supporting e t .
We assume that m < 3, and restrict the functions f i and g on C ⊂ R 2 . Then, Theorem 1 implies that there exists a line ℓ ⊂ R 2 through (ζ * s , ζ * t ) such that g is constant on ℓ ∩ C. Moreover, Lemma 4 implies that at least one of the f i is constant on ℓ∩C. Assume that only f 1 is constant on ℓ∩C. Then, by the same argument as above for Case (II), for any point (ζ s , ζ t ) ∈ ℓ ∩ C with (ζ s , ζ t ) = (ζ * s , ζ * t ), we have a strict inequality g(ζ s , ζ t ) = f 1 (ζ s , ζ t ) < f i (ζ s , ζ t ) for i ≥ 2, leading to a contradiction: for any arbitrarily small neighborhood U ⊂ C of (ζ * s , ζ * t ), g(ζ s , ζ t ) = f 1 (ζ s , ζ t ) < f i (ζ s , ζ t ) for (ζ s , ζ t ) ∈ ∂U ∩ ℓ and thus there exist a sufficiently small neighborhood U ′ ⊂ C of (ζ s , ζ t ) and (ζ ′ s , ζ ′ t ) ∈ U ′ such that
f 1 (ζ s , ζ t ) < f 1 (ζ ′ s , ζ ′ t ) < f i (ζ ′ s , ζ ′ t ), which implies that d(s * , t * ) = g(ζ * s , ζ * t ) = g(ζ s , ζ t ) < g(ζ ′ s , ζ ′ t ).
Hence, both f 1 and f 2 are constant on ℓ ∩ C. Then, by Lemma 7, u 1 , u 2 , and s * are collinear and v 1 , v 2 , and t * are collinear. Since m < 3, this situation violates the second part of Lemma 2. Thus, we get a contradiction again, concluding that m = |Π(s * , t * )| ≥ 3 for Case (BB) when both s * and t * lie on B.
Case (BI): When s * ∈ B and t * ∈ intP. We assume that s * ∈ e s ∈ E and t * ∈ intP. Define s(ζ s ) as done in Case (BB) with s(ζ * s ) = s * . We redefine the function f i :
D i → R to be f i (ζ s , t x , t y ) := h s i (s(ζ s )) + d(u ′ i , v ′ i ) + h t i (t x , t y ), where D i := s −1 ((VP(u ′ i ) ∪ VP(u i )) ∩ e s ) × (VP(v ′ i ) ∪ VP(v i )) is a subset of R 3 . Let g(ζ s , t x , t y ) := min i f i (ζ s , t x , t y ) for any (ζ s , t x , t y ) ∈ D i .
Analogously to Lemmas 3 and 5, each f i is convex on any convex subset of D i and there exists a convex neighborhood C ⊂ R 3 of (ζ * s , t * x , t * y ) with C ⊂ D i such that the four requirements are satisfied. Suppose that m = |Π(s * , t * )| < 4. Then, Theorem 1 implies that there exists a line ℓ ⊂ R 3 through (ζ * s , t * x , t * y ) ∈ R 3 such that g is constant on ℓ ∩ C, thus at least one of the f i is constant on ℓ ∩ C by Lemma 4. If only one of the f i is constant on ℓ ∩ C, then we have a contradiction as done in Cases (II) and (BB). Thus, assume that f 1 and f 2 are constant on ℓ ∩ C. By Lemmas 4 and 7, v 1 , v 2 , and t * should be collinear. Further, by the second part of Lemma 2, t * must lie in the interior of the convex hull of V t * . In order to have an interior point of the convex hull on the segment between v 1 and v 2 , we need at least two more points. Nonetheless, we have |V t * | ≤ |Π(s * , t * )| < 4, a contradiction. Thus, we have m = |Π(s * , t * )| ≥ 4 for Case (BI), as claimed.
Finally, we complete a proof of Theorem 2: The claimed bounds on |V s * | and |V t * | are shown by Lemma 2, and the bounds on |Π(s * , t * )| are shown case by case as above.
Computing the Geodesic Diameter
Since a diametral pair is in fact maximal, it falls into one of the cases shown in Theorem 2. In order to find a diametral pair we examine all possible scenarios accordingly.
Cases (V-), where at least one point is a corner in V , can be handled in O(n 2 log n) time by computing SPM(v) for every v ∈ V and traversing it to find the farthest point from v, as discussed in Section 2. We thus focus on Cases (BB), (BI), and (II), where a diametral pair consists of two non-corner points.
From the computational point of view, the most difficult case corresponds to Case (II) of Theorem 2; in particular, |Π(s * , t * )| = 5 in which 10 corners of V are involved, resulting in |V s * | = |V t * | = 5 (see Appendix A.3). Note that we do not need to take a special care for the case of |Π(s * , t * )| > 5. By Theorem 2 and its proof, it is guaranteed that there are five distinct pairs (u 1 , v 1 ), . . . , (u 5 , v 5 ) of corners in V such that len u i ,v i (s * , t * ) = d(s * , t * ) for any i ∈ {1, . . . , 5} and the system of equations len u 1 ,v 1 (s, t) = · · · = len u 5 ,v 5 (s, t) indeed determines a 0-dimensional zero set, corresponding to a constant number of candidate pairs in intP × intP. Moreover, each path-length function len u,v is an algebraic function of degree at most 4. Thus, given five distinct pairs (u i , v i ) of corners, we can compute all candidate pairs (s, t) in O(1) time by solving the system. 2 Then, for each candidate pair we compute the geodesic distance between the pair to check its validity. Since the geodesic distance between any two points s, t ∈ P can be computed in O(n log n) time [12], we obtain a brute-force O(n 11 log n)-time algorithm, checking O(n 10 ) candidate pairs obtained from all possible combinations of 10 corners in V .
As a different approach, one can exploit the SPM-equivalence decomposition of P, which subdivides P into regions such that the shortest path map of any two points in a common region are "topologically equivalent" [7]. It is not difficult to see that if (s, t) is a pair of points that equalizes any five path-length functions, then both s and t appear as vertices of the decomposition. However, the currently best upper bound on the complexity of the SPM-equivalence decomposition is O(n 10 ) [7], and thus this approach hardly leads to a remarkable improvement.
Instead, we do the following for Case (II) with |V s * | = 5. We choose any five corners u 1 , . . . , u 5 ∈ V (as a candidate for the set V s * ) and overlay their shortest path maps SPM(u i ). Since each SPM(u i ) has O(n) complexity, the overlay consists of O(n 2 ) cells. Then, any cell of the overlay is the intersection of five cells associated with v 1 , . . . , v 5 ∈ V in SPM(u 1 ), . . . , SPM(u 5 ), respectively. Choosing a cell of the overlay, we get five (possibly, not distinct) v 1 , . . . , v 5 and thus a constant number of candidate pairs by solving the system len u 1 ,v 1 (s, t) = · · · = len u 5 ,v 5 (s, t). We iterate this process for all possible tuples of five corners u 1 , . . . , u 5 , obtaining a total of O(n 7 ) candidate pairs in O(n 7 log n) time. Note that the other subcases with |V s * | ≤ 4 can be handled similarly, resulting in O(n 6 ) candidate pairs.
In order to test the validity of each candidate pair (s, t), we check the geodesic distance d(s, t) using a two-point query structure of Chiang and Mitchell [7]: for a fixed parameter 0 < δ ≤ 1 and any fixed ǫ > 0, we can construct, in O(n 5+10δ+ǫ ) time, a data structure that supports O(n 1−δ log n)-time two-point shortest path queries. Then, the total running time is O(n 7 log n) + O(n 5+10δ+ǫ ) + O(n 7 ) × O(n 1−δ log n). We set δ = 3 11 to optimize the running time to O(n 7+ 8 11 +ǫ ). Also, we can use an alternative two-point query data structure whose performance is sensitive to the number h of holes [7]: after O(n 5 ) preprocessing time using O(n 5 ) storage, two-point queries can be answered in O(log n + h) time. 3 Using this alternative structure, the total running time of our algorithm becomes O(n 7 (log n + h)). Note that this method outperforms the previous one when h = O(n 8 11 ). The other cases can be handled analogously with strictly better time bound. For Case (BI), we handle only the case of |Π(s * , t * )| = 4 with |V t * | = 3 or 4. For the subcase with |V t * | = 4, we choose any four corners from V as v 1 , . . . , v 4 as a candidate for V t * and overlay their shortest path maps SPM(v i ). The overlay, together with V , decomposes ∂P into O(n) intervals. Then, each such interval determines u 1 , . . . , u 4 as above, and the side e s ∈ E on which s * should lie. Now, we have a system of four equations on four variables: three from the corresponding path-length functions len u i ,v i which should be equalized at (s * , t * ) and the fourth from the supporting line of e s . Solving the system, we get a constant number of candidate maximal pairs, again by Theorem 2 and its proof. In total, we obtain O(n 5 ) candidate pairs. The other subcase with |V t * | = 3 can be handled similarly, resulting in O(n 4 ) candidate pairs. As above, we can exploit two different structures for two-point queries. Consequently, we can handle Case (BI) in O(n 5+ 10 11 +ǫ ) or O(n 5 (log n + h)) time. In Case (BB) when s * , t * ∈ B, we handle the case of |Π(s * , t * )| = 3 with |V s * | = 2 or 3. For the subcase with |V s * | = 3, we choose three corners as a candidate of V s * and take the overlay of their shortest path maps SPM(u i ). It decomposes ∂P into O(n) intervals. Then, each such interval determines three corners v 1 , v 2 , v 3 forming V t * and a side e t ∈ E on which t * should lie. Note that we have only three equations so far; two from the three path-length functions and the third from the line supporting to e t . Since s * also should lie on a side e s ∈ E with e s = e t , we need to fix such a side e s that 1≤i≤3 VP(u i ) intersects e s . In the worst case, the number of such sides e s is Θ(n). Thus, we have O(n 5 ) candidate pairs for Case (BB); again, the other subcase with |V s * | = 2 contributes to a smaller number O(n 4 ) of candidate pairs. Testing each candidate pair can be performed as above, resulting in O(n 5+ 10 11 +ǫ ) or O(n 5 (log n + h)) total running time. For Case (BB), however, one can exploit a two-point query structure only for boundary points on ∂P. The two-point query structure by Bae and Okamato [5] indeed builds an explicit representa-tion of the graph of the lower envelope of the path-length functions len u,v restricted on ∂P × ∂P in O(n 5 log n log * n) time. 4 Since |Π(s * , t * )| ≥ 3 in Case (BB), such a pair appears as a vertex on the lower envelope. Hence, we are done by traversing all the vertices of the lower envelope.
The following table summarizes the discussion so far. As Case (II) being a bottleneck, we conclude the following.
Theorem 3 Given a polygonal domain having n corners and h holes, the geodesic diameter and a diametral pair can be computed in O(n 7+ 8 11 +ǫ ) or O(n 7 (log n + h)) time in the worst case, where ǫ is any fixed positive number.
We can avoid some difficult cases when h is a small constant based on a simple observation: If there are two distinct shortest paths between s and t in P, then we know that there is at least one hole in the region closed by the two paths. In general, if k shortest paths exist between any two points of P, we can conclude h ≥ k − 1. By contraposition, if h < k − 1, then there cannot exist two points that have k or more distinct shortest paths between them.
Theorem 4 Given a polygonal domain having n corners and h holes, the geodesic diameter and a diametral pair can be computed in the following worst-case time bound, depending on h.
• O(n) time, if h = 0 (by Hershberger and Suri [11]),
• O(n 2 log n) time, if h = 1, • O(n 5 log n) time, if h = 2 or 3, • O(n 7 (log n + h)) time, if 4 ≤ h = O(n 8 11 ),
• O(n 7+ 8 11 +ǫ ) time, otherwise.
Concluding Remarks
We have presented first algorithms that compute the geodesic diameter of a given polygonal domain.
They are based on our new geometric observations on local maxima of the geodesic distance function, which show tight lower bounds on the number of shortest paths between a maximal pair. It is worth noting that with analysis in Section 5 the number of geodesic-maximal pairs is shown to be at most O(n 7 ). On the other hand, one can easily construct a simple polygon in which the number of maximal pairs is Ω(n 2 ). An interesting question would be what the maximum possible number of maximal pairs in a polygonal domain is. Though in this paper we have focused on exact geodesic diameters only, an efficient algorithm for finding an approximate geodesic diameter would be also interesting. Notice that any point s ∈ P and its farthest point t ∈ P yield a 2-approximate diameter; that is, diam(P) ≤ 2 max t∈P d(s, t) for any s ∈ P. Also, based on a standard technique using a rectangular grid with a specified parameter 0 < ǫ < 1, one can easily obtain a (1 + ǫ)-approximate diameter in O(( n ǫ 2 + n 2 ǫ ) log n) time. 5 Breaking the quadratic bound in n for the (1 + ǫ)-approximate diameter seems a challenge at this stage. Thus, we pose the following problem: For any or some 0 < ǫ < 1, is there any algorithm that finds a (1 + ǫ)-approximate diametral pair in O(n 2−δ · poly(1/ǫ)) time for some positive δ > 0?
APPENDIX A More Examples and Remarks
In this section, we show more constructions of polygonal domains and their diametral pairs with interesting remarks. In the figures, we keep the following rules: the boundary ∂P is depicted by dark gray segments and the interior of holes by light gray region. A diametral pair is given as (s * , t * ) and shortest paths between s * and t * are described as black dashed polygonal chains. Figure 4: (a-c) Polygonal domains whose geodesic diameter is determined by a corner s * and (d-g) variations of the construction (c). (a) When both s * and t * are corners; (b) When t * is a point on ∂P; (c) When t * ∈ intP. This polygonal domain consists of two holes, forming a narrow corridor and three shortest paths between s * and t * . Here, we have d(s * , v 1 ) = d(s * , v 2 ) = d(s * , v 3 ) and t * is indeed the vertex of SPM(s * ) defined by v 1 , v 2 , v 3 ; (d) Variation of (c) with all convex holes; (e) Three shortest paths are not enough to determine a boundaryinterior diametral pair; (f) If we add one more hole, then the diameter is determined by s * ∈ B and t * ∈ intP with four shortest paths; (g) A polygonal domain made by attaching two copies of (e) and modifying it to have
A.1 Examples where at least one point of a diametral pair lies on ∂P
s * t * s * t * v 3 v 2 v 1 t * s * (c) (a) (b) v 3 v 2 t * s * v 1 v 3 v 2 t * s * v 1 (e) (f) (d) v 1 u 1 t * s * u 2 u 3 v 3 v 2 (g) t * s * v 1 v 3 v 2d(u 1 , v 1 ) = d(u 2 , v 2 ) = d(u 3 , v 3 ).
Observe that, in this polygonal domain, the diameter is determined by two boundary points with three shortest paths.
Note that, as expected, every example in Figure 4 satisfies Theorem 2. An interesting construction is Figure 4(g), where neither of the two centers of △u 1 u 2 u 3 and of △v 1 v 2 v 3 appears in any diametral pair. Also note that Figure 4(d) consists of convex holes only. We think that any complicated construction can be "convexified" in a similar fashion. This would suggest that computing the diameter in polygonal domains with convex holes only might be as difficult as the general case. Claim 1 In the polygonal domain described in Figure 1(c), (s * , t * ) is the unique diametral pair. Proof of Claim. Recall that by construction of the problem instance, the triangles △u 1 u 2 u 3 and △v 1 v 2 v 3 are regular and d(
u 1 u 2 u 3 v 3 v 2 v 1 u 1 u 2 u 3 v 3 v 2 v 1 s * t *u 1 , v 1 ) = d(u 1 , v 2 ) = d(u 2 , v 2 ) = d(u 2 , v 3 ) = d(u 3 , v 3 ) = d(u 3 , v 1 ) = L,
for some arbitrarily large value L > 0. Also, s * and t * are the centers of △u 1 u 2 u 3 and △v 1 v 2 v 3 , respectively.
We assume that both triangles △u 1 u 2 u 3 and △v 1 v 2 v 3 are inscribed in a unit circle. Then, we have d(s * , t * ) = 2 + L. For any point s on any shortest path between u i and v j , it is easy to see that d(s, t) ≤ √ 3 + L < d(s * , t * ) for every point t ∈ P. Thus, any point on those paths cannot contribute to the diameter.
(1) First, observe that max t∈△v 1 v 2 v 3 d(s * , t) = max s∈△u 1 u 2 u 3 d(s, t * ) = d(s * , t * ).
(2) For any s ∈ △u 1 u 2 u 3 , its farthest point t ∈ △v 1 v 2 v 3 is on the angle bisector of some v i . Consider any s ∈ △u 1 u 2 u 3 . Without loss of generality we assume that s − u 1 ≤ min i { s − u i }. Then, both the shortest paths to v 1 and to v 2 from s pass through u 1 . We thus have d(s, v 1 ) = d(s, v 2 ) by construction and its farthest point t ∈ △v 1 v 2 v 3 must be in the angle bisector of v 3 . By symmetry, the same property holds when the closest corner from s * is either u 2 or u 3 .
Conversely, for any t, its farthest point s ∈ △u 1 u 2 u 3 must be on a bisector of some u i . In any diametral pair (s, t), we have that t is the farthest point of s (and vice versa), so both must be on one of the angle bisectors.
(3) If (s, t) is a diametral pair, then s ∈ u i s * and t ∈ v j t * , for some i and j. Suppose that s lies on the bisector of u 1 but not in between u 1 and s * . We then have s − u 2 = s − u 3 < s − u 1 and d(s, v 1 ) = d(s, v 2 ) = d(s, v 3 ) = s − u 2 + L by construction. This implies that t * is the farthest point of such s. Since s − u 2 < 1 and thus d(s, t * ) < 2 + L, (s, t * ) is not a diametral pair.
(4) Now, pick any point s ∈ u 1 s * with s = s * . Suppose that t ∈ △v 1 v 2 v 3 is the farthest point from s. We know that t ∈ v 3 t * by above discussions. In this case, we have four shortest paths between s and t through (u 1 , v 1 ), (u 1 , v 2 ), (u 2 , v 3 ), and (u 3 , v 3 ); the other two are strictly longer unless s = s * . Thus, by Theorem 2, such s ∈ u 1 s * with s = s * and its farthest point t cannot form a maximal pair. By symmetry, the other cases where s ∈ u i s * can be handled.
Hence, (s * , t * ) is a unique diametral pair and the geodesic diameter is 2 + L.
A.3 Diametral pair of Case (II) with exactly 5 shortest paths
Here, we present a polygonal domain in which the diameter is determined by two interior points and exactly five shortest paths between them. This proves the tightness of Case (II) in Theorem 2.
u 1 =u 2 =u 3 v 3 =v 4 v 2 v 1 =v 5 s * t * π 1 π 2 π 3 π 4 π 5 u 4 u 5
c u c v Figure 6: A schematic diagram of a polygonal domain in which |V s * | = |V t * | = 3 and |Π(s * , t * )| = 5. Figure 6 shows a schematic description of a polygonal domain P. We assume that only the position of the vertices u i and the v i are geometrically precise. Thus, we construct the problem instance such that we have u 1 = u 2 = u 3 , v 1 = v 5 , and v 3 = v 4 , and the convex hulls of the u i and of the v i form isosceles triangles △ u and △ v . Each of △ u and △ v is inscribed in a unit circle centered at c u and c v . Moreover, the bases of both triangles are horizontal and the angles opposite to the bases are 18 • and 112 • , respectively. Note that the side lengths of the triangles △ u and △ v are as follows: u 1 − u 4 = 1.97537 · · · and u 4 − u 5 = 0.61803 · · · ; v 2 − v 1 = 1.11833 · · · and v 1 − v 3 = 1.85436 · · · .
In this configuration, we set the constants as follows: letting L := d(u 1 , v 1 ) = d(u 3 , v 3 ) be some sufficiently large number, we set d(u 2 , v 2 ) = L + 0.5 and d(u 4 , v 4 ) = d(u 5 , v 5 ) = L + 0.2. Note that this configuration can be realized with four obstacles in a similar way as Figure 1(c).
Since we have fixed all necessary parameters, we have a fully explicit description of the len u i ,v i . Due to the difficulty of finding an exact analytical solution, we used numerical methods to solve the system of equations len u 1 ,v 1 (s, t) = · · · = len u 5 ,v 5 (s, t). We have found that there is a unique solution (s * , t * ) such that s * ∈ △ u and t * ∈ △ v ; we obtained s * = c u +(0, −0.102795 · · · ), t * = c v +(0, 0.555361 · · · ) and d(s * , t * ) = 2.047433734 · · · + L. (See Figure 6.)
We first checked that (s * , t * ) is indeed a maximal pair based on the following lemma, which can be shown using elementary linear algebra together with the convexity of the path-length functions.
Lemma 8
Suppose that (s, t) is a solution to the system len u 1 ,v 1 (s, t) = · · · = len u 5 ,v 5 (s, t). If any four of the five gradients ∇len u i ,v i at (s, t) are linearly independent (as vectors in a 4-dimensional space) and one of them is represented as a linear combination of the other four with all "negative" coefficients, then (s, t) is a local maximum of the pointwise minimum of the five functions len u i ,v i . Next, to see that (s * , t * ) is a diametral pair, we have run our algorithm for each of Cases (BB), (BI), and (II); as a result, there are 44 candidate pairs, including (s * , t * ), falling into those cases among which at most 11 are maximal and only (s * , t * ) is diametral. Note that the pair (s * , t * ) is the only candidate pair of Case (II). Also, observe that any point on the shortest path between u i and v i cannot belong to a diametral pair. This implies that none of the u i and the v i belongs to a diametral pair and thus that there is no diametral pair in Cases (V-). In addition, we also sampled about 350,000 points uniformly from each of △ u and △ v , and evaluated the geodesic distances of the 350,000 2 pairs. Note that one can modify the construction to have |V s * | = |V t * | = |Π(s * , t * )| = 5. For the purpose, we can split u 1 , u 2 , u 3 into three close corners (analogously for corners, v 1 , v 5 and v 3 , v 4 ). The splitting process should preserve the differences between the distances d(u i , v i ) for all i = 1, . . . , 5. We also have tested such an example in the same way as above and concluded that a solution equalizing the five path-length functions is indeed a diametral pair.
and the second corner are collinear; s * and the first three corners are collinear (b) For points in a small disk B centered at s * with B ⊂ VP(u ′ i ) ∪ VP(u i ), the function h s i measures the length of the shortest path from u ′ i to each.
f 1 , . . . , f m and g on C. Then, each f i is convex by Lemma 3 and, by Lemma 5, the m functions f i and their pointwise minimum g satisfy the conditions of Theorem 1 with open convex domain C ⊂ R 4 .
Dependent on h (VV), (VB), (VI) O(n 2 log n) (BB) O(n 5 log n log * n) O(n 5 (log n + h)) (BI) O(n 5+ 10 11 +ǫ ) O(n 5 (log n + h)) (II) O(n 7+ 8 11 +ǫ ) O(n 7 (log n + h))
Figure 5 :
5A schematic diagram corresponding to the polygonal domain shown inFigure 1(c).
Here, we assume that fundamental operations on a constant number of polynomials of constant degree with a constant number of variables can be performed in constant time.
If h is relatively small, one could use the structure of Guo, Maheshwari and Sack[10] which answers a two-point query in O(h log n) time after O(n 2 log n) preprocessing time using O(n 2 ) storage, or another structure by Chiang and Mitchell[7] that supports a two-point query in O(h log n) time, spending O(n + h 5 ) preprocessing time and storage.
More precisely, in O(n 4 λ65(n) log n) time, where λm(n) stands for the maximum length of a Davenport-Schinzel sequence of order m on n symbols.5 We thank Hee-Kap Ahn for giving us the idea of this approximation algorithm.
Proof of Lemma 7The lemma immediately follows from Lemma 4 and its proof.vi
Proof of Lemma 4There are four cases according to the position of s and t: either s ∈ VP(u ′ i ) or s ∈ VP(u i ) \ VP(u ′ i ); either t ∈ VP(v ′ i ) or t ∈ VP(v i ) \ VP(v ′ i ). We give a proof only for the case of s ∈ VP(u i ) \ VP(u ′ i ) and t ∈ VP(v i ) \ VP(v ′ i ) but the proofs for the other cases are almost identical. Recall that s = u i and t = v i . Any ray (or half-line) γ ⊂ R 4 with endpoint (s, t) can be determined by three parameters (θ s , θ t , λ) with 0 ≤ θ s , θ t ≤ π and λ ≥ 0 as follows: Let γ s and γ t be the projections of γ onto the (s x , s y )-plane and the (t x , t y )-plane, respectively. Note that γ s is a ray in the (s x , s y )-plane with endpoint s and γ t is a ray in the (t x , t y )-plane with endpoint t. Let θ s be the smaller angle at s made by γ s and another ray starting from s in direction away from u i . Define θ t analogously. Then, the derivative of f i at (s, t) along γ is represented as c(cos θ s + λ cos θ t ) for some constants λ ≥ 0 and c > 0. Also, the second derivative of f i at (s, t) along γ is derived as c sin 2 θs s−u i + λ sin 2 θt t−v i . Suppose that f i is constant along γ locally around (s, t). Then, its first and second derivatives should be zero in a small neighborhood U ⊂ R 4 of (s, t) with U ⊂ D i . First, we observe that λ should be positive; if λ = 0, then t is fixed while s moves along γ s so that f i does not stay constant. Since every term of the second derivative is nonnegative and λ > 0, we get (θ s , θ t ) = (0, π) or (π, 0). We hence obtain only two solutions (θ s , θ t , λ) = (0, π, 1) or (π, 0, 1). The two rays γ corresponding to these two solutions form a unique line, proving the second part of the lemma.Remark that ℓ i ∩ intD i may consist of more than one connected components; a typical situationFigure 3for more intuitive and geometric description on ℓ i .Proof of Lemma 5 (i) In this proof, we define leni (s, t)} = min u,v∈π i ∩V len u,v (s, t) for all (s, t) ∈ U i . We claim that there exists a convex neighborhood C ⊂ i U i such that for any (s, t) ∈ C d(s, t) = min 1≤i≤m f i (s, t) = g(s, t).To prove our claim, assume to the contrary that for every convex neighborhood C ⊂ R 4 of (s * , t * ) ∈ R 4 there exist a pair (u, v) of corners and (s, t) ∈ C such that d(s, t) = len u,v (s, t) < min i f i (s, t). Note that none of the shortest paths π i ∈ Π(s * , t * ) between s * and t * passes through both of such u and v since otherwise len u,v (s, t) = min i f i (s, t).Consider a sequence C 1 , C 2 , . . . of neighborhoods of (s * , t * ) ∈ R 4 that converges to the singleton {(s * , t * )}. Since there are only n 2 pairs of corners, there exist a fixed pair (u 0 , v 0 ) of corners and a subsequence C k 1 , C k 2 , . . . converging to the singleton {(s * , t * )} such that none of the π i passes through both of u 0 , v 0 and for any integerSince lim j→∞ (s j , t j ) = (s * , t * ), it holds that lim j→∞ d(s j , t j ) = lim j→∞ min i f i (s j , t j ) = d(s * , t * ). By the sandwich theorem, we thus have lim j→∞ len u 0 ,v 0 (s j , t j ) = len u 0 ,v 0 (s * , t * ) = d(s * , t * ).This implies the existence of the (m+1)-st shortest path between s * and t * since none of the π i ∈ Π(s * , t * ) contains both u 0 and v 0 , a contradiction.v Proof of Lemma 6 Let ℓ i ⊂ R 4 be the unique line through (s * , t * ) such that f i is constant on ℓ i ∩ C. The uniqueness of ℓ i is proven byLemma 4.Observe from the proof of Lemma 4 that the projection of ℓ i onto the (s x , s y )-plane is the line through s * and u i . Also, the projection of ℓ i onto the (t x , t y )-plane is the line through t * and v i . Hence, ℓ i = ℓ j implies that u i , u j , s * are collinear and v i , v j , t * are collinear. First, since the pairs (u i , v i ) are all distinct, we have u i = u j or v i = v j . If u i = u j and v i = v j , one can easily check that ℓ i = ℓ j . We thus have u i = u j and v i = v j . Moreover, s * must lie in between u i and u j and t * must lie in between v i and v j by definition; if u j lies in between u i and s * , then the first corner of π i from s * becomes u j since the three are collinear. Therefore, for each i ∈ {1, . . . , m}, there is at most one index j ∈ {1, . . . , m} such that i = j and ℓ i = ℓ j , completing the proof.
Star unfolding of a polytope with applications. P K Agarwal, B Aronov, J O'rourke, C A Schevon, SIAM J. Comput. 266P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26(6):1689-1713, 1997.
The furthest-site geodesic Voronoi diagram. B Aronov, S Fortune, G Wilfong, Discrete Comput. Geom. 9B. Aronov, S. Fortune, and G. Wilfong. The furthest-site geodesic Voronoi diagram. Discrete Comput. Geom., 9:217-255, 1993.
Computing the geodesic center of a simple polygon. T Asano, G Toussaint, SOCS-85.32McGill UniversityTechnical ReportT. Asano and G. Toussaint. Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University, 1985.
The geodesic farthest-site Voronoi diagram in a polygonal domain with holes. S W Bae, K.-Y Chwa, Proc. 25th Annu. Sympos. 25th Annu. SymposS. W. Bae and K.-Y. Chwa. The geodesic farthest-site Voronoi diagram in a polygonal domain with holes. In Proc. 25th Annu. Sympos. Comput. Geom. (SoCG), pages 198-207, 2009.
Querying two boundary points for shortest paths in a polygonal domain. S W Bae, Y Okamoto, arXiv:0911.5017Proc. 20th Annu. Internat. Sympos. Algo. Comput. (ISAAC). 20th Annu. Internat. Sympos. Algo. Comput. (ISAAC)5878A longer version is available asS. W. Bae and Y. Okamoto. Querying two boundary points for shortest paths in a polygonal domain. In Proc. 20th Annu. Internat. Sympos. Algo. Comput. (ISAAC), volume 5878 of LNCS, pages 1054-1063, 2009. A longer version is available as arXiv:0911.5017.
A theorem on polygon cutting with applications. B Chazelle, Proc. 23rd Annu. Sympos. Found. Comput. Sci. (FOCS). 23rd Annu. Sympos. Found. Comput. Sci. (FOCS)B. Chazelle. A theorem on polygon cutting with applications. In Proc. 23rd Annu. Sympos. Found. Comput. Sci. (FOCS), pages 339-349, 1982.
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Shortest paths and networks. J S B Mitchell, Handbook of Discrete and Computational Geometry, chapter 27. CRC Press, Inc2nd editionJ. S. B. Mitchell. Shortest paths and networks. In Handbook of Discrete and Computational Geometry, chapter 27, pages 607-641. CRC Press, Inc., 2nd edition, 2004.
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(ii) The claim follows from the fact that f i (s * , t * ) = len u i. S Suri, Proc. 3rd Annu. Sympos. 3rd Annu. Sympos64The all-geodesic-furthest neighbors problem for simple polygons. v i (s * , t * )S. Suri. The all-geodesic-furthest neighbors problem for simple polygons. In Proc. 3rd Annu. Sympos. Comput. Geom. (SoCG), page 64, 1987. (ii) The claim follows from the fact that f i (s * , t * ) = len u i ,v i (s * , t * ).
From (i), g(s, t) = d(s, t) for all (s, t) ∈ C. Since d attains a local maximum at (s * , t * ). so does gFrom (i), g(s, t) = d(s, t) for all (s, t) ∈ C. Since d attains a local maximum at (s * , t * ), so does g.
f i (s, t) = len u i ,v i (s, t) < f i (s * , t * ). Hence, there is no neighborhood U ⊂ R 4 of (s * , t * ) such that f i (s * , t * ) ≤ f i (s, t) for any (s, t) ∈ U. * Then, implying that (s * , t * ) is not a local minimum of any f it = t * . Then, f i (s, t) = len u i ,v i (s, t) < f i (s * , t * ). Hence, there is no neighborhood U ⊂ R 4 of (s * , t * ) such that f i (s * , t * ) ≤ f i (s, t) for any (s, t) ∈ U , implying that (s * , t * ) is not a local minimum of any f i .
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[] |
[
"CONJUGACY OF LOCAL HOMEOMORPHISMS VIA GROUPOIDS AND C*-ALGEBRAS",
"CONJUGACY OF LOCAL HOMEOMORPHISMS VIA GROUPOIDS AND C*-ALGEBRAS"
] |
[
"Becky Armstrong ",
"TOKEKevin Aguyar Brix ",
"AND SØREN EILERSMeier Carlsen "
] |
[] |
[] |
We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalises recent work of Matsumoto and of the second-and third-named authors.
|
10.1017/etds.2022.50
|
[
"https://export.arxiv.org/pdf/2105.00479v2.pdf"
] | 233,481,691 |
2105.00479
|
21915c16eedb131370e32d493f746b18626dbc57
|
CONJUGACY OF LOCAL HOMEOMORPHISMS VIA GROUPOIDS AND C*-ALGEBRAS
Becky Armstrong
TOKEKevin Aguyar Brix
AND SØREN EILERSMeier Carlsen
CONJUGACY OF LOCAL HOMEOMORPHISMS VIA GROUPOIDS AND C*-ALGEBRAS
arXiv:2105.00479v2 [math.OA] 9 Aug 2022
We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalises recent work of Matsumoto and of the second-and third-named authors.
Introduction
The tradition of constructing operator algebras from dynamical systems originated with the seminal work of Murray and von Neumann on the group von Neumann algebra construction [MvN]. This approach has not only produced a plethora of interesting examples of operator algebras, but has also led to interesting results pertaining to topological dynamics. Prime examples of such results are those of Giordano, Putnam, and Skau [GPS95, Theorems 2.1 and 2.2] that use C*-crossed products to show that Cantor minimal systems can be classified up to (strong) orbit equivalence by K-theory. C*-algebras constructed from dynamical systems have mainly been obtained as C*crossed products of actions of locally compact groups on topological spaces. However, in the last forty years or so, C*-algebras constructed from non-invertible actions on topological spaces have also attracted considerable attention. This approach inspired Krieger's dimension group [K80a, K80b], which has had a tremendous impact on the study of symbolic dynamical systems. A particularly interesting class of these kinds of C*-algebras is the class of C*-algebras arising from a dynamical system consisting of a local homeomorphism acting on a locally compact Hausdorff space [D95,Re00]. These C*-algebras come equipped with a family of symmetries induced by functions on the space, and by using a topological groupoid approach, we show how to recover the conjugacy class of the underlying system from the C*-algebra and its family of symmetries.
A system consisting of a locally compact Hausdorff space X together with a local homeomorphism σ X between open subsets of X is called a Deaconu-Renault system (see, for example, [D95,Re00,CRST,ABS]). Examples of Deaconu-Renault systems include self-covering maps [D95,EV06], one-sided shifts of finite type [Wi73,LM95,Ki98], the boundary-path space of a directed graph together with the shift map [We14,BCW17], and more generally, the boundary-path space of a topological graph together with the shift map [KL17], the one-sided edge shift space of an ultragraph together with the restriction of the shift map to points with nonzero length [GR19], the full one-sided shift over an infinite alphabet together with the restriction of the shift map to points with nonzero length [OMW14], the cover of a one-sided shift space constructed in [BC20b], and more generally, the canonical local homeomorphism extension of a locally injective map constructed in [Th11].
A C*-algebra is naturally associated to a Deaconu-Renault system via a groupoid construction (see, for example, [D95,Re00,CRST]), and the class of such C*-algebras includes crossed products by actions of Z on locally compact Hausdorff spaces, Cuntz-Krieger algebras [CK80], graph C*-algebras [Ra05], and, via Katsura's topological graphs [Ka04], all Kirchberg algebras (i.e. all purely infinite, simple, nuclear, separable C*-algebras) satisfying the universal coefficient theorem (see [Ka08]), C*-algebras associated with one-sided shift spaces [BC20b], and C*-algebras of locally injective surjective maps [Th11].
It is natural to ask how much information from a dynamical system can be recovered by C*-algebraic data associated with it. It is known that an action of a locally compact group on a topological space can be recovered up to conjugacy from its C*-crossed product together with the corresponding dual action (see [KOQ18,Proposition 4.3]), and similar results have been obtained in more specialised settings; see, for example, [ In recent years, a similar approach has been used to encode dynamical relations between irreducible shifts of finite type into Cuntz-Krieger algebras. Matsumoto has been a driving force in this endeavour with his characterisations of continuous orbit equivalence [Ma10] and one-sided eventual conjugacy [Ma17], among others. Together with Matui, Matsumoto characterised flow equivalence as diagonal-preserving * -isomorphism of stabilised Cuntz-Krieger algebras using groupoids [MM14] (see also [CEOR]), and this led the third-named author and Rout to prove similar characterisations for two-sided conjugacy of shifts of finite type [CR17]. Complementing the work of two of the authors [BC20a], Matsumoto recently proved that a one-sided conjugacy of irreducible shifts of finite type can be encoded into Cuntz-Krieger algebras using gauge actions [Ma21a]. Matsumoto's methods immediately inspired us and helped shape the present work. Since then, Matsumoto has released three other works on related topics [Ma20,Ma21b,Ma21c].
In this paper we provide a characterisation of conjugacy of a pair of Deaconu-Renault systems in terms of isomorphisms of their groupoids and their C*-algebras. Our results are summarised in Theorem 3.1. Since we work in the general framework of Deaconu-Renault systems, our work complements (and applies) the groupoid reconstruction theory of [CRST], which is based on the pioneering work of Renault [Re80,Re08] and Kumjian [Ku86]. We prove in Proposition 3.12 that a * -isomorphism of the C*-algebras of Deaconu-Renault groupoids that intertwines a sufficiently rich collection of automorphisms induces a conjugacy between the underlying systems. It is noteworthy that we do not require the * -isomorphism to be diagonal-preserving. By restricting to the case of one-sided shifts of finite type, we therefore not only recover, but strengthen Matsumoto's [Ma21a, Theorem 1] characterisation of one-sided conjugacy (see Corollary 3.3). The proof requires a technical result (Lemma 3.10) which relates actions of the C*-algebra to cocycles on the groupoid, and we believe that this may be of independent interest.
In future work [ABCE], we shall approach conjugacy of directed graphs from an algorithmic and combinatorial point of view related to [ER].
This paper is organised as follows. In Section 2 we introduce the necessary notation and preliminaries and establish some basic facts about Deaconu-Renault systems, conjugacy of Deaconu-Renault systems, and Deaconu-Renault groupoids and their C*-algebras. We also provide several examples of Deaconu-Renault systems and Deaconu-Renault groupoids and their C*-algebras and relate them to previous work. In Section 3 we prove our main results relating conjugacy of Deaconu-Renault systems to the associated groupoids and C*-algebras. Our results are summarised in Theorem 3.1, which follows from Propositions 3.4, 3.8, and 3.12.
Acknowledgements
The first-named author would like to thank the other three authors for their hospitality during a visit to the University of Copenhagen in 2019. This visit was funded by an Australian Mathematical Society Lift-Off Fellowship. The second-named author is supported by the Carlsberg Foundation through an Internationalisation Fellowship, and is grateful to the people at the University of Wollongong for providing a hospitable and stimulating environment during his stay. The third-named author is supported by Research Council Faroe Islands, and is grateful to the Department of Mathematical Sciences at the University of Copenhagen for their hospitality during a visit in 2019. The fourth-named author was supported by the DFF-Research Project 2 "Automorphisms and Invariants of Operator Algebras", no. 7014-00145.
Preliminaries
Here we introduce the basic concepts and notation used throughout the paper. We denote the real numbers by R, the integers by Z, the nonnegative integers by N, and the positive integers by N + . We denote the cardinality of a set A by |A|. Given a locally compact Hausdorff space X, we write C b (X) for the set of continuous bounded functions from X to the complex numbers, we write C 0 (X) for the subset of C b (X) consisting of functions that vanish at infinity, and we write C c (X) for the subset of C 0 (X) consisting of functions that have compact support. The spaces C 0 (X) and C b (X) are abelian C*algebras, and C b (X) is (isomorphic to) the multiplier algebra of C 0 (X) (see, for instance, [Murphy, Example 3.1.3]).
2.1. Deaconu-Renault systems. A Deaconu-Renault system is a pair (X, σ X ) consisting of a locally compact Hausdorff space X and a partially defined local homeomorphism σ X : dom(σ X ) → ran(σ X ), where both dom(σ X ) and ran(σ X ) are open subsets of X. Let σ 0 X := id X , and inductively define dom(σ k X ) := σ −1 X (dom(σ k−1 X )) for k ∈ N. Then for every k ∈ N, the map σ k X : dom(σ k X ) → ran(σ k X ) defined by σ k X (x) := σ k−1 X (σ X (x)) is a local homeomorphism onto an open subset of X. Whenever we write σ k X (x) it is to be understood that x ∈ dom(σ k X ). The orbit of a point x ∈ X is the subset orb X (x) := k,l∈N
σ −l X σ k X (x) ,
and a pair of points x, y ∈ X are in the same orbit if and only if σ k
X (x) = σ l X (y) for some k, l ∈ N. A point x ∈ X is periodic with period p ∈ N + if x = σ p X (x), it is eventually periodic if σ n X (x)
is periodic for some n ∈ N, and it is aperiodic if it is not eventually periodic. We say that a Deaconu-Renault system (X, σ X ) is topologically free if the set {x ∈ X : x is not periodic} is dense in X, and that the system is second-countable if X is second-countable.
Similar systems were studied independently by Deaconu in [D95] and by Renault in [Re00] (as singly generated dynamical systems). Here we follow the terminology of [CRST, Section 8].
Definition 2.1. Let (X, σ X ) and (Y, σ Y ) be Deaconu-Renault systems. We call a homeomorphism h : X → Y a conjugacy if h(σ X (x)) = σ Y (h(x)) and h −1 (σ Y (y)) = σ X (h −1 (y)) for all x ∈ dom(σ X ) and y ∈ dom(σ Y ). We say that the systems (X, σ X ) and (Y, σ Y ) are conjugate if there exists a conjugacy h : X → Y. that φ is also a conjugacy between the Deaconu-Renault systems (Σ A , σ Σ A ) and (Σ B , σ Σ B ).
Example 2.4. Let G be an ultragraph satisfying the condition (RFUM) introduced in [GR19], let (X G , σ G ) be the one-sided edge shift of G constructed in [GR19], and let σ X G be the restriction of σ G to X G \ p 0 , where p 0 is the set of ultrapaths of length 0 (see [GR19, Section 2.1]). It follows from [GR19, Propositions 3.7, 3.12, and 3.16] that (X G , σ X G ) is a Deaconu-Renault system.
If G 1 and G 2 are two ultragraphs satisfying condition (RFUM), then a map φ : X G 1 → X G 2 is a conjugacy between the Deaconu-Renault systems (X G 1 , σ X G 1 ) and (X G 2 , σ X G 2 ) if and only if it is a length-preserving conjugacy, as defined in [GR19, Definition 3.19].
The following is an example of a homeomorphism h : X → Y between two Deaconu-Renault systems (X, σ X ) and (Y, σ Y ) that is not a conjugacy, even though h(σ X (x)) = σ Y (h(x)) for all x ∈ dom(σ X ).
Example 2.5. Consider the graph E consisting of a single vertex, and the graph F consisting of a single loop. The boundary-path spaces ∂E and ∂F are both singletons, and so there is a homeomorphism h : ∂E → ∂F that trivially satisfies h(σ E (x)) = σ F (h(x)) for all x ∈ dom(σ E ) (since dom(σ E ) = ∅). However, (∂E, σ E ) and (∂F, σ F ) are not conjugate systems, because dom(σ F ) = ∂F , whereas dom(σ E ) = ∅.
It will be convenient later to have slight reformulations of the conjugacy condition at our disposal. Note that the conditions in item (3) below are set equalities.
Lemma 2.6. Let (X, σ X ) and (Y, σ Y ) be Deaconu-Renault systems, and let h : X → Y be a homeomorphism. The following statements are equivalent.
(1) h : X → Y is a conjugacy.
(2) h(dom(σ X )) = dom(σ Y ), and h • σ X = σ Y • h on dom(σ X ). (3) h(σ −1 X (x)) = σ −1 Y (h(x)) and h −1 (σ −1 Y (y)) = σ −1 X (h −1 (y))
, for all x ∈ X and y ∈ Y. In particular, if the two systems have globally defined dynamics (i.e. if dom(σ X ) = X and dom(σ Y ) = Y), then the condition h • σ X = σ Y • h is equivalent to h being a conjugacy.
Proof. (1) ⇐⇒ (2): Assume first that h : X → Y is a conjugacy. If x ∈ dom(σ X ), then h(σ X (x)) = σ Y (h(x)), and so h(dom(σ X )) ⊆ dom(σ Y ). For the reverse inclusion, fix y ∈ dom(σ Y ). Then σ X (h −1 (y)) = h −1 (σ Y (y)), and so y ∈ h(dom(σ X )).
For the converse, we need to verify that h −1 • σ Y = σ X • h −1 on dom(σ Y ). Fix y ∈ dom(σ Y ), and let x := h −1 (y) ∈ dom(σ X ). Then h(σ X (x)) = σ Y (h(x)) = σ Y (y), and so h −1 (σ Y (y)) = σ X (x) = σ X (h −1 (y)), as required.
(1) ⇐⇒ (3): Assume first that h is a conjugacy, and fix x ∈ X. Suppose that σ −1 X (x) is nonempty, and fix z ∈ σ −1 X (x). Then σ Y (h(z)) = h(σ X (z)) = h(x), and thus
h(z) ∈ σ −1 Y (h(x)). It follows that h(σ −1 X (x)) ⊆ σ −1 Y (h(x)). For the reverse inclusion, fix w ∈ σ −1 Y (h(x)). Then σ X (h −1 (w)) = h −1 (σ Y (w)) = x, and so w ∈ h(σ −1 X (x)). There- fore, σ −1 Y (h(x)) ⊆ h(σ −1 X (x)). Suppose instead that σ −1 X (x) is empty. We claim that σ −1 Y (h(x)) is also empty. Suppose for contradiction that there exists w ∈ σ −1 Y (h(x)). Then σ X (h −1 (w)) = h −1 (σ Y (w)) = x, and thus h −1 (w) ∈ σ −1 X (x), which contradicts the hypothesis that σ −1 X (x) = ∅. A similar argument shows that if h is a conjugacy, then h −1 (σ −1 Y (y)) = σ −1 X (h −1 (y)) for all y ∈ Y.
For the converse, fix x ∈ dom(σ X ), and let w := h(σ X (x)).Then
x ∈ σ −1 X (σ X (x)) = σ −1 X (h −1 (w)) = h −1 (σ −1 Y (w)) by hypothesis, and so σ Y (h(x)) = w = h(σ X (x)). A similar argument shows that h −1 (σ Y (y)) = σ X (h −1 (y)) for all y ∈ dom(σ Y )
, and so h is a conjugacy.
Remark 2.7. As Example 2.5 shows, there are some subtleties involved in the definition of a conjugacy of arbitrary Deaconu-Renault systems: it is important that we have conditions on both h and its inverse h −1 . However, if the two systems have globally defined dynamics (i.e. if dom(σ X ) = X and dom(σ Y ) = Y), then Lemma 2.6 implies that the condition h • σ X = σ Y • h is sufficient. In the case of directed graphs, the condition of having globally defined dynamics is equivalent to there being no singular vertices.
We now introduce two maps (σ X ) * : C 0 (X) → C b (dom(σ X )) and (σ X ) * : C c (dom(σ X )) → C c (X) that we will use in Theorem 3.1 and Proposition 3.4 to characterise conjugacy of Deaconu-Renault systems.
Suppose that (X, σ X ) is a Deaconu-Renault system. We define the map (σ X ) * :
C 0 (X) → C b (dom(σ X )) by (σ X ) * (f )(x) := f (σ X (x))
, for all f ∈ C 0 (X) and x ∈ dom(σ X ). The following example shows that the range of (σ X ) * is in general larger than the collection of functions vanishing at infinity.
Example 2.8. Let E be the graph with two vertices v and w such that v emits infinitely many edges {e n : n ∈ N} to w. Then 1 Z(w) ∈ C 0 (∂E), and 1 Z(w) • σ E is defined and nonzero (and bounded) on the entire non-compact set dom(σ E ) = {e n : n ∈ N}.
Since σ X is a local homeomorphism, there is a map (σ X ) * :
C c (dom(σ X )) → C c (X) given by (σ X ) * (f )(x) := z∈σ −1 X (x) f (z),
for all f ∈ C c (dom(σ X )) and x ∈ X.
2.2. Deaconu-Renault groupoids. Every Deaconu-Renault system (X, σ X ) gives rise to a Deaconu-Renault groupoid X by (x, p, y)(y, q, z) := (x, p + q, z), and inversion is defined on G X by (x, p, y) −1 := (y, −p, x). The range and source maps of G X are r : (x, p, y) → x and s : (x, p, y) → y, respectively. The unit space of G X is G
G X := m,n∈N (x, m − n, y) ∈ dom(σ m X ) × {m − n} × dom(σ n X ) : σ m X (x) = σ n X (y) , with composable pairs G (2) X := (x, p, y), (w, q, z) ∈ G X × G X : y = w (cf. [CRST,(0) X = {(x, 0, x) ∈ G X :
x ∈ X}, and we identify it with X via the map (x, 0, x) → x. Given m, n ∈ N and open subsets U and V of X, we define
Z(U, m, n, V ) := (x, m − n, y) ∈ G X : x ∈ U, y ∈ V, σ m X (x) = σ n X (y) ,(2.1)
and the collection of all such sets forms a basis for a topology on G X . Equivalently, the topology is generated by sets of the form (2.1), where, in addition, σ m X | U and σ n X | V are homeomorphisms onto their images, and σ m X (U) = σ n X (V ). Under this topology, G X is an amenable locally compact Hausdorff groupoid which isétale, in the sense that the range and source maps are local homeomorphisms. A set B ⊆ G X is called a bisection of G X if there is an open subset U of G X such that B ⊆ U, and r| U and s| U are homeomorphisms onto open subsets of G (0) X . Everyétale groupoid has a basis consisting of open bisections, and by choosing U, V ⊆ X sufficiently small, the sets of the form (2.1) become open bisections of G X . Moreover, G X is second-countable when X is second-countable, and G X is ample (meaning it has a basis of compact open bisections) when X is totally disconnected. Given Deaconu-Renault groupoids G X and G Y and a map ψ :
G X → G Y , we write ψ (0) : X → Y for the map induced by the restriction ψ| G (0) X : G (0) X → G (0) Y . The isotropy of a groupoid G is the subgroupoid Iso(G) := {γ ∈ G : r(γ) = s(γ)}. If
G is a second-countable locally compact Hausdorffétale groupoid, then so is the interior Iso(G) • of the isotropy of G. Since the unit space of anétale groupoid is open, we have
G (0) ⊆ Iso(G) • if G isétale.
We now prove a well known result that we were unable to find in the literature.
Lemma 2.9. If (X, σ X ) is a topologically free Deaconu-Renault system, then Iso(G X ) • = X.
Proof. Suppose for contradiction that there exist x ∈ X and k, l ∈ N with k = l and σ k
X (x) = σ l X (x) such that (x, k − l, x) ∈ Iso(G X ) • . Then there are nonempty open subsets U, V ⊆ X such that σ k X | U and σ l X | V are homeomorphisms onto their images, σ k X (U) = σ l X (V ), and Z(U, k, l, V ) ⊆ Iso(G X ). We therefore have that σ k X (x ′ ) = σ l X (x ′ ) for any x ′ ∈ U.
It follows that any element of the open subset σ k X (U) is periodic with period |k − l|, but this contradicts the assumption that (X, σ X ) is topologically free. Thus,
Iso(G X ) • = X.
Given a group Γ and a function f : X → Γ, for each k ∈ N and x ∈ dom(σ k X ), we write
f (k) (x) := k−1 i=0 f (σ i X (x)),
where it is understood that f (0) = 0. We use additive notation because in all of our examples of interest Γ will be abelian, but of course a similar expression using multiplicative notation makes sense. We call a continuous groupoid homomorphism from G X into a topological group Γ a continuous cocycle. Any continuous map f : X → Γ gives rise to a continuous cocycle
c f : G X → Γ defined by c f (x, m − n, y) := f (m) (x) − f (n) (y) = m−1 i=0 f (σ i X (x)) − n−1 j=0 f (σ j X (y)), (2.2)
for (x, m − n, y) ∈ G X satisfying σ m X (x) = σ n X (y). When Γ = Z and f ≡ 1, we obtain the canonical continuous cocycle c f : (x, p, y) → p, which we denote by c X .
2.3. Deaconu-Renault C*-algebras. Given a locally compact Hausdorffétale groupoid G, there are associated full and reduced groupoid C*-algebras C * (G) and C * r (G) encoding the structure of G; see, for instance, [S20,Re80] for details. An argument similar to the one used in [SW16, Lemma 3.5] shows that Deaconu-Renault groupoids are amenable, so the two C*-algebras C * (G X ) and C * r (G X ) are canonically * -isomorphic, and we shall not distinguish between them: we let C * (G X ) denote the C*-algebra associated to G X . Since G X isétale, the unit space G (0) X ∼ = X is open in G X , and we view the diagonal subalgebra C 0 (X) as a subalgebra of C * (G X ). Note that C 0 (X) need not be a C*-diagonal (in the sense of Kumjian [Ku86]), nor a Cartan subalgebra (in the sense of Renault [Re08]).
The Pontryagin dual of a locally compact abelian group Γ is the locally compact group Γ consisting of continuous group homomorphisms from Γ to the circle group T, endowed with the compact-open topology. A continuous cocycle c : G X → Γ induces an action
γ c : Γ C * (G X ) satisfying γ c χ (ξ)(η) = χ(c(η)) ξ(η), (2.3)
for χ ∈ Γ, ξ ∈ C c (G), and η ∈ G; cf. [Re80, Chapter 2, Section 5]. In particular, there is a weighted action γ X,f : Γ C * (G X ) associated to each f ∈ C(X, Γ), satisfying
γ X,f χ (ξ)(x, m − n, y) = χ f (m) (x) − f (n) (y) ξ(x, m − n, y), (2.4) for χ ∈ Γ, ξ ∈ C c (G X ), and (x, m − n, y) ∈ G X satisfying σ m X (x) = σ n X (y). The canonical gauge action γ X : T C * (G X ) is induced by the canonical continuous cocycle c X on G X .
Example 2.10. Let X be a locally compact Hausdorff space, and let σ : X → X be a homeomorphism. Then (X, σ) is a Deaconu-Renault system. The Deaconu-Renault groupoid G X of (X, σ) is isomorphic to the transformation groupoid X × σ Z (see, for instance, [Re80, Examples I.1.2(a)]), and C * (G X ) is isomorphic to the crossed product C 0 (X) ⋊ σ Z by an isomorphism that restricts to the identity map on C 0 (X) and intertwines the canonical gauge action of C * (G X ) and the dual action of C 0 (X) ⋊ σ Z (see, for instance, [S20, Example 9.2.6]).
Example 2.11. If X is a locally compact Hausdorff space and σ : X → X is a covering map (that is, σ is continuous and surjective, and for each x ∈ X, there is an open neighbourhood V of x such that T −1 (V ) is a disjoint union of open sets (U i ) i∈I such that σ| U i is a homeomorphism onto V for each i ∈ I), then (X, σ) is a Deaconu-Renault system. The Deaconu-Renault groupoid G X of (X, σ) is the groupoid Γ considered in [D95]. If X is compact and each x ∈ X has exactly p preimages under σ, then according to [EV06,Theorem 9.1], the C*-algebra C * (G X ) is isomorphic to the crossed product C(X) ⋊ α,L N, where α is the * -homomorphism from C(X) to C(X) given by α(f ) := f • σ, and L is the transfer operator from C(X) to C(X) given by L(f )(x) := 1 p y∈σ −1 (x) f (y). The proof of [EV06, Theorem 9.1] goes through with minor modifications if the assumption that each x ∈ X has exactly p preimages under σ is dropped and the definition of L(f )(x) is changed to
1 |σ −1 (x)| y∈σ −1 (x) f (y).
Example 2.12. Let E be a directed graph. The Deaconu-Renault groupoid G ∂E of the Deaconu-Renault system (∂E, σ E ) is the graph groupoid G E described, among other places, in [BCW17, Section 2.3], and C * (G ∂E ) is isomorphic to the graph C*-algebra C * (E) of E by an isomorphism that maps C 0 (∂E) onto the diagonal C*-subalgebra D(E) of C * (E) and intertwines the canonical gauge action γ ∂E : T C * (G ∂G ) and the gauge action γ E : T C * (E) (see, for instance, [BCW17, Proposition 2.2]). In [CR17, Section 3] the third-named author and Rout consider generalised gauge actions. A function k : E 1 → R extends to a function k : E * → R by setting k| E 0 ≡ 0 and k(e 1 · · · e n ) := k(e 1 ) + · · · + k(e n ) for e 1 · · · e n ∈ E * \ E 0 . There is a continuous cocycle c k : G E → R given by c k (µx, |µ| − |ν|, νx) := k(µ) − k(ν), and this cocycle induces a generalised gauge action γ E,k : Example 2.14. Let (X, σ X ) be a one-sided shift of finite type (see, for instance, [LM95, Section 13.8]). It follows from [IT74, Theorem 1] that (X, σ X ) is a Deaconu-Renault system. The Deaconu-Renault groupoid G X of (X, σ X ) is identical to the groupoid G X described in [CEOR, Section 2.6].
R C * (G E ),
If A is an n × n {0, 1}-matrix and (X A , σ X A ) is the one-sided topological Markov shift defined by A (see, for instance, [Ki98, Page 3]), then the Deaconu-Renault groupoid G X A of (X A , σ X A ) is the groupoid G A described in [MM14, Section 2.2], and C * (G X A ) is isomorphic to the Cuntz-Krieger algebra O A [CK80] with generators s 1 , . . . , s n , via an isomorphism that maps C(X A ) onto the C*-subalgebra D A generated by the projections s i s * i , and intertwines the canonical gauge action γ X A : T C * (G X A ) and the gauge action λ : T O A (see, for instance, [MM14, Section 2.3]). Moreover, the isomorphism between C * (G X A ) and O A can be constructed such that it has the property that if g ∈ C(X A , Z), then it intertwines the weighted gauge action γ X A ,g : T C * (G X A ) and the action ρ A,g : T O A considered in [Ma21a].
The map (σ A ) * :
C(X A ) → C(X A ) coincides with the map φ A : D A → D A given by φ A (x) := n i=1 s i xs * i for x ∈ D A .
This map φ A appeared in Cuntz and Krieger's original paper as an invariant of one-sided conjugacy; cf. [CK80, Proposition 2.17]. On the other hand, the map (σ A ) * : C(X A ) → C(X A ) coincides with the adjacency operator λ A on D A , given by λ A (x) := n i=1 s * i xs i for x ∈ D A . Proposition 3.4 shows that these maps can be used to characterise conjugacy.
Example 2.15. Let (X, σ X ) be a one-sided shift space, and let ( X, σ X ) be the cover of (X, σ X ) constructed in [BC20b, Section 2.1]. It follows from [BC20b, Lemma 2.3] that ( X, σ X ) is a Deaconu-Renault system. If two one-sided shift spaces (X, σ X ) and (Y, σ Y ) are conjugate, then the corresponding Deaconu-Renault systems ( X, σ X ) and ( Y, σ Y ) are conjugate (see [BC20b,Lemma 4.1])l however, there are examples of non-conjugate onesided shift spaces (X, σ X ) and (Y, σ Y ) for which ( X, σ X ) and ( Y, σ Y ) are conjugate (for example, consider a one-sided strictly sofic shift (X, σ X ) and the one-sided edge shift (Y, σ Y ) of its left Krieger cover, cf. [Ki98, Exercise 6.1.9]). The Deaconu-Renault groupoid G X of ( X, σ X ) is the groupoid G X described in [BC20b, Section 2.2]. It is shown in [C04] that there is an isomorphism from C * (G X ) to the C*algebra O X studied in [C08] that maps C 0 ( X) onto the C*-subalgebra D X and intertwines the canonical gauge actions of C * (G X ) and O X .
Example 2.16. Let X be a compact metrisable space, let φ : X → X be a continuous locally injective surjection, and let ( D, ψ) be the canonical extension of (X, φ) constructed in [Th11, Section 4]. It follows from [Th11, Proposition 4.1] that ( D, ψ) is a Deaconu-Renault system. If two continuous locally injective surjective maps φ : X → X and φ ′ : X ′ → X ′ between compact metrisable spaces are conjugate, then the corresponding Deaconu-Renault systems ( D, ψ) and ( D ′ , ψ ′ ) are conjugate (see [Th11, Section 4]), but there are, as in Example 2.15, examples of non-conjugate maps φ : X → X and φ ′ : X ′ → X ′ for which ( D, ψ) and ( D ′ , ψ ′ ) are conjugate.
The Deaconu-Renault groupoid G D of ( D, ψ) is the groupoid Γ ψ studied in [Th11]. It therefore follows from [Th11, Theorem 5.4] that there is an isomorphism from C * (G D ) to the C*-algebra C * r (Γ φ ) constructed in [Th10] that maps C( D) onto the C*-subalgebra D Γ φ .
Characterising conjugacy via groupoids and C*-algebras
In this section, we investigate the conditions that must be imposed on isomorphisms of Deaconu-Renault groupoids or their C*-algebras in order to ensure that the underlying Deaconu-Renault systems are conjugate. Our results are summarised in the following theorem.
Theorem 3.1. Let (X, σ X ) and (Y, σ Y ) be second-countable Deaconu-Renault systems. The following statements are equivalent.
(1) The systems (X, σ X ) and (Y, σ Y ) are conjugate.
(2) There exists a * -isomorphism ϕ : C 0 (X) → C 0 (Y) satisfying the following three equivalent conditions:
(i) there is a conjugacy h : X → Y satisfying ϕ(f ) = f • h −1 for f ∈ C 0 (X); (ii) ϕ (σ X ) * (f )g = (σ Y ) * ϕ(f ) ϕ(g) for all f, g ∈ C 0 (X); (iii) ϕ C c (dom(σ X )) = C c (dom(σ Y )), and ϕ • (σ X ) * = (σ Y ) * • ϕ| Cc(dom(σ X )) . (3)
There exists a groupoid isomorphism ψ : G X → G Y satisfying the following three equivalent conditions:
(i) there is a conjugacy h : X → Y such that ψ(x, p, y) = (h(x), p, h(y)) for (x, p, y) ∈ G X ; (ii) c g•ψ (0) = c g • ψ for all g ∈ C(Y, R); (iii) there is a homeomorphism h : X → Y that satisfies c g•h = c g • ψ for all g ∈ C(Y, R). (4)
There is a * -isomorphism ϕ : C * (G X ) → C * (G Y ) satisfying the following two equivalent conditions: (i) ϕ(C 0 (X)) = C 0 (Y), and there is a conjugacy h :
X → Y such that ϕ(f ) = f • h −1 for f ∈ C 0 (X) and ϕ • γ X,g•h t = γ Y,g t
• ϕ for all t ∈ R and g ∈ C(Y, R); (ii) there is a homeomorphism h : X → Y (which is not necessarily a conjugacy) such that ϕ • γ X,g•h t = γ Y,g t • ϕ for all t ∈ R and g ∈ C(Y, R).
Remark 3.2. Theorem 3.1 follows from Propositions 3.4, 3.8, and 3.12, which we prove below (Proposition 3.4 gives us that the three conditions in Theorem 3.1(2) are equivalent, and that (1) and (2) are equivalent; Proposition 3.8 gives us that the three conditions in Theorem 3.1(3) are equivalent, and that (1) and (3) are equivalent; and Proposition 3.12 gives us that the two conditions in Theorem 3.1(4) are equivalent, and that (1) and (4) are equivalent). It follows from Propositions 3.8 and 3.12 that in (3) and (4) in Theorem 3.1 we can replace R by any group that is separating for X and Y, in the sense of Definition 3.5. In particular, if X and Y are totally disconnected, then we can replace R by Z. We therefore obtain the following corollary, which is a generalisation and a strengthening of [Ma21a, Theorem 1].
Corollary 3.3. Let E and F be countable directed graphs. If there exist a homeomorphism h : ∂E → ∂F and a * -isomorphism ϕ :
C * (E) → C * (F ) satisfying ϕ • γ E,g•h z = γ F,g z • ϕ
for all z ∈ T and g ∈ C(∂F, Z), then the boundary-path spaces (∂E, σ E ) and (∂F, σ F ) are conjugate. Conversely, if h : ∂E → ∂F is a conjugacy, then there is a * -isomorphism
ϕ : C * (E) → C * (F ) satisfying ϕ(D(E)) = D(F ), ϕ(f ) = f • h −1 for f ∈ D(E), and ϕ • γ E,g•h z = γ F,g z • ϕ
for all z ∈ T and g ∈ C(∂F, Z).
We now prove Proposition 3.4, from which it follows that the three conditions (i), (ii), and (iii) in Theorem 3.1(2) are equivalent, and that (1) and (2) in Theorem 3.1 are equivalent.
Proposition 3.4. Let (X, σ X ) and (Y, σ Y ) be Deaconu-Renault systems, and let h : X → Y be a homeomorphism. The map ϕ : f → f • h −1 is a * -isomorphism from C 0 (X) to C 0 (Y), and the following statements are equivalent.
(1) h : X → Y is a conjugacy.
(2) For all f, g ∈ C 0 (X),
ϕ (σ X ) * (f )g = (σ Y ) * ϕ(f ) ϕ(g). (3.1) (3) ϕ C c (dom(σ X )) = C c (dom(σ Y )), and ϕ • (σ X ) * = (σ Y ) * • ϕ| Cc(dom(σ X )) . (3.2)
Proof. A routine argument shows that the map ϕ :
f → f • h −1 is a * -isomorphism from C 0 (X) to C 0 (Y).
(1) =⇒ (2) and (3) : Assume first that h : X → Y is a conjugacy. It follows from Lemma 2.6 that h(dom(σ X )) = dom(σ Y ), and hence ϕ C c (dom(σ X )) = C c (dom(σ Y )).
Fix f, g ∈ C 0 (X). Since h is a conjugacy, we have
f σ X (h −1 (y)) g(h −1 (y)) = f h −1 (σ Y (y)) g(h −1 (y))
for all y ∈ dom(σ Y ), and equation (3.1) follows. By Lemma 2.6, we have σ −1 X (h −1 (y)) = h −1 (σ −1 Y (y)) for all y ∈ Y, and hence ϕ (σ X ) * (f ) (y) =
z∈σ −1 X (h −1 (y)) f (z) = w∈σ −1 Y (y) f (h −1 (w)) = (σ Y ) * ϕ(f ) (y),
and so equation (3.2) holds.
(2) =⇒ (1): Suppose that equation (3.1) holds. Fix y ∈ dom(σ Y ). We claim that σ X (h −1 (y)) = h −1 (σ Y (y)). To see this, first choose g ∈ C 0 (X) such that g(h −1 (y)) = 1. Then for all f ∈ C 0 (X), we have
f σ X (h −1 (y)) = ϕ (σ X ) * (f )g = (σ Y ) * ϕ(f ) ϕ(g) = f h −1 (σ Y (y)) ,
and so σ X (h −1 (y)) = h −1 (σ Y (y)), as claimed. Since the assumption of equation (3.1) is equivalent to the assumption that
ϕ −1 (σ Y ) * (f ′ )g ′ = (σ X ) * ϕ −1 (f ′ ) ϕ −1 (g ′ )
for all f ′ , g ′ ∈ C 0 (Y), a similar argument shows that σ Y (h(x)) = h(σ X (x)) for all x ∈ dom(σ X ). Therefore, h is a conjugacy.
(3) =⇒ (1): Suppose that ϕ C c (dom(σ X )) = C c (dom(σ Y )) and that equation (3.2) holds. We will use the implication (3) =⇒ (1) of Lemma 2.6 to show that h is a conjugacy. Fix y ∈ Y. By equation (3.2), we have
z∈σ −1 X (h −1 (y)) f (z) = w∈σ −1 Y (y) f (h −1 (w)) = t∈h −1 (σ −1 Y (y)) f (t), (3.3)
for all f ∈ C c (dom(σ X )). Suppose for contradiction that h −1 (σ −1 Y (y)) = σ −1 X (h −1 (y)). Then there exists x ∈ σ −1 X (h −1 (y)) such that x / ∈ h −1 (σ −1 Y (y)). Since X is a locally compact Hausdorff space, it is regular, and thus since h −1 (σ −1 Y (y)) is closed, there is an open neighbourhood U ⊆ X of x such that U ∩ h −1 (σ −1 Y (y)) = ∅. By Urysohn's lemma, there exists a function f ∈ C c (X, [0, 1]) such that f (x) = 1 and f (z) = 0 for all z ∈ h −1 (σ −1 Y (y)). But this contradicts equation (3.3), and so we must have h −1 (σ −1 Y (y)) = σ −1 X (h −1 (y)). Since the assumption of equation (3.2) is equivalent to the assumption that
ϕ −1 • (σ Y ) * = (σ X ) * • ϕ −1 | Cc(dom(σ Y )) , a similar argument shows that h(σ −1 X (x)) = σ −1 Y (h(x))
for all x ∈ X. Therefore, Lemma 2.6 implies that h is a conjugacy.
We now introduce the notion of a separating group, which we use in Propositions 3.8 and 3.12.
Definition 3.5. Let X be a locally compact Hausdorff space, and let Γ be a locally compact group with identity element id Γ . We say that Γ is separating for X if, for any finite set F ⊆ X and x ∈ X\F , there exists f ∈ C(X, Γ) such that f (x) has infinite order in Γ and f | F ≡ id Γ .
Example 3.6. Urysohn's lemma for locally compact Hausdorff spaces ensures that R is separating for any locally compact Hausdorff space X. If X is totally disconnected, then Z is separating for X.
Lemma 3.7. Let (X, σ X ) be a Deaconu-Renault system and suppose that Γ is a locally compact group that is separating for X. If k, l ∈ N and
k i=0 f (a i ) = l j=0 f (b j ) (3.4)
for some (not necessarily distinct) elements a 0 , . . . , a k , b 0 , . . . , b l ∈ X and all f ∈ C(X, Γ), then k = l and {a 0 , . . . , a k } = {b 0 , . . . , b k }. Moreover, if x, x ′ ∈ X satisfy σ k X (x) = σ l X (x ′ ) and f (k) (x) = f (l) (x ′ ) for all f ∈ C(X, Γ), then k = l and x = x ′ .
Proof. Let A := {a 0 , . . . , a k } and B := {b 0 , . . . , b l }. For x ∈ X, we may choose f ∈ C(X, Γ) such that f (x) has infinite order and f | (A∪B)\{x} ≡ id Γ . By equation (3.4), we then have that i ∈ {0, . . . , k} : a i = x = j ∈ {0, . . . , l} : b j = x . By applying this observation for all x ∈ X, we see that k = l and A = B.
For the second part, the hypothesis that f (k) (x) = f (l) (x ′ ) for all f ∈ C(X, Γ) means that
k−1 i=0 f (σ i X (x)) = l−1 j=0 f (σ j X (x ′ ))
for all f ∈ C(X, Γ). It follows from the first part that k = l and that
{σ i X (x) : i = 0, . . . , k − 1} = {σ j X (x ′ ) : j = 0, . . . , k − 1}.
This means that x = σ p X (x ′ ) and x ′ = σ q X (x) for some p, q ∈ N. By choosing n ∈ N + such that k n(p + q), the fact that σ k
X (x) = σ k X (x ′ ) implies that x = σ n(p+q) X (x) = σ n(p+q) X (x ′ ) = x ′ ,
as required.
We now prove Proposition 3.8, from which it follows that the three conditions (i), (ii), and (iii) in Theorem 3.1(3) are equivalent, and that (1) and (3) in Theorem 3.1 are equivalent.
Proposition 3.8. Let (X, σ X ) and (Y, σ Y ) be Deaconu-Renault systems. A conjugacy h : X → Y induces a groupoid isomorphism ψ : G X → G Y satisfying ψ(x, p, y) = (h(x), p, h(y)),
for (x, p, y) ∈ G X . Moreover, if ψ : G X → G Y is a groupoid isomorphism and Γ is a locally compact group that is separating for Y, then the following three conditions are equivalent.
(1) There is a conjugacy h : X → Y such that ψ(x, p, y) = (h(x), p, h(y)) for (x, p, y) ∈ G X .
(2) c g•ψ (0) = c g • ψ for g ∈ C(Y, Γ). (3) There is a homeomorphism h : X → Y that satisfies c g•h = c g •ψ for all g ∈ C(Y, Γ).
Proof. A routine argument shows that if h : X → Y is a conjugacy, then the map ψ : G X → G Y given by ψ(x, p, y) = (h(x), p, h(y)) is a groupoid isomorphism.
We now prove the implication (1) =⇒ (2). Suppose that condition (1) holds. Let Γ be a locally compact group, and fix g ∈ C(Y, Γ). We claim that c g•h = c g • ψ. It suffices to prove the relation for groupoid elements in Z(X, 1, 0, σ X (X)) ⊆ G X , so fix x ∈ dom(σ X ). Then
c g•h x, 1, σ X (x) = g(h(x)) = c g h(x), 1, σ Y (h(x)) = c g ψ(x, 1, σ X (x)) ,
which proves the claim. The implication (2) =⇒ (3) is obvious.
It remains to prove (3) =⇒ (1). Suppose that ψ : G X → G Y is a groupoid isomorphism and that h : X → Y is a homeomorphism such that c g•h = c g • ψ for all g ∈ C(Y, Γ). Consider the homeomorphism h := ψ (0) and note that h and h need not be equal. We will show that h is a conjugacy.
Since Γ is separating for Y, it contains an element of infinite order, and this element generates a copy of Z in Γ. By choosing g ∈ C(Y, Γ) to be constantly equal to such an element, we see that ψ intertwines the canonical cocycles. Therefore, ψ(x, p, y) = ( h(x), p, h(y))
for (x, p, y) ∈ G X , and it follows from [CRST,Theorem 8.10] that h and h −1 are eventual conjugacies (cf. [CRST,Definition 8.9]). In particular, h(dom(σ X )) = dom(σ Y ). For x ∈ dom(σ X ), we let k(x) be the minimal nonnegative integer satisfying
σ k(x)+1 Y • h (x) = σ k(x) Y • h • σ X (x), (3.5)
noting that such an integer exists because h is an eventual conjugacy. By Lemma 2.6 it only remains to show that h(σ X (x)) = σ Y ( h(x)) for x ∈ dom(σ X ). This is equivalent to showing that k(x) = 0 for x ∈ dom(σ X ). Fix x ∈ dom(σ X ), and suppose for contradiction that k(x) > 0. The hypothesis c g•h = c g • ψ implies that
g(h(x)) = c g ψ(x, 1, σ X (x)) = k(x) i=0 g (σ i Y • h)(x) − k(x)−1 j=0 g (σ j Y • h • σ X )(x) , (3.6)
for any g ∈ C(Y, Γ). Since Γ is separating for Y, it follows from Lemma 3.7 that the sets
A := (σ i Y • hσ i Y • h (x) = σ k(x)−1 Y • h • σ X (x).
(3.7)
If i = k(x), then equation (3.7) contradicts the minimality of k(x), so we must have i < k(x).
We will now show that k(x) = 0. Let us first consider the case when h(x) is aperiodic, i.e. there is no pair of distinct nonnegative integers m and n such that (σ m
Y • h)(x) = (σ n Y • h)(x). If k(x) > 0, then σ k(x)+1 Y • h (x) = σ k(x) Y • h • σ X (x) = σ i+1 Y • h (x)
, and this together with the assumption that h(x) is aperiodic implies that i = k(x), which we have already seen cannot be the case. Therefore, k(x) = 0 when h(x) is aperiodic.
We now consider the case when h(x) is eventually periodic. In this case, there is a nonnegative integer n such that (σ n Y • h)(x) is periodic, and we proceed by induction on n. For n = 0 (i.e. h(x) is periodic), we choose i as in equation (3.7), and observe that
σ i Y • h (x) = σ k(x)−1 Y • h • σ X (x) is periodic. If p is a period of σ i Y • h (x) = σ k(x)−1 Y • h • σ X (x)
, then the eventual conjugacy condition equation (3.5) implies that
σ k(x) Y • h (x) = σ k(x)+p Y • h (x) = σ k(x)−1+p Y • h • σ X (x) = σ k(x)−1 Y • h • σ X (x),
which again contradicts the minimality of k(x). Therefore, k(x) = 0 when h(x) is periodic.
Assume now that n ∈ N and that k(
x ′ ) = 0 whenever x ′ ∈ dom(σ X ) and σ n Y ( h(x ′ )) is periodic. Suppose that x ∈ dom(σ X ) with σ n+1 Y ( h(x)
) periodic, and that n is the minimal nonnegative integer for which σ n+1 Y ( h(x)) is periodic. Assuming for contradiction that k(x) > 0, we again choose i according to equation (3.7). Since i < k(x), we have σ i+1 Y ( h(x)) ∈ A = B. There are two cases to consider. For the first case, suppose that
σ i+1 Y ( h(x)) = σ j Y h(σ X (x))
, for some j ∈ {0, . . . , k(x) − 1}. From equation (3.7), we see that
σ j Y • h • σ X (x) = σ i+1 Y • h (x) = σ k(x) Y • h • σ X (x)
is periodic. In particular, σ
k(x)−1 Y • h• σ X (x) is periodic (because j k(x) −1). Since i < k(x), we also see that σ k(x) Y • h (x) is periodic with the same period as σ k(x)−1 Y • h•σ X (x).
It now follows from equation (3.5) that if p is a common period of σ
k(x) Y • h (x) and σ k(x)−1 Y • h • σ X (x), then σ k(x) Y • h (x) = σ k(x)+p Y • h (x) = σ k(x)−1+p Y • h • σ X (x) = σ k(x)−1 Y • h • σ X (x),
which contradicts the minimality of k(x).
For the second case, suppose that σ i+1
Y ( h(x)) = h(x). Choose x ′ ∈ X such that h(x ′ ) = h(x). If n i, then σ n−i Y ( h(x ′ )) = σ n+1 Y ( h(x)) is periodic, and if n < i, then h(x ′ ) = σ i+1 Y ( h(x)
) is periodic. In both cases it follows from the inductive hypothesis that k(x ′ ) = 0. The assumption that c g•h = c g • ψ then implies that g(h(x ′ )) = g( h(x ′ )) for all g ∈ C(Y, Γ), and hence h(x ′ ) = h(x ′ ). Since h(x ′ ) = h(x) and h is a homeomorphism, we have x = x ′ . This means that either (σ n−i Y • h)(x) is periodic (if n i), or that h(x) is periodic (if n < i), but this contradicts the assumption that n is the minimal nonnegative integer for which σ n+1 Y ( h(x)) is periodic. We conclude that k(x) = 0 for all x ∈ dom(σ X ), and this implies that h is a conjugacy.
For the proof of Proposition 3.12, we need the following two lemmas. Given an automorphism γ of C * (G X ), we define
C * (G X ) γ := {f ∈ C * (G X ) : γ(f ) = f }.
Lemma 3.9. Let (X, σ X ) be a Deaconu-Renault system, and let Γ be a locally compact abelian group that is separating for X. Then
C 0 (X) = f ∈C(X,Γ) C * (G X ) γ X,f .
Proof. Every function in C 0 (X) is fixed by all the weighted automorphisms, so one containment is clear. For the reverse containment, take ξ ∈ C c (G X ) such that ξ is fixed by γ X,f for all f ∈ C(X, Γ). If (x, k − l, y) ∈ G X with σ k X (x) = σ l X (y) and ξ(x, k − l, y) = 0, then
ξ(x, k − l, y) = γ X,f χ (ξ)(x, k − l, y) = χ f (k) (x) − f (l) (y) ξ(x, k − l, y)
, for all χ ∈ Γ. Since the characters of an abelian group separate points, it follows that f (k) (x) = f (l) (y) for every f ∈ C(X, Γ). Since Γ is separating for X, it follows that k = l and x = y, by Lemma 3.7. Thus ξ is only supported on the unit space of G X , and so ξ ∈ C 0 (X), and the result follows.
The following technical lemma is actually the main bulk of the proof of Proposition 3.12 below. It uses the groupoid reconstruction theory of [CRST]. We state and prove Lemma 3.10 in a more general setting than we need here, as we believe it may be of independent interest.
Recall that if c is a cocycle from G into a group G with identity element id G , then c −1 (id G ) is a subgroupoid of G. We refer the reader to [CRST] for relevant details on the coactions δ c 1 and δ c 2 . The reader is invited to let c 1 and c 2 be the canonical continuous cocycles on G X and G Y , respectively; in which case, the coaction condition (equation (3.8)) reduces to the condition that ϕ intertwines the canonical gauge actions.
Note that in the statement below, we are not assuming that h and h are equal.
Lemma 3.10. Let (X, σ X ) and (Y, σ Y ) be second-countable Deaconu-Renault systems, and let h : X → Y be a homeomorphism. Let G be a discrete group with identity element id G . Let c 1 : G X → G and c 2 : G Y → G be continuous cocycles such that Iso(c −1
1 (id G )) • = X and Iso(c −1 2 (id G )) • = Y. Suppose that ϕ : C * (G X ) → C * (G Y ) is a * -isomorphism such that ϕ(C 0 (X)) = C 0 (Y) and ϕ(f ) = f • h −1 for all f ∈ C 0 (X), and that ϕ satisfies the coaction condition δ c 2 • ϕ = (ϕ ⊗ id) • δ c 1 . (3.8)
Then there is a groupoid isomorphism ψ : G X → G Y satisfying ψ (0) = h and c 1 = c 2 • ψ. Moreover, this ψ has the property that c g•h = c g •ψ whenever Γ is a locally compact abelian group, g ∈ C(Y, Γ), h : X → Y is a homeomorphism, and
ϕ • γ X,g•h χ = γ Y,g χ • ϕ (3.9)
for all χ ∈ Γ.
Proof. The first part of the lemma follows from [CRST, Theorem 6.2], but since we need an explicit description of the groupoid isomorphism ψ : G X → G Y in order to prove the second half of the lemma, we begin by recalling the construction of ψ.
For this, let us first establish some notation. The open support of ξ ∈ C c (G X ) is the set osupp(ξ) := {γ ∈ G X : ξ(γ) = 0}.
The extended Weyl groupoid H X := H(C * (G X ), C 0 (X), δ c 1 ) of the triple (C * (G X ), C 0 (X), δ c 1 ) consists of equivalence classes [n, x] of pairs (n, x), where n is a normaliser of C 0 (X) in C * (G X ) and x ∈ osupp(n); cf. [CRST,Section 4]. Let H Y := H(C * (G Y ), C 0 (Y), δ c 2 ) be the extended Weyl groupoid of (C * (G Y ), C 0 (Y), δ c 2 ). Let θ X : G X → H X and θ Y : We now observe that if [n, x] ∈ H Y and θ Y (η) = [n, x], then n(η) = 0. Indeed, following the proof of the fact that θ Y is surjective in [CRST, proof of Proposition 6.5], there exist η ′ ∈ Iso(c 2 −1 (id G )) • and γ ∈ G Y with s(η ′ ) = r(η ′ ) = s(γ) = x and n(γ) = 0 such that θ Y (γ(η ′ ) −1 ) = [n, x]. Our assumption that Iso(c 2 −1 (id G )) • = Y implies that η ′ = x, and since θ Y is injective, it follows that γ = η, and, in particular, n(η) = 0. Now, let h : X → Y be a homeomorphism (which need not be the same as h), let Γ be a locally compact abelian group, and fix g ∈ C(Y, Γ) such that ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ. We need to show that c g•h = c g • ψ. Since G X is generated by elements either belonging to G (0) X or to the compact open set Z(X, 1, 0, σ X (X)) = { x, 1, σ X (x) :
G Y → H Y beψ := θ −1 Y • ϕ * • θ X : G X → G Y is
x ∈ dom(σ X )}, it suffices to verify that (g • h)(r(η)) = c g (ψ(η)), (3.10)
for η ∈ Z(X, 1, 0, σ X (X)).
Choose n ∈ C c (G X ) with n(η) = 1 such that osupp(n) is a bisection contained in Z(X, 1, 0, σ X (X)) ∩ c 1 −1 (c 1 (η)). By the construction of ψ, we have
θ Y ψ(η) = ϕ * θ X (η) = [ϕ(n), h(s(η))],
and so the observation above implies that ϕ(n)(ψ(η)) = 0. Moreover, since osupp(n) ⊆ Z(X, 1, 0, σ X (X)), we have γ X,g•h
χ (n) = (χ • g • h)n with χ • g • h ∈ C b (X).
Here, we view C b (X) as a subalgebra of the multiplier algebra M(C * (G X )).
There is a * -isomorphism of multiplier algebras ϕ : M(C * (G X )) → M(C * (G Y )) which extends ϕ, and since the diagonal subalgebras contain approximate units of the ambient C*-algebras, we have ϕ(
C b (X)) = C b (Y) with ϕ(f ) = f • (β h) −1 ,ϕ γ X,g•h χ (n) (ψ(η)) = ϕ(χ • g • h)ϕ(n) (ψ(η)) = χ (g • h • h −1 • ψ (0)
)(r(η)) ϕ(n)(ψ(η)) = χ (g • h)(r(η)) ϕ(n)(ψ(η)).
Applying this observation together with equation (3.9), we see that χ c g (ψ(η)) ϕ(n)(ψ(η)) = γ Y,g χ ϕ(n) (ψ(η)) = ϕ γ X,g•h χ (n) (ψ(η)) = χ (g • h)(r(η)) ϕ(n)(ψ(η)), for all χ ∈ Γ. Since ϕ(n)(ψ(η)) = 0, it follows that χ (g • h)(r(η)) = χ c g (ψ(η)) , for all χ ∈ Γ. Since the characters of an abelian group separate points, equation (3.10) follows.
Before we get to Proposition 3.12, we point out that if (X, σ X ) and (Y, σ Y ) are secondcountable topologically free Deaconu-Renault systems, then the coaction condition equation (3.8) in Lemma 3.10 is superfluous. Although we do not need this fact in this paper, we believe it is worth recording in a corollary.
Corollary 3.11. Suppose that (X, σ X ) and (Y, σ Y ) are second-countable and topologically free Deaconu-Renault systems. If ϕ : C * (G X ) → C * (G Y ) is a * -isomorphism satisfying ϕ(C 0 (X)) = C 0 (Y), then there is a groupoid isomorphism ψ : G X → G Y such that ϕ(f ) = f •(ψ (0) ) −1 , for f ∈ C 0 (X). If, moreover, Γ is a locally compact abelian group, g ∈ C(Y, Γ), h : X → Y is a homeomorphism, and ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ, then c g•h = c g • ψ. Proof. By Lemma 2.9, we have Iso(G X ) • = X and Iso(G Y ) • = Y, and so the result follows immediately from Lemma 3.10 by letting G be the trivial group and taking c 1 : G X → G and c 2 : G Y → G to be the trivial cocycles.
Finally, we prove Proposition 3.12, from which it follows that the two conditions (i) and (ii) in Theorem 3.1(4) are equivalent, and that (1) and (4) in Theorem 3.1 are equivalent.
Proposition 3.12. Let (X, σ X ) and (Y, σ Y ) be second-countable Deaconu-Renault systems.
(i) If h : X → Y is a conjugacy, then there is a * -isomorphism ϕ : C * (G X ) → C * (G Y ) satisfying ϕ(C 0 (X)) = C 0 (Y), ϕ(f ) = f • h −1 for all f ∈ C 0 (X), and ϕ • γ X,g•h χ = γ Y,g χ • ϕ whenever Γ is a locally compact abelian group, g ∈ C(Y, Γ), and χ ∈ Γ. (ii) Conversely, suppose that ϕ : C * (G X ) → C * (G Y ) is a * -isomorphism, h : X → Y is a homeomorphism (which is not necessarily a conjugacy), and Γ is a locally compact abelian group that is separating for X and Y such that ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ and g ∈ C(Y, Γ). Then ϕ(C 0 (X)) = C 0 (Y), and there exists a conjugacy h : X → Y such that ϕ(f ) = f • h −1 for all f ∈ C 0 (X) and ϕ • γ X,g• h χ = γ Y,g χ • ϕ for all χ ∈ Γ and g ∈ C(Y, Γ).
Proof. For part (i), suppose that h : X → Y is a conjugacy. By Proposition 3.8, there is a groupoid isomorphism ψ : G X → G Y satisfying ψ(x, p, y) = (h(x), p, h(y)),
for (x, p, y) ∈ G X . This isomorphism induces a * -isomorphism ϕ : C * (G X ) → C * (G Y ) satisfying ϕ(ξ) = ξ • ψ −1 for ξ ∈ C c (G X ) and ϕ(C 0 (X)) = C 0 (Y) with ϕ(f ) = f • h −1 for f ∈ C 0 (X). Suppose that Γ is a locally compact abelian group and g ∈ C(Y, Γ). It follows from Proposition 3.8 that c g•h = c g • ψ. (Note that the proof of the relevant part of Proposition 3.8 does not require Γ to be separating for Y.) This implies that ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ. For part (ii), suppose that ϕ : C * (G X ) → C * (G Y ) is a * -isomorphism, h : X → Y is a homeomorphism, and Γ is a locally compact abelian group that is separating for X and Y such that ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ and g ∈ C(Y, Γ). Since Γ is separating for both X and Y, it follows from Lemma 3.9 that ϕ(C 0 (X)) = C 0 (Y). Let h : X → Y be the induced homeomorphism satisfying ϕ(f ) = f • h −1 for f ∈ C 0 (X) from Gelfand duality.
Since Γ is separating for X and Y, it contains an element ζ of infinite order. Choose g ∈ C(Y, Γ) to be constantly equal to ζ. Then γ Y,g χ = γ Y χ(ζ) and γ X,g•h χ = γ X χ(ζ) for χ ∈ Γ. Since ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ, it follows that if we let G = Z, c 1 = c X , and c 2 = c Y , then equation (3.8) in Lemma 3.10 holds. An application of Lemma 3.10 thus gives us a groupoid isomorphism ψ : G X → G Y with ψ (0) = h such that c g•h = c g • ψ for all g ∈ C(Y, Γ). It now follows from Proposition 3.8 that h = ψ (0) is a conjugacy, and that c g• h = c g • ψ for all g ∈ C(Y, Γ).
Fix g ∈ C(Y, Γ). Then c g• h = c g • ψ = c g•h . It follows that γ X,g• h = γ X,g•h , and thus
ϕ • γ X,g• h χ = ϕ • γ X,g•h χ = γ Y,g χ • ϕ for all χ ∈ Γ.
)(x) : i = 0, . . . , k(x) and B := h(x), (σ j Y • h•σ X )(x) : j = 0, . . . , k(x)−1 are equal. Therefore, there exists i ∈ {0, . . . , k(x)} such that
then a groupoid isomorphism that satisfies ψ (0) = h and c 1 = c 2 • ψ; cf. [CRST, proof of Theorem 6.2].
Example 2.2. Let E be a directed graph, let ∂E be the boundary-path space of E introduced in [We14], and let σ E : ∂E 1 → ∂E be the shift map described in [BCW17, Section 2.2]. Then (∂E, σ E ) is a Deaconu-Renault system (see [BCW17, Section 2.2]). In [BCW17, Section 6.1], two directed graphs E and F are defined to be conjugate precisely when the corresponding Deaconu-Renault systems (∂E, σ E ) and (∂F, σ F ) are conjugate; cf. Lemma 2.6. Example 2.3. Let A be an infinite countable set, let (Σ A , σ) be the one-sided full shift over A defined in [OMW14, Definitions 2.1 and 2.22], and let σ Σ A be the restriction of σ to Σ A \ { 0}. It follows from [OMW14, Proposition 2.5 and Proposition 2.23] that (Σ A , σ Σ A )is a Deaconu-Renault system.
If A and B are infinite countable sets and φ : Σ A → Σ B is a conjugacy as defined in
[OMW14, Definition 4.8], then it follows from [OMW14, Proposition 4.2 and Remark 4.9]
as defined in equation (2.3). If f : ∂E → R is the continuous map given by f | Z(e) ≡ k(e) for all e ∈ E 1 , then the induced cocycle c f defined in equation (2.2) agrees with c k on G E , and thus our weighted actions include all generalised gauge actions. Example 2.13. Let E be a topological graph, let ∂E be the boundary-path space of E defined in [KL17, Definition 3.1], and let σ : ∂E \ E 0 sg → ∂E be the one-sided shift mapmentioned in [KL17, Lemma 6.1]. It follows from [KL17, Proposition 3.6 and Lemma 6.1]
and [Ye07, Theorem 3.16 and Proposition 4.4] that (∂E, σ) is a Deaconu-Renault system.
The Deaconu-Renault groupoid G ∂E of (∂E, σ) is the groupoid Γ(∂E, σ) defined in [KL17,
Definition 6.6]. It follows from [KL17, Theorem 6.7] that C * (G (∂E,σ) ) is isomorphic to the
C*-algebra O E introduced in [Ka04]. One can check that the isomorphism given in [KL17,
Theorem 6.7] intertwines the canonical gauge actions of C * (G ∂E ) and O E .
the groupoid isomorphisms of [CRST, Proposition 6.5], and let ϕ * : H X → H Y be the groupoid isomorphism given by ϕ * ([n, x]) = [ϕ(n), h(x)] for [n, x] ∈ H X ; cf. [CRST, proof of Theorem 6.2]. The composition
where β h: βX → βY is the unique extension of h to the Stone-Čech compactifications; c.f. for example, [Ped, Proposition 3.12.10, and 3.12.12]. It now follows that
Remark 3.13. In[ERS20], Ruiz, Sims, and the fourth-named author show that a pair of amplified graphs (i.e. graphs in which every vertex emits either infinitely many or no edges to any other vertex) are graph-isomorphic if and only if there is a * -isomorphism of their graph C*-algebras that intertwines the canonical gauge actions. It follows from this and Proposition 3.12 that the boundary-path spaces of two amplified graphs are conjugate if and only if there is a * -isomorphism of their graph C*-algebras that intertwines the canonical gauge actions. This is an interesting result which we cannot expect to hold for larger classes of graphs. In fact, it is known that if the boundary-path spaces of two directed graphs are eventually conjugate, then there is a * -isomorphism of their graph C*-algebras that intertwines the canonical gauge actions (see[CR17,Theorem 4.1]), and [BC20a, Example 3.6] provides an example of two (finite) directed graphs (with no sinks and no sources) with boundary-path spaces that are eventually conjugate, but not conjugate.
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Einsteinstr. 62, 48149 Münster, Germany Email address: becky.armstrong@uni-muenster. Armstrong) Mathematical Institute, Münster, deArmstrong) Mathematical Institute, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany Email address: [email protected]
|
[] |
[
"A Continuation Method for Fitting a Bandlimited Curve to Points in the Plane",
"A Continuation Method for Fitting a Bandlimited Curve to Points in the Plane"
] |
[
"Mohan Zhao ",
"Kirill Serkh [email protected] ",
"\nDept. of Computer Science\nDept. of Math. and Computer Science\nUniversity of Toronto\nUniversity of Toronto\nM5S 2E4TorontoON\n",
"\nUniversity of Toronto\nM5S 2E4TorontoON\n"
] |
[
"Dept. of Computer Science\nDept. of Math. and Computer Science\nUniversity of Toronto\nUniversity of Toronto\nM5S 2E4TorontoON",
"University of Toronto\nM5S 2E4TorontoON"
] |
[] |
In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in R 2 . The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc length function and the tangential angle of the curve, and applying smooth perturbations, after each filtering step, until the curve is represented by a reasonably small number of coefficients. The algorithm produces a curve passing through the set of points to an accuracy of machine precision, after a limited number of iterations. It costs O(N log N ) operations at each iteration, provided that the number of discretization nodes is N . The resulting curves are smooth and visually appealing, and do not exhibit any ringing artifacts. The bandwidths of the constructed curves are much smaller than those of curves constructed by previous methods. We demonstrate the performance of our algorithm with several numerical experiments.
|
10.48550/arxiv.2301.04241
|
[
"https://export.arxiv.org/pdf/2301.04241v2.pdf"
] | 255,595,802 |
2301.04241
|
db9b17096439f646744a09021d97e2dc70abeae8
|
A Continuation Method for Fitting a Bandlimited Curve to Points in the Plane
May 25, 2023 24 May 2023
Mohan Zhao
Kirill Serkh [email protected]
Dept. of Computer Science
Dept. of Math. and Computer Science
University of Toronto
University of Toronto
M5S 2E4TorontoON
University of Toronto
M5S 2E4TorontoON
A Continuation Method for Fitting a Bandlimited Curve to Points in the Plane
May 25, 2023 24 May 2023⋄ This author's work was supported in part by the NSERC Discovery Grants RGPIN-2020-06022 and DGECR-2020-00356. †parametrizationbandlimited functionsC ∞ functionsapproximation theoryfilteringBézier splinessmooth interpolation
In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in R 2 . The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc length function and the tangential angle of the curve, and applying smooth perturbations, after each filtering step, until the curve is represented by a reasonably small number of coefficients. The algorithm produces a curve passing through the set of points to an accuracy of machine precision, after a limited number of iterations. It costs O(N log N ) operations at each iteration, provided that the number of discretization nodes is N . The resulting curves are smooth and visually appealing, and do not exhibit any ringing artifacts. The bandwidths of the constructed curves are much smaller than those of curves constructed by previous methods. We demonstrate the performance of our algorithm with several numerical experiments.
1 Introduction
The construction of smooth curves passing through data points has uses in many areas of applied science, including boundary integral equation methods, computer graphics and geometric modeling. While much of the time, C k continuity is sufficient, there are certain applications for which C ∞ continuity is essential. One such example is the high accuracy solution of partial differential equations on general geometries. In CAD/CAM systems, C ∞ smooth curves can be used as primitives to construct arbitrary smooth objects. Solving partial differential equations on these smooth objects prevents the loss of accuracy due to imperfect smoothness of C k shapes.
Countless methods have been proposed for fitting a spline or a C k curve to a given set of data points. Most interpolation techniques use piecewise polynomials and impose constraints to ensure global C k smoothness of the curve (see, for example, [1], [2], [3], [4]). In CAD/CAM systems, non-uniform rational B-splines (NURBS) are commonly used to construct a curve which approximates a set of control points, by defining the curve as a linear combination of the control points multiplied by C k and compactly supported B-spline basis functions. The contribution of each control point to the overall curve is determined by the corresponding weight, and the B-spline basis functions are normalized to ensure that the approximating curve remains affine invariant [6]. A generalization of NURBS, called partition of unity parametrics (PUPs) was first introduced by Runions and Samavati ( [5]). The PUP curves are constructed by replacing the weighted B-spline basis functions with arbitrary normalized weight functions (WFs), so that the resulting curves still exhibit the desired properties, including compact support and C k smoothness. In [5], the authors specifically discuss uniform B-spline WFs, to illustrate that each WF can be adjusted independently to fine-tune various shape parameters of the curve. Additionally, they observe that it is possible to choose the WFs to generate a PUP curve that interpolates the control points without solving a system of equations.
Another method proposed by Zhang and Ma ([7]) employs products of the sinc function and Gaussian functions as basis functions for constructing C ∞ interpolating curves that pass through all the given data points exactly. The resulting curves are almost affine invariant and almost compactly supported, and their shapes can be adjusted locally by directly adding or moving control points. Subsequently, Runions and Samavati ([8]) designed CINPACT-splines, by employing C ∞ and compactly supported bump functions as the WFs in a PUP curve, optionally multiplied by the normalized sinc function. When the WFs are chosen to be products of bump functions and the normalized sinc function, the resulting C ∞ curve interpolates the control points exactly, and when the WFs are bump functions, the resulting C ∞ curve approximates a uniform B-spline with the given control points. In addition to the properties inherited from PUP curves, CINPACT-splines possess C ∞ smoothness and the ability to specify tangents at control points. To increase the accuracy of the approximation to uniform B-splines, Akram, Alim and Samavati ( [9]) further proposed CINAPACT-splines, by successively convolving a CINPACT-spline with B-splines of order one, ensuring any finite order of approximation to uniform B-splines, as well as to other compactly supported kernels with maximal order and minimal support ( [10]), while preserving C ∞ smoothness and compact support. Zhu ([11]) proposed curves that share similarities with CINPACT-splines in terms of affine invariance, compact support, and C ∞ smoothness. In [11], a class of non-negative blending functions is constructed by designing basis functions which combine bump functions with the sinc function. The resulting interpolating curves are defined by three local shape parameters, with one of the parameters determining whether the curve approximates or interpolates the given control points.
One notable distinction of the approach of Zhang and Ma ( [7]) from the other methods we have discussed is that, since Gaussian functions are utilized in the basis functions, the interpolating curves produced by [7] are not only C ∞ smooth, but also are analytic. This paper mainly compares our method with [7], as the interpolating curves in [7] have a smaller bandwidth, compared to methods based on C ∞ compactly supported bump functions. The approach in [7] (as well as [5], [8], [9], [11]) necessitates a more specially chosen distribution of data points to achieve a visually smooth curve, as it only guarantees smoothness in the curve parameter, which does not necessarily correspond to smoothness of the curve in R 2 . However, our method directly smooths the tangential angle of the curve and the first derivative of the arc length function, yielding a significantly smoother curve which is also more visually appealing.
Among all the methods for constructing a C ∞ interpolating curve, the algorithm described by Beylkin and Rokhlin ([12]) bears the closest resemblance to our method, generating a bandlimited closed curve through a set of data points. The bandlimited curve is constructed by filtering the Fourier coefficients of the tangential angle of the curve, parametrized by arc length. However, the number of coefficients required to represent the curve can be large, which appears to be a major drawback of the algorithm in practical applications.
In this paper, we describe an algorithm for fitting a bandlimited closed or open curve to pass through a collection of points. The main idea is to iteratively filter the tangential angle and the first derivative of the arc length function of the curve, and apply small corrections after each filtering step, until the desired bandwidth of the curve is reached, to the required precision. Our algorithm produces an analytic and affine invariant curve with far fewer coefficients, and the curve is visually appealing and free of ringing artifacts.
The structure of this paper is as follows. Section 2 describes the mathematical preliminaries. Section 3 describes the algorithm to construct the bandlimited approximation to a closed curve, and to an open curve. Finally, Section 4 presents several numerical examples to show the performance of our algorithm, as well as some comparisons between our algorithm and the methods proposed in [7] and [12].
Preliminaries
In this section, we describe the mathematical and numerical preliminaries.
Geometric properties of a curve
Let γ : [a, b] → R 2 be a smooth curve parametrized by the curve parameter t, such that
γ(t) = (x(t), y(t)), t ∈ [a, b],(1)
where x(t) and y(t) are the x and y coordinates.
Assuming γ ∈ C 1 ([a, b]), we define the tangent vector T (t),
T (t) = (x ′ (t), y ′ (t)), t ∈ [a, b],(2)
and the arc length s(t), which is the length of the curve from the point (x(a), y(a)) to the point (x(t), y(t)),
s(t) = t a ∥T (τ )∥dτ, t ∈ [a, b].(3)
It is obvious that
s ′ (t) = ∥T (t)∥, t ∈ [a, b].(4)
Thus, we have
s ′ (b) = s ′ (a)(5)
when the curve is closed. The tangential angle θ(t) of the curve at the point (x(t), y(t)) measures the angle between the tangent vector T (t) at that point and the x-axis, defined by the formula
θ(t) = atan2(y ′ (t), x ′ (t)), t ∈ [a, b],(6)
where atan2 : R 2 → (−π, π] is the arctangent at the point (x(t), y(t)). As a result, θ(t) ∈ (−π, π]. Since the function atan2 has a branch cut at θ = −π, it is possible for θ(t) to have ω jump discontinuities of size 2π, where ω ∈ Z is the winding number. The curve (x(t), y(t)) can be constructed from θ(t) and s ′ (t) by the formulas
x(t) = t a s ′ (τ ) cos θ(τ ) dτ + x(a), t ∈ [a, b],(7)y(t) = t a s ′ (τ ) sin θ(τ ) dτ + y(a), t ∈ [a, b],(8)
and (x(a), y(a)) = γ(a). If the curve is closed, we require x(a) = x(b) and y(a) = y(b), which means that b a s ′ (τ ) cos θ(τ ) dτ = 0 (9) and b a s ′ (τ ) sin θ(τ ) dτ = 0.
Cubic Bézier Interpolation
A Bézier curve is a function B : [0, 1] → R 2 defined by a set of control points P 0 , . . . , P m ∈ R 2 . The Bézier curve is designed to go through the first and and the last control point P 0 and P m , and the shape of the curve is determined by the intermediate control points P 1 , . . . , P m−1 . A mth order Bézier curve is a polynomial of degree m, defined by
B(t) = m i=0 m i (1 − t) m−i t i P i , = (1 − t) m P 0 + m 1 (1 − t) m−1 tP 1 + · · · + m m − 1 (1 − t)t m−1 P m−1 + t m P m ,
where t ∈ [0, 1]. A continuous Bézier spline connecting all the given points C 0 , . . . , C n can be constructed by combining n cubic Bézier curves
B i (t) = (1 − t) 3 P i0 + 3(1 − t) 2 tP i1 + 3(1 − t)t 2 P i2 + t 3 P i3 , i = 1, . . . , n, 5
where t ∈ [0, 1] and B i (t) is the ith Bézier curve, with controls points
P i0 = C i−1 ,(11)P i3 = C i ,(12)
for i = 1, . . . , n. We define the spline S : [0, n] → R 2 from the cubic Bézier curves B i by letting
S(t) = B i (t − i + 1) for t ∈ [i − 1, i], for i = 1, . . . , n.
It is easy to see that S ∈ C 0 ([0, n]), however, in general, S / ∈ C 1 ([0, n]). It is possible to ensure S ∈ C 2 ([0, n]) by imposing additional conditions on the intermediate control points, which we derive as follows. Note that the following derivation is similar to the one presented in [16]. First, we observe that the first and second derivatives of a cubic Bézier curve are
B ′ i (t) = −3(1 − t) 2 P i0 + 3(3t 2 − 4t + 1)P i1 + 3t(2 − 3t)P i2 + 3t 2 P i3 , B ′′ i (t) = 6(1 − t)P i0 + 6(3t − 2)P i1 + 6(1 − 3t)P i2 + 6tP i3 ,
for i = 1, . . . , n. In order for S ∈ C 2 ([0, n]), we require that
B ′ i−1 (1) = B ′ i (0), i = 1, . . . , n,(13)B ′′ i−1 (1) = B ′′ i (0), i = 1, . . . , n.(14)
Then, (13) implies that
P (i−1)2 = 2C i−1 − P i1 , i = 1, . . . , n.(15)
Likewise, it is possible to show that (14) implies that
P (i−1)1 + 2P i1 = P i2 + 2P (i−1)2 , i = 1, . . . , n.(16)
Substituting (15) into (16),we get
P (i−1)1 + 4P i1 + P (i+1)1 = 2C i + 4C i−1 , i = 1, . . . , n.(17)
Solving for control points for an open curve
When the curve is open, we have (17) must hold for i = 2, . . . , n − 1, and we need two boundary conditions in order to solve a linear system of n equations for the values of P 11 , . . . , P n1 . Assume that users specify the slope at two end points of the curve, c left and c right , we have
B ′ 1 (0) = c left ,(18)
and
B ′ n (1) = c right .(19)
It is possible to show that (18) implies that
P 11 = c left + 3C 0 3(20)
6 and (19) implies that
P n2 = 3C n − c right 3 .(21)
Substituting (15) and (21) into (16),we get
P (n−1)1 + 4P n1 = 4C n−1 + C n − c right 3 .(22)
With (17), (20) and (22), we build a system of n equations to calculate P 11 , . . . , P n1 and use (15), (21) and the values of P 11 , . . . , P n1 to calculate P 12 , . . . , P n2 . This system of equations is tridiagonal, and so can be solved in O(n) operations.
Solving for control points for a closed curve
When the curve is closed, we require n + 1 cubic Bézier curves instead of n cubic Bézier curves to connect the points C 0 , . . . , C n , where the (n + 1)th curve connects the points C n and C 0 . We have that the conditions (17) must hold for i = 2, . . . , n, and we need the following two boundary conditions,
B ′ 1 (0) = B ′ n+1 (1),(23)B ′′ 1 (0) = B ′′ n+1 (1),(24)
to solve a linear system of (n + 1) equations for the values of P 11 , . . . , P (n+1)1 . It is possible to show that (23) implies that
P 11 + P (n+1)2 = 2C 0(25)
and (24) implies that
−2P 11 + P 12 = P (n+1)1 − 2P (n+1)2 .(26)
Substituting (15) and (16) into (25), we get
P 11 + P n1 + 4P (n+1)1 = 2C 0 + 4C n .(27)
Substituting (15) and (16) into (26), we get
−2P 11 − P 21 + 2P n1 + 7P (n+1)1 = −2C 1 + 8C n .(28)
Similarly, with (17), (27) and (28), we build a system of (n + 1) equations to calculate P 11 , . . . , P (n+1)1 and use (15), (25) and the values of P 11 , . . . , P (n+1)1 to calculate P 12 , . . . , P (n+1)2 . This system of equations is cyclic tridiagonal, and thus we can solve it in O(n) operations.
Chebyshev Polynomial Interpolation
A smooth function f (x) on the interval [−1, 1] can be approximated by a (n − 1)th order Chebyshev expansion with the formula
f (x) ≈ n−1 k=0 f k T k (x),(29)
where T k (x) is the Chebyshev polynomial of the first kind of degree k, defined by
T k (x) = cos(k arccos x), x ∈ [−1, 1].(30)
It is known that the Chebyshev coefficients { f k } decay like O(n −k+ 1 2 ) when f ∈ C k ([−1, 1]), when the coefficients f k are chosen to satisfy the n collocation equations
f (x i ) = n−1 k=0 f k T k (x i ), i = 0, . . . , n − 1,(31)
for the practical Chebyshev nodes {x i },
x i = − cos iπ n − 1 , i = 0, . . . , n − 1.(32)
Alternatively, one can compute f k for k = 0, . . . , n − 1 using the Discrete Chebyshev Transform,
f 0 = 1 n − 1 1 2 (f (x 0 ) + f (x n−1 ) + n−2 i=1 f (x i )T 0 (x i ),(33)
and
f k = 2 n − 1 1 2 (f (x 0 )(−1) k + f (x n−1 ) + n−2 i=1 f (x i )T k (x i ),(34)
Spectral Differentiation and Integration
Assuming that k ≥ 1 is an integer, the formula
2T k (x) = T ′ k+1 (x) k + 1 − T ′ k−1 (x) k − 1 ,(35)
can be used to spectrally differentiate the Chebyshev expansion of f (x), as follows. Suppose that
f (x) ≈ n−1 k=0 f k T k (x)(36)
and that
f ′ (x) ≈ n−1 k=0 f ′ k T k (x).(37)
The coefficients f ′ k can be computed from f k by iterating from k = n − 1, n − 2, . . . , 2 and, at each iteration, assigning f ′ k−1 the value 2k f k , and assigning f k−2 the value
k k−2 f k + f k−2 .
8 Similarly, the formula
2 t −1 T k (x) dx = T k+1 (t) k + 1 − T k−1 (t) k − 1 − (−1) k+1 k + 1 + (−1) k−1 k − 1 (38)
can be used to spectrally integrate the Chebyshev expansion of f (x). Suppose that
t −1 f (x) dx ≈ n k=0 f k T k (t).(39)
Since
t −1 f (x)dx ≈ n−1 k=0 f k t −1 T k (x) dx = n−1 k=1 f k 1 2 T k+1 (t) k + 1 − T k−1 (t) k − 1 − (−1) k+1 k + 1 + (−1) k−1 k − 1 + f 0 (t + 1),(40)
one can compute the coefficients f k from f k by firstly assigning f 1 the value f 1 + f 0 , then iterating from k = n−1, . . . , 1, and at each iteration, assigning f k+1 the value f k+1
+ f k 2(k+1) , assigning f k−1 the value f k−1 − f k 2(k−1) , and assigning f 0 the value f 0 − f k ( (−1) k+1 2(k+1) − (−1) k−1 2(k−1)
). Finally, f k takes the value f k , for k = n, . . . , 0.
The Discrete Fourier Transform (DFT)
A periodic and smooth function f (x) on the interval [0, 1] can be approximated by a n-term Fourier series using the Discrete Fourier Transform. The Discrete Fourier Transform defines a transform from a sequence of n complex numbers f 0 , . . . , f n−1 to another sequence of n complex numbers f 0 , . . . , f n−1 , by
f k = n−1 j=0 f j e − 2πi n kj , k = 0, . . . , n − 1,(41)
The sequence { f k } consists of the Fourier coefficients of {f k }. The Inverse Discrete Fourier Transform (IDFT) is given by
f j = 1 n n−1 k=0 f k e 2πi
n kj , j = 0, . . . , n − 1.
Another representation of the DFT which is usually used in applications is given by a shift in the index k, and a change in the placement of the scaling by 1 n ,
f k = 1 n n−1 j=0 f j e − 2πi n kj , k = − n 2 , . . . , n 2 − 1.(43)
Thus, the corresponding IDFT is
f j = n 2 −1 k=− n 2 f k e 2πi n kj , j = 0, . . . , n − 1.(44)
Suppose that f : [0, 1] → C is a smooth and periodic function, and that f j = f (t j ) for j = 0, . . . , n − 1, where {t j } are the equispaced points on [0, 1]. Observing that
f k = 1 n n−1 j=0 f j e − 2πi n kj , ≈ 1 0 f (x)e −2πikx dx,(45)
for k = − n 2 , . . . , n 2 − 1, we obtain the approximation to f (x) by a truncated Fourier series,
f (x) ≈ n 2 −1 k=− n 2 f k e 2πikx , x ∈ [0, 1].(46)
It is known that the Fourier coefficients
{ f k } decay like O(n −k+ 1 2 ) when f ∈ C k (S 1 ), where S 1 = [0, 1] is the circle.
Spectral Differentiation and Integration
The spectral differentiation of the truncated Fourier series approximation to f (x) on [0, 1] is as follows. Suppose that f (x) is given by (46) and that
f ′ (x) ≈ 1 n n 2 −1 k=− n 2 f ′ k e 2πikx .(47)
Since
f ′ (x) ≈ 1 n n 2 −1 k=− n 2 f k e 2πikx · 2πik,(48)
the coefficients f ′ k can be computed from f k by assigning f ′ k the value f k · 2πik for k = − n 2 , . . . , n 2 − 1. Similarly, the spectral integration of the truncated Fourier series approximation to f (x) is as follows. Suppose that
t 0 f (x)dx ≈ 1 n n 2 −1 k=− n 2 f k e 2πikt .(49)
Since
t 0 f (x)dx ≈ 1 n k̸ =0 f k 2πik e 2πikt − 1 n k̸ =0 f k 2πik + 1 n f 0 t,(50)
it is easy to see that, for t 0 f (x)dx to be periodic, it must be the case that f 0 = 0. Then, we have
t 0 f (x)dx ≈ 1 n k̸ =0 f k 2πik e 2πikt − 1 n k̸ =0 f k 2πik . (51)
We can compute the Fourier coefficients f k from f k by assigning
f k the value f k 2πik for k = − n 2 , . . . , n 2 − 1, k ̸ = 0 and assigning f 0 the value − k̸ =0 f k 2πik .
Gaussian filter
A low-pass filter is commonly used in signal processing to construct a bandlimited function.
In this paper, we use the Gaussian filter, which is a popular low-pass filter whose impulse response is a Gaussian function,
g(x) = ae −πa 2 x 2 ,(52)
where a determines the bandwidth of g(x). The Gaussian filter g 0 , . . . , g n−1 is defined to be the IDFT of the sequence
g k = e −π k 2 a 2 , k = − n 2 , . . . , n 2 − 1,(53)
and coincides with the discrete values of g(x) at the equispaced nodes x j = j n , j = 0, . . . , n − 1.
To filter the Fourier coefficients f 0 , . . . , f n−1 in (43), we take the product
h k = g k f k , k = − n 2 , . . . , n 2 − 1.(54)
It is easily to obtain h 0 , . . . , h n−1 by the IDFT,
h j = 1 n j k=0 g k f j−k , j = 0, . . . , n − 1.(55)
This can be considered to be a smoothing of f (x) by a convolution of f (x) with the Gaussian function g(x).
Filtering the Chebyshev coefficients f 0 , . . . , f n−1 defined in (31) is very similar to filtering the Fourier coefficients, which we describe as follows. Substituting
x = cos(θ), where x ∈ [−1, 1], into (29), we have f (cos(θ)) ≈ n−1 k=0 f k T k (cos(θ)) = n−1 k=0 f k cos(kθ),(56)where θ ∈ [−π, π]. Letting f −k = f k , k = 1, . . . , n − 1, we have f (cos(θ)) ≈ 1 2 n−1 k=−n+1 f k e ikθ + 1 2 f 0 , θ ∈ [−π, π].(57)
Hence, by defining ϕ by the formula θ = 2πϕ,
f (cos(2πϕ)) ≈ 1 2 n−1 k=−n+1 f k e 2πikϕ + 1 2 f 0 , ϕ ∈ [− 1 2 , 1 2 ].(58)
Since (58) can be viewed as a Fourier Transform in ϕ with the Fourier coefficients { f k }, we follow the equation (54) to filter { f k }, and apply the IDFT to obtain the filtered values of {f j }. Therefore, f (x) is smoothed by a convolution with the Gaussian function
g(ϕ) in the ϕ-domain, where x = cos(2πϕ).
Alternatively, there are other low-pass filters that can be used, such as the Butterworth filter (see, for example, Chapter 14 of [13]) which resembles the Gaussian filter but is flatter in the passband. The brick-wall filter also preserves signals with lower frequencies and excludes signals with higher frequencies. However, after applying the brick-wall filter, the resulting functions tend to oscillate at the cutoff frequency (this phenomenon is known as ringing).
The Algorithm
In this section, we give an overview of our algorithm for fitting a C ∞ curve to pass through a collection of points C 0 , . . . , C n . We begin with a C 2 cubic Bézier spline connecting the points C 0 , . . . , C n . Given that the curve is at least C 2 , we interpolate the tangential angle θ(t) and the first derivative of the arc length vector s ′ (t), which are both C 1 , using Chebyshev expansions when the curve is open, or using truncated Fourier series when the curve is closed. We then iteratively filter the coefficients of θ(t) and s ′ (t) by applying a Gaussian filter, whose bandwidth decreases with each iteration. If the curve is closed before filtering, we impose constraints on θ(t) and s ′ (t) to ensure that the curve remains closed. We then reconstruct the curve with the filtered values of θ(t) and s ′ (t) at discretization nodes.
While filtering leads to small discrepancies between the reconstructed curve and the points C 0 , . . . , C n , it also improves the bandwidth of the curve. To fix the discrepancies after each filtering step, we rotate and rescale the curve to minimize the total distance between the curve and the points, and add small, smooth perturbations, which do not negatively affect the smoothness of the curve. We stop filtering when the desired bandwidths of the Chebyshev or Fourier approximations to θ(t) and s ′ (t) are achieved. This algorithm gives us a C ∞ smooth curve that can be represented by a reasonably small number of coefficients.
Initial Approximation
To initialize our algorithm, we require a C 2 curve, the reasons for which are described in Section 3.3.
Given a set of data points C 0 , . . . , C n ∈ R 2 , we fit a cubic Bézier spline by solving for the intermediate control points {P i1 } and {P i2 } described in Section 2.2.1 for an open curve, or in Section 2.2.2 for a closed curve. We define the Bézier spline S : [0, L] → R 2 connecting all the points C 0 , . . . , C n by
S(t) = B i (t − i + 1), t ∈ [0, n] and i = 1, . . . , n,(59)
if the curve is open, or
S(t) = B i (t − i + 1), t ∈ [0, n + 1] and i = 1, . . . , n + 1,(60)
if the curve is closed.
Representations of the Curve
In this section, we denote the curve by
γ(t) = (x(t), y(t)),(61)
where γ : [0, L] → R 2 is at least C 2 .
Representation of an Open Curve
When the curve is open, we discretize x(t) and y(t) at N ≫ n practical Chebyshev nodes
{t j } on the interval [0, L] (see formula (32)) to obtain {x j } and {y j }, where x j = x(t j )
and y j = y(t j ). We use (N − 1)th order Chebyshev expansions to approximate x(t) and y(t), constructing the coefficients from {x j } and {y j } using the Discrete Chebyshev Transform, and then spectrally differentiate x(t) and y(t) to derive the Chebyshev expansions approximating x ′ (t) and y ′ (t). By (4) and (6), we can compute the values of s ′ (t) and θ(t) sampled at nodes {t j }, and then construct the corresponding Chebyshev expansions, again using the Discrete Chebyshev Transform. However, performing the Chebyshev Transform on θ(t) requires θ(t) to be continuous, and as discussed in Section 2.1, θ(t) can have jump discontinuities of size 2π. These can be fixed by adding or subtracting multiples of 2π to θ(t) wherever a discontinuity is detected.
Representation of a Closed Curve
When the curve is closed, we discretize x(t) and
y(t) at N ≫ n equispaced nodes {t j } on the interval [0, L], where t j = j N L, j = 0, . . . , N − 1,(62)
to obtain {x j } and {y j } by x j = x(t j ) and y j = y(t j ). We then approximate x(t) and y(t) by an N -term Fourier series, separately, and spectrally differentiate x(t) and y(t) to approximate x ′ (t) and y ′ (t). Following the same procedures in Section 3.2.1, we ensure that θ(t) is continuous, and approximate s ′ (t) by a truncated Fourier series. Recall that, in order to approximate functions by their Fourier series, the functions must be both smooth and periodic. The sequence {θ j }, which are the discrete values of θ(t) at {t j }, is not periodic after shifting by multiples of 2π to remove the discontinuities. Defining c by
c = θ(n + 1) − θ(0),(63)
13 we have that
θ j = θ j − c L t j , t j ∈ [0, L],(64)
transforms {θ j } into a periodic sequence { θ j } on the interval [0, L], which can be approximated by a truncated Fourier series. To recover the true values of {θ j } after filtering, we can add c L t j to θ j . In an abuse of notation, we denote { θ j } by {θ j } wherever the meaning is clear.
Filtering the Curve
In this section, we describe the process of iteratively filtering θ(t) and s ′ (t) using a Gaussian filter. Given γ(t) ∈ C 2 , we have θ(t) ∈ C 1 and s ′ (t) ∈ C 1 . It is known that the decay rate of the Chebyshev coefficients or the Fourier coefficients of a C 1 function is
O(N − 1 2 ),
where N is the order of the expansion. By iteratively decreasing the bandwidth of the Gaussian filter, we construct a sequence of bandlimited representations of θ(t) and s ′ (t). The decay rate of the Fourier coefficients or the Chebyshev coefficients in the expansions of θ(t) and s ′ (t) increases with each iteration. This filtering process smooths both the curve itself and the parameterization of the curve.
Filtering the Open Curve
Let {t j } denote the practical Chebyshev nodes translated to the interval [0, L] (see formula (32)). Using the Chebyshev expansions of θ(t) and s ′ (t) computed in Section 3.2.1, we discretize θ(t) and s ′ (t) at the points {t j } to obtain the sequences {θ j } and {s ′ j }, where θ j = θ(t j ) and s ′ j = s ′ (t j ). We filter the Chebyshev coefficients { θ k } of θ(t), and { s ′ k } of s ′ (t) using the Gaussian filter in (53), and obtain the filtered coefficients { θ
(f ) k } and { s ′ (f ) k }, θ (f ) k = e −π k 2 a 2 θ k , k = 0, . . . , N − 1,(65)
and
s ′ (f ) k = e −π k 2 a 2 s ′ k , k = 0, . . . , N − 1.(66)
Applying the IDFT to { θ
(f ) k } and { s ′ (f ) k }, we obtain θ (f ) j = N −1 k=0 θ (f ) k T k (t j ),t j ∈ [−1, 1],(67)wheret j = 2 L t j − 1, t j ∈ [0, L], and s ′(f ) j = N −1 k=0 s ′ (f ) k T k (t j ).(68)
We can then use the values of {θ (7) and (8).
(f ) j } and {s ′(f ) j } to recover {x (f ) j } and {y (f ) j } using
Filtering the Closed Curve
Assume that {θ j } and {s ′ j } are the values of θ(t) and s ′ (t) discretized at the equispaced nodes {t j } in (62), where θ j = θ(t j ) and s ′ j = s ′ (t j ). We apply the DFT to derive the Fourier coefficients { θ k } of θ(t) and { s ′ k } of s ′ (t). Using the Gaussian filter, we filter the
Fourier coefficients { θ k } and { s ′ k } to obtain the filtered Fourier coeffcients { θ (f ) k } and { s ′ (f ) k }, θ (f ) k = e −π k 2 a 2 θ k , k = − N 2 , . . . , N 2 − 1,(69)
and
s ′ (f ) k = e −π k 2 a 2 s ′ k , k = − N 2 , . . . , N 2 − 1.(70)
We recover the filtered sequences {θ
(f ) j } and {s ′(f ) j } by applying the IDFT to the filtered Fourier coefficients { θ (f ) k } and { s ′ (f ) k }, θ (f ) j = N 2 −1 k=− N 2 θ (f ) k e 2πi N kj + c L t j , j = 0, . . . , N − 1,(71)
and
s ′(f ) j = N 2 −1 k=− N 2 s ′ (f ) k e 2πi N kj , j = 0, . . . , N − 1.(72)
Similarly, the curve can be reconstructed from {θ
Closing the Curve
Applying a filter to θ(t) and s ′ (t) for a closed curve, in general, makes the curve become open. To close the curve, we require that
L 0 s ′ (t) cos θ(t) dt = 0,(73)and L 0 s ′ (t) sin θ(t) dt = 0.(74)
The process of orthogonalizing s ′ (t) to cos θ(t) and sin θ(t) using the trapezoidal rule is as follows. Supposing that we have the values {s ′ j }, {cos θ j } and {sin θ j } of s ′ (t), cos θ(t) and sin θ(t) sampled at the points {t j } defined in (62). We ensure that {s ′ j } is orthogonal to cos θ j by setting {s ′ j } to the values
s ′ j − cos θ j 1 N N −1 j=0 s ′ j cos θ j 1 N N −1 j=0 cos 2 θ j , j = 0, . . . , N − 1.(75)
We let {λ j } be the vector defined by the formula
λ j = sin θ j − cos θ j 1 N N −1 j=0 sin θ j cos θ j 1 N N −1 j=0 cos 2 θ j , j = 0, . . . , N − 1.(76)
Finally, we orthogonalize {s ′ j } to {λ j } by setting {s ′ j } to the values
s ′ j − λ j 1 N N −1 j=0 s ′ j λ j 1 N N −1 j=0 λ 2 j , j = 0, . . . , N − 1.(77)
The sequence {s ′ j } is now orthogonal to both {cos θ j } and {sin θ j }. Thus, the conditions (73) and (74) are satisfied to within the accuracy of the trapezoidal rule.
Repositioning the Curve
In general, the curve will not pass through the original data points after filtering. Moreover, filtering θ(t) and s ′ (t) changes the tangential vector T (t), which results in changes in the orientation and position of the curve. In this section, we describe how to rotate the reconstructed curve so that the sum of squares of the distances between the curve and the original data points is minimized.
Given the original data points
{C i }, where C i = (C ix , C iy ), i = 0, . . . , n, we find t 0 , . . . , t n ∈ [0, L] such that, if ( x i , y i ) = (x( t i ), y( t i )), then ( x i , y i )
is the closest point on the curve to (C ix , C iy ) for i = 0, . . . , n. We determine t 0 , . . . , t n only once, described in
(x + ∆x + r i cos (ϕ i + ψ) − C ix ) 2 + (ȳ + ∆y + r i sin (ϕ i + ψ) − C iy ) 2 ,(78)
where (x,ȳ) is the average of {( x i , y i )}. We use Newton's method to obtain the values of ψ, ∆x and ∆y which minimize f (ψ, ∆x, ∆y).
Remark 3.1. One might also think to rescale the curve by multiplying {r i } by a constant c, since filtering s ′ (t) changes the length of the curve. However, rescaling the curve distorts the structure of the closest points ( x i , y i ) on the curve. Large perturbations, as described in Section 3.6, are sometimes needed as a result, and therefore the smoothness of the curve after adding perturbations can be reduced.
Notice that each point on the curve is, in some sense, equivalent. The procedure of repositioning ensures that the resulting curve is affine invariant.
Adding Perturbations to the Curve
Since the curve does not pass through the original data points {C i } after filtering θ(t) and s ′ (t), we introduce a set of Gaussian functions, {g i (t)}, which we use as smooth perturbations that can be added to the curve to ensure that the curve passes through the points {C i }. We define g i (t) by
g i (t) = e −σ i t− t i L 2 , i = 0, . . . , n,(79)
for t ∈ [0, L], where t i is the curve parameter of the closest point ( x i , y i ) = (x( t i ), y( t i )) to C i , and σ i determines the bandwidth of the perturbation. When the curve is closed, g i (t) is modified to be a periodic function with period L, given by the formula
g i (t) = ∞ k=−∞ e −σ i t− t i L +k 2 , i = 0, . . . , n.(80)
It is obvious that g i (t) = g i (t + L). We construct {(x j ,ȳ j )} from {(x j , y j )} by adding g i (t) at the discretized points {t j },
x j = x j + n i=0 c ix g i (t j ), j = 0, . . . , N − 1,(81)andȳ j = y j + n i=0 c iy g i (t j ), j = 0, . . . , N − 1,(82)
where {c ix } and {c iy } are the coefficients of perturbations in x and y, separately, which are reasonably small since the curve is filtered slightly at each iteration. Let {( x j , ȳ j )} denote the points on the perturbed curve corresponding to t 0 , . . . , t n . We require
x i = C ix , i = 0, . . . , n,(83)
and
ȳ i = C iy , i = 0, . . . , n,(84)
and solve two linear systems of n + 1 equations to compute the values of {c ix } and {c iy }. We observe that, since the perturbations g i (t) are Gaussians, they are each, to finite precision, compactly supported. Thus, the linear system that we solve is effectively banded, and the number of bands is determined by min i σ i . An O(n + 1) solver can be used to speed up the computations.
Remark 3.2. We only calculate { t i } once, at the first iteration before filtering, and use the same set of { t i } at each iteration. Although it seems more natural to recalculate { t i } at each iteration, so that the discrepancies are fixed by smaller perturbations, the resulting perturbations are always orthogonal to the curve. The effect of the changes in the length of the curve due to filtering can not be eliminated by adding such perturbations, with the effect that the length of the curve grows if the points { t i } are calculated at each iteration. By using the same set of closest points for all iterations, the perturbations can be oblique, which results in nice control over the total length of the curve during the filtering process.
The Termination Criterion of the Algorithm
Since the bandwidths of the coefficients of θ(t) and s ′ (t) are reduced at each iteration, and adding small, smooth perturbations has a negligible effect on the bandwidth of the curve, one can expect to achieve the desired bandwidth of the representations of θ(t) and s ′ (t) by iteratively filtering the coefficients. However, we note that there is a minimum number of coefficients that are necessary to represent a curve, as determined by the sample data points. When fewer than this number of coefficients are used, the curve reconstructed by these overfiltered coefficients may deviate drastically from the sample data points. The resulting large perturbations required to fix the discrepancies can harm the smoothness of the curve. The purpose of this section is to set up a termination criterion, so that the algorithm will terminate if the coefficients of θ(t) and s ′ (t), beyond a user-specified number of terms, are filtered to zero, to the requested accuracy. We denote the desired accuracy of the approximation by ϵ, which is often set to be machine precision, and the number of coefficients representing the curve that are larger than ϵ by n coefs . Due to the potentially large condition number of spectral differentiation, some accuracy is lost when computing the coefficients of x ′ (t) and y ′ (t), and thus θ(t) and s ′ (t), at each iteration. Thus, we measure thresholds for the coefficients of θ(t) and s ′ (t), below which they are considered to be zero, and denote them by δ θ and δ s ′ . We consider first the open curve case. Since the condition number of the Chebyshev differentiation matrix is bounded by approximately N 3 2 , where N is the number of coefficients, the error induced by differentiating x(t) and y(t) is approximately
ϵN 3 2 ∥x(t)∥ 2 L 2 [0,L] + ∥y(t)∥ 2 L 2 [0,L] (85) ≈ ϵN 3 2 j x 2 j w j + j y 2 j w j ,(86)
where {w j } denotes the Chebyshev weights on [0, L]. Considering the way θ(t) is calculated, the error in θ(t) is proportional to the error in x ′ (t) and y ′ (t), divided by the norm of the tangential vector (x ′ (t), y ′ (t)). Thus, we set
δ θ = ϵN 3 2 ∥x(t)∥ 2 L 2 [0,L] + ∥y(t)∥ 2 L 2 [0,L] · 1 x ′ (t) 2 + y ′ (t) 2 L ∞ [0,L] ≈ ϵN 3 2 j x 2 j w j + j y 2 j w j min x ′2 j w j + y ′2 j w j ,(87)where x ′ i , y ′ i are the discretized values of x ′ (t), y ′ (t).
Similarly, the error in s ′ (t) is proportional to the error in x ′ (t) and y ′ (t). Thus, we set
δ s ′ = ϵN 3 2 ∥x(t)∥ 2 L 2 [0,L] + ∥y(t)∥ 2 L 2 [0,L] , ≈ ϵN 3 2 j x 2 j w j + j y 2 j w j .(88)
The thresholds δ θ and δ s ′ for the closed curve case are almost identical, except that the condition number of spectral differentiation matrix is approximately N , where N is the number of coefficients, from which it follows that N 3 2 is replaced by N , and the weights w j are replaced by L N . Suppose that we have the desired accuracy of the approximation, ϵ, the threshold, δ θ , and the number of coefficients larger than ϵ, n coefs . We consider first the coefficients of θ(t). Our goal is to determine the number of coefficients, n δ θ coefs , that we expect to be larger than δ θ , when there are only n coefs terms larger than ∥ θ∥ ∞ ϵ. In order to approximate n δ θ coefs , we assume that the coefficients { θ k } decay exponentially, like ∥ θ∥ ∞ e −Ck , from the maximum value ∥ θ∥ ∞ to ∥ θ∥ ∞ ϵ. This implies that C = log (1/ϵ) n coefs . Thus,
e − log (1/ϵ) n δ θ coefs n coefs = δ θ ,(89)
so,
n δ θ coefs = n coefs log (1/δ θ ) log (1/ϵ) .(90)
We compute n δ s ′ coefs in exactly the same way. At each iteration, if only n δ θ coefs and n δ s ′ coefs numbers of terms are larger than δ θ and δ s ′ , respectively, then the algorithm terminates. Eventually, n coefs coefficients are returned to the user to represent the curve, up to the precision ϵ.
Remark 3.3.
Since the values of δ θ , δ s ′ , n δ θ coefs and n δ s ′ coefs are fairly consistent in each iteration, we only calculate these values once, at the first iteration.
Summary and Cost of the Algorithm
The algorithm can be summarized as follows:
1. Given n + 1 points C 0 , . . . , C n , fit a C 2 Bézier spline to connect the points. 6. In the case of a closed curve, modify {s ′ j } to satisfy the constraints (73) and (74) in order to close the curve after filtering. 7. Reconstruct the curve from {θ j } and {s ′ j } by equations (7) and (8). 8. Rotate the curve to minimize the sum of squares of the distances between the curve and the points C 0 , . . . , C n .
9. Add smooth Gaussian perturbations to make the curve pass through the points C 0 , . . . , C n .
Solving for the control points of the Bézier spline in Step 1 costs O(n + 1) operations, and discretizing the spline at N points in
Numerical Results
In this section, we demonstrate the performance of our algorithm with several numerical examples, and present both the analytic curves produced by the algorithm and filtered coefficients of the functions θ(t) and s ′ (t) representing the curves, where θ(t) is the tangential angle and s ′ (t) is the first derivative of the arc length. We implemented our algorithm in Fortran 77, and compiled it using the Gfortran Compiler, version 9.4.0, with -O3 flag. All experiments were conducted on a laptop with 16 GB of RAM and an Intel 11th Gen Core i7-1185G7 CPU. Furthermore, we use FFTW library (see [14]) for the implementations of the FFT and the Fast Cosine Transform. The latter is used to implement the Fast Chebyshev Transform.
The following variables appear in this section:
− N : the number of discretization nodes.
− n: the number of sample data points.
− n iters : the maximum number of iterations.
− n stop : the number of iterations needed for the algorithm to terminate.
− h filter : the proportion of the coefficients that are filtered to zero at each iteration.
− ϵ: the desired accuracy of the approximation to the curve. As ϵ is dependent on the size of the curve, for consistency, we scale the sample data points, so that either the width or height of the collection of data points, whichever is closer to 1, is 1. − E samp : the maximum l 2 norm of the distance between the curve, defined by n coefs Chebyshev or Fourier coefficients, and the sample data points.
While there is no strict rule on how to choose these variables, we assume that the users pick a reasonable combination of inputs, so that the algorithm terminates before reaching the maximum number of iterations, n iters .
Open Curve Examples
We sample some points from a spiral with the polar representation (r(t) cos φ(t), r(t) sin φ(t)), where φ(t) = 6π log 2 log t,
r(t) = φ(t),(91)
with t ∈ [1,2], and construct the initial Bézier spline passing through the data points, as shown in Figure 1(a). The sample data points are scaled so that their width is 1. We set N = 1000, n = 50, x ′ left = 0.05, y ′ left = 0.05, x ′ right = 0.05, y ′ right = 0.05, n iters = 60 h filter = 1 25 , ϵ = 10 −16 , n coefs = 500, n bands = 8. After n stop = 16 iterations, the algorithm terminates and returns a curve represented by only 500 Chebyshev coefficients. We display the Chebyshev coefficients that are necessary to represent both the initial and final curve in Figure 3. We can see that the shape of the final curve in Figure 1(b) is smoother, especially at the center of the spiral. Moreover, the resulting curve curve passes through the sample data points with an error of E samp = 0.11548 · 10 −13 . The magnitudes of the Chebyshev coefficients of s ′ (t) and θ(t) before and after filtering are displayed in Figure 2.
Another example depicted in Figure 4(a) is obtained by sampling from the curve
γ(t) = {5t, 3 cos (10tπ) 3 }, t ∈ [0, 1].(92)
The sample data points are scaled so that their height is 1. We run the algorithm by choosing n = 70, N = 4500, x ′ left = 0.25, y ′ left = 0.25, x ′ right = 0.25, y ′ right = 0.25, n iters = 70, h filter = 1 45 , ϵ = 10 −16 , n coefs = 3620, n bands = 6. The curve before smoothing is observed to bend unnaturally when zooming in on some details, for example, those shown in Figure 5(a). Thus, a reasonably large number of Chebyshev coefficients are required to represent s ′ (t) and θ(t), as shown in Figure 6. By looking at Figure 4(b) and Figure 5(b), the curve appears more like a manually drawn smooth curve after n stop = 60 iterations. The coefficients returned by the algorithm represent a curve passing through the sample data points to within an error of E samp = 0.16875 · 10 −13 . We display the magnitudes of the Chebyshev coefficients of both the initial and final curve in Figure 7. Figure 8(a) shows a roughly sketched shape resembling a snake. We scale the sample data points so that their height is 1, and run the algorithm by choosing N = 4000, n = 44, x ′ left = 0.05, y ′ left = −0.02, x ′ right = −0.06, y ′ right = 0.02, n iters = 80, h filter = 1 50 , ϵ = 10 −16 , n coefs = 1780, n bands = 6. The algorithm terminates at the n stop = 71st iteration, and the resulting curve passes through the sample data points to within an error of E samp = 0.35056 · 10 −14 . We present the magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering in Figure 9, and the magnitudes of the coefficients of x(t) and y(t) of both the initial and final curve in Figure 10.
We illustrate some damping oscillations, as displayed in Figure 11(a). The sample data points are scaled so that their width is 1. The initial curve has some sharp corners, and is distorted unnaturally. We set N = 4500, n = 40, x ′ left = 0.20, y ′ left = −0.20, x ′ right = −0.20, y ′ right = 0.40, n iters = 70, h filter = 1 40 , ϵ = 10 −16 , n coefs = 1830, n bands = 8. After n stop = 69 iterations, the algorithm terminates and returns a curve passing through the sample data points to within an error of E samp = 0.22649 · 10 −13 . The resulting curve in Figure 11(b) resembles a curve drawn by hand, with a completely smooth shape that naturally bends to pass through all the sample data points to exhibit those damping oscillations. We present the magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering in Figure 12, and the magnitudes of the coefficients of x(t) and y(t) of both the initial and final curve in Figure 13. We apply the algorithm in [7] to the same data points, by setting a = 0.2, which produces the smoothest shape of the curve, as displayed in Figure 14(a). Although the curve in Figure 14(a) requires fewer coefficients to represent x(t) and y(t) compared to the curve in Figure 11(b), it requires a much larger number of coefficients for s ′ (t) and θ(t), as shown in 14(b). This results in a curve with a high level of curvature. With our algorithm, any high curvature areas are effectively smoothed, yielding a visually smoother curve and requiring much fewer coefficients to represent s ′ (t) and θ(t).
The runtimes per iteration for the open curve case are displayed in Table 1. Since we use the library [14] for the implementation of the FFT, and the speed of the FFT routines in the library depends in a complicated way on the input size, we observed that the runtimes in Table 1 are not strictly proportional to the number of discretization points, N .
Closed Curve Examples
The first closed curve example is obtained by sampling from the polar representation (r(t) cos φ(t), r(t) sin φ(t)), where 23 The result of algorithm in [7] applied to the same data points in Figure 11.
φ(t) = 2πt, r(t) = (1 + 1 α cos (18φ(t)) sin (4φ(t)),(93)
with t ∈ [0, 1], and α is a tuning parameter. The sample data points are scaled so that both their width and height are 1. We sample the curve (93) with α = 2, N = 8000 and n = 100 to obtain the initial curve in Figure 15(a). Applying the algorithm with n iters = 70, h filter = 1 35 , ϵ = 10 −16 , n coefs = 5200, n bands = 12, we obtained the filtered coefficients of θ(t) and s ′ (t), as displayed in Figure 16. We find that, after n stop = 67 iterations, 5200 coefficients of x(t) and y(t) are necessary to represent the smooth curve displayed in Figure 15(b), to within an error of E samp = 0.22453 · 10 −14 . The magnitudes of the coefficients of both the initial and final curve are displayed in Figure 17.
X k = X −k , k = − N 2 , . . . , N 2 − 1,(94)
we only show the magnitudes of the coefficients, for k = 0, . . . , N 2 − 1.
Another example is shown in Figure 18, by sampling the curve (93) with α = 8, N = 2000 and n = 60. The sample data points are scaled so that both their width and height are 1. It is obvious that the curve has fewer wobbles than the previous curve. In this case, we set n iters = 60, h filter = 1 35 , ϵ = 10 −16 , n coefs = 1560, n bands = 8. The algorithm terminates after n stop = 34 iterations, and the error between the final curve and the sample data points is E samp = 0.11008 · 10 −14 . The magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering are displayed in Figure 19, and the magnitudes of the coefficients of both the initial and final curve are displayed in Figure 20. While the shapes of the curves appear similarly before and after filtering, the coefficients change dramatically.
The example shown in Figure 21 has the shape of a cat, corresponding to n = 50 sample points. We scale the sample data points so that their height is 1, and discretize the curve with N = 4000, and set n iters = 100, h filter = 1 45 , ϵ = 10 −15 , n coefs = 1360, n bands = 4. After n stop = 98 iterations, 1360 coefficients are sufficient to represent the curve, and the error between the final curve and the sample data points is E samp = 0.64403 · 10 −14 . The magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering are displayed in Figure 22, and the magnitudes of the coefficients of both the initial and final curve are displayed in Figure 23. We observe that the sharp edges on the curve in Figure 21(a) becomes more rounded, and the resulting curve more closely resembles the shape of a cat. Meanwhile, Figure 24(a) is the result of the algorithm in [7] applied to the same data points, with a = 0.4. The number of coefficients that are necessary to represent s ′ (t) and θ(t) are displayed in Figure 24(b). As we can observe from Figure 24(a), the curve contains sharp corners, and its visual smoothness is similar to the initial curve displayed in Figure 21(a), prior to applying our algorithm. In contrast, our algorithm eliminates high curvature areas, resulting in a much smoother appearance. Inspired by Figure 4.5 and Figure 4.3 in [12], we apply our algorithm to the same sample data points to show that our algorithm produces a smoother curve and represents the curve with fewer coefficients. We start with the example in Figure 4.5. The sample data points are scaled so that their width is 1. With the parameters N = 1600, n = 13, n iters = 80, h filter = 1 35 , ϵ = 10 −16 , n coefs = 840, and n bands = 4, our algorithm produces a curve represented by only 840 coefficients, while the algorithm of [12] produces a curve represented by 2 · 25, 000 = 50, 000 coefficients. The algorithm terminates at the n stop = 58th iteration, and the resulting curve passes through the sample data points within an error of E samp = 0.15713 · 10 −13 . Notice that the corners in Figure 4.5 are eliminated and the final curve in Figure 25(b) looks much smoother. We display the magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering in Figure 27, and the magnitudes of the coefficients of both the initial and final curve in Figure 28.
Since the initial curve is constructed by using smooth splines to connect the sample data points, and our algorithm filters the curve further during the filtering process, this causes the shape of the curve to deviate from that of Figure 4.5 in [12]. In order to preserve the shape of the curve in Figure 4.5, we increase the number of sample data points. The sample data points are scaled so that their width is 1. Applying the algorithm to the curve displayed in Figure 29(a), with N = 4000, n = 59, n iters = 80, h filter = 1 45 , ϵ = 10 −16 , n coefs = 1700, n bands = 4, we obtain E samp = 0.83564 · 10 −13 at the n stop = 72nd iteration. The magnitudes of the coefficients of s ′ (t) and θ(t) before and after filtering are displayed in Figure 30, and the magnitudes of the coefficients of both the initial and final curve are displayed in in Figure 31. It is noteworthy that the shape of the curve has been preserved, while achieving a smoother curve than in [12]. Moreover, the number of coefficients required to represent the curve increases only moderately when compared to those shown in Figure 27.
For Figure 4.3 in [12], we scale the sample data points so that their height is 1, and apply the algorithm to the reproduced curve in Figure 32(a), with N = 2000, n = 41, n iters = 70, h filter = 1 40 , ϵ = 10 −16 , n coefs = 680, n bands = 4. Although the difference can not be distinguished visually, after n stop = 55 iterations, 680 coefficients are necessary to represent the curve, to within an error of E samp = 0.15102 · 10 −13 . The magnitudes of the coefficients of s ′ (t) and θ(t) are displayed in in Figure 34, and the magnitudes of the coefficients of both the initial and final curve are displayed in in Figure 35. Notice that the algorithm of [12] requires approximately 2 · 7000 = 14, 000 coefficients to fit a curve passing through the same sample data points.
The runtimes per iteration for the first two closed curve cases are displayed in Table 2. We observe that, as in the open curve case, the runtimes are not strictly proportional to N .
Conclusion
Our algorithm produces a bandlimited curve passing through a set of points, up to machine precision. It first constructs a C 2 Bézier spline passing through the points, and then recursively applies a Gaussian filter to both the derivative of the arc length function and the tangential angle of the curve, to control the bandwidth of the coefficients, followed by smooth corrections. The resulting curve can be represented by a small number of coefficients, and resembles a smooth curve drawn naturally by hand, free of ringing artifacts. The algorithm costs O(N log N ) operations at each iteration, and the cost can be further reduced by calling the FFT in the FFTW library [14], in which the speed of the FFT routines is optimized for inputs of certain sizes.
One possible extension of this paper is to design an algorithm for curves and surfaces in R 3 . The main methodology is still applicable, if we parametrize a curve in R 3 by a function γ(t) : I → R 3 , where I ⊂ R, in terms of the same parameter t as in this paper, and a surface in R 3 by a function γ(s, t) : I 1 × I 2 → R, where I 1 , I 2 ⊂ R, in terms of both s and t. We can apply the Chebyshev or the Fourier approximation in each parameter, depending on whether the curve or surface is periodic in that parameter, filter the coefficients and add smooth perturbations in a similar way. Another application is to implement the algorithm of this paper as a geometric primitive in CAD/CAM systems.
Since primitives are generally defined as level sets of polynomials (see Chapter 2 of [15]), the techniques in this paper could be used for the constructions of more general C ∞ shapes in CAD/CAM systems.
properties of a curve . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Cubic Bézier Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Solving for control points for an open curve . . . . . . . . . . . . . 6 2.2.2 Solving for control points for a closed curve . . . . . . . . . . . . . 7 2.3 Chebyshev PolynomialInterpolation . . . . . . . . . . . . . . . . . . . . . 7 2.3.1 Spectral Differentiation and Integration . . . . . . . . . . . . . . . 8 2.4 The Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . 9 2.4.1 Spectral Differentiation and Integration . . . . . . . . . . . . . . . 10 2.5 Gaussian filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The Algorithm 12 3.1 Initial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Representations of the Curve . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.1 Representation of an Open Curve . . . . . . . . . . . . . . . . . . . 13 3.2.2 Representation of a Closed Curve . . . . . . . . . . . . . . . . . . . 13 3.3 Filtering the Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.1 Filtering the Open Curve . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.2 Filtering the Closed Curve . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Closing the Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Repositioning the Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Adding Perturbations to the Curve . . . . . . . . . . . . . . . . . . . . . . 17 3.7 The Termination Criterion of the Algorithm . . . . . . . . . . . . . . . . . 18 3.8 Summary and Cost of the Algorithm . . . . . . . . . . . . . . . . . . . . . 19 4 Numerical Results 20 4.1 Open Curve Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Closed Curve Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
for k = 1, . . . , n − 1. Once the coefficients { f k } are computed, we can use the expansion n−1 k=0 f k T k (x) to evaluate f (x) everywhere on the interval [−1, 1].
Remark 3.2. Suppose that {ϕ i } are the values of the angle between {( x i , y i )} and (x,ȳ), and that {r i } are the distances between {( x i , y i )} and (x,ȳ), where (x,ȳ) is the center of all the closest points {( x i , y i )}. We shift the center of all the closest points {( x i , y i )} by (∆x, ∆y), and rotate the curve by an angle of ψ around the center. Observed that the sum of squares of the distances between the closest points and the original data points is given by f (ψ, ∆x, ∆y) = n i=0
2 .
2Discretize the curve at N ≫ n+1 Chebyshev nodes if the curve is open, or N ≫ n+1 equispaced nodes if the curve is closed, and compute {θ j } and {s ′ j }. Repeat the steps 3, . . . , 9 until a C ∞ smooth curve can be represented by the requested number of coefficients, n coefs : 3. Obtain the Chebyshev coefficients or the Fourier coefficients of {θ j } and {s ′ j }.4. Determine the number of coefficients of {θ j } and {s ′ j } larger than δ θ and δ s ′ . If there are fewer than n δ θ coefs and n δ s ′ coefs , respectively, then return the first n coefs coefficients of x(t) and y(t).
5 .
5Apply the filter to the coefficients of {θ j } and {s ′ j } to compute the filtered values of {θ j } and {s ′ j }.
Step 2 costs O(N ) operations. Step 3 involves spectral differentiation and the Discrete Chebyshev Transform in the open curve case, or the DFT in the closed curve case, where the Discrete Chebyshev Transform can be replaced by the Fast Chebyshev Transform and the DFT can be replaced by the FFT. The cost of step 3 is thus reduced to O(N log N ). Checking the termination condition in Step 4 costs approximately O(N ) operations. Applying the filter and reconstructing {θ j } and {s ′ j } in Step 5 has the same cost as applying the inverse Fast Chebyshev Transform or the IFFT, which costs O(N log N ) operations. If the curve is closed, we must modify {s ′ j } so that the curve remains closed. The cost of closing the curve by looping through {s ′ j } in Step 6 is O(N ). Step 7 involves spectral integration, and the inverse Fast Chebyshev Transform in the open curve case, or the IFFT in the closed curve case, which has the same O(N log N ) cost as Step 3. The cost of using Newton's method to rotate the curve in Step 8 is O(n + 1), and the cost of solving for the coefficients of the smooth perturbations added to the curve in Step 9 is O(n + 1). The total cost is thus O(N log N ) per iteration.
−
n coefs : the requested number of the coefficients representing the curve to precision ϵ. − n bands : the bandwidth of the matrix describing the effect of the Gaussian perturbations centered at each sample point. − x ′ left , y ′ left : the derivative of the initial curve specified at the left end point, in the x coordinate and y coordinate separately. This variable only exists in the open curve case. Notice that the filtering process can potentially alter the value of this variable. − x ′ right , y ′ right : the derivative of the initial curve specified at the right end point, in the x coordinate and y coordinate separately. This variable only exists in the open curve case. Notice that the filtering process can potentially alter the value of this variable.
Figure 1 0
1.14308 · 10 −02 0.20470 · 10 −02 0.29810 · 10 −02 0.51040 · 10 −02
Figure 1 :Figure 2 :
12The result of algorithm applied to (91). The red dots mark the sample points. Chebyshev coefficients of s ′ (t) and θ(t) corresponding toFigure 1. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 355th coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 380th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line.
Figure 3 :
3Chebyshev coefficients of x(t) and y(t) corresponding toFigure 1. The value of n coefs is indicated by a vertical dashed line.
Figure 4 :
4The result of algorithm applied to (92). The red dots mark the sample points.
Figure 6 :Figure 7 :Figure 8 :
678Chebyshev coefficients of s ′ (t) and θ(t) corresponding toFigure 4. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 2472nd coefficients of s ′ (t) decay to δ s ′ , indicated by a vertical solid line. The 2148th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Chebyshev coefficients of x(t) and y(t) corresponding toFigure 4. The value of n coefs is indicated by a vertical dashed line. A hand-drawn depiction of a snake shape. The red dots mark the sample points.
Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :
910111213Chebyshev coefficients of s ′ (t) and θ(t) corresponding toFigure 8. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 1214th coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 1171st coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Chebyshev coefficients of x(t) and y(t) corresponding toFigure 8. The value of n coefs is indicated by a vertical dashed line.(a) The curve before smoothing (b) The curve after smoothing A shape of oscillations that exhibit damping. The red dots mark the sample points. Chebyshev coefficients of s ′ (t) and θ(t) corresponding toFigure 11. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 1222nd coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 1088th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Chebyshev coefficients of x(t) and y(t) corresponding toFigure 11. The value of n coefs is indicated by a vertical dashed line.(a) The interpolating curve. The red dots mark the sample points. Chebyshev coefficients of s ′ (t) and θ(t) corresponding toFigure 14(a).
Figure 14 :
14Figure 14: The result of algorithm in [7] applied to the same data points in Figure 11.
Remark 4 . 1 .
41Note that, since the DFT, X − N 2 , . . . , X N 2 −1 , of a real sequence, x 0 , . . . , x N −1 , satisfies the relation:
Figure 15 0
15.44050 · 10 −03 0.72667 · 10 −03 0.13412 · 10 −02 0.26530 · 10 −02Figure 180.36162 · 10 −03 0.61837 · 10 −03 0.11924 · 10 −02 0.24492 · 10 −02
Figure 15 :Figure 16 :
1516The result of algorithm applied to (93) with α = 2. The red dots mark the sample points. Fourier coefficients of s ′ (t) and θ(t) corresponding toFigure 15. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 1901st coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 1785th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line.
Figure 17 :Figure 18 :
1718Fourier coefficients of x(t) and y(t) corresponding to Figure 15. The value of n coefs is indicated by a vertical dashed line. The result of algorithm applied to (93) with α = 8. The red dots mark the sample points.
Figure 19 :Figure 20 :Figure 22 :Figure 23 :
19202223Fourier coefficients of s ′ (t) and θ(t) corresponding to Figure 18. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 588th coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 588th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Fourier coefficients of x(t) and y(t) corresponding to Figure 18. The value of n coefs is indicated by a vertical dashed line. (a) The curve before smoothing (b) The curve after smoothing Figure 21: A hand-drawn depiction of a cat shape. The red dots mark the sample points. Fourier coefficients of s ′ (t) and θ(t) corresponding to Figure 21. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 525th coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 492nd coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Fourier coefficients of x(t) and y(t) corresponding to Figure 21. The value of n coefs is indicated by a vertical dashed line. (a) The interpolating curve. The red dots mark the sample points.
) Fourier coefficients of s ′ (t) and θ(t) corresponding toFigure 24(a).
Figure 24 : 21 Figure 25 :
242125The result of algorithm in[7] applied to the same data points in Figure The result of algorithm applied toFigure 4.5 in[12]. The red dots mark the sample points.
Figure 26 :Figure 27 :Figure 28 :Figure 29 :Figure 30 :Figure 31 :Figure 32 :
26272829303132Figure Fouriercoefficients of s ′ (t) and θ(t) corresponding to Figure 25. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 303rd coefficients of s ′ (t) decays to the δ s ′ , indicated by a vertical solid line. The 283rd coefficients of θ(t) decays to the δ θ , indicated by a vertical dashed line. Fourier coefficients of x(t) and y(t) corresponding to Figure 25. The value of n coefs is indicated by a vertical dashed line. (a) The curve before smoothing (b) The curve after smoothing The result of algorithm applied to Figure 4.5 in [12], with more sample data points. Due to the large quantity and non-uniform distribution of the sample data points, we choose not to display them in the plot. Fourier coefficients of s ′ (t) and θ(t) corresponding to Figure 29. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 545th coefficients of s ′ (t) decays to the δ s ′ , indicated by a vertical solid line. The 494th coefficients of θ(t) decays to the δ θ , indicated by a vertical dashed line. Fourier coefficients of x(t) and y(t) corresponding to Figure 29. The value of n coefs is indicated by a vertical dashed line. The result of algorithm applied to Figure 4.3 in [12]. The red dots mark the sample points.
Figure 33 :Figure 34 :Figure 35 :
333435Figure Fouriercoefficients of s ′ (t) and θ(t) corresponding to Figure 32. The value of δ s ′ is indicated by a horizontal solid line and the value of δ θ is indicated by a horizontal dashed line. The 237th coefficients of s ′ (t) decays to δ s ′ , indicated by a vertical solid line. The 225th coefficients of θ(t) decays to δ θ , indicated by a vertical dashed line. Fourier coefficients of x(t) and y(t) corresponding to Figure 32. The value of n coefs is indicated by a vertical dashed line.
Table 1 :
1Average runtime per iteration, for an open curve, calculated by determining the total runtime for 100 iterations and dividing by the number of iterations.
Table 2 :
2Average runtime per iteration, for the first two closed curves, calculated by determining the total runtime for 250 iterations and dividing by the number of iterations.
A new method of interpolation and smooth curve fitting based on local procedures. H Akima, J. Assoc. Comput. Mach. 17Akima, H. "A new method of interpolation and smooth curve fitting based on local procedures." J. Assoc. Comput. Mach. 17.4 (1970): 589-602.
General cubic curve fitting algorithm using stiffness coefficients. U Björkenstam, S Westberg, Computer-Aided Design. 19Björkenstam, U., and S. Westberg. "General cubic curve fitting algorithm using stiffness coefficients." Computer-Aided Design. 19.2 (1987): 58-64.
Optimizing at the end-points the Akima's interpolation method of smooth curve fitting. M A Bica, Computer Aided Geometric Design. 31Bica, M.A. "Optimizing at the end-points the Akima's interpolation method of smooth curve fitting." Computer Aided Geometric Design. 31.5 (2014): 245-257.
Interpolating Cubic Spilnes. G D Knott, Birkhäuser BostonKnott, G.D. Interpolating Cubic Spilnes. Birkhäuser Boston, 2000.
Partition of Unity parametrics: A framework for meta-modeling. A Runions, F F Samavati, The Visual Comput. 27Runions, A., and F.F. Samavati. "Partition of Unity parametrics: A framework for meta-modeling." The Visual Comput. 27 (2011): 495-505.
The NURBS Book. L Piegl, W Tiller, Springer-BerlinPiegl, L., and W. Tiller. The NURBS Book. Springer-Berlin, 1995.
An Efficient Scheme for Curve and Surface Construction based on a Set of Interpolatory Basis Functions. R Zhang, W Ma, ACM T. Graphic. 30Zhang, R., and W. Ma. "An Efficient Scheme for Curve and Surface Construction based on a Set of Interpolatory Basis Functions." ACM T. Graphic. 30.2 (2011): 1-11.
CINPACT-splines: A class of C ∞ Curves with Compact Support. A Runions, F F Samavati, Curves and Surfaces 2014: Curves and Surfaces. Runions, A., and F.F. Samavati. "CINPACT-splines: A class of C ∞ Curves with Compact Support." Curves and Surfaces 2014: Curves and Surfaces. 2015: 384-398.
CINAPACT-Splines: A Family of Infinitely Smooth, Accurate and Compactly Supported Splines. B Akram, U R Alim, F F Samavati, ISVC 2015: Advances in Visual Computing. Akram, B., U.R. Alim, and F.F. Samavati. "CINAPACT-Splines: A Family of In- finitely Smooth, Accurate and Compactly Supported Splines." ISVC 2015: Advances in Visual Computing. 2015: 819-829.
MOMS: Maximal-order interpolation of minimal support. T Blu, P Thévenaz, M Unser, IEEE. T. Image Process. 10Blu, T., P. Thévenaz, and M. Unser. "MOMS: Maximal-order interpolation of minimal support." IEEE. T. Image Process. 10.7 (2001): 1069-1080.
A class of blending functions with C ∞ smoothness. Y Zhu, Numer. Algorithms. 88Zhu, Y. "A class of blending functions with C ∞ smoothness." Numer. Algorithms 88 (2021): 555-582.
Fitting a bandlimited curve to points in a plane. D Beylkin, V Rokhlin, SIAM J. Sci. Comput. 36Beylkin, D., and V. Rokhlin. "Fitting a bandlimited curve to points in a plane." SIAM J. Sci. Comput. 36.3 (2014): 1048-1070.
Intuitive Analog Circuit Design. M T Thompson, Newnes2nd ed.Thompson, M.T. Intuitive Analog Circuit Design. 2nd ed. Newnes, 2014.
The Design and Implementation of FFTW3. M Frigo, S G Johnson, Proc. IEEE. 93. IEEE. 932Frigo, M., and S.G. Johnson. "The Design and Implementation of FFTW3." Proc. IEEE. 93.2 (2005).
Geometric and Solid Modeling. C M Hoffmann, Hoffmann, C.M. Geometric and Solid Modeling. 2002. See https://www.cs.purdue. edu/homes/cmh/distribution/books/geo.html.
Cubic Bézier Splines. M Joost, Joost, M. "Cubic Bézier Splines." Notes. 2011. See https://www.michael-joost. de/bezierfit.pdf.
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"Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking",
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"Daniel Murnane \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Paolo Calafiura \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Nicholas Choma \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Sean Conlon \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Steven Farrell \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Yaoyuan Xu \nLawrence Berkeley National Laboratory\nBerkeleyCAUSA\n",
"Maria Spiropulu \nCalifornia Institute of Technology\nPasadenaCAUSA\n",
"Jean-Roch Vlimant \nCalifornia Institute of Technology\nPasadenaCAUSA\n",
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"Jim Kowalkowski \nFermi National Accelerator Laboratory\nBataviaILUSA\n",
"Markus Atkinson \nUniversity of Illinois at Urbana-Champaign\nUrbanaILUSA\n",
"Mark Neubauer \nUniversity of Illinois at Urbana-Champaign\nUrbanaILUSA\n",
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"Aditi Chauhan \nUniversity of Washington\nSeattleWAUSA\n",
"Alex Schuy \nUniversity of Washington\nSeattleWAUSA\n",
"Shih-Chieh Hsu \nUniversity of Washington\nSeattleWAUSA\n",
"Alex Ballow \nYoungstown State University\nYoungstownOHUSA\n",
"Alina Lazar \nYoungstown State University\nYoungstownOHUSA\n"
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The Exa.TrkX project has applied geometric learning concepts such as metric learning and graph neural networks to HEP particle tracking. Exa.TrkX's tracking pipeline groups detector measurements to form track candidates and filters them. The pipeline, originally developed using the TrackML dataset (a simulation of an LHC-inspired tracking detector), has been demonstrated on other detectors, including DUNE Liquid Argon TPC and CMS High-Granularity Calorimeter. This paper documents new developments needed to study the physics and computing performance of the Exa.TrkX pipeline on the full TrackML dataset, a first step towards validating the pipeline using ATLAS and CMS data. The pipeline achieves tracking efficiency and purity similar to production tracking algorithms. Crucially for future HEP applications, the pipeline benefits significantly from GPU acceleration, and its computational requirements scale close to linearly with the number of particles in the event.arXiv:2103.06995v2 [physics.data-an] 21 Sep 2021
|
10.1140/epjc/s10052-021-09675-8
|
[
"https://export.arxiv.org/pdf/2103.06995v2.pdf"
] | 237,604,960 |
2103.06995
|
4b8b941a85b564d0f943f4fbddbc6e83d42f7c86
|
Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking
Xiangyang Ju
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Daniel Murnane
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Paolo Calafiura
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Nicholas Choma
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Sean Conlon
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Steven Farrell
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Yaoyuan Xu
Lawrence Berkeley National Laboratory
BerkeleyCAUSA
Maria Spiropulu
California Institute of Technology
PasadenaCAUSA
Jean-Roch Vlimant
California Institute of Technology
PasadenaCAUSA
Adam Aurisano
University of Cincinnati
CincinnatiOHUSA
V Hewes
University of Cincinnati
CincinnatiOHUSA
Giuseppe Cerati
Fermi National Accelerator Laboratory
BataviaILUSA
Lindsey Gray
Fermi National Accelerator Laboratory
BataviaILUSA
Thomas Klijnsma
Fermi National Accelerator Laboratory
BataviaILUSA
Jim Kowalkowski
Fermi National Accelerator Laboratory
BataviaILUSA
Markus Atkinson
University of Illinois at Urbana-Champaign
UrbanaILUSA
Mark Neubauer
University of Illinois at Urbana-Champaign
UrbanaILUSA
Gage Dezoort
Princeton University
PrincetonNJUSA
Savannah Thais
Princeton University
PrincetonNJUSA
Aditi Chauhan
University of Washington
SeattleWAUSA
Alex Schuy
University of Washington
SeattleWAUSA
Shih-Chieh Hsu
University of Washington
SeattleWAUSA
Alex Ballow
Youngstown State University
YoungstownOHUSA
Alina Lazar
Youngstown State University
YoungstownOHUSA
Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking
the date of receipt and acceptance should be inserted laterEur. Phys. J. C manuscript No. (will be inserted by the editor)
The Exa.TrkX project has applied geometric learning concepts such as metric learning and graph neural networks to HEP particle tracking. Exa.TrkX's tracking pipeline groups detector measurements to form track candidates and filters them. The pipeline, originally developed using the TrackML dataset (a simulation of an LHC-inspired tracking detector), has been demonstrated on other detectors, including DUNE Liquid Argon TPC and CMS High-Granularity Calorimeter. This paper documents new developments needed to study the physics and computing performance of the Exa.TrkX pipeline on the full TrackML dataset, a first step towards validating the pipeline using ATLAS and CMS data. The pipeline achieves tracking efficiency and purity similar to production tracking algorithms. Crucially for future HEP applications, the pipeline benefits significantly from GPU acceleration, and its computational requirements scale close to linearly with the number of particles in the event.arXiv:2103.06995v2 [physics.data-an] 21 Sep 2021
Introduction
Charged particle tracking plays an essential role in High-Energy Physics (HEP), including particle identification and kinematics, vertex finding, lepton reconstruction, and flavor jet tagging. At the core of particle tracking there is a pattern recognition algorithm that must associate a list of 2D or 3D position measurements from a E-mail: [email protected] a tracking detector (known as hits or spacepoints in literature) to a list of particle track candidates (or tracks. A track is defined as a list of spacepoints associated by the pattern recognition to a charged particle).
The number of particle track candidates varies significantly from one experiment setup to another. For example, in a High-Luminosity LHC (HL-LHC) [1] collision event, due to the pile-up of multiple proton-proton collision per bunch crossing, there are typically 5,000 charged particles and 100,000 spacepoints, about 50 % of which are associated to particles of interest.
A typical HEP offline tracking algorithm [3,4,5] has four stages: spacepoint formation, track seeding, track following, and track fitting. The spacepoint formation stage combines the detector readout cell raw data in clusters from which the spacepoint 3D coordinates, and their uncertainties, are determined. Track seeding combines spacepoints in doublet or triplet seeds. Each seed provides an initial track direction, origin, and possibly a curvature, with associated uncertainties. The track following stage adds more spacepoints to the seed by looking for matching spacepoints along the extrapolated trajectory. Finally a track fitting stage, which may be combined with the track following, fits a trajectory through the track spacepoints to assess the track quality and measure the particle's physical and kinematic properties (charge, momentum, origin, etc). To avoid biasing physics results, each stage of the algorithm must have high efficiency, meaning it must identify e.g. > 90% of the charged particles within a fiducial Fig. 1 A simulated HL-LHC collision event (top) as seen by the TrackML tracking detector [2]. The detector schematic (bottom) shows the top half of the detector projected on the r-z plane. The z-axis is along the beam direction.
region (e.g. p T > 1 GeV, |η| < 4) as track candidates. Track seeding and track filtering must also have high purity, meaning that e.g. > 60% of the track seeds and track candidates must correspond to charged particles. High purity allows to keep the number of track candidates, and the associated computational costs, under control.
Online tracking algorithms may use different pattern recognition algorithms 1 to create and filter track seeds and candidates, but share the same high efficiency requirements. Online application also have stringent computing requirements (e.g. latency O(10) µs for LHC triggers).
The computational cost of current tracking algorithms grows worse than linearly with beam intensity and detector occupancy, as demonstrated in Figure 2. Given the order-of-magnitude increase for beam intensity at HL-LHC, charged particle pattern recognition algorithms might well limit the discovery potential of HL-LHC experiments.
Over the last two decades, tracking computational challenges arising from the increased number of combinations have been addressed by tightening fiducial regions for charged particles, developing highly opti-1 including Hough transforms [6,7] and cellular automata [8,9] Average pileup [10]. Bottom: CMS time spent in tracking sequence for 2016 tracking, 2017 tracking with conventional seeding, and 2017 tracking with Cellular Automaton (CA) seeding [11]. mized tracking algorithms [4,5], and even optimizing the geometry of tracking detectors. These optimizations brought order-of-magnitude gains in tracking computational performance with limited impact on physics. While these efforts continue [12], it is unlikely that another order of magnitude can be gained through incremental optimization without impacting physics performance. Furthermore, given the computational complexity and iterative nature of current track following and filtering algorithms, it is challenging to run them efficiently on data parallel architectures like GPUs.
The TrackML challenge [2] jump-started the application of deep learning pattern recognition methods applied to HEP tracking. The HEP.TrkX pilot project [13] proposed the use of graph networks to filter track doublet and triplet seeds [14]. Building on that work, the Exa.TrkX project [15] has demonstrated the applicabil-ity of Geometric Deep Learning (GDL) methods [16] specifically metric learning and Graph Neural Networks (GNN) -to particle tracking [17]. GDL is concerned with learning representations of data that have complex geometrical relationships and no natural ordering, like detector spacepoints. GDL models are computationally regular, naturally parallel and therefore well-suited to run on hardware accelerators.
This work describes new developments that enabled the first study of the computing and physics performance of the Exa.TrkX pipeline on the entire TrackML detector at HL-LHC design luminosity, a step towards the validation of the pipeline on ATLAS and CMS data.
Related work
Early on, the Hep.TrkX pilot project attempted to assign and regress track parameters to single spacepoints using image processing models. Subsequent attempts at estimating track parameters using image processing and recurrent networks showed promising results [18] in a simplified environment. A similar realization of the method is reported in [19] where a model processing image from successive pixel detector layers is used to produce tracklets, seeds to classical pattern recognition. The method yields superior seeding efficiency for tracks within jets in dense environments. The concept of using LSTM [20] to supplement the Kalman Filter method for track following developed by HEP.TrkX [18,21,14] was later found in one of the promising solutions of the accuracy phase [22] of the TrackML challenge. The task of particle tracking was addressed with a hit-to-track assignment method using gated recurrent unit [23] (GRU), producing promising result in sparse environments [21]. This approach was constrained computationally due to the use of recurrent models.
Ref. [24] applies the track finding approach developed in Ref. [25] to the whole detector by exploiting a new data-driven graph construction method and large model support in Tensorflow [26]. Ref. [27] applies a similar GNN model to the task of particle-flow reconstruction. The model has a classification objective, followed by a partial regression of generator-level particle candidate kinematics. The method performs at least as well as a classical particle-flow algorithm in HL-LHC-like collision conditions. As part of the Exa.TrkX project, graph networks are used for LArTPC track reconstruction [28]. Ref. [29] explores the opportunity to implement Exa.TrkX-inspired graph networks on FP-GAs. Starting from the input stage of the Exa.TrkX pipeline, Ref. [30] studies the impact of cluster shape information on track seeding performance. In Ref. [31], metric learning is used to improve the purity in spacepoints buckets formed using similarity hashing. With the advent of quantum computer of increasing size came the development of quantum machine learning techniques, also applied in particle physics [32]. In particular, inspired by the use of GNN for charged particle tracking of the Exa.TrkX team, quantum graph networks have been tested on the same problem [33].
Methodology
Input Data
This study is based on the TrackML dataset that uses a Montecarlo simulation of top quark pair production from proton-proton collisions at the HL-LHC. To simulate the effect of event pileup and produce realistic detector occupancy, a Poisson random number (with µ = 200) of QCD "minimum bias" events are overlaid on top of the tt collisions.
The TrackML detector is a set of concentric cylindrical layers of pixelated sensors (the barrel ) complemented by a set of circular disks (the endcaps) to ensure nearly 4π coverage in solid angle, as pictured in Figure 1. Figure 3 shows the spatial distribution of the spacepoints of a typical event. One notable feature of this dataset is the inclusion of "noise" spacepoints, added as a proxy for various low-momentum particle interactions and detector effects which would otherwise require more expensive and detailed simulations.
The Geometric Deep Learning Pipeline
This paper updates the methodology previously presented in Ref. [17] to a fully-learned pipeline, where both graph construction and graph classification are trained. This section describes the pipeline (represented schematically in Figure 4) used to obtain the results in § 4. Details of the latest model design, parameter choices, and technical optimizations are discussed in § 5.
The pipeline currently used to reconstruct tracks from a pointcloud of spacepoints requires six discrete stages of processing and inference. These broadly consist of a preprocessing stage, three stages required to construct a spacepoint graph, and two stages required to classify the graph edges and partition them into track candidates. Each stage is trained independently (due to memory constraints) on the output of the previous stage's inference.
First, the dataset is processed into a format suitable for model training. This includes calculating directional . These values are appended to the cylindrical coordinates of each spacepoint to form an input feature vector to the pipeline. To apply a graph neural network to this set of data, it is necessary to arrange them into a graph. One can apply various geometric heuristics to define which spacepoints are likely to be connected by an edge (i.e. belong to the same track), but a useful technique is to train a model on the geometry of connected tracks. Thus, our second stage is to train an Embedding Network -a multi-layer perceptron (MLP) which embeds each spacepoint into an N-dimensional latent space. The graph is constructed by connecting neighboring spacepoints within a radius r embedding , in the latent space. We train this embedding with a pairwise hinge loss, to encourage spacepoints that belong to the same track to be close in the embedded space, according to the Euclidean metric. This allows for a highly efficient edge construction, since we do not rely on any heuristics of the detector geometry that may lead to missed edges.
The edge selection at this stage is close to 100% efficient but O(1)% pure, with a graph size of O(10 5 ) nodes and O(10 7 ) edges (the purity-efficiency trade-off can be tuned with the choice of r embedding ). Before running training or inference on the memory-intensive GNN, we filter these edges down with another MLP. The input to this third stage is the concatenated features on either side of each edge. That is, the Filter Network is a binary classifier applied to the set of edges. Constraining edge efficiency to remain high (above 96%) leads to much sparser graphs, of O(10 6 ) edges.
The fourth stage of the pipeline is the training and inference of the graph neural network. The results presented in this work are predominantly obtained from the Interaction Network architecture, first proposed in Ref. [34]. This varietal of GNN includes hidden features on both nodes and edges, which are propagated around the graph (called "message passing") with consecutive concatenations along edges and aggregations of messages at receiving nodes. In the final layer of the network, a binary classification is obtained for each edge as true or fake, and trained on a cross-entropy loss.
The final stage of the TrackML pipeline involves task-specific post-processing. If our goal is track formation, we can place a threshold on the edge scores produced by the GNN and partition the graph into connected components. If our goal is track seeding, we can directly sample the classified edges for high likelihood combinations of connected triplets, or convert the entire graph to a triplet graph and train this on a second GNN to classify the triplets. A triplet graph is formed by taking all edges in the original (doublet) graph and assigning them as nodes in the new triplet graph. The nodes in this triplet graph are connected if they share a hit in the doublet graph. Applying a GNN to this structure produces highly pure sets of seeds as shown in Ref. [17].
Many of these techniques are common to other applications being explored in the Exa.TrkX collaboration. The pattern of nearest-neighbor graph-building and GNN edge classification has shown its potential for neutrino experiments [28] and CMS High Granularity Calorimeter [25]. Indeed, these applications build on the TrackML pipeline and extend it, for example by adding the particle type as an edge feature.
Results
Tracking Performance of the TrackML pipeline
Tracking Efficiency and Purity
The performance of a tracking pipeline is mainly characterized by tracking efficiency and purity. For efficiency calculations, only charged particles that satisfy |η| < 4.0 and p T > 100 MeV are considered. These selected particles, N particles (selected), are hereafter referred to as particles.
The overall tracking efficiency, known as physics efficiency phys (Eq. 1), is defined as the fraction of particles that are matched to at least one reconstructed track. A particle is considered to be matched to a reconstructed track when 1) the majority of spacepoints in the reconstructed track belong to the same true track, and 2) the majority of spacepoints in the matched true particle track are found in the reconstructed track 2 .
To measure the efficiency of the tracking pipeline itself, we also define the technical efficiency tech (Eq. 2) as the fraction of reconstructable particles matching at least one reconstructed track. Reconstructable particles have a trajectory that leaves at least five spacepoints in the detector. Tracking purity (Eq. 3) is defined as the fraction of reconstructed tracks that match a selected particle 3 .
phys = N particles (selected, matched) N particles (selected) (1) tech = N particles (selected, reconstructable, matched) N particles (selected, reconstructable) (2) Purity = N tracks (selected,matched) N tracks (selected)(3)
Averaged over 50 testing events from the TrackML dataset, the physics efficiency for particles with p T > 500 MeV is 88.7 ± 0.3% and the technical efficiency is 97.6±0.3%. Without any fiducial p T cut, the physics efficiency becomes 67.2±0.1% and the technical efficiency 91.3 ± 0.2%. The tracking purity is 58.3 ± 0.6%. Using the TrackML challenge scoring system and all tracks in the event, we obtained a score of 0.877 ± 0.005 4 . The errors quoted are statistical only. Figure 5 shows the p T distribution of particles as well as the tracking efficiency as a function of particle p T . The physics efficiency for particles with p T of [100, 300] MeV is 43%, therefore, is not displayed in the plot. The physics efficiency for particles with p T > 700 MeV is above 88%. The technical efficiency is 82% for particles with p T of [100, 300] MeV, and increases to above 97% for particles with p T > 700 MeV. Figure 5 also shows the η distribution of particles with p T > 500 MeV as well as the tracking efficiency as a function of the particle η. The physics efficiency is higher in the barrel region of the detector (volumes 8,13,17 in Figure 1), while the technical efficiency is almost flat across the η range. In Figure 5 the p T and η of the matched truth particle were used, rather than the p T and η of the reconstructed track. We leave a study of track quality and detector resolution effects for future work.
Systematic Studies
Before using a tracking algorithm in production, it is necessary to measure its sensitivity to systematic effects, including pile-up, noise and digitization errors, and uncertainties in the measurement of detector properties (alignment, rotation, magnetic field map, etc.).
Measuring precisely the impact of pile-up collisions on tracking performance is beyond the scope of this work, but we can estimate pile-up's impact on tracking performance by plotting efficiency and purity as a function of the number of spacepoints in the detector. Figure 6 shows that the effect of the increased detector occupancy is a smooth performance degradation O(%). In future work, we will study the origin of this degradation to achieve the stable performance of traditional algorithms [36].
The impact of noise spacepoints can be estimated using the TrackML dataset by studying the inference performance of the tracking pipeline, trained without any noise spacepoints, as a function of the fraction of noise spacepoints (up to a maximum of 20% of the total). Table 1 shows the technical tracking efficiency and purity for different noise levels. The efficiency decreases by 1.6% and the purity by 5.4% when 20% of noise spacepoints are presented. The loss of efficiency happens primarily for particles with p T < 500 MeV (Figure 7). Detector misalignment effects are approximated by shifting by up to 1 mm the x-axis of all spacepoints in the inner-most TrackML barrel detector layer or the four innermost layers (volume 8 in Figure 1). In both cases, the impact on the tracking efficiency is less than 0.1%. However, studying in depth misalignments, and other detector effects, requires access to experiment detailed detector simulation data. We leave these studies as future work to be performed in collaboration with each experiment.
Distributed Training Performance
Our training sample consists of 7500 pileup events from the TrackML dataset. It takes about 1.5 days to train the Exa.TrkX pipeline on a Nvidia A100 GPU for a set of hyper-parameters. It is therefore desirable to use distributed training to parallelize model training and hyper-parameter optimization (HPO). This study relied on data parallel training [37] implemented using Horovod [38] and Tensorflow's tf.distributed framework [39]. Horovod supports distributed training across multiple nodes, while tf.distributed allows to use the same code across CPUs, TPUs, and GPUs.
For this study, the TrackML pipeline is trained on up to 64 Nvidia V100 GPUs across eight NERSC Cori-GPU computing nodes. Using the Horovod framework (Figure 8), training time is reduced from 22 minutes, with 1 GPU, to 0.5 minutes with 64 GPUs 5 . The strong scaling efficiency 6 is about 90% with 2 GPUs and 75% with 8 GPUs. This deviation from ideal scaling is due to the model setup time and data movement costs. Figure 8 also shows the scaling behaviour of the tf.distributed implementation. Since this implementation requires all input data to be of the same size, we have to pad all input graphs to a fixed size. This essentially doubles the time needed to train one epoch, that increases from 22 minutes for dynamic input graph sizes to 41 minutes for constant graph sizes. Leaving aside this fixed overhead, tf.distributed appears to scale better than Horovod, achieving 85% strong scaling efficiency with 8 GPUs.
Inference performance on CPU and GPU
It is crucial to characterize the computational cost of the end-to-end learned tracking algorithm. We rely on the Pytorch and TensorFlow libraries to optimize our inference pipeline on CPU and GPU. The execution time for the inference pipeline has been measured on two hardware platforms: Nvidia V100 GPUs with 16 GB on-board memory, and Intel Xeon 6148s (Skylake) CPUs with 40 cores and 192 GB memory per node. The inputs to the filtering step do not fit into the GPU 5 All measurements in this section were taken training on spacepoints from the barrel region of the TrackML detector. For comparison, training with spacepoints from the whole detector takes 70 minutes per epoch on one Nvidia A100 GPU 6 defined as t 1 /(N × t N ) * 100% where t N is the time to train on a fixed total number of events across N GPUs. Top row: selected, reconstructable, and matched particles (left) and tracking efficiency (right) as a function of p T for particles with |η| < 4. Bottom row: selected, reconstructable, and matched particles (left) and tracking efficiency (right) as a function of η for p T > 0.5 GeV. The definition of "selected", "reconstructable", and "matched" can be found in § 4.1.1 memory. Therefore, edge filtering for one event is executed in mini-batches with a fixed batch size of 800k edges. Typically, the inputs to the filtering from one event are split into seven batches, leading to additional computational cost for moving data from host to GPU. The peak GPU memory consumption is about 15.7 GB as obtained from the Nvidia profiling tool.
Averaging over 500 events, it takes 2.2 ± 0.3 wallclock seconds per event (as measured by measured by the python module time) to run the inference pipeline on the GPU and 202 ± 35 seconds to run it on a single CPU core. This total execution time includes every step of the calculation, and in particular the time needed to move data from host to GPU. Table 2 breaks down the wall-clock time for the most significant steps of the pipeline. The results show how the graph creation and filtering steps are the biggest targets for further optimization in order to surpass traditional algorithms in terms of inference time [40].
In addition, Figure 9 shows how the total inference time depends almost linearly on the number of spacepoints in the event for both CPUs and GPUs. The steplike dispersion in the GPU case is due to the splitting of the inputs to the filtering step into mini-baches. A step-like jump indicates one more mini-batch is added. Many optimizations were introduced to the pipeline in order to achieve these GPU timings, which before optimization took over 20 seconds per event. These improvements include porting all data processing to the GPU-accelerated CuPy library [41], writing custom sparse operations for graph processing (e.g. doublet-totriplet conversion [42], graph intersection methods), using FAISS [43] ments are specific to the inference stage; training optimizations will be discussed in the following section, and ongoing developments in § 6. No CPU-specific optimization was performed in this work.
Discussion
The performance given above is the result of experimentation across various feature sets, architectures, model configurations and hyperparameters. It has also been necessary to overcome a variety of training hurdles in terms of memory and computational availability. We describe here training and inference details that should allow a reader to reproduce these results on the provided codebase.
Feature Set
The input dataset includes both spatial coordinates and highly granular pixel cluster shape information. Graph construction (the second pipeline step in Figure 4, that includes learned embedded space model and edge filter model) appears to benefit significantly from the cluster shape information, approximately doubling the purity for a held fixed high efficiency. The summary cluster shape statistics include the number of channels and the total charge deposited, as well as local and global representations of the cluster as a high-level feature vector.
Details about the calculation of this feature vector as well as a thorough exploration of the effect of cluster shape information on seeding performance are provided in Ref. [30]. Cluster shape information does not appear to improve the performance of the GNN, and in fact seems to degrade it. This suggests that the width of the GNN hidden layers is not great enough to capture the functional relationship of cluster information between nodes. Scaling to a width that properly explores this question would require more memory than available on the Nvidia A100 GPUs used for this study. Total time (s) Depending on the final goal of the pipeline, further features can be included in the loss calculation in order to bias the model towards desired regions. For example, if our aim is to maximize the TrackML score (described in Ref. [2]) -a weighting function s i that places more importance on a spacepoint i from a longer and higher p T track, and in the first and last sets of detector layers -we can weight-up true edges by this function, normalized to have a mean of weight = 1. To measure the performance of models trained to this goal, we introduce a weighted purity measure. Weighted purity is defined as a function the TrackML weights w ij and the truth y ij ∈ {0, 1} of each edge connecting spacepoint i and spacepoint j,
Purity weighted = ij w ji y ij ij w ij , w ij = 1 2 (s i + s j ), if y ij = 1 1, if y ij = 0(4)
We see significant improvements in this metric when validating on the weighted model: the Embedding Network improves from a weighted purity of 1.7%±0.2% to 2.0% ± 0.3%, while the Filter Network improves from a weighted purity of 8.4% ± 0.6% to 11.7% ± 1.0%. Given this weighting, the model learns to prioritize higher p T and longer tracks, while disregarding less informative tracks. Using this bias, we can achieve the same TrackML score with a constructed graph size reduced by approximately 25%. Using this technique to improve the TrackML score is an ongoing work.
Graph Construction
Having chosen a feature set, to train the learned embedding space we use a training paradigm commonly referred to as a Siamese Network [46], where a particular spacepoint -called the source -is run through an MLP, here 6 layers each with 512 hidden channels, hyperbolic tan activations, and layer normalization. The final layer of the MLP takes the features to an 8-dimensional latent space. A different, comparison spacepoint -called the target -is also run through this same Embedding Network, and the L2 norm distance d in the latent space between the source and target enters a comparative hinge loss
L hinge = d p , if y ij = 1 max(0, 1 − d p ), if y ij = 0 (5)
where p is a hyperparameter that we choose to be 2. If the source i and target j spacepoints share an edge in the event's truth graph 8 , we designate them as neighbours with y ij = 1, otherwise they are designated y ij = 0. In this way, the hinge loss draws together truth graph neighbors and repels non-neighbors.
Training performance of the Embedding Network is highly dependent on choice of source-target example pairs. In early epochs, it is enough to choose random pairs. However, at some point, many random pairs will contribute no gradient to the loss, as they will be separated by a distance greater than the margin. At that point, it is useful to implement hard negative mining [47]. We run a GPU-optimised k-nearestneighbor (KNN) algorithm 9 to mine examples around each source vector, within the hinge margin d = 1. The computational overhead of the KNN step is significantly offset by the examples mined which all contribute to the loss.
A similar technique is used in the Filter Network, where the vast majority of the edges produced from the graph construction in the embedded space are easy to classify as fake. This is already a highly imbalanced dataset, with around 98.5% of edges fake. Again, within several epochs, the Filter Network is able to classify many of these as fake, so we balance each batch with all true edges, the same number of hard negatives (i.e. negatives the filter is unsure of) and the same number of easy negatives (to maintain performance on these edges). The Filter Network is a MLP that takes the 24feature concatenated edge features and feeds forward through 3 layers of 1024 hidden channels, to a binary cross-entropy loss function.
GNN Edge Classification
In choosing the best GNN architecture, memory usage remains a significant constraint. The Interaction Network (IN) [34] presented in these results does appear to marginally attain the best performance against Attention Graph Neural Networks (AGNN) [49,14] -the other class of GNN considered for the pipeline. However, both of these networks require gradients to be retained in memory for every graph edge. Indeed, this anisotropic treatment of edges (i.e. a node is able to receive the messages of each of its neighbors in a nonuniform way) is what allows these two architectures to be so expressive. Depending on hardware availability, we have found two solutions to the memory constraint. Access to next-generation Nvidia A100 GPUs allowed an IN to be trained with 8 steps of message passing, aggregating edge features at each node, and each node and concatenated edge features passing through twolayer MLPs of [128,64] hidden features and ReLU activations [50]. Choice of aggregation function should be permutation invariant. In this work, we take it to be a summation.
For lower-memory GPUs, such as the Nvidia V100, we attained similar performance training the AGNN architecture, with [64,64,64]-channel MLPs applied to each edge and node. Adding residuals [51] across the 8 message passing steps greatly improved performance in this case. To fit full-event training on a single V100, it was necessary to employ various techniques, such as mixed precision training and gradient checkpointing. The latter stores only the input of each layer, not the gradients. On the backward pass, gradients are re-calculated on the fly, allowing for a 4x reduction in memory usage for an 8-iteration GNN. Another technique explored is to split the events piecemeal and train on each piece as a standalone batch. There is a noticeable impact on performance due to messages being interrupted at the graph edges. In future work, we will present ongoing efforts to parallelise these graph pieces across multiple GPUs, retaining the high performance that full-event training allows.
Physics-inspired data augmentation
Preliminary work on using coordinate transforms to augment the training data has been explored with varying degrees of success. In this study, focused on track seeding, only the innermost detector layers (volumes 7-9 in Figure 1) were used.
One promising approach is to make a copy of each graph in the training set that has been reflected across the phi-axis [52]. The phi reflection creates the charge conjugate graph and helps to balance any asymmetry between positive and negatively charged particles within the training set. Using the phi-reflected graphs boosts efficiency by 2% and purity by 1% in the barrel. This performance boost comes at the cost of doubling the training time. In future work, we will investigate the opportunity of integrating charge conjugation symmetry into the network itself.
A second promising trick is to use a Hough Transform [6,7] on the graph to create edge features. Using the Hough parameters as edge features boosts efficiency by 2% and purity by 1%. A further efficiency boost of 3% (and 2% to purity) comes from using the Hough accumulator to extract an edge weight. This edge weight effectively pools information from every node, and therefore comes at a large computational cost (filling the accumulator in Hough space). On the other hand, the Hough parameters can be computed quickly from the two nodes that define the edge.
Conclusions and Future Work
This works shows how a tracking pipeline based on geometric deep learning can achieve state-of-the-art com-puting performance that scales linearly with the number of spacepoints, showing great promise for the next generation of HEP experiments. The inference pipeline has been optimized on GPU systems, on the assumption that the next generation of HEP experiments will have widespread access to accelerators either locally in heterogeneous systems [27,53] or remotely [54,55].
Within the simplifying assumptions of the TrackML dataset, we have shown how the Exa.TrkX pipeline could meet the tracking performance requirements of current collider experiments. Preliminary studies suggest that this performance should be robust against systematic effects like detector noise, misalignment, and pile-up.
Much remains to be done to validate these promising results. To this end, the Exa.TrkX project is collaborating with physicists from ATLAS [56], CMS [57], DUNE [58], ICARUS [59], and MuonE [60].
The goal is to adapt the Exa.TrkX pipeline to each experiment's needs and simulated datasets, measure its performance and robustness against systematic effects according to the experiment metrics. For example, it is crucial for HL-LHC experiments to study the performance of tracking algorithms in dense environments, like high-p T jets. Given the interest in long-lived particle observation at the HL-LHC, it will also be important to study the performance of the Exa.TrkX pipeline for tracks coming from a displaced vertex 10 .
On the computational side, there are several optimization opportunities to explore systematically, including mixed precision training, multi-GPU training and inference with graph data parallelisation (that is, one event spread across multiple GPUs) [61]; locality sensitive hashing to speed-up KNN/graph construction stage [62], model quantization, operator fusion and other improvements with TensorRT [63], clustering of final node embeddings rather than hard connected components method with GravNet-style architectures [64].
The distributed training results presented in this work are promising but still preliminary. To fully exploit the capabilities of upcoming HPC systems and to further reduce training time while potentially pushing further on model size, it will be beneficial to perform further studies on large scale training of GNNs for track reconstruction. Given the size of the input graphs, this problem may be amenable to training techniques which parallelise the processing of input graphs across multiple GPUs in training.
Finally, it will be interesting to measure the computing performance of (parts of) the Exa.TrkX pipeline on domain-specific accelerators like Google TPU [65] and GraphCore IPU [66], comparing power consumption, latency and throughput with "traditional" GPUs.
Software availability
A growing number of groups are studying the application of graph networks to HEP reconstruction (see [67] for a recent review). Some of these works [24,27,28,29,30,31,33] have strong connections with the Exa.TrkX project. To promote collaboration and reproducibility, the Exa.TrkX software is available from the HEP Software Foundation's Trigger and Reconstruction GitHub 11 . A pipeline of re-usable modules is implemented within the Pytorch Lightning system, which allows for uncluttered and simple model definitions. As each stage of the pipeline is dependent, logging utilities are integrated that allow a specific combination of stages and hyperparameters to be trackable and reproducible. Extensive documentation is provided to help track reconstruction groups start exploring geometric learning. The roadmap for this repository includes adding performance metrics to the codebase; a taxonomy of model features; and short tutorials in each of the available applications.
Fig. 2
2Reconstruction wall time per event as a function of the average number of interactions per bunch crossing µ . Top: ATLAS Run 2 Inner Detector reconstruction with default configurations
Fig. 3
3A typical event distribution of spacepoints projected on the x-z plane, parallel to the beam direction (left), and the x-y plane, orthogonal to the beam direction (right).
Fig. 4
4Stages of the TrackML track formation inference pipeline. Light red boxes are trainable stages. information and summary statistics from the charge deposited in each spacepoint, i.e. the cell features in Figure 4
Fig. 5
5Fig. 5 Top row: selected, reconstructable, and matched particles (left) and tracking efficiency (right) as a function of p T for particles with |η| < 4. Bottom row: selected, reconstructable, and matched particles (left) and tracking efficiency (right) as a function of η for p T > 0.5 GeV. The definition of "selected", "reconstructable", and "matched" can be found in § 4.1.1
Fig. 6
6Mean and standard deviation of the technical efficiency (top) and purity (bottom) as a function of the total number of spacepoints in an event.
Fig. 8
8Time per training epoch (left) and Strong scaling efficiency (right) for GNN's distributed training. The top row refers to the Horovod implementation, the bottom row to the tf.distributed one. The first bin in the bottom left diagram refers to the serial case, in which the input graph is not padded.
Fig. 9
9Total inference time as a function of number of spacepoints in each event for CPUs (top) and GPUs (bottom).
for large-k NN graph construction, and performing track labelling with CuGraph's connected component algorithm on GPU [44]7 . These improve-7 on CPU, track labeling uses the DBSCAN algorithm[45]. Average inference time for synchronous execution of the TrackML pipeline benchmarked on CPUs and GPUs. For these step-by-step measurements, we force the pipeline to execute serially by calling torch.cuda.synchronize after each step. The total inference time comprises all the steps including ones not listed in the table.0
1
2
3
4
5
pT [GeV]
0.97
0.98
0.99
1.00
Ratio of Track Efficiencies
Noise Level
4 %
8 %
12 %
16 %
20 %
Fig. 7 Relative technical efficiency as a function of p T . Each
curve shows the ratio of eff(noise = N %)/eff(noise = 0).
Wall time [s] on Wall time [s] on
Xeon 6148s
Nvidia V100
single core
synchronous
Data Loading 0.0049 ± 0.0153 0.0023 ± 0.0003
Embedding
3.02 ± 0.39
0.024 ± 0.003
Build Edge
66 ± 13
0.76 ± 0.10
Filtering
99 ± 19
1.57 ± 0.34
GNN
27 ± 2
0.45 ± 0.06
Labeling
3.23 ± 0.34
0.08 ± 0.01
Total (sync)
202 ± 35
3.3 ± 0.5
Table 2
This nomenclature and the associated definitions broadly follow[35,2].3 HEP tracking literature often quotes fake rate = 1 − purity4 We obtained a score of 0.914 ± 0.006 by training the pipeline with a dataset that includes noise hits, that we otherwise removed from our training dataset to facilitate the noise impact studies of section 4.1.2
one can also designate y ij = 1 for source and target in the same track, rather than immediate neighbors in the track. This does lead to similar performance in later stages of the pipeline, but the more lax concept of truth leads to graphs around three times more dense than the strict track neighbor definition.
We use two high-performance libraries, FAISS[43] and Py-torch3D[48], depending on number of nearest neighbors k. Fastest performance is obtained with FAISS for k > 35, Py-torch3D for k <= 35.
it may be worth noticing that in LArTPC applications[28] all tracks come from a displaced vertex.
https://hsf-reco-and-software-triggers.github.io/ Tracking-ML-Exa.TrkX
AcknowledgementsThis research was supported in part by: Finally, we thank Marcin Wolter (IFJ PAN), Ben Nachman, Alex Sim and Kesheng Wu (LBNL) for the useful discussions.
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"https://github.com/rapidsai/cugraph,"
] |
[
"Interaction-induced magnetoresistance in a two-dimensional electron gas",
"Interaction-induced magnetoresistance in a two-dimensional electron gas"
] |
[
"I V Gornyi \nInstitut für Nanotechnologie\nForschungszentrum Karlsruhe\n76021KarlsruheGermany\n",
"A D Mirlin \nInstitut für Nanotechnologie\nForschungszentrum Karlsruhe\n76021KarlsruheGermany\n\nInstitut für Theorie der kondensierten Materie\nUniversität Karlsruhe\n76128KarlsruheGermany\n"
] |
[
"Institut für Nanotechnologie\nForschungszentrum Karlsruhe\n76021KarlsruheGermany",
"Institut für Nanotechnologie\nForschungszentrum Karlsruhe\n76021KarlsruheGermany",
"Institut für Theorie der kondensierten Materie\nUniversität Karlsruhe\n76128KarlsruheGermany"
] |
[] |
We study the interaction-induced quantum correction δσ αβ to the conductivity tensor of electrons in two dimensions for arbitrary T τ (where T is the temperature and τ the transport scattering time), magnetic field, and type of disorder. A general theory is developed, allowing us to express δσ αβ in terms of classical propagators ("ballistic diffusons"). The formalism is used to calculate the interaction contribution to the longitudinal and the Hall resistivities in a transverse magnetic field in the whole range of temperature from the diffusive (T τ ≪ 1) to the ballistic (T τ 1) regime, both in smooth disorder and in the presence of short-range scatterers. Further, we apply the formalism to anisotropic systems and demonstrate that the interaction induces novel quantum oscillations in the resistivity of lateral superlattices.
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10.1103/physrevb.69.045313
|
[
"https://export.arxiv.org/pdf/cond-mat/0306029v1.pdf"
] | 119,085,599 |
cond-mat/0306029
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cadb3f1dc460da0fedd91f388e4c1520697324b8
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Interaction-induced magnetoresistance in a two-dimensional electron gas
2 Jun 2003
I V Gornyi
Institut für Nanotechnologie
Forschungszentrum Karlsruhe
76021KarlsruheGermany
A D Mirlin
Institut für Nanotechnologie
Forschungszentrum Karlsruhe
76021KarlsruheGermany
Institut für Theorie der kondensierten Materie
Universität Karlsruhe
76128KarlsruheGermany
Interaction-induced magnetoresistance in a two-dimensional electron gas
2 Jun 2003(Dated: January 9, 2022)numbers: 7210-d7323Ad7110-w7343Qt
We study the interaction-induced quantum correction δσ αβ to the conductivity tensor of electrons in two dimensions for arbitrary T τ (where T is the temperature and τ the transport scattering time), magnetic field, and type of disorder. A general theory is developed, allowing us to express δσ αβ in terms of classical propagators ("ballistic diffusons"). The formalism is used to calculate the interaction contribution to the longitudinal and the Hall resistivities in a transverse magnetic field in the whole range of temperature from the diffusive (T τ ≪ 1) to the ballistic (T τ 1) regime, both in smooth disorder and in the presence of short-range scatterers. Further, we apply the formalism to anisotropic systems and demonstrate that the interaction induces novel quantum oscillations in the resistivity of lateral superlattices.
I. INTRODUCTION
The magnetoresistance (MR) in a transverse field B is one of the most frequently studied characteristics of the two-dimensional (2D) electron gas 1,2 . Within the Drude-Boltzmann theory, the longitudinal resistivity of an isotropic degenerate system is B-independent,
ρ xx (B) = ρ 0 = (e 2 νv 2 F τ ) −1 , (1.1)
where ν is the density of states per spin direction, v F the Fermi velocity, and τ the transport scattering time. Deviations from the constant ρ xx (B) are customarily called a positive or negative MR, depending on the sign of the deviation. There are several distinct sources of a nontrivial MR, which reflect the rich physics of the magnetotransport in 2D systems. First of all, it has been recognized recently that even within the quasiclassical theory memory effects may lead to strong MR 3,4,5,6,7,8,9 . The essence of such effects is that a particle "keeps memory" about the presence (or absence) of a scatterer in a spatial region which it has already visited. As a result, if the particle returns back, the new scattering event is correlated with the original one, yielding a correction to the resistivity (1.1). Since the magnetic field enhances the return probability, the correction turns out to be B-dependent. As a prominent example, memory effects in magnetotransport of composite fermions subject to an effective smooth random magnetic field explain a positive MR around half-filling of the lowest Landau level 7 . Another type of memory effects taking place in systems with rare strong scatterers is responsible for a negative MR in disordered antidot arrays 3,4,5,8,9 . However, such effects turn out to be of a relatively minor importance for the low-field quasiclassical magnetotransport in semiconductor heterostructures with typical experimental parameters, while at higher B they are obscured by the development of the Shubnikovde Haas oscillation (SdHO).
Second, the negative MR induced by the suppression of the quantum interference by the magnetic field is a famous manifestation of weak localization 1 . While the weak-localization correction to conductivity is also related to the return probability, it has (contrary to the quasiclassical memory effects) an intrinsically quantum character, since it is governed by quantum interference of time-reversed paths. As a result it is suppressed already by a classically negligible magnetic field, which changes relative phases of the two paths. Consequently, the corresponding correction to ρ xx in high-mobility structures is very small and restricted to the range of very weak magnetic fields.
Finally, another quantum correction to MR is induced by the electron-electron interaction. While this effect is similar to those discussed above in its connection with the return probability (see Sec. IV below), it is distinctly different in several crucial aspects. In contrast to the memory effects, this contribution is of quantum nature and is therefore strongly T -dependent at low temperatures. On the other hand, contrary to the weak localization, the interaction correction to conductivity is not destroyed by a strong magnetic field. As a result, it induces an appreciable MR in the range of classically strong magnetic fields. This effect will be the subject of the present paper.
It was discovered by Altshuler and Aronov 1 that the Coulomb interaction enhanced by the diffusive motion of electrons gives rise to a quantum correction to conductivity, which has in 2D the form (we set k B = = 1)
δσ xx ≃ e 2 2π 2 1 − 3 2 F ln T τ, T τ ≪ 1. (1.2)
The first term in the factor (1 − 3 2 F ) originates from the exchange contribution, and the second one from the Hartree contribution. In the weak-interaction regime, κ ≪ k F , where κ = 4πe 2 ν is the inverse screening length, the Hartree contribution is small, F ∼ (κ/k F ) ln(k F /κ) ≪ 1. The conductivity correction (1.2) is then dominated by the exchange term and is negative. The condition T τ ≪ 1 under which Eq. (1.2) is derived 1 implies that electrons move diffusively on the time scale 1/T and is termed the "diffusive regime". Subsequent works 10,11 showed that Eq. (1.2) remains valid in a strong magnetic field, leading (in combination with δσ xy = 0) to a parabolic interaction-induced quantum MR,
δρ xx (B) ρ 0 ≃ 1 − 3 2 F (ω c τ ) 2 − 1 πk F l ln T τ, T τ ≪ 1,(1.
3) where ω c = eB/mc is the cyclotron frequency and l = v F τ the transport mean free path.
Indeed, a T -dependent negative MR was observed in experiments 12,13,14,15,16 and attributed to the interaction effect. However, the majority of experiments 12,13,14 cannot be directly compared with the theory 1,10,11 since they were performed at higher temperatures, T τ 1. (In high-mobility GaAs heterostructures conventionally used in MR experiments, 1/τ is typically ∼ 100 mK and becomes even smaller with improving quality of samples.) In order to explain the experimentally observed T -dependent negative MR in this temperature range the authors of Refs. 12,13 conjectured various ad hoc extensions of Eq. (1.3) to higher T . Specifically, Ref. 12 conjectures that the logarithmic behavior (1.3) with τ replaced by the quantum time τ s is valid up to T ∼ 1/τ s , while Ref. 13 proposes to replace ln T τ by −π 2 /2T τ . These proposals, however, were not supported by theoretical calculations. There is thus a clear need for a theory of the MR in the ballistic regime, T 1/τ .
In fact, the effect of interaction on the conductivity at T 1/τ has been already considered in the literature 17,18,19,20,21,22,23 . Gold and Dolgopolov 17 analyzed the correction to conductivity arising from the T -dependent screening of the impurity potential. They obtained a linear-in-T correction δσ ∼ e 2 T τ . In the last few years, this effect attracted a great deal of interest in a context of low-density 2D systems showing a seemingly metallic behavior 24,25 , dρ/dT > 0. Recently, Zala, Narozhny, and Aleiner 18,19,20 developed a systematic theory of the interaction corrections valid for arbitrary T τ . They showed that the temperature-dependent screening of Ref. 17 has in fact a common physical origin with the Altshuler-Aronov effect but that the calculation of Ref. 17 took only the Hartree term into account and missed the exchange contribution. In the ballistic range of temperatures, the theory of Refs. 18,19,20, predicts, in addition to the linear-in-T correction to conductivity σ xx , a 1/T correction to the Hall coefficient 19 ρ xy /B at B → 0, and describes the MR in a parallel field 20 .
The consideration of Ref. 18,19,20 is restricted, however, to classically weak transverse fields, ω c τ ≪ 1, and to the white-noise disorder. The latter assumption is believed to be justified for Si-based and some (those with a very large spacer) GaAs structures, and the results of Refs. 18,19,20 have been by and large confirmed by most recent experiments 26,27,28,29,30,31,32 on such systems. On the other hand, the random potential in typical GaAs heterostructures is due to remote donors and has a longrange character. Thus, the impurity scattering is predominantly of a small-angle nature and is characterized by two relaxation times, the transport time τ and the single-particle (quantum) time τ s governing damping of SdHO, with τ ≫ τ s . Therefore, a description of the MR in such systems requires a more general theory valid also in the range of strong magnetic fields and for smooth disorder. [A related problem of the tunneling density of states in this situation was studied in Ref. 33.] In this paper, we develop a general theory of the interaction-induced corrections to the conductivity tensor of 2D electrons valid for arbitrary temperatures, transverse magnetic fields, and range of random potential. We further apply it to the problem of magnetotransport in a smooth disorder at ω c τ ≫ 1. In the ballistic limit, T τ ≫ 1 (where the character of disorder is crucially important), we show that while the correction to ρ xx is exponentially suppressed for ω c ≪ T , a MR arises at stronger B where it scales as B 2 T −1/2 . We also study the temperature-dependent correction to the Hall resistivity and show that it scales as BT 1/2 in the ballistic regime and for strong B. We further investigate a "mixed-disorder" model, with both short-range and longrange impurities present. We find that a sufficient concentration of short-range scatterers strongly enhances the MR in the ballistic regime.
The outline of the paper is as follows. In Sec. II we present our formalism and derive a general formula for the conductivity correction. We further demonstrate (Sec. II C) that in the corresponding limiting cases our theory reproduces all previously known results for the interaction correction. In Sec. III we apply our formalism to the problem of interaction-induced MR in strong magnetic fields and smooth disorder. Section IV is devoted to a physical interpretation of our results in terms of a classical return probability. In Sections V and VI we present several further applications of our theory. Specifically, we analyze the interaction effects in systems with short-range scatterers and in magnetotransport in modulated systems (lateral superlattices). A summary of our results, a comparison with experiment, and a discussion of possible further developments is presented in Sec. VII. Some of the results of the paper have been published in a brief form in the Letter 34 .
II. GENERAL FORMALISM
A. Smooth disorder
We consider a 2D electron gas (charge −e, mass m, density n e ) subject to a transverse magnetic field B and to a random potential u(r) characterized by a correlation function u(r)u(r ′ ) = w(|r − r ′ |) (2.1) with a spatial range d. The total (τ −1 s ) and the transport (τ −1 ) scattering rates induced by the random potential are given by
1 τ s = 2πν 2π 0 dφ 2π W (φ), (2.2) 1 τ = 2πν 2π 0 dφ 2π W (φ)(1 − cos φ), (2.3)
where W (φ) =w(2k F sin φ 2 ) is the scattering crosssection. We begin by considering the case of smooth disorder, k F d ≫ 1, when τ /τ s ∼ (k F d) 2 ≫ 1; generalization onto systems with arbitrary τ /τ s will be presented in Sec. II B. We assume that the magnetic field is not too strong, ω c τ s ≪ 1, so that the Landau quantization is destroyed by disorder. Note that this assumption is not in conflict with a condition of classically strong magnetic fields (ω c τ ≫ 1), which is a range of our main interest in the present paper.
We consider two types of the electron-electron interaction potential U 0 (r): (i) point-like interaction, U 0 (r) = V 0 , and (ii) Coulomb interaction, U 0 (r) = e 2 /r. In order to find the interaction-induced correction δσ αβ to the conductivity tensor, we make use of the "ballistic" generalization of the diffuson diagram technique of Ref. 1. We consider the exchange contribution first and will discuss the Hartree term later on. Within the Matsubara formalism, the conductivity is expressed via the Kubo formula through the current-current correlation function, σ αβ (iΩ k ) = n e e 2 mΩ k δ αβ − 1 Ω k 1/T 0 dτ d 2 r T τĵα (r, τ )ĵ β (0, 0) e iΩ k τ , (2.4) where Ω k = 2πkT is the bosonic Matsubara frequency. Diagrams for the leading-order interaction correction are shown in Fig. 1 and can be generated in the following way. First, there are two essentially different ways to insert an interaction line into the bubble formed by two electronic Green's function. Second, one puts signs of electronic Matsubara frequencies in all possible ways. On the third step, one connects lines with opposite signs of frequencies ǫ n > 0, ǫ m < 0 by impurity-line ladders (which are not allowed to cross each other). Finally, in the case of the diagram a, where four electronic lines form a "box", one should include two additional diagrams, b and c, with an extra impurity line ("Hikami box"). The impurity-line ladders are denoted by shaded blocks in Fig. 1; we term them "ballistic diffusons". Formally, the ballistic diffuson is defined as an impurity average (denoted below as . . . imp ) of a product of a retarded and advanced Green's functions, D(iǫ m , iǫ n ; r 1 , r 2 ; r 3 , r 4 ) = θ(−ǫ m ǫ n ) G(r 1 , r 2 ; iǫ m )G(r 3 , r 4 ; iǫ n ) imp .
(2.5) Exchange diagrams for the interaction correction to σ αβ . The wavy (dashed) lines denote the interaction (impurity scattering), the shaded blocks are impurity ladders, and the +/− symbols denote the signs of the Matsubara frequencies. The diagrams obtained by a flip and/or by an exchange + ↔ − should also be included. "Inelastic" part of the diagrams f, g is canceled by a contribution of the Coulomb-drag type, see Appendix A.
Following the standard route of the quasiclassical formalism 35,36,37 , we perform the Wigner transformation,
D(iǫ m − iǫ n ; R 1 , p 1 ; R 2 , p 2 ) = drdr ′ ×e −i[p1−(e/c)A(R1)]r e −i[p2−(e/c)A(R2)]r ′ ×D(iǫ m , iǫ n ; R 1 , r; R 2 , r ′ ),(2.6)
where R 1 = (r 4 + r 1 )/2, R 2 = (r 2 + r 3 )/2, r = r 4 − r 1 , and r ′ = r 2 − r 3 . Note that the factors depending on the vector potential make the ballistic diffuson (2.6) gaugeinvariant. Finally, we integrate out the absolute values of momenta p 1,2 and get the final form of the ballistic diffuson D(iω l ; R 1 , n 1 ; R 2 , n 2 ) = 1 2πν p 1 dp 1 2π
p 2 dp 2 2π D(iω l ; R 1 , p 1 ; R 2 , p 2 ),(2.7)
which describes the quasiclassical propagation of an electron in the phase space from the point R 2 , n 2 to R 1 , n 1 .
Here n is the unit vector characterizing the direction of velocity on the Fermi surface. The ballistic diffuson satisfies the quasiclassical Liouville-Boltzmann equation
|ω l | + iv F q cos(φ − φ q ) + ω c ∂ ∂φ +Ĉ D(iω l , q; φ, φ ′ ) = 2πδ(φ − φ ′ ), (2.8)
where φ (φ q ) is the polar angle of n (q) andĈ is the col- W (n, n ′ ). For the case of a smooth disorder, the collision integral is given byĈ
= − 1 τ ∂ 2 ∂φ 2 . (2.9)
In contrast to the diffusive regime, where D has a universal and simple structure D(iω l , q) = 1/(Dq 2 + |ω l |) determined by the diffusion constant D only, its form in the ballistic regime is much more complicated. We are able, however, to get a general expression for δσ αβ in terms of the ballistic propagator D(ω l , q; n, n ′ ). The temperature range of main interest in the present paper is restricted by T τ s ≪ 1, since at higher T the MR will be small in the whole range of the quasiclassical transport ω c τ s ≪ 1 (see below). In this case the ladders are dominated by contributions with many (≫ 1) impurity lines. We will assume this situation when evaluating diagrams in the present subsection. A general case of arbitrary T τ s and τ s /τ will be addressed in Sec. II B.
We start with the diagrams d and e that give rise to the logarithmic correction in the diffusive regime 1 . Let us fix the sign of the external frequency, Ω k > 0. Each of the diagrams d and e generates four diagrams by a flip with respect to the horizontal line or by exchange + ↔ −, see Fig. 2. Consider first the diagram d ++ . There are two triangular boxes containing each a current vertex and three electron Green's functions (Fig. 3). In the quasiclassical regime ω c τ s ≪ 1 one may neglect the effect of magnetic field on the Green's functions (keeping ω c in the ballistic propagators only). Furthermore, using T τ s ≪ 1, we neglect the difference in momenta and frequencies in the Green's functions, since typical values of frequencies iΩ k , iω l and momenta q carried by the ballistic diffusons are set by the temperature. Each triangle then reads
Γ α (n) = e m pdp 2π p n α (−ξ p + i/2τ s ) 2 (−ξ p − i/2τ s ) ≃ i2πντ 2 s ev F n α ,(2.10)
where ξ p = p 2 /2m − µ. Combining the triangles with the three ballistic propagators separated by the impurity lines (see Fig. 3), we obtain the following expression for the electronic part of the diagram d ++ ,
(2πν) 3 6 i=1 dφ i 2π D(iω l , q; φ, φ 1 )W (φ 1 − φ 2 )Γ α (φ 2 ) ×W (φ 2 − φ 3 )D(iω l − iΩ k , q; φ 3 , φ 4 )W (φ 4 − φ 5 ) ×Γ β (φ 5 )W (φ 5 − φ 6 )D(iω l , q; φ 6 , φ ′ ) ≡ 4πσ 0 τ B d αβ (iω l , −iΩ k , q; φ, φ ′ ). (2.11)
In what follows we will use for brevity a short-hand notation
(2πν) 3 DW Γ α W DW Γ β W D
for the l.h.s. of (2.11) and analogous notations for other structures of this type. Making use of the smallangle nature of scattering in a smooth random potential, we can replace the
W (φ i − φ j ) factors in (2.11) by (ντ s ) −1 δ(φ i − φ j ), yielding B d αβ (q, iω l , iΩ k ; φ, φ ′ ) ≃ −D(iω l , q)n α D(iω l + iΩ k , q)n β D(iω l , q).
In the exchange term (calculated in the present subsection) this structure is further integrated over the angles φ and φ ′ ,
B d αβ (iω l , iΩ k , q) = B d αβ (iω l , iΩ k , q; φ, φ ′ ) . (2.12)
The angular brackets . . . denote averaging over velocity directions, e.g.
n x Dn x ≡ dφ 1 2π dφ 2 2π cos φ 1 D(ω l , q; φ 1 , φ 2 ) cos φ 2 .
The fermionic frequencies obey the inequalities ǫ m > 0, ǫ m − ω l < 0, and ǫ m − Ω k > 0, which implies ω l > ǫ m > Ω k , so that the summation over ǫ m gives the factor (ω l − Ω k )/2πT . The diagram d −− has the same structure (both triangles have opposite signs, thus the total sign remains unchanged), but the frequency summation is restricted by ǫ m < 0, ǫ m − ω l > 0, and ǫ m − Ω k < 0, yielding the factor −ω l /2πT in the conductivity correction. The diagrams d ′ ++ and d ′ −− obtained from d ++ and d −− by a flip (or, equivalently, by reversing all arrows) double the result. Combining the four contributions and changing sign of the summation variable, ω l → −ω l in d −− and d ′ −− terms, we have where U (iω, q) is the interaction potential equal to a constant V 0 for point-like interaction and to
δσ d αβ (iΩ k ) = − 8πσ 0 τ T 2 Ω k d 2 q (2π) 2 × ω l >Ω k ω l − Ω k 2πT U (iω l , q)B d αβ (−iω l , iΩ k , q) + ω l >0 ω l 2πT U (iω l , q)B d αβ (iω l , iΩ k , q) ,(2.U (iω l , q) = 1 2ν κ q + κ[1 − |ω l | D(q, iω l ) ]
(2.14)
for screened Coulomb interaction. In (2.13) we used the fact that U (q, −iω l ) = U (q, iω l ) and D(q, −iω l ) = D(q, iω l ). Equation (2.14) is a statement of the randomphase approximation (RPA), with the polarization operator given by
Π(iω l , q) = 2ν[1 − |ω l | D(iω l , q) ]. (2.15)
The first term (unity) in square brackets in (2.15) comes from the ++ and −− contributions to the polarization bubble, while the second term is generated by the +− contribution (ballistic diffuson).
The diagrams e are evaluated in a similar way. In all four diagrams of this type one of the electron triangles is the same as in diagrams d while another one has an opposite sign. The structures arising after integrating out fast momenta in electron bubbles coincide with those of d-type (B d αβ ). Summation over the fermionic frequency ǫ m is constrained by the condition ω l > ǫ m > Ω k for all the e-type diagrams. The correction due to the diagrams e therefore reads
δσ e αβ (iΩ k ) = 8πσ 0 τ T 2 Ω k d 2 q (2π) 2 ω l >Ω k ω l − Ω k 2πT U (iω l , q) × B d αβ (−iω l , iΩ k , q) + B d αβ (iω l , iΩ k , q) . (2.16)
We see that the first term in square brackets in (2.16) cancels the first term in (2.13). Thus, the sum of the contributions of diagrams d and e takes the form
δσ d+e αβ (iΩ k ) = − 4σ 0 τ T Ω k Ω k ω l =0 ω l Φ d αβ (iω l , iΩ k ) + ω l >Ω k Ω k Φ d αβ (iω l , iΩ k ) ,(2.17)
where we introduced a notation
Φ µ αβ (iω l , iΩ k ) = d 2 q (2π) 2 U (iω l , q)B µ αβ (iω l , iΩ k , q), (2.18) with the index µ labeling the diagram.
Similarly, we obtain for the diagram h
δσ h αβ (iΩ k ) = − 4σ 0 τ T Ω k Ω k ω l =0 ω l Φ h αβ (iω l , iΩ k ) + ω l >Ω k Ω k Φ h αβ (iω l , iΩ k ) , (2.19) with B h αβ (iω l , iΩ k , q) = −2T αγ n γ D(iω l +iΩ k , q)n β D(iω l , q) .
(2.20) The tensor T αγ appearing in (2.20) describes the renormalization of a current vertex connecting two electronic lines with opposite signs of frequencies,
T αβ = 2 n α Dn β | q=0,ω→0 = σ αβ e 2 v 2 F ν = τ 1 + ω 2 c τ 2 1 −ω c τ ω c τ 1 . (2.21)
We turn now to diagrams f and g. The expressions for the corresponding contributions read
δσ f αβ (iΩ k ) = − 4σ 0 τ T Ω k ω l ≥0 Ω k Φ f αβ (iω l , iΩ k ) + −Ω k <ω l <0 (Ω k + ω l )Φ f αβ (−iω l , iΩ k ) , (2.22) δσ g αβ (iΩ k ) = 4σ 0 τ T Ω k Ω k ω l =0 (Ω k − ω l )Φ f αβ (iω l , iΩ k ) + −Ω k <ω l <0 (Ω k + ω l )Φ f αβ (−iω l , iΩ k ) , (2.23) with B f αβ (iω l , iΩ k , q) = T αγ n γ D(iω l + iΩ k , q)n δ T δβ . (2.24)
The sum of the contributions f and g is therefore given by
δσ f +g αβ (iΩ k ) = − 4σ 0 τ T Ω k Ω k ω l =0 ω l Φ f αβ (iω l , iΩ k ) + ω l >Ω k Ω k Φ f αβ (iω l , iΩ k ) . (2.25)
We see that when the diagrams f and g are combined, the same Matsubara structure as for other diagrams [Eqs. (2.17), (2.19)] arises. In other words, the role of the diagrams g is to cancel the extra contribution of diagrams f, which has a different Matsubara structure. A word of caution is in order here. In our calculation we have set the value of velocity coming from current vertices to be equal v F , thus neglecting a particle-hole asymmetry. If one goes beyond this approximation and takes into account the momentum-dependence of velocity (violating the particle-hole symmetry), the above cancellation ceases to be exact and an additional term with a different Matsubara structure arises in δσ f +g αβ . After the analytical continuation is performed, the corresponding correction to the conductivity has a form
δσ inel αβ = − 2σ 0 τ ∞ −∞ dω 2π ω 2T sinh 2 (ω/2T ) × d 2 q (2π) 2 δB f αβ (ω, q) Im U (ω, q), (2.26)
characteristic for effects governed by inelastic scattering. This contribution is determined by real inelastic scattering processes with an energy transfer ω T and behaves (in zero magnetic field) as e 2 (T τ ) 2 . This implies that the corresponding resistivity correction, δρ ∼ (T /eE F ) 2 is independent of disorder. However, such a correction should not exist because of total momentum conservation. Indeed, an explicit calculation (see Appendix A) shows that this term is canceled by the Aslamazov-Larkin-type diagrams analogous to those describing the Coulomb drag. Finally, we consider the diagrams a,b, and c. Already taken separately, each of them has the expected Matsubara structure (contrary to the diagrams d,e and f,g, which should be combined to get this structure). However, another peculiarity should be taken into account. The diagrams a,b, and c form together the Hikami box, so that their sum is smaller by a factor ∼ τ s /τ than separate terms. Therefore, some care is required: subleading terms of order τ s /τ should be retained when contributions of individual diagrams are calculated. The result reads
δσ a+b+c αβ (iΩ k ) = − 4σ 0 τ T Ω k Ω k ω l =0 ω l Φ a+b+c αβ (iω l , iΩ k ) + ω l >Ω k Ω k Φ a+b+c αβ (iω l , iΩ k ) . (2.27)
Here the contributions of individual diagrams a, b, and c have the form B a αβ (iω l , iΩ k , q)
= 1 2 T αγ 1 τ s δ γδ + (T −1 ) γδ T δβ D(iω l , q)D(iω l , q) + 1 2 T αγ T γβ D(iω l , q) ,(2.28)
where the matrixT αβ has the same form as T αβ with a replacement τ → τ s ,
B b αβ (iω l , iΩ k , q) = − 1 2τ s T αγ T γβ D(iω l , q)D(q, iω l ) ,
(2.29) and
B c αβ (iω l , iΩ k , q) = − 1 2 1 τ s − 1 τ T αγ T γβ D(iω l , q)D(iω l , q) . (2.30)
We see that although each of the expressions (2.28), (2.29), and (2.30) depends on τ s , the single-particle time disappears from the total contribution of the Hikami-box,
B a+b+c αβ (iω l , iΩ k , q) = 1 2 T αβ D(iω l , q)D(iω l , q) + 1 2 T αγ D(iω l , q) T γβ . (2.31)
The total correction to the conductivity tensor is obtained by collecting the contributions (2.17), (2.19), (2.25), and (2.27). Carrying out the analytical continuation to real frequencies, we get
δσ αβ (Ω) = σ 0 iπτ Ω ∞ −∞ dω ω coth ω 2T × [Φ αβ (ω + Ω, Ω) − Φ αβ (ω, Ω)], (2.32) where Φ αβ (ω, Ω) = Φ a+b+c αβ (ω, Ω) + Φ d αβ (ω, Ω) + Φ f αβ (ω, Ω) + Φ h αβ (ω, Ω). (2.33)
We are interested in the case of zero external frequency, Ω → 0, when Eq. (2.32) can be rewritten as
δσ αβ = − σ 0 iπτ ∞ −∞ dω × Φ αβ (ω, 0) ∂ ∂ω ω coth ω 2T . (2.34)
Recalling the definition (2.18) of Φ µ , we finally arrive at the following result
δσ αβ = −2e 2 v 2 F ν ∞ −∞ dω 2π ∂ ∂ω ω coth ω 2T × d 2 q (2π) 2 Im [ U (ω, q) B αβ (ω, q) ] ,(2.35)
where the tensor B αβ (ω, q) is given by
B αβ (ω, q) = T αβ 2 DD + T αγ δ γδ 2 D − n γ Dn δ T δβ − 2T αγ n γ Dn β D − Dn α Dn β D . (2.36)
The first term in (2.36) originates from the diagrams a,b,c, the second term from a,f,g, the third term from h, and the last one -from d and e. We remind the reader that this result has been obtained under the assumption τ s ≪ τ, T −1 ; generalization to arbitrary τ s /τ and τ s T will be considered in Sec. II B. It will be shown there that the conductivity correction retains the form (2.35) in the general case but the expression (2.36) for B αβ (ω, q) is slightly modified.
B. General case
In the previous subsection we have derived the formula for the correction to the conductivity tensor for the case of a smooth disorder (with τ s ≪ τ ) assuming τ s ≪ T −1 . Since characteristic momenta q and frequencies ω are set by the temperature, this assumption implies ql s ≪ 1 and ωτ s ≪ 1. This allowed us to simplify the calculation by neglecting the q and ω dependence of Green's functions connecting ballistic diffusons and by considering only the ladders with many impurity lines. Furthermore, we have used the small-angle nature of scattering when calculating the Hikami box contribution (2.31). We are now going to discuss the general case of arbitrary τ s /τ and T τ s .
It turns out that the expressions (2.17), (2.19), and (2.25) for the contribution of the diagrams d − h derived in the case of a smooth disorder remain valid in the general situation. The simplest way to show this is to use the following technical trick (cf. Refs. 38,39). One can add to the system an auxiliary weak smooth random potential with a long transport scattering timeτ ≫ τ but short single-particleτ s ≪ τ s , such that Tτ s ≪ 1. This potential will not affect the quasiclassical dynamics and thus should not change the result. On the other hand, it allows us (in view of the condition Tτ s ≪ 1) to perform the gradient and frequency expansion in Green's functions as was done in Sec. II A. Adding such an auxiliary disorder amounts to a re-distribution between quantum and quasiclassical degrees of freedom: all the information about the real disorder is now contained in the ballistic propagators. It can be verified by a direct calculation (without using the additional disorder) that the above procedure yields the correct result.
It remains to consider the Hikami-box contribution (2.27). When calculating it in Sec. II A, we used the small-angle nature of scattering implying that a single scattering line inserted between two ballistic propagators approximately preserves the direction of velocity, DW D → (2πν) −1 DD /τ s and Dn α W n β D → (2πν) −1 DD (1/τ s − 1/τ )δ αβ . In the more general situation, when the scattering is at least partly of the largeangle character, this is no longer valid and Eq. (2.31) acquires a slightly more complicated form,
B a+b+c αβ (q, iω l , iΩ k ) = πνT αγ [ DS γδ D − 2 Dn γ W n δ D ]T δβ + 1 2 T αγ D T γβ ,(2.37)
where S xx = S yy = W (n, n ′ ), S xy = −S yx = ω c /2πν. Summarizing the consideration in this subsection, in the general situation the interaction correction retains the form (2.35) with the tensor B αβ (ω, q) given by
B αβ (ω, q) = T αγ πν[ DS γδ D − 2 Dn γ W n δ D ]T δβ +T αγ δ γδ 2 D − n γ Dn δ T δβ −2T αγ n γ Dn β D − Dn α Dn β D . (2.38)
The correction δρ αβ to the resistivity tensor is then immediately obtained by using δρ = −ρ δσρ. This yields
δρ αβ = 2 e 2 v 2 F ν ∞ −∞ dω 2π ∂ ∂ω ω coth ω 2T × d 2 q (2π) 2 Im U (ω, q) B (ρ) αβ (ω, q) ,(2.αβ = (T −1 ) αγ B γδ (T −1 ) δβ . (2.40)
Explicitly, corrections to the components of the resistivity tensor are expressed through δσ xx = δσ yy and δσ xy = −δσ yx as follows
δρ xx = ρ 2 0 [(ω 2 c τ 2 − 1)δσ xx + 2ω c τ δσ xy ], (2.41) δρ xy = ρ 2 0 [(ω 2 c τ 2 − 1)δσ xy −
C. Limiting cases
Having obtained the general formula, we will now demonstrate that it reproduces, in the appropriate limits, the previously known results for the interaction correction. Specifically, in Sec. II C 1 we will consider the diffusive limit T τ ≪ 1 studied in Refs. 1,10,11, while Sec. II C 2 is devoted to the B → 0 case with a white-noise disorder addressed in Refs. 18,19. In Sec. II C 3 we will analyze how the linear-in-T asymptotics of δσ(B = 0) in the ballistic regime obtained in Ref. 18 for a whitenoise disorder depends on the character of the random potential.
Diffusive limit
We begin by considering the diffusive limit T τ ≪ 1 in which we reproduce (for arbitrary B and disorder range) the logarithmic correction (1.2), (1.3) determined by the diagrams a-e. Let us briefly outline the corresponding calculation. The propagator for ql, ωτ ≪ 1 can be decomposed as D = D s + D reg , where D s is singular, while D reg is finite (regular) at q, ω → 0, see e.g. Refs. 40,41. The singular contribution is governed by the diffusion mode and has the form [see Eq. (D.5)]
D s (ω, q; φ, φ ′ ) ≃ Ψ R (φ, q)Ψ L (φ ′ , q) Dq 2 − iω , (2.44) Ψ ν (φ, q) = 1 − ic (1) ν cos(φ − φ q ) − ic (2) ν sin(φ − φ q ), where D = v 2 F τ /2(1 + ω 2 c τ 2 )
is the diffusion constant in the presence of a magnetic field and
c (1) R (q) = c (1) L (q) = qv F τ 1 + ω 2 c τ 2 , (2.45) c (2) R (q) = −c (2) L (q) = qv F ω c τ 2 1 + ω 2 c τ 2 .
(2.46)
The leading-order contribution of the diagrams a, b and c (that containing two singular diffusons D s ) is exactly canceled by the part of the diagrams d and e with the structure D s n α D reg n β D s , i.e. with one regular part of the propagator inserted between two singular diffusons, D s = (Dq 2 − iω) −1 . Indeed, in view of n α D reg n β = 1 2 T αβ , the latter contribution reduces to − 1 2 D s 2 T αβ , while the diagrams a, b and c yield
1 2 D s 2 T αγ δ γδ τ + ω c ǫ γδ T δβ = 1 2 D s 2 T αβ ,(2.47)
where ǫ αβ is the antisymmetric tensor, ǫ xx = ǫ yy = 0, ǫ xy = −ǫ yx = 1. It remains thus to calculate only the contribution of the diagrams d + e with three singular diffusons,
δσ αβ = e 2 v 2 F 2π ∞ −∞ dω ∂ ∂ω ω coth ω 2T × d 2 q (2π) 2 Im D s n α D s n β D s 1 + iω D s ≃ 2e 2 v 2 F π(1 + ω 2 c τ 2 ) 2 1/τ T dω × d 2 q (2π) 2 Im (−iq α l)(−iq β l) Dq 2 (Dq 2 − iω) 2 = e 2 2π 2 ln(T τ )δ αβ ,(2.48)
in agreement with 1,10,11 . The result for a point-like interaction differs only by a factor νV 0 .
2. B → 0, white-noise disorder
We allow now for arbitrary T τ but consider the limit of zero magnetic field assuming a white-noise disorder (τ = τ s and W (n, n ′ ) = 1/2πντ ), which is the limit studied in Refs. 18,19. The contribution (2.37) of the diagrams a,b,c takes for the white-noise disorder the form 3. B = 0, ballistic limit T τ ≫ 1
B a+b+c αβ = 1 2 T αγ D D δ γδ τ + ω c DD ǫ γδ − 2 τ Dn γ n δ D + δ γδ D T δβ .
In the ballistic limit T τ ≫ 1 and for white-noise disorder the result of Ref. 18 (recovered in Sec. II C 2 and Appendix B) yields a linear-in-T conductivity correction, δσ = (2νV 0 e 2 /π)T τ for the point-like interaction and δσ = (e 2 /π)T τ for the Coulomb interaction. The question we address in this subsection is how this behavior depends on the nature of disorder [i.e. on the scattering cross-section W (φ)].
In order to get the T τ ballistic asymptotics, it is sufficient to keep contributions to (2.38) with a minimal number of scattering processes. Specifically, the propagator D in the first and the third terms of (2.38) can be replaced by the free propagator,
D f (ω, q; φ, φ ′ ) = 2πδ(φ − φ ′ ) −i(ω + i0) + iqv F cos(φ − φ q )
, (2.50) while in the second term it should be expanded up to the linear term in the scattering cross-section W [the second term produces then the same contribution as the first term in (2.38)]. The last (fourth) term in (2.38) does not contribute to the T τ asymptotics. We get therefore
B xx ≃ 2πντ 2 ( D f W D f − 2 D f n x W n x D f ) − 2τ n x D f n x D f . (2.51)
Let us consider first the case of a short-range interaction, U 0 (r) = V 0 . The structure of Eqs. (2.35), (2.51) implies that the interaction correction is governed by returns of a particle to the original point in a time t T −1 ≪ τ after a single scattering event. It follows that the coefficient in front of the linear-in-T term is proportional to the backscattering probability W (π) =w(2k F ),
δσ xx = 2νV 0 e 2 π 2πνW (π)T τ 2 . (2.52)
As shown in Appendix C, this result remains valid in the case of Coulomb interaction, with the factor 2νV 0 replaced by unity. This shows that in the ballistic limit the Coulomb interaction is effectively reduced to the statically screened form, U (r) = 1/2ν when the leading contribution to δσ xx is calculated. According to (2.52), in a smooth disorder with a correlation length d ≫ k −1 F the T τ contribution is suppressed by an exponentially small factor 2πντw(2k F ) ∼ e −kF d . In fact, for a smooth disorder the linear term represents the leading contribution for T τ s ≫ 1 only. In the intermediate range τ −1 ≪ T ≪ τ −1 s the dominant return processes are due to many small-angle scattering events. However, the corresponding return probability is also exponentially suppressed ∼ exp(−const τ /t) for relevant (ballistic) times t ≪ τ , yielding a contribution δσ xx ∼ exp[−const(T τ ) 1/2 ]. Thus, the interaction correction in the ballistic regime is exponentially small at B = 0 for the case of smooth disorder. Moreover, the same argument applies to the case of a non-zero B, as long as 43 ω c ≪ T . In any realistic system there will be a finite concentration of residual impurities located close to the electron gas plane and inducing large-angle scattering processes. In other words, a realistic random potential can be thought as a superposition of a smooth disorder with a transport time τ sm and a white-noise disorder characterized by a time τ wn . Neglecting the exponentially small contribution of the smooth disorder to the linear term, we then find that the ballistic asymptotics (2.52) of the interaction correction takes the form
δσ = e 2 π τ τ wn T τ × 2νV 0 , point − like, 1, Coulomb, (2.53) where τ −1 = τ −1 sm + τ −1
wn is the total transport scattering rate. If the transport is dominated by the smooth disorder, τ wn ≫ τ sm , the coefficient of the T τ term is thus strongly reduced as compared to the white-noise result of Ref. 18.
Finally, it is worth mentioning that in addition to the T τ term corresponding to the lower limit ω ∼ T of the frequency integral in (2.35), there is a much larger but T -independent contribution δσ ∝ E F τ governed by the upper limit ω ∼ E F . This contribution is just an interaction-induced Fermi-liquid-type renormalization of the bare (noninteracting) Drude conductivity.
III. STRONG B, SMOOTH DISORDER
A. Quasiclassical dynamics
We have shown in Sec. II C that due to small-angle nature of scattering in a smooth disorder the interaction correction is suppressed in the ballistic regime T τ ≫ 1 in zero (or weak) magnetic field. The situation changes qualitatively in a strong magnetic field, ω c τ ≫ 1 and ω c ≫ T . The particle experiences then within the time t ∼ T −1 multiple cyclotron returns to the region close to the starting point. The corresponding ballistic propagator satisfies the equation (2.8) with the collision term (2.9).
The solution of this equation in the limit of a strong magnetic field, ω c τ ≫ 1, is presented in Appendix D. For calculation of the leading order contribution to δσ xx and δρ xx , the following approximate form is sufficient:
D(ω, q; φ, φ ′ ) = exp[−iqR c (sin φ − sin φ ′ )] × χ(φ)χ(φ ′ ) Dq 2 − iω + n =0 e in(φ−φ ′ ) Dq 2 − i(ω − nω c ) + n 2 /τ , ≡ D s (ω, q; φ, φ ′ ) + D reg (ω, q; φ, φ ′ ), (3.1)
where χ(φ) = 1−iqR c cos φ/ω c τ and D ≃ R 2 c /2τ, and the polar angles of velocities are counted from the angle of q.
Equation (3.1) is valid under the assumption (qR c ) 2 ≪ ω c τ .
We will see below that the characteristic momenta q are determined by the condition Dq 2 ∼ ω ∼ T , so that the above assumption is justified in view of ω c ≫ T . Furthermore, this condition allows us to keep only the first (singular) term D s in square brackets in (3.1) when calculating D ,
D = J 2 0 (qR c ) Dq 2 − iω , (3.2)
where J 0 (z) is the Bessel function. Moreover, the formula (2.36) for B xx can be cast in a form linear in D by using
DD = −i ∂ ∂ω D , (3.3) n α Dn β D = i v F ∂ ∂q β n α D , (3.4) Dn x Dn x D = − 1 2v 2 F ∂ 2 ∂q 2 x D . (3.5)
Therefore, it is again sufficient to take into account only the first term in
B xx (ω, q) = J 2 0 (qR c ) (ω c τ ) 2 Dτ q 2 (Dq 2 − iω) 3 = 4τ 3 β 2 J 2 0 (Q)Q 2 (Q 2 − iΩ) 3 . (3.6)
In the second line we introduced dimensionless variables Q = qR c , Ω = 2ωτ, β = ω c τ . Note that Eqs. (3.2), (3.6) differ from those obtained in the diffusive regime by the factor J 2 0 (qR c ) only. This is related to the fact that the motion of the guiding center is diffusive even on the ballistic time scale t ≪ τ (provided t ≫ ω −1 c ), while the additional factor corresponds to the averaging over the cyclotron orbit (see Sec. IV below).
We turn now to the calculation of B xy . Substituting (3.1) in (2.36), we classify the obtained contributions according to powers of the small parameter 1/β. The leading contributions are generated by the first and the last terms in (2.36) and are of order 1/β, i.e. larger by factor β as compared to B xx , Eq. (3.6). (This extra factor of β is simply related to |σ xy |/σ xx = β.) However, these leading contributions cancel,
T xy 2 DD − Dn x Dn y D order 1/β = τ 2β D s D s − D s n x D reg n y D s (3.7) = − 2τ 3 β J 2 0 (Q) (Q 2 − iΩ) 2 + 2τ 3 β J 2 0 (Q) (Q 2 − iΩ) 2 = 0,
as in the diffusive limit, see the text above Eq. (2.47).
To evaluate terms of higher order in 1/β, we need a more accurate form of the propagator (3.1). Since the contributions of order 1/β 2 to B xy turn out to cancel as well, we have to know the propagator with the accuracy allowing to evaluate the terms of order 1/β 3 . To simplify the calculation, we use again the identities (3.3) and (3.4). As to Eq. (3.5), it cannot be generalized onto xy-component of the tensor, and we use instead
Dn x Dn y D = i v F ∂D ∂q x n y D . (3.8)
It is then sufficient to calculate the propagator D up to the 1/β 2 order. This is done in Appendix D, see Eqs. (D.14)-(D.17). Substituting this result for D in Eq. (3.6) and combining all terms, we get after some algebra
B xy (ω, q) = − τ 3 β 3 7Q 2 J 2 0 (Q) + 4QJ 0 (Q)J 1 (Q) (Q 2 − iΩ) 2 + 4QJ 0 (Q)J 1 (Q)) Q 2 − iΩ . (3.9)
We see that similarly to (3.6) the kernel B xy (ω, q) has a diffusive-type structure with Q 2 − iΩ in denominator reflecting the diffusion of the guiding center, while the Bessel functions describe the averaging over the cyclotron orbit. Clearly, both kernels (3.6) and (3.9) vanish at q = 0, as required by (2.43).
B. Point-like interaction
To find the interaction correction to the conductivity, we have to substitute Eqs. (3.6) and (3.9) in the formula (2.35). We consider first the simplest situation, when the interaction U (ω, q) in (2.35) is of point-like character, U (ω, q) = V 0 . Using v 2 F qdq = ω 2 c QdQ, we see that all the B-dependence drops out from δσ xx , and the exchange contribution reads
δσ xx = −8e 2 νV 0 ∞ 0 dΩ 2π ∂ ∂Ω Ω coth Ω 4T τ × ∞ 0 QdQ 2π Im J 2 0 (Q)Q 2 (Q 2 − iΩ) 3 .
(3.10)
To simplify the result (3.10), it is convenient to perform a Fourier transformation with respect to Ω (which corresponds to switching to the time representation)
Im ∞ 0 dω 2π F (ω) ∂ ∂ω ω coth ω 2T = ∞ 0 dt πT 2 t sinh 2 (πT t)F (t). (3.11)
The integral over Q is then easily evaluated, yielding
δσ xx = − e 2 2π 2 νV 0 G 0 (T τ ), (3.12) G 0 (x) = π 2 x 2 ∞ 0 du exp(−1/u) sinh 2 (πxu) (3.13) × [(u − 1)I 0 (1/u) + I 1 (1/u)] ,
where I 0 (z) and I 1 (z) are modified Bessel functions. The Hartree term in this case is of the opposite sign and twice larger due to the spin summation (we neglect here the Zeeman splitting and will return to it later). It follows from Eqs. (3.6) and (3.9) that the correction to the Hall conductivity is smaller by the factor (ω c τ ) −1 as compared to (3.12). This implies, according to (2.41) that in a strong magnetic field the correction to the longitudinal resistivity is governed by δσ xx ,
δρ xx ρ 0 = (ω c τ ) 2 δσ xx σ 0 , (3.14)
similarly to the diffusive limit (1.3). In fact, it turns out that the relation (3.14) holds in a strong magnetic field, ω c ≫ T, for arbitrary disorder and interaction, see below.
On the other hand, as is seen from (2.42), contributions of both δσ xx and δσ xy to δρ xy are of the same order in (ω c τ ) −1 . We will return to the calculation of δρ xy in Sec. III G. The MR ρ xx (B) is thus quadratic in B, with the temperature dependence determined by the function G 0 (T τ ), which is shown in Fig. 4a. In the diffusive (x ≪ 1) and ballistic (x ≫ 1) limits the function G 0 (x) has the following asymptotics
G 0 (x) ≃ − ln x + const, x ≪ 1, c 0 x −1/2 , x ≫ 1,(3.15)
with (here ζ(z) is the Riemann zeta-function). Let us note that the crossover between the two limits takes place at numerically small values T τ ∼ 0.1 (a similar observation was made in Refs. 18,19). This can be traced back to the fact that the natural dimensionless variable in (3.12) is 2πT τ .
c 0 = 3ζ(3/2) 16 √ π ≃ 0.276 (3.16)
C. Coulomb interaction, exchange
For the case of the Coulomb interaction the result turns out to be qualitatively similar. Substituting (3.2) in (2.14) and neglecting the first term q ∼ (T /D) 1/2 ≪ κ in the denominator of (2.14), we obtain the effective in-
teraction U (ω, q) = 1 2ν Q 2 − iΩ Q 2 − iΩ[1 − J 2 0 (Q)]
.
(3.17)
Inserting (3.17) and (3.6) into (2.35), we get the exchange (Fock) contribution
δσ F xx = − e 2 π 2 ∞ 0 dΩ ∂ ∂Ω Ω coth Ω 4T τ × Im ∞ 0 QdQ Q 2 J 2 0 (Q) (Q 2 − iΩ[1 − J 2 0 (Q)])(Q 2 − iΩ) 2 . (3.18)
Using (3.14) we find the MR
δρ F xx (B) ρ 0 = − (ω c τ ) 2 πk F l G F (T τ ), (3.19) G F (x) = 32π 2 x 2 ∞ 0 dQQ 3 J 2 0 (Q) (3.20) × ∞ n=1 n(12πxn[1 − J 2 0 (Q)] + [3 − J 2 0 (Q)]Q 2 ) (4πxn + Q 2 ) 3 (4πxn[1 − J 2 0 (Q)] + Q 2 ) 2 .
Note that in contrast to the case of a point-like interaction, a transformation to the time representation does not allow us to simplify (3.18), since the resulting Qintegral can not be evaluated analytically. We have chosen therefore to perform the Ω-integration, which results in an infinite sum (3.20). This amounts to returning to the Matsubara (imaginary frequency) representation and is convenient for the purpose of numerical evaluation of G F (x). In the diffusive (x ≪ 1) and ballistic (x ≫ 1) limits this function has the asymptotics
G F (x) ≃ − ln x + const, x ≪ 1, c 0 2 x −1/2 , x ≫ 1, (3.21)
and is shown in Fig. 4b.
D. Coulomb interaction, Hartree contribution
We turn now to the Hartree contribution. The corresponding diagrams can be generated in a way similar to exchange diagrams (Sec. II A) but in this case one should start from two electron bubbles connected by an interaction line. There are again two distinct ways to generate a skeleton diagram: two current vertices can be inserted either both in the same bubble or in two different bubbles. Then one puts signs of Matsubara frequencies in all possible ways and insert ballistic diffusons correspondingly. The obtained set of diagrams is shown in Fig. 5 There is one-to-one correspondence between these Hartree diagrams and the exchange diagrams of Fig. 1, which is reflected in the labeling of diagrams.
As seen from comparison of Figs. 1 and 5, the electronic part B µ αβ (φ, φ ′ ) of each Hartree diagram is identical to that of its exchange counterpart. The only difference is in the arguments of the interaction propagator,
+ + + + + + + _ _ _ _ _ _ _ _ _ _ + + + + + + + _ + _ + + _ + _ a b c d e f g hU (ω, q) → U (0, 2k F sin φ−φ ′ 2 )
, where φ and φ ′ are polar angles of the initial and final velocities [cf. Eqs. (2.11), (2.12)]. Therefore, in the first order in the interaction, the Hartree correction to conductivity has a form very similar to the exchange correction (2.35),
δσ H αβ = 4e 2 v 2 F ν ∞ −∞ dω 2π ∂ ∂ω ω coth ω 2T × d 2 q (2π) 2 dφ 2π dφ ′ 2π (3.22) × Im [ U H (φ, φ ′ )B αβ (ω, q; φ, φ ′ ) ] , where U H (φ, φ ′ ) = U 0, 2k F sin φ − φ ′ 2 (3.23)
is the Hartree interaction and B αβ (ω, q; φ, φ ′ ) is given by Eqs. (2.36), (2.38) without angular brackets (denoting integration over φ and φ ′ ), see Eq. (2.12). Clearly, for a point-like interaction U (ω, q) = V 0 this yields
δσ H αβ = −2δσ F αβ ,(3.24)
as has already been mentioned in Sec. III B.
In the case of the Coulomb interaction the situation is, however, more delicate 44 . To analyze this case, it is convenient to split the interaction into the singlet and triplet parts 1,18,44 . For the weak interaction, κ ≪ k F , the conductivity correction in the triplet channel is then given by Eq. (3.22) with an extra factor 3 4 . As to the singlet part, it is renormalized by mixing with the exchange term. The effective interaction U s in the singlet channel is therefore determined by the equation
U s (φ, φ ′ ) = U 0 − 1 2 U H (φ, φ ′ ) − dφ 1 2π dφ 2 2π × [U 0 − 1 2 U H (φ, φ 1 )] P(φ 1 , φ 2 )U s (φ 2 , φ ′ ), (3.25)
where U 0 = 2πe 2 /q is the bare Coulomb interaction, and
P(ω, q; φ 1 , φ 2 ) = 2ν[2πδ(φ 1 − φ 2 ) + iωD(ω, q; φ 1 , φ 2 )] (3.26)
describes the electronic bubble. Solving (3.25) to the first order in U H , we get
U s (ω, q; φ, φ ′ ) = U (ω, q) − U s H (ω, q; φ, φ ′ ), (3.27)
where U (ω, q) is the RPA-screened Coulomb interaction (2.14) which has already been considered in Sec. III C, while the second term describes the renormalized Hartree interaction in the singlet channel,
U s H (φ, φ ′ ) = 1 2 U H (φ, φ ′ ) − 1 2Π dφ 1 2π dφ 2 2π × [U H (φ, φ 1 )P(φ 1 , φ 2 ) + P(φ 1 , φ 2 )U H (φ 2 , φ ′ )] + 1 2Π 2 dφ 1 2π dφ 2 2π dφ 3 2π dφ 4 2π × P(φ 1 , φ 2 )U H (φ 2 , φ 3 )P(φ 3 , φ 4 ).
(3.28)
Here Π = P is the polarization operator (2.15), and we have used the singular nature of the bare Coulomb interaction implying |Π|U 0 ≫ 1 for all relevant momenta. Taking into account that the angular dependence of leading contributions to B xx (ω, q; φ, φ ′ ) and D(ω, q; φ, φ ′ ) is of the form exp[−iQ(sin φ − sin φ ′ )], we find that the singlet Hartree correction to σ xx is given by Eq. (3.22) with a replacement
U H (φ, φ ′ ) → U H (φ, φ ′ ) − U H (φ, φ ′ ) 4 [1 + iω D(ω, q) ] 2 . (3.29)
Note that in the diffusive limit B αβ is independent of φ, φ ′ , so that only the zero angular harmonic of the interaction contributes. On the other hand, the zero angular harmonic is suppressed in the effective singlet-channel interaction (3.29). Therefore, the singlet channel does not contribute to the Hartree correction in the diffusive limit, in agreement with Refs. 1,44. The situation changes, however, in the ballistic regime, when B αβ becomes angle-dependent. After the angular integration, the triplet Hartree conductivity correction takes the form (3.10) with the replacement V 0 → 1/2ν, and
J 2 0 (Q) → −3y π 0 dφ 2π J 0 (2Q sin φ) y + 2 sin φ ,(3.30)
where y = κ/k F . For the singlet part we have a result similar to (3.18) with a slightly different Q-integral,
∞ 0 QdQ J (y, Q)Q 2 (Q 2 − iΩ[1 − J 2 0 (Q)]) 2 (Q 2 − iΩ) , where J (y, Q) = −y π 0 dφ 2π J 0 (2Q sin φ) − J 2 0 (Q) y + 2 sin φ . (3.31)
This yields for the total Hartree contribution
δρ H xx (B) ρ 0 = (ω c τ ) 2 πk F l [G s H (T τ, y) + 3G t H (T τ, y)], (3.32)
where G s H and G t H governing the temperature dependence of the singlet and triplet contributions have the form
G s H (x, y) = 32π 2 x 2 ∞ 0 dQQ 3 J (y, Q) × ∞ n=1 n(12πxn[1 − J 2 0 (Q)] + [3 − 2J 2 0 (Q)]Q 2 ) (4πxn + Q 2 ) 2 (4πxn[1 − J 2 0 (Q)] + Q 2 ) 3 , (3.33) G t H (x, y) = πx 2 y 4 ∞ 0 du sinh 2 (πxu) × π 0 dφ exp[−2 sin 2 φ/u] y + 2 sin φ u − 2 sin 2 φ . (3.34) Figure 6a shows G H (x, y) = G s H (x, y) + 3G t H (x, y) as a function of x ≡ T τ for several values of y ≡ κ/k F . The asymptotic behavior of δρ H xx is as follows: δρ H xx (B) ρ 0 = (ω c τ ) 2 πk F l × y ln y [ 3 4 ln(T τ ) + ln y], T τ ≪ 1, y ln 2 [ y (T τ ) 1/2 ], 1 ≪ T τ ≪ 1/y 2 , πc 0 (T τ ) −1/2 , T τ ≫ 1/y 2 ,(3.35)
We see that at κ/k F ≪ 1 a new energy scale T H ∼ τ −1 (k F /κ) 2 arises where the MR changes sign. Specifically, at T ≪ T H the MR, δρ xx = δρ F xx + δρ H xx , is dominated by the exchange term and is therefore negative, while at T ≫ T H the interaction becomes effectively point-like and the Hartree term wins, δρ H xx = −2δρ F xx , leading to a positive MR with the same (T τ ) −1/2 temperature-dependence, see Fig. 6a.
E. Hartree contribution for a strong interaction
In Sec. III D we have assumed that κ/k F ≪ 1, or, in other words, the interaction parameter r s = √ 2e 2 /εv F (where ε is the static dielectric constant of the material) is small. This condition is, however, typically not met in experiments on semiconductor structures. If κ/k F is not small, the exchange contribution (3.19) remains unchanged, while the Hartree term is subject to strong Fermi-liquid renormalization 1,18 and is determined by angular harmonics F σ,ρ m of the Fermi-liquid interaction F σ,ρ (θ) in the triplet (σ) and singlet (ρ) channels.
The effective interaction U σ,ρ
eff replacing U H (φ, φ ′ ) in (3.22) is then given by an equation of the type (3.25) but with F σ,ρ (φ − φ ′ )/ν substituted for U H (φ, φ ′ ) (and without U 0 in the triplet channel), U ρ eff (φ, φ ′ ) = U 0 + F ρ (φ − φ ′ ) 2ν − dφ 1 2π dφ 2 2π × U 0 + F ρ (φ − φ 1 ) 2ν P(φ 1 , φ 2 )U ρ eff (φ 2 , φ ′ ), (3.36) U σ eff (φ, φ ′ ) = F σ (φ − φ ′ ) 2ν − dφ 1 2π dφ 2 2π × F σ (φ − φ 1 ) 2ν P(φ 1 , φ 2 )U σ eff (φ 2 , φ ′ ), (3.37)
A general solution of these equations requires inversion of integral operators with the kernels I − F σ P and I − (U 0 + F ρ )P and is of little use for practical purposes. The situation simplifies, however, in both diffusive and ballistic limits.
In the diffusive regime, T ≪ 1/τ , the second term in the polarization bubble (3.26) and B αβ are independent of angles φ, φ ′ . As discussed in Sec. III D, this leads to the suppression of the Hartree contribution in the singlet channel, while in the triplet channel only the zero angular harmonic survives,
U σ eff (ω, q) = 1 2ν F σ 0 (Dq 2 − iω) (1 + F σ 0 )Dq 2 − iω (3.38)
We then reproduce the known result 1,
18 G H (T τ ) = 3G t H (T τ ) with G t H (T τ ) = 1 − ln(1 + F σ 0 ) F σ 0 ln T τ.
(3.39)
In the ballistic limit, T ≫ 1/τ , the first term is dominant in (3.26), since D is suppressed by a factor J 2 0 (Q) ≪ 1, according to (3.2). The angular harmonics then simply decouple in Eqs. (3.36)
, (3.37), yield- ing effective Hartree interaction constants U ρ 0,eff = 0, U ρ m,eff = (2ν) −1 F ρ m /(1 + F ρ m ), m = 0, and U σ m,eff = (2ν) −1 F σ m /(1 + F σ m )
. Therefore, the Hartree contribution reads
G H (T τ ) = − c 0 2 m =0 F ρ m 1 + F ρ m + 3 m F σ m 1 + F σ m 1 √ T τ .
(3.40) From a practical point of view, it is rather inconvenient to describe the interaction by an infinite set of unknown parameters F σ,ρ m . For this reason, one often assumes that the interaction is isotropic and thus characterized by two coupling constants F σ 0 and F ρ 0 only. Within this frequently used (though parametrically uncontrolled) approximation, the singlet part of the Hartree term is completely suppressed. The Hartree contribution is then determined solely by the triplet channel with the effective interaction
U σ eff (ω, q) = 1 2ν F σ 0 1 + F σ 0 Q 2 − iΩ Q 2 − iΩ 1 − F σ 0 1 + F σ 0 J 2 0 (Q)
.
(3.41) The Hartree correction to the resistivity takes the form of Eq. (3.19) with an additional overall factor of 3 and with J 2 0 (Q) multiplied by F σ 0 /(1 + F σ 0 ),
J 2 0 (Q) → J 2 0 (Q) F σ 0 1 + F σ 0 ≡ 1 − J σ (Q) ,(3.42)
everywhere in (3.20); the result is shown in Fig. 6b for several values of F σ 0 .
F. Effect of Zeeman splitting
Until now we assumed that the temperature is much larger than the Zeeman splitting E Z , T ≫ E Z . In typical semiconductor structures this condition is usually met in non-quantizing magnetic fields in the ballistic range of temperatures, allowing one to neglect the Zeeman term. If, however, this condition is violated, T E Z , the Zeeman splitting suppresses the triplet contributions with the z-projection of the total spin S z = ±1, while the triplet with S z = 0 and singlet parts remain unchanged.
In the case of a weak interaction, κ/k F ≪ 1, the triplet contribution 3G t H (T τ, κ/k F ) in Eq.(3.32) is modified in the following manner,
3G t H (x, y) → G t H (x, y) + 2ReG t H (x, y; ǫ z ),(3.43)
where ǫ z = 2τ E Z , and the functionG t H (x, y; ǫ z ) describing the temperature dependence of the contribution with ±1 projection of the total spin is given bỹ
G t H (x, y; ǫ z ) = πx 2 y 4 ∞ 0 du exp[iǫ z u] sinh 2 (πxu) × π 0 dφ exp[−2 sin 2 φ/u] y + 2 sin φ u − 2 sin 2 φ .
(3. 44) We see that at T τ ≪ ǫ z , the contributions of ±1components of the triplet saturate at the value given by (3.34) with a replacement T τ → ǫ z , i.e. at ∼ G t H (ǫ z , y). In the opposite limit, T τ ≫ ǫ z , we haveG t H (x, y; ǫ z ) ≃ G t H (x, y), and the result (3.32) is restored. The triplet contribution for strong isotropic interaction (i.e. determined by F σ 0 only) in the presence of Zeeman splitting reads
δρ H xx (B) ρ 0 = − (ω c τ ) 2 πk F l [G σ H (T τ, 0) + 2Re G σ H (T τ, ǫ z )].
(3.45)
The function G σ H (T τ, ǫ z ) is given by a formula similar to (3.20),
G σ H (x, ǫ z ) = 32π 2 x 2 ∞ 0 dQQ 3 [1 − J σ (Q)] × ∞ n=1 n(12πxnJ σ (Q) + [2 + J σ (Q)][Q 2 + iǫ z ]) (4πxn + [Q 2 + iǫ z ]) 3 (4πxnJ σ (Q) + [Q 2 + iǫ z ]) 2 , (3.46)
with J σ (Q) as defined in (3.42). Again, for high temperatures T τ ≫ ǫ z , all the triplet components contribute, so that the overall factor of 3 (as in the absence of the Zeeman splitting) restores. On the other hand, for T τ ≪ ǫ z , the contributions with ±1 projection of the spin saturate at low temperatures, and therefore the triplet contribution is partly suppressed, see Fig. 7.
G. Hall resistivity
As discussed in Sec. III B, calculation of the correction δρ xy to the Hall resistivity requires evaluation of both δσ xx and δσ xy . In fact, as we show below, the temperature dependence of δρ xy in a strong magnetic field is governed by δσ xx in the diffusive limit and by δσ xy in the ballistic limit.
Since δσ xx has been studied above, it remains to calculate δσ xy . Using the result (3.9) for the corresponding kernel B xy , we get the exchange contribution for the case of a point-like interaction
δσ xy = − e 2 2π 2 νV 0 ω c τ G (xy) 0 (T τ ) + G (xy) UV ,(3.47)
where the temperature dependence of the correction is governed by the function
G (xy) 0 (x) = −π 2 ∞ 0 du u exp(−1/u) × x 2 sinh 2 (πxu) − 1 (πu) 2 (3.48) × [ (9u − 3)I 0 (1/u) + (3 − 2u)I 1 (1/u) ] .
When writing (3.48), we subtracted a temperature independent but ultraviolet-divergent (i.e. determined by the upper limit in frequency integral) contribution G
UV ; we will return to it in the end of this subsection.
The function G (xy) 0 (x) has the following asymptotics:
G (xy) 0 (x) ≃ 9πx, x ≪ 1, 11c 1 x 1/2 , x ≫ 1,(3.δρ xy ρ xy = νV 0 πk F l G ρxy 0 (T τ ),(3.51)
where
G ρxy 0 (x) = 2G 0 (x) − G (xy) 0 (x) ≃ −2 ln x + const, x ≪ 1, −11c 1 x 1/2 , x ≫ 1. (3.52)
The function G ρxy 0 (x) is shown in Fig. 8. As usual, the Hartree term in the case of point-like interaction has an opposite sign and is twice larger in magnitude, if the Zeeman splitting can be neglected.
An analogous consideration for the Coulomb interaction yields a similar result for the exchange correction
δρ F xy ρ xy = G ρxy F (T τ ) πk F l , (3.53) G ρxy F (x) = 2G F (x) − G (xy) F (x) ≃ −2 ln x + const, x ≪ 1, − 11 2 c 1 x 1/2 , x ≫ 1. (3.54)
The function G (x). To analyze the Hartree contribution, we restrict ourselves to the isotropic-interaction approximation. Then, similarly to the consideration in the end of Sec. III E, only the triplet part contributes, and, in order to calculate G In the diffusive limit the Hartree correction to the Hall resistivity is determined by (3.39), while in the ballistic limit we have again effectively point-like interaction We return now to the T -independent contribution G (xy) UV that was subtracted in Eq. (3.48). In view of the divergency of this term at u → 0, it is determined by the short-time cut-off u min = t min /2τ
with νV 0 = 3 2 F σ 0 /(1 + F σ 0 ), implying that G (xy) H (x) ≃ −3G (xy) 0 (x)F σ 0 /2(1 + F σ 0 ). This yields δρ H xy ρ xy = − G ρxy H (T τ ) πk F l , (3.55) G ρxy H (x) ≃ 3 × 2 1 − ln(1 + F σ 0 ) F σ 0 ln x, x ≪ 1, 11 2 c 1 F σ 0 1 + F σ 0 x 1/2 , x ≫ 1.G (xy) UV ∝ umin du u 3/2 ∼ u −1/2 min .
(3.57)
Since the correction we are discussing is governed by cyclotron returns, the cut-off t min corresponds to a single cyclotron revolution, u min ∼ π/ω c τ . [On a more formal level, this is related to the assumption ω ≪ ω c used for derivation of (3.48); see the text below Eq. (3.1).] We have, therefore, G
UV = c (xy) (ω c τ ) 1/2 , with a constant c (xy) of order unity 45 . For the point-like interaction, the considered term produces a temperature-independent correction to the Hall resistivity of the form
δρ xy ρ xy = νV 0 c (xy) πk F l (ω c τ ) 1/2 . (3.58)
In the case of Coulomb interaction, this correction (with both, exchange and Hartree, terms included) has the same form with
νV 0 → 1 2 [1 − 3F σ 0 /(1 + F σ 0 )
]. Finally, let us discuss the expected experimental manifestation of the results of this subsection. Equations (3.54), (3.55) predict that in the presence of interaction the temperature-dependent part of the Hall resistivity ρ xy (B) in a strong magnetic field ω c ≫ τ −1 , T is linear in B at arbitrary T , with the T -dependence crossing over from ln T in the diffusive regime to T 1/2 in the ballistic regime. More specifically, if the interaction is not too strong, the exchange contribution (3.54) wins and the slope decreases with increasing temperature, while in the limit of strong interaction the slope increases due to the Hartree term (3.55). In an intermediate range of F σ 0 the slope is a non-monotonous function of temperature. Surprisingly, this behavior of the slope of the Hall resistivity is similar to the behavior of σ xx obtained in Ref. 18 for B = 0 and white-noise disorder. This is a very non-trivial similarity, since the correction to ρ xy at weak fields 19 shows a completely different behavior, vanishing as T −1 in the ballistic regime. In addition to the temperaturedependent linear-in-B contribution, the interaction gives rise to a T -independent correction (3.58), which scales as δρ xy ∝ B 3/2 (assuming again that ω c ≫ τ −1 , T ) Let us recall that these results are governed by multiple cyclotron returns and thus are valid under the assumption ω c ≫ T . In the opposite case, ω c ≪ T, the correction is suppressed in the ballistic regime (similarly to δρ xx , see Secs. II B3 and III A), and the Hall resistance takes its Drude value.
IV. QUALITATIVE INTERPRETATION: RELATION TO RETURN PROBABILITY
It was argued in Ref. 46 by using the Gutzwiller trace formula and Hartree-Fock approximation that the interaction correction to conductivity is related to a classical return probability. The aim of this section is to demonstrate how this relation follows from the explicit formulas for σ xx .
We begin by considering the case of smooth disorder, when the kernel B xx (ω, q) is given by Eq. (2.36). For simplicity, we will further assume a point-like interaction, when only the first two terms in (2.36) give nonzero contributions. In fact, we know that the result for the Coulomb interaction is qualitatively the same [cf. Eqs. (3.15) and (3.21)].
We will concentrate on the first term in (2.36); the second one yields a contribution of the same order in the ballistic regime and is negligible in the diffusive limit. Therefore, for the purpose of a qualitative discussion it is sufficient to consider the first term. Using (3.3), the corresponding contribution can be estimated as
δσ xx σ xx ∼ V 0 ∞ −∞ dω ∂ ∂ω ω coth ω 2T (dq)Re ∂ D(ω, q) ∂ω ∼ V 0 ∞ 0 dt (πT ) 2 t sinh 2 (πT t) t D(r = 0, t) ∼ V 0 T −1 0 dt D(r = 0, t) ,(4.1)
where σ xx is the Drude conductivity in magnetic field and we performed in the second line the Fourier transformation of D to the coordinate-time representation (3.11).
The return probability in a strong magnetic field, ω c τ ≫ 1, is shown schematically in Fig. 9. In the diffusive time range, t ≫ τ , it is given by R(t) = 1/4πDt (where D is the diffusion constant in the magnetic field, D ≃ R 2 c /2τ ). Equation (4.1) thus yields in the diffusive regime, T τ ≪ 1,
R(t) ≡ D(r = 0, t) ,(4.δσ xx σ xx ∼ V 0 D ln(T τ ),(4.3)
in agreement with (1.2), (2.48). At short (ballistic) time, t ≪ τ , the return probability is governed by multiple cyclotron returns after n = 1, 2, . . . revolutions,
R(t) = n ω c τ 4 √ 3π 2 nR 2 c exp − [t − 2πn/ω c ] 2 ω 3 c τ 12πn
.
(4.4) Since T ≪ ω c , the conductivity correction (4.1) is in fact determined by the smoothened return probability,
R(t) = 1 (2π) 3/2 1 R 2 c τ t 1/2 .
(4.5)
Substituting (4.5) in (4.1) we find that in the ballistic limit, T τ ≫ 1, the conductivity correction scales as
δσ xx σ xx ∼ V 0 D (T τ ) −1/2 ,(4.6)
in agreement with the exact results (3.12), (3.15). As to the diffusive regime, T τ ≪ 1, the contribution of short times t τ to the integrand in (4.1) yields a subleading T -independent correction ∼ V 0 /D to (4.3).
It is worth emphasizing that the ballistic behavior (4.5) of the return probabilityR(t) corresponds to a onedimensional diffusion. Consequently the ballistic result (4.6) has the same form as the diffusive Altshuler-Aronov correction in the quasi-one-dimensional geometry. To clarify the reason for emergence of the one-dimensional diffusion, we illustrate the dynamics of a particle subject to a strong magnetic field and smooth disorder in Fig. 10 Let us assume that the velocity is in y direction at t = 0. As is clear from Fig. 10, the return probability R 1 after the first cyclotron revolution (the integral of the first peak in Fig. 9) is determined by the shift δ x of the guiding center in the cyclotron period t 1 = 2π/ω c . In view of the diffusive dynamics of the guiding center, this shift has a Gaussian distribution with
δ 2 1 ≡ δ 2 x = 2Dt 1 = 2πR 2 c ω c τ ,(4.7)
yielding
R 1 = (ω c τ ) 1/2 2πv F R c . (4.8)
Furthermore, we have δ 2 x = nδ 2 1 after n revolutions, yielding the return probability R n = R 1 / √ n. As to the y-component δ y of the guiding center shift, it only governs the width of the peaks in Eq. (4.4) and Fig. 9 without affectingR(t). Therefore, the smoothened return probability isR
(t) = R n t 1 n=t/t1 ,(4.9)
which reproduces Eq. (4.5).
As mentioned in Sec. III A, the emergence of the onedimensional diffusion in the ballistic regime is reflected by the factor J 2 0 (Q) ∼ 1/πQ in the formula (3.6) for the kernel B xx (ω, q). This factor effectively reduces the dimensionality of the q-integral, d 2 q → R −1 c dq.
In the above we considered a system with smooth disorder, for which δσ xx at B = 0 vanishes exponentially in the ballistic limit. Now we turn to the opposite case of a white-noise disorder. We will show that the linear-in-T correction 17,18 (Sec. II C3) is again related to the return probability but the relation is different from (4.1). Indeed, according to (2.49), we have now the structure D D instead of DD that was relevant for smooth disorder. On the other hand, the return probability at ballistic times t ≪ τ is clearly dominated by processes with a single back-scattering event, implying
D(r = 0, t) ∼ 1 τ (dq)dω D f (ω, q) 2 e iωt .
(4.10)
Therefore, the contribution of the first term in (2.49) can be cast in the form
δσ xx σ xx ∼ V 0 τ T −1 E −1 F dt t D(r = 0, t) , ∼ V 0 τ [ const − D(0, t ∼ T −1 ) ]. (4.11)
It is easy to see that the probability of a ballistic return after a single scattering event is
D(r = 0, t) ∼ 1 τ d 2 r 1 δ(t − 2r 1 /v F ) (v F r 1 ) 2 ∼ 1 v 2 F τ t . (4.12)
Substituting (4.12) in (4.11), we reproduce the linear-in-T correction (2.52),
δσ xx (T ) ∼ e 2 νV 0 T τ (4.13)
The constant term in (4.11) comes from the lower limit of the time integral, which is of the order of E −1 F . This constant merely renormalizes the bare value of the Drude conductivity.
On the diffusive time scale D D ≃ DD , so that there is no difference between white-noise and smooth disorder. Therefore, in the diffusive limit the result (4.1) applies, yielding the usual logarithmic correction (4.3). In fact the contribution of the type (4.1) arises also in the ballistic regime when all terms in (2.38) are taken into account. According to (4.12), it has the form
δσ xx σ xx ∼ V 0 D [ ln(T τ ) − const],(4.14)
which is a subleading correction to the linear-in-T term (2.52), (4.13).
In the ballistic regime, T τ ≫ 1, the above qualitative arguments for a white-noise disorder can be reformulated in terms of the interaction-induced renormalization of the differential scattering cross-section on a single impurity. Specifically, the renormalization occurs due to the interference of two waves, one scattered off the impurity and another scattered off the Friedel oscillations created by the impurity 18,47 . The interference contribution is proportional to the probability W (π) of backscattering off the impurity (see Appendix C) and hence, to the return probability after a single-scattering event, as discussed above.
On the other hand, this implies that the scattering cross-section around φ ∼ π is itself modified by the Friedel oscillations (in other words, the impurities are seen by electrons as composite scatterers with an anisotropic cross-section). The renormalization of the bare impurity depends on the energy of the scattered waves, which after the thermal averaging translates into the T -dependence of the effective transport scattering time, 18 τ (T ) [this corresponds to setting t ∼ T −1 in the return probability, see Eq. (4.11)]. This mechanism provides a systematic microscopic justification of the concept of temperature-dependent screening 17 .
We recall that, in addition to the linear-in-T term, the conductivity correction contains a T -independent contribution determined by the ultraviolet frequency cutoff ∼ E F . In the case of strong interaction this term can be of the same order as the bare (non-interacting) Drude conductivity. The coefficient in front of this term cannot be calculated within the quasiclassical approach because it is governed by short-distance physics at scales of the order of λ F . At the same time, according to the above picture, this T -independent correction also modifies the impurity scattering cross-section around φ = π. The corresponding correction δW (φ) may thus be comparable to the bare isotropic scattering probability W 0 . An interesting consequence of this fact is a possible situation when the total relaxation rate
τ −1 s ∝ dφ[W 0 + δW (φ)] is smaller than the transport relaxation rate τ −1 ∝ dφ[W 0 + δW (φ)](1 − cos φ).
In smooth disorder (small-angle scattering), the backscattering amplitude vanishes exponentially with k F d, and so does the amplitude of Friedel oscillations. This leads to the suppression of the T τ -contribution to the conductivity [see Sec. II C3; this fact was realized within the T -dependent screening picture already in Ref. 17 for the case of scattering on long-range interface roughness]. We note, however, that the understanding of the interaction effects in terms of scattering off Friedel oscillations is only possible in the ballistic regime. Indeed, the diffusive correction in a smooth random potential is not exponentially small and is related to random (having no 2k F -oscillating structure) fluctuations of the electron density, as was pointed out in Refs. 1,46. The correlations of these fluctuations (which reduce to the Friedel oscillations on the ballistic scales) are described by the return probability at arbitrary scales.
Finally, we use the interpretation of the interaction correction in terms of return probability to estimate the MR in the white-noise random potential and at sufficiently weak magnetic fields, ω c ≪ T . Note that the zero-B ballistic correction (4.13) does not imply any dependence of resistivity on magnetic field. Indeed, as follows from (1.1), a temperature dependence of the transport time τ (T ) is not sufficient to induce any non-trivial MR,
∆ρ xx (B, T ) ≡ ρ xx (B, T ) − ρ xx (0, T ) = 0, if τ is B-independent.
In order to obtain the B-dependence of the resistivity, we thus have to consider the influence of the magnetic field on the return probability determining the correction to the transport time. Since in the ballistic regime the characteristic length of relevant trajectories is L ∼ v F /T ≪ l, their bending by the magnetic field modifies only slightly the return probability for ω c ≪ T . The relative correction to the return probability is thus of the order of (L/R c ) 2 ∼ ω 2 c /T 2 independently of the relation between ω c and τ −1 . Therefore, to estimate the MR in the white-noise potential for ω c , τ −1 ≪ T , one can simply multiply the result (4.13) for B = 0 by a factor (ω c /T ) 2 , yielding
∆ρ xx ρ 0 ∼ (ω c τ ) 2 k F l 1 T τ , ω c ≪ T. (4.15)
A formal derivation of this result is presented in Sec. V B.
In a stronger magnetic field, ω c ≫ T , the situation changes dramatically due to multiple cyclotron returns, see above. This regime is considered in Sec. V A below.
V. MIXED DISORDER MODEL
A. Strong B
In Sec. III, we studied the interaction correction for a system with a small-angle scattering induced by smooth disorder with correlation length d ≫ k −1 F . This is a typical situation for high-mobility GaAs structures with sufficiently large spacer d. It is known, however, that with further increasing width of the spacer the large-angle scattering on residual impurities and interface roughness becomes important and limits the mobility. Furthermore, in Si-based structures the transport relaxation rate is usually governed by scattering on short-range impurities.
This motivates us to analyze the situation when resistivity is predominantly due to large-angle scattering. We thus consider the following two-component model of disorder ("mixed disorder"): white-noise random potential with a mean free time τ wn and a smooth random potential with a transport relaxation time τ sm and a single particle relaxation time τ sm,s [τ sm /τ sm,s ∼ (k F d) 2 ≫ 1]. We will further assume that while the transport relaxation rate τ −1 = τ −1 wn + τ −1 sm is governed by short-range disorder, τ wn ≪ τ sm , the damping of SdHO is dominated by smooth random potential, τ sm,s ≪ τ wn . This allows us to consider the range of classically strong magnetic fields, ω c τ wn ≫ 1, neglecting at the same time Landau quantization (which is justified provided ω c τ sm,s /π ≪ 1).
To calculate the interaction corrections, we have to find the corresponding kernel B αβ (ω, q) determined by the classical dynamics. Naively, one could think that under the assumed condition τ wn ≪ τ sm the smooth disorder can simply be neglected in the expression for the classical propagator. While this is true in diffusive limit, the situation in the ballistic regime is much more nontrivial.
To demonstrate the problem, let us consider the kernel B (ρ) xx in the ballistic limit T τ ≃ T τ wn ≫ 1 and in a strong magnetic field ω c ≫ T ≫ τ −1 . If the smooth random potential is completely neglected in classical propagators, we have [see Appendix B; the second term in Eq. (B.39) can be neglected for ω c ≫ T ]
B (ρ) xx ≃ 1 τ wn i ∂g 0 ∂ω + g 2 0 − 1 4 ∂g 0 ∂Q 2 ,(5.1)
where g 0 (ω, q) is the angle-averaged propagator with only out-scattering processes included,
g 0 (ω, q) = iπ ω c J µ (qR c )J −µ (qR c ) sin πµ ,(5.2)
and µ = (ω + iτ −1 wn )/ω c . If characteristic frequencies satisfy ω ≪ ω c (which is the case for T ≪ ω c ), Eq. (5.2) can be further simplified,
g 0 = J 2 0 (Q) −iω + τ −1 wn . (5.3)
Substituting (5.1) and (5.3) in (2.39), we see that momentum-and frequency-integrations decouple and that the first term in (5.1) generates a strongly ultraviolet-divergent q-integral ∼ dQ.
The physical meaning of this divergency is quite transparent. The contribution of the first term in (5.1) to δρ xx is proportional to the time-integrated return probability dtg 0 (r = 0, t), similarly to (4.1). For t ≪ τ wn the propagator g 0 (r, t) describes the ballistic motion in the absence of scattering, which is merely the undisturbed cyclotron rotation in the case of a strong magnetic field. Since at t = 2πn/ω c the particle returns exactly to the original point, the integral dtg 0 (r = 0, t) diverges.
The encountered divergency signals that the neglect of smooth disorder is not justified, even though τ wn ≪ τ sm . Indeed, with smooth disorder taken into account, the particle does not return exactly to the original point after a cyclotron revolution, see Sec. IV. The return probability is then described by Eqs. (4.4), (4.5) with τ replaced by τ sm . It is worth mentioning a similarity with the problem of memory effects in a system with strong scatterers, where even a weak smooth disorder turns out to be crucially important 8,9 .
To demonstrate the role of the smooth disorder on a more formal level, we write down the angle-averaged propagator in the ballistic regime, T τ wn ≫ 1, for the mixed-disorder model,
D(ω, q) = J 2 0 (Q) Q 2 /2τ sm − iω + τ −1 wn . (5.4)
Clearly, in both limits τ sm = ∞ and τ wn = ∞ this formula reduces to (5.3) and (3.2), respectively. In view of ωτ wn ≫ 1 the last term in the denominator of (5.4) can be neglected, and we return to the expression for solely smooth disorder. The presence of the term Q 2 /2τ sm regularizes the Q-integrals, thus solving the problem of ultraviolet-divergencies discussed above. The characteristic momenta are thus determined by Q 2 ∼ T τ sm . Therefore, despite the weakness of the smooth disorder, τ sm ≫ τ wn , it is the first (Q-dependent) rather than the third term which has to be retained in the denominator of (5.4). In other words, in the ballistic regime and in a strong magnetic field the dynamics in the considered model is governed by smooth disorder. The above discussion demonstrates that at ω c ≫ T ≫ τ −1 wn the kernel B αβ (ω, q) for the mixed-disorder model is given by (2.38) with propagators D calculated in smooth random potential (i.e. with white-noise disorder neglected). The time τ wn enters the result only through the matrices T αβ (determined by the transport time τ ≃ τ wn ) and S αβ . Using τ sm /τ wn ≫ 1, we find then that the resulting expression,
B xx ≃ 1 2ω 2 c τ 1 + τ τ wn DD − 1 2ω 2 c τ wn [ D D − 2 Dn y n y D ] − 1 2ω 2 c [ D − 2 n y Dn y ] + 2 ω c n y Dn x D − Dn x Dn x D (5.5)
is dominated by the first term corresponding to the first term in Eq. (5.1). This yields for Q ≡ qR c ≫ 1
B xx (ω, q) ≃ 1 ω 2 c τ DD ≃ 4τ 2 sm ω 2 c τ J 2 0 (Q) (Q 2 − iΩ) 2 , (5.6)
where Ω = 2ωτ sm . As in previous sections, we first calculate the conductivity correction for a point-like interaction. Substituting (5.6) in (2.35), we get
δσ xx = − e 2 2π 2 νV 0 τ sm τ 1/2 4c 0 (T τ ) 1/2 , T ≫ 1/τ wn , (5.7)
with the constant c 0 as defined in Eq. (3.16). For an arbitrary (not necessarily small) value of the ratio τ /τ sm the coefficient 4 in (5.7) is replaced by 4−3τ /τ sm . For τ = τ sm (i.e. without white-noise disorder) we then recover the ballistic asymptotics of Eq. (3.15).
As in the case of purely smooth disorder, the resistivity correction δρ xx is related to δσ xx via Eq. (3.14). Comparing (5.7) with (3.15), we see that the correction δρ xx is enhanced in the mixed-disorder model by a factor ∼ 4(τ sm /τ ) 1/2 ≫ 1 as compared to the purely smoothdisorder case. On the other hand, the scaling with temperature and magnetic field, δρ xx ∝ B 2 T −1/2 , remains the same.
Let us analyze now the crossover from the ballistic to the diffusive regime. Setting T τ ∼ 1 in (5.7), we find that the correction is parametrically large, δσ xx ∼ (τ sm /τ wn ) 1/2 . Clearly, this does not match the diffusive contribution (2.48), yielding δσ xx ∼ 1 at T τ ∼ 1. This indicates that returns without scattering on white-noise disorder continue to govern the correction in certain temperature window below T ∼ 1/τ , which normally belongs to the diffusive regime.
To find the corresponding contribution, one should take into account the scattering-out term τ −1 wn in the denominator of (5.4), which yields
G 1 (x, γ) = 2 π γ 2x 1/2 ∞ 0 dzz 3/2 exp[−z/πx] sinh 2 z = (2γ) 1/2 , x ≪ 1, 4c 0 γ 1/2 x −1/2 , x ≫ 1, (5.8)
where γ = τ sm /τ ≫ 1 and x = T τ . To describe the temperature dependence of the interaction correction for all T , we have to add here the diffusive contribution, which has the form (2.48) for T τ ≪ 1 and vanishes for T τ ≫ 1. This contribution corresponds to long times t ≫ τ and describes the trajectories multiply scattered off white-noise disorder. Since at T τ ∼ 1 the sum of the ballistic and diffusive contributions will be dominated by G 1 (1, γ) ∼ γ 1/2 ≫ 1, the precise way of vanishing of the diffusive contribution at T τ ∼ 1 is inessential. Therefore, we can describe it by the function G 0 (x), Eq. (3.12). The resistivity correction for a system with mixed disorder and point-like interaction has thus the following form:
δρ xx (B) ρ 0 = − (ω c τ ) 2 πk F l νV 0 G mix 0 (T τ, τ sm /τ ), (5.9) where G mix 0 (x, γ) = G 1 (x, γ) + G 0 (x) (5.10) = − ln x + (2γ) 1/2 , x ≪ 1, 4c 0 γ 1/2 x −1/2 , x ≫ 1,
This result is illustrated in Fig. 11a.
In the case of Coulomb interaction, we have as usual a similar result for the exchange contribution
δρ F,mix xx (B) ρ 0 = − (ω c τ ) 2 πk F l G mix F (T τ, τ sm /τ ), (5.11) with G mix F (x, γ) = 1 2 G 1 (x, γ) + G F (x) (5.12) = − ln x + (γ/2) 1/2 , x ≪ 1, 2c 0 γ 1/2 x −1/2 , x ≫ 1.
This function is shown in in Fig. 11b. In fact, here the diffusive contribution can be described either by the function G F (x) or by G 0 (x) because in the diffusive limit they coincide up to a small constant. Since in the intermediate and ballistic regimes [where G F (x) and G 0 (x) differ] the contribution 1 2 G 1 (x, γ) is dominant, the behavior of the diffusive contribution is of no importance, as in the case of point-like interaction. Note that the ballistic contribution corresponds to the point-like interaction with νV 0 = 1 2 , yielding a factor 1 2 in front of G 1 (x, γ) as compared to (5.10). This is because the dynamical part of screening is suppressed for all relevant Q ∼ T τ sm ≫ 1 in the whole range of temperatures, even for T τ < 1, where this contribution is important.
This also applies to the Hartree contribution to the resistivity. Within the "F σ 0 -approximation" we have again an effectively point-like interaction with νV 0 ≃ 3 2 F σ 0 /(1 + F σ 0 ) in the ballistic term. The result thus reads
δρ H,mix xx (B) ρ 0 = 3 (ω c τ ) 2 πk F l G mix H (T τ, τ sm /τ ), (5.13) where G mix H (x, γ) = 1 2 F σ 0 (1 + F σ 0 ) G 1 (x, γ) + G t H (x) = 1 − ln(1 + F σ 0 ) F σ 0 ln x + F σ 0 1 + F σ 0 γ 2 1/2 , x ≪ 1, −2c 0 F σ 0 (1 + F σ 0 ) γ 1/2 x −1/2 , x ≫ 1.
(5.14)
Before closing this subsection, we briefly discuss the Hall resistivity in the mixed disorder model. Repeating the steps described above, we find that the ballistic contribution to ρ xy also contains an extra factor (τ sm /τ ) 1/2 , similarly to ρ xx . For an arbitrary (not necessarily small) value of the ratio τ /τ sm the coefficient 11 in Eqs. (3.52), (3.54) is replaced by [6 + 5τ /τ sm ](τ sm /τ ) 1/2 .
B. Weak B
In the case of a purely smooth disorder (Sec. III) the resistivity correction in the ballistic regime is exponentially suppressed for ω c ≪ T because the particle cannot return to the origin. When the short-range potential is present, the situation changes and the return probability is determined for T τ ≫ 1 by the single-backscattering processes. The interaction-induced MR arises then due to the influence of the magnetic field on the probability of such return, as discussed in the end of Sec. IV. In this case, there is no need to take the smooth potential into account and the MR is determined solely by the white-noise disorder. Let us calculate the corresponding correction using the ballistic form (B.40) of the kernel ∆B
∆ρ xx ρ 0 = − (ω c τ ) 2 νV 0 πk F l π 72 T τ . (5.15)
In the case of Coulomb interaction, ∆B The corresponding Hartree term also scales as B 2 /T . It is worth noting that there is another contribution to the MR in this regime, which comes from the suppression of the triplet channel due to Zeeman splitting E Z rather than from the orbital effects. This contribution is identical to that found in Ref. 20 for the ballistic magnetoresistance in a parallel magnetic field. It also scales as B 2 /T in a weak magnetic field; however, it contains an extra factor (E Z /ω c ) 2 , as compared to (5.16). This factor is small in typical experiments on semiconductor heterostructures where the effective mass of the carriers is much smaller that the bare electron mass. We now turn to the Hall resistivity. Using (B.38) and (B.41), we find for ω c , τ −1 ≪ T and for arbitrary relation between ω c and τ −1
δρ xy ρ xy = νV 0 πk F l π 12 T τ (5.17)
for the point-like interaction, and For the corresponding Hartree-correction to δρ xy calculated within the "F σ 0 -approximation", we refer the reader to Ref. 19.
δρ F xy ρ xy = 1 πk F l 1 − 49(ω c τ )
VI. ANISOTROPIC SYSTEMS
A. Qualitative discussion
In the preceding consideration, we assumed that the 2D system is isotropic. While this is true for the majority of magnetotransport experiments we have in mind, there exists a number of important situations when the transport is anisotropic, σ xx = σ yy . First, such an anisotropy can be induced by the orientation of the 2D electron gas plane with respect to the crystal axes, see e.g. Ref. 48 for a measurement of the quantum correction for the (110) surface of the Si-MOSFET. Second, the electronelectron interaction may lead to spontaneous formation of a charge-density wave. Finally, the anisotropy may be induced by a one-dimensional periodic modulation (lateral superlattice). The latter example is of special interest in view of emergence of commensurability oscillations (known as Weiss oscillations) 49 , and we will discuss it in more detail in Sec. VI C.
The interaction-induced correction to the conductivity tensor of an anisotropic system was calculated for the diffusive regime and B = 0 by Bhatt, Wölfle, and Ramakrishnan 41 . They showed, in particular, that the quantum correction preserves the anisotropy of the quasiclassical (Boltzmann) conductivity. Below we will generalize their result onto the case of a classically strong magnetic field, and, furthermore, will extend the consideration to the ballistic regime.
We begin by presenting a simple argument allowing one to estimate the conductivity correction in an anisotropic system; we will confirm it by a formal calculation in Sec. VI B. According to Eq. (4.1), the relative correction to a diagonal component σ µµ (µ = x, y) of the conductivity tensor is determined by the return probability (and is, thus, the same for µ = x and µ = y). This implies, in the diffusive regime 6)], so that we have instead of (6.2),
δσ µµ σ µµ ∼ −Re 1 ν (dq) 1 D αβ q α q β − iωδσ xx ∼ e 2 K σ xx σ yy (T τ ) −1/2 . (6.3)
The explicit form of the function K(x) will be calculated below. Since the conductivity corrections (6.2) and (6.3) are only determined by the anisotropic diffusion, we expect that they do not depend on the particular source of anisotropy, in analogy with Ref. 41. An important feature of the results (6.2) and (6.3) is that they mix the components σ xx and σ yy of the conductivity tensor. This will play a central role in our analysis of the interaction effect on the magnetoresistivity of modulated systems in Sec. VI C. It is worth mentioning that the validity of the formula (6.3) for the ballistic regime is restricted on the hightemperature side by the condition T T ad , where T −1 ad is the time scale on which the anisotropic diffusion of the guiding center sets in. The value of T ad depends on the particular microscopic mechanism of anisotropy. We will estimate T ad and the behavior of the conductivity correction at T ≫ T ad for a modulated system in Sec. VI C.
B. Calculation of the interaction-induced correction to resistivity
We proceed now with a formal calculation of the quantum correction to the conductivity of an anisotropic system in a strong magnetic field. As a model of anisotropy, we will assume anisotropic impurity scattering, with a cross-section W (φ, φ ′ ) = W (φ − φ ′ ). Repeating the derivation performed in Secs. II A and II B, we find that the result (2.35), (2.36) remains valid in the anisotropic case, with the matrix T αβ proportional to the corresponding (anisotropic) diffusion tensor D αβ ,
T αβ = 2D αβ v 2 F = 1 1 + ω 2 c τ x τ y τ x −ω c τ x τ y ω c τ x τ y τ y ,(6.
4) where τ x and τ y are the relaxation times for the corresponding components of the momentum. We begin by considering the diffusive limit, when the leading contribution comes from three-diffuson diagrams, Fig. 1d and e (see Sec. II C1), which are represented by the last term in Eq. (2.36). The singular contribution to the propagator D, governed by the diffusion mode, has a form analogous to (2.44),
D s (ω, q; φ, φ ′ ) ≃ Ψ R (φ, q)Ψ L (φ ′ , q) D αβ q α q β − iω ,(6.5)
see Appendix E for the derivation of (6.5) and explicit expressions of Ψ R,L . Using (6.5) and (E.3), we get
D ≃ D s = 1 D αβ q α q β − iω (6.6)
and
B xx (ω, q) ≃ − D s n α D s n β D s = 4 v 2 F D 2 xx q 2 x (D xx q 2 x + D yy q 2 y − iω) 3 . (6.7)
The result (6.7) can also be obtained with making use of the identity (3.5); then it is sufficient to keep only the leading term (unity) in the expressions for functions Ψ R,L entering (6.5). Substituting (6.7), (6.6), (2.14) in (2.35), we obtain the final result for the conductivity correction in the diffusive regime,
δσ xx = e 2 2π 2 D xx D yy 1/2 ln T τ, (6.8) δσ yy = e 2 2π 2 D yy D xx 1/2
ln T τ, (6.9) in full agreement with a qualitative consideration of Sec. VI A [Eq. (6.
2)]. The correction to the Hall conductivity is zero in the leading (ln T τ ) order, as in the isotropic case. For the point-like interaction, the result remains the same, up to a factor νV 0 . We now extend the consideration beyond the diffusive limit (thus allowing for qR c 1), assuming first the smooth disorder and concentrating on longitudinal components of the conductivity and resistivity tensors. In analogy with (3.1), the singular contribution D s to the propagator acquires then the form (see Appendix E)
D(ω, q; φ, φ ′ ) = exp{−iqR c [sin(φ − φ q ) − sin(φ ′ − φ q )]} × χ(φ)χ(φ ′ ) D αβ q α q β − iω (6.10)
where
χ(φ) = 1 − iqv F ω 2 c 1 τ x cos φ cos φ q + 1 τ y sin φ sin φ q . (6.11) This yields D = J 2 0 (qR c ) D xx q 2 x + D yy q 2 y − iω (6.12)
and
B xx (ω, q) = 4 v 2 F J 2 0 (qR c )D 2 xx q 2 x (D xx q 2 x + D yy q 2 y − iω) 3 ,(6.13)
which differs from (6.6), (6.7) by the factor J 2 0 (qR c ) only. In the ballistic limit T τ x , T τ y ≫ 1 the relevant values of qR c are large, qR c ≫ 1, so that the screening is effectively static and the interaction is effectively point-like with V 0 = 1/2ν. Substituting then (6.13) in (2.35) and rescaling the integration variables q
x = D −1/2 xxqx , q y = D −1/2 yyqy , we find δσ yy = − e 2 4π 2 c 0 (T τ y ) −1/2 × 2 π K 1 − D xx /D yy ,(6.
14)
δσ xx = D yy D xx δσ yy ,(6.15)
where K is the elliptic integral, π/2 0 dx cos 2 x + q 2 sin 2 x = K( 1 − q 2 ), 0 < q < 1, (6.16) and we assumed that y is the easy-diffusion axis, D yy > D xx .
Let us analyze the obtained results in the limits of weak and strong anisotropy. It is convenient to set D xx = D 0 , τ x = τ 0 , D yy = D 0 + ∆D, and to introduce a dimensionless anisotropy parameter α = ∆D/D 0 . Using the asymptotics of the elliptic integral,
K(s) ≃ π 2 1 + s 2 4 , s ≪ 1, ln 4 √ 1 − s , 1 − s ≪ 1,(6.17)
we find
δσ xx ≃ − e 2 4π 2 c 0 (T τ y ) −1/2 × 1 − α 4 , α ≪ 1, ln(16α) πα 1/2 , α ≫ 1,(6.18)
and δσ yy = (1 + α)δσ xx Equations (6.14),(6.15) and (6.18) confirm the qualitative arguments in Sec. VI A (based on the behavior of the return probability) which led to Eq. (6.3).
C. Modulated systems
In this subsection, we apply the results of Sec. VI B to a particularly important class of anisotropic 2D systems, namely, 2D electron gas subject to a periodic potential varying in one direction. Such systems (lateral superlattices) have been intensively investigated experimentally during the last fifteen years. In a pioneering work 49 , Weiss, von Klitzing, Ploog and Weimann discovered that even a weak one-dimensional periodic modulation with a wave vector k e x may induce strong oscillations of the magnetoresistivity ρ xx (B) [while showing almost no effect on ρ yy (B) and ρ xy (B)], with the minima satisfying the condition 2R c /a = n − 1/4. Here n = 1, 2, . . . and a = 2π/k is the modulation period. The quasiclassical nature of these commensurability oscillations was demonstrated by Beenakker 50 , who showed that the interplay of the cyclotron motion and the superlattice potential induces a drift of the guiding center along y axis, with an amplitude squared oscillating as cos 2 (kR c − π/4) [this is also reproduced by a quantum-mechanical calculation, see Refs. 51,52,53]. While Ref. 50 assumed isotropic impurity scattering (white-noise disorder), it was shown later that the character of impurity scattering affects crucially the dependence of the oscillation amplitude on the magnetic field. The theory of commensurability oscillations in the situation of smooth disorder characteristic for high-mobility 2D electron gas was worked out in Ref. 54 (see also numerical solution of the Boltzmann equation in Ref. 55) and provided a quantitative description of the experimentally observed oscillatory magnetoresistivity ∆ρ xx (B). The result has the form 54
∆ρ xx ρ 0 = πη 2 k 2 lR c 4 sinh(πλ) J iλ (kR c )J −iλ (kR c ),(6.19)
where η is the dimensionless amplitude of the modulation potential [V (x) = ηE F cos(kx)], and
λ = 1 ω c τ s 1 − 1 + τ s τ (kR c ) 2 −1/2 . (6.20)
In the range of sufficiently strong magnetic fields Eq. (6.19) describes the commensurability oscillations with an amplitude proportional to B 3 ,
∆ρ xx ρ 0 ≃ η 2 (ω c τ ) 2 πkR c cos 2 (kR c − π/4),(6.21)
For precise conditions of validity of (6.21), as well as for an analysis of the result (6.19) in the whole range of magnetic fields, the reader is referred to Ref. 54.
As to the modulation-induced corrections ∆ρ xy , ∆ρ yy to the other components of the resistivity tensor, they are exactly zero within the quasiclassical (Boltzmann equation) approach, independently of the form of the impurity collision integral 50,54,55 . Such corrections appear in the quantum-mechanical treatment of the problem 56,57 and are related to the de Haas-van Alphen oscillations of the density of states induced by the Landau quantization of spectrum. As a consequence, these oscillations are exponentially damped by disorder, with the damping factor ∼ exp[−2π/ω c τ s ]. The phase of such quantum oscillations ∆ρ (DOS) yy is opposite to that of quasiclassical commensurability oscillations in ∆ρ xx , Eqs. (6.19), (6.21). Indeed, oscillations in ∆ρ yy that are much weaker than those in ∆ρ xx , have the opposite phase, and vanish much faster with decreasing B, were observed in Ref. 49. We will neglect these oscillations, which are exponentially weak in the range of magnetic fields considered in the present paper, ω c τ s /π ≪ 1. We are going to show that the interaction-induced correction to resistivity also generates oscillations in ρ yy , which are, however, unrelated to the DOS oscillations of a non-interacting system and become dominant with lowering temperature.
To demonstrate this, we apply the result of Sec. VI B for the interaction-induced correction in an anisotropic system. The anisotropy parameter is governed by the quasiclassical correction to ρ xx due to modulation,
α = σ yy σ xx − 1 ≃ ρ xx ρ yy − 1 = ∆ρ xx ρ 0 (6.22)
and is given by Eq. (6.19). For simplicity we will assume that the effect of modulation is relatively weak, α ≪ 1. (Generalization to the large-α case with making use of the corresponding formulas of Sec. VI B is completely straightforward.) Using (6.8) and (6.15), we find the oscillatory correction to ρ yy as a combined effect of the modulation and the Coulomb interaction,
δρ yy ρ 0 = (ω c τ ) 2 2πk F l ∆ρ xx ρ 0 × − ln T τ, T τ ≪ 1, c 0 4 (T τ ) −1/2 , T τ ≫ 1. (6.23)
In the presence of strong scatterers (mixed disorder model), the result for the ballistic regime is enhanced by the factor 4(τ sm /τ ) 1/2 ≫ 1, as discussed in Sec. V. Let us remind the reader that the result (6.23) is valid for temperatures T ≪ T ad , where T −1 ad ≪ τ is the characteristic time on which the motion of the guiding center takes the form of anisotropic diffusion (see Sec. VI A). For the case of a modulated system with a smooth random potential we find T −1 ad ∼ τ (a/R c ) 2 . This is because on a scale shorter than T −1 ad the modulation leads to a drift of the guiding center along y axis with the velocity depending on the coordinate X of the guiding center 50 ,
v d (X) = − ηv F 2 kR c J 0 (kR c ) sin(kX) (6.24) ≃ − ηv F √ 2πkR c cos(kR c − π/4) sin(kX).
In a time a 2 /D ≡ T −1 ad the position X of the guiding center is shifted by a distance of the order of the modulation period a due to the small-angle impurity scattering. Therefore, the drift velocity v d typically changes sign on this time scale, so that the drift is transformed to an additional diffusion process, with ∆D yy ∼ v 2 d T −1 ad , in agreement with (6.21). To estimate the resistivity correction δρ yy in the ultra-ballistic regime T ≫ T ad , we use the relation between the conductivity correction and the return probability (Sec. IV). The return probability R n after n revolutions (introduced in Sec. IV) is modified by the modulation-induced drift in the following way:
R mod n = R n 1 − n ω c τ π v 2 d 2v 2 F . (6.25)
According to (4.1), this yields an oscillatory correction to resistivity suppressed by a factor ∼ T ad /T as compared to the second line (ballistic regime) of Eq. (6.23). Let us summarize the results obtained in this subsection. We have shown that in a periodically modulated system the interaction induces, in addition to the quadratic MR studied in Secs. III and IV, an oscillatory contribution to the component ρ yy of the resistivity tensor, which is not affected by modulation (and thus shows no oscillations) within the Boltzmann theory. When the parabolic MR is negative (meaning that the exchange contribution dominates), which is the case under typical experimental conditions and for not too high temperatures, these quantum interaction-induced oscillations in ρ yy are in phase with classical oscillations in ρ xx , as follows immediately from Eq. (6.23) [see Fig. 12]. In other words, their phase is opposite to that of the abovementioned contribution ∆ρ (DOS) yy induced by the DOS oscillations.
We come therefore to the following conclusion concerning the phase of the total oscillatory contribution to ρ yy . While at sufficiently high temperatures the ρ yyoscillations have, due to the contribution ∆ρ (DOS) yy [and possible due to the Hartree counterpart of Eq. (6.23)], the phase opposite to ∆ρ xx , with lowering temperature the exchange contribution Eq. (6.23) starts to dominate, implying that ρ yy oscillates in phase with ρ xx . Furthermore, the both contributions are damped differently by disorder: the high-temperature out-of-phase oscillations ∆ρ (DOS) yy vanish with lowering B much faster that the low-temperature in-phase interaction-induced oscillations δρ yy .
Our results are in qualitative agreement with a recent experiment 58 . It was observed there that at sufficiently high temperature, T 2.5 K, the oscillations in ρ yy have the opposite phase with respect to ρ xx , in accord with earlier experimental findings 49 . However, when the temperature was lowered, the phase has changed and ρ yy started to oscillate in phase with ρ xx , with an amplitude increasing with decreasing T . In addition to these novel oscillations, a smooth negative MR was seen to develop in the same temperature range. The authors of Ref. 58 emphasized a puzzling character of the temperature dependence of the observed oscillations, which cannot be explained by earlier theories 50,51,52,53,54 discarding the interaction effects. Our theory leading to Eq. vides a plausible explanation of these experimental findings. Quantitative comparison of the theory and experiment requires, however, a more systematic experimental study of the temperature dependence of the amplitude of ρ yy -oscillations in a broader temperature range.
VII. CONCLUSIONS
A. Summary of main results
Let us summarize the key results of the present paper. We have derived a general formula (2.35), (2.38) for the interaction-induced quantum correction δσ αβ to the conductivity tensor of 2D electrons valid for arbitrary temperature, magnetic field and disorder range. It expresses δσ αβ in terms of classical propagators in random potential ("ballistic diffusons"). In the appropriate limiting cases, it reproduces all previously known results on the interaction correction (see Sec. II C).
Applying this formalism, we have calculated the interaction contribution to the MR in strong B in systems with various types of disorder and for arbitrary T τ . In the diffusive limit, T τ ≪ 1, the result does not depend on the type of disorder, as expected. Specifically, the MR scales with magnetic field and temperature as follows, δρ xx ∝ B 2 ln(T τ ) and δρ xy ∝ B ln T τ , in agreement with Refs. 10 and 11.
In the ballistic limit, T τ ≫ 1, the result is strongly affected by the character of disorder. In Sec. III we have performed a detailed study of the case of smooth disorder characteristic for high-mobility GaAs heterostructures. We have found that the temperature-dependent MR scales at ω c ≫ T as δρ xx ∝ B 2 (T τ ) −1/2 and δρ xy ∝ B(T τ ) 1/2 . In addition, there is a temperatureindependent (but larger) contribution ∝ B 3/2 to the Hall resistivity. In the opposite limit ω c ≪ T the MR is suppressed.
We have further considered a mixed disorder model, with strong scatterers (modeled by white-noise disorder) superimposed on a smooth random potential (Sec. V). A qualitatively new situation arises when the momentum relaxation rate τ −1 sm due to smooth disorder is much less than the total momentum relaxation rate τ −1 . Such a model is believed to be relevant to Si-based structures, as well as to GaAs structures with very large spacer. We have shown that in the ballistic limit and at ω c ≫ T the corrections to both longitudinal and Hall resistivities are enhanced (as compared to the case of smooth disorder) by a factor ∼ (τ sm /τ ) 1/2 ≫ 1. In the range of weaker magnetic fields, ω c ≪ T , the interaction-induced MR scales in the ballistic regime as ∆ρ xx ∝ B 2 (T τ ) −1 and
δρ xy ∝ B(T τ ) −1 [1 − const (ω c τ ) 2 ].
For a weak interaction (κ ≪ k F ) the correction is dominated by the exchange contribution, implying that ∆ρ xx is negative and that the slope of ρ xy decreases with increasing temperature. This is true up to a temperature T H ≫ τ −1 (defined in Sec. III D) where the sign changes. In the case of a strong interaction the magnitude of the Hartree contribution (and thus the sign of the total correction) depends on angular harmonics F ρ,σ n of the Fermi-liquid interaction (Sec. III E). It is worth emphasizing that in contrast to the diffusive limit where only F σ 0 is relevant, in the ballistic regime all the Fermi-liquid parameters are, strictly speaking, impor-tant, see Eq. (3.40). Therefore, predictions of the "F σ 0approximation" (with only one Fermi-liquid parameter retained) should be treated with caution.
We have further applied our formalism to anisotropic systems (Sec. VI) and demonstrated that the correction mixes the components ρ xx and ρ yy of the resistivity tensor. This result is of special interest in the case of systems subject to a one-dimensional periodic modulation (lateral superlattice; wave vector k e x ). Specifically, we have shown that the interaction induces novel oscillations in ρ yy , which are in phase with quasiclassical commensurability (Weiss) oscillations in ρ xx .
B. Comparison with experiment
Our results for ρ xx in the case of smooth disorder (published in a brief form in the Letter 34 ) have been confirmed by a recent experiment on n-GaAs system 14 , which was performed in the broad temperature range, from the diffusive to the ballistic regime. Specifically, Li et al. 14 found that the MR scales as ∆ρ xx ∝ B 2 in strong magnetic fields. The obtained temperature-dependence of the proportionality coefficient G(T τ ) was in good agreement with our predictions.
Very recently, Olshanetsky et al. 59 studied the MR in the ballistic regime in a Si/SiGe structure of n-type, where both short-and long-range potential are expected to be present. They found the interaction-induced correction to ρ xx larger by a factor ∼ 5 as compared to our prediction 34 for the case of smooth disorder. This conforms with the results of the present paper for the mixed-disorder model, where we find an enhancement of ∆ρ xx by a factor 4(τ sm /τ ) 1/2 ≫ 1.
As has been mentioned in Introduction, the interaction-induced MR in the ballistic regime was measured for the first time as early as in 1983, by Paalanen, Tsui, and Hwang 12 , who studied GaAs structures. Again, a parabolic temperature-dependent MR ∆ρ xx was obtained, in agreement with our findings. However, its magnitude was considerably (roughly an order of magnitude) larger compared with our theoretical result for the case of smooth disorder, as well as with the recent experiment 14 . We speculate that samples used in Ref. 12 probably contained an appreciable concentration of background impurities, which has led to an enhancement of the interaction-induced contribution to resistivity, similarly to the recent work 59 . (Indeed, the results for the mixed-disorder model shown in Fig. 11 may create an impression that the log T behavior extends up to T τ ∼ 10, as was concluded in Ref. 12.) Remarkably, the interaction-induced quantum correction to conductivity may serve as an indicator of the dominant type of disorder.
To the best of our knowledge, no experimental study of the interaction effects on Hall resistivity ρ xy has been published. This part of our predictions therefore awaits its experimental verification.
Finally, our results for systems with one-dimensional periodic modulation are in qualitative agreement with the recent work by Mitzkus et al. 58 , as we discussed in detail in Sec. VI C. Quantitative comparison of the theory and experiment requires an experimental study of the temperature-dependence of novel oscillations (found experimentally in Ref. 58 and theoretically in the present paper) in a broader temperature range.
C. Outlook
Before closing the paper, we list a few further applications of our formalism and its possible generalizations. First, our results can be generalized to frequencydependent (rather than temperature-dependent) MR. Second, the interaction effects in systems of other dimensionality, as well as in macroscopically inhomogeneous systems, can be investigated by our general method. Third, the formalism can be used to calculate the phonon-induced contribution to resistivity, which becomes larger than that due to Coulomb interaction at sufficiently high temperatures. Further, thermoelectric phenomena in the full range of magnetic fields and temperatures can be studied in a similar way. Finally, our approach can be generalized to the regime of still stronger magnetic fields, where the Landau quantization can not be neglected anymore; the work in this direction is in progress 60 .
VIII. ACKNOWLEDGMENTS
We thank P.T. Coleridge, A.V. Germanenko, Z.D. Kvon, G.M. Minkov, C. Mitzkus, A.K. Savchenko, and D. Weiss for informing us on experimental results prior to publication and for stimulating discussions. This work was supported by the Schwerpunktprogramm "Quanten-Hall-Systeme" and the SFB195 der Deutschen Forschungsgemeinschaft, and by the RFBR.
APPENDIX A: CANCELLATION OF THE INELASTIC TERM
As discussed in Sec. II A, diagrams f and g give rise, in addition to the contribution (2.25), to a term of the type (2.26), characteristic for inelastic effects. This yields at B = 0 a disorder-independent correction to resistivity δρ ∼ (T /eE F ) 2 , see below. Note that such a contribution to resistivity would be obtained if one substitutes the inelastic relaxation rate of a clean 2D electron gas, τ −1 inel ∼ T 2 /E F in the Drude formula (1.1). However, such a procedure clearly makes no sense. Indeed, in a translationally invariant system electron-electron collisions conserve the total momentum and thus give no contribution to resistivity. Therefore, the correction (2.26) ought to be canceled by some other contribution. Below we show FIG. 13: Aslamazov-Larkin-type diagrams describing the Coulomb-drag contribution to the resistivity, which cancels the "inelastic" part of the diagrams f, g from Fig. 1.
explicitly that this is indeed the case, and that this second contribution is of the Coulomb-drag type, described by the diagrams in Fig. 13. For simplicity, we restrict our consideration here to the case of zero B and white-noise disorder, which allows us to use the results of Ref. 61 for the Coulomb drag. Note that while Ref. 61 considered the drag between two layers, we refer to the "self-drag" within a single layer. As we will see below, the characteristic momenta q determining the contribution (2.26) are large, q ∼ k F . For this reason, there is no need to take into account impurity-line ladders while evaluating this term, similarly to the calculation of drag in Ref. 61 for a small inter-layer distance.
We thus have
δB f xx (ω, q) = 1 2πνv 2 F d 2 p (2π) 2 × Re 2p 2 x G 2 R (ǫ, p)G R (ǫ − ω, p − q)G A (ǫ, p) + p x (p x − q x )G R (ǫ, p)G A (ǫ, p) × G R (ǫ − ω, p − q)G A (ǫ − ω, p − q)] , (A.1)
where G R,A (ǫ, p) = (E F + ǫ − p 2 /2m ± i/2τ ) −1 are the disorder-averaged retarded and advanced Green's functions. Using the identity
G R (ǫ, p)G A (ǫ, p) = i τ [ G R (ǫ, p) − G A (ǫ, p) ],
we reduce (A.1) to the form
δB f xx (ω, q) = − τ 2 πv 2 F ν d 2 p (2π) 2 p x q x × Re[G R (ǫ, p)G A (ǫ − ω, p − q)] = − τ 2 q 2 x 2v 2 F νω ImΠ(ω, q), (A.2)
where Π(ω, q) is the polarization operator (2.15),
ImΠ(ω, q) = ω π d 2 p (2π) 2 G R (ǫ, p)G A (ǫ − ω, p − q) ≃ 2ν ω qv F θ(qv F − ω), (A.3)
where θ(x) is the step function. Furthermore, the imaginary part of the interaction propagator within the RPA is proportional to ImΠ(ω, q)
ImU (ω, q) = −|U (ω, q)| 2 ImΠ(ω, q).
δσ inel xx = − e 2 τ 2 m 2 ∞ −∞ dω 2π 1 2T sinh 2 (ω/2T ) d 2 q (2π) 2 × q 2 x |U (ω, q)| 2 [ImΠ(ω, q)] 2 ,(A.δσ inel + δσ drag = 0. (A.6)
Using the explicit form of ImΠ(ω, q), Eq. (A.3), and of U (ω, q), Eq. (2.14), in the ballistic regime, it is easy to estimate δσ inel (we assume here κ ∼ k F for simplicity),
δσ inel ∼ −e 2 τ 2 T T 0 dω kF qdq k 2 F ∼ −e 2 (T τ ) 2 . (A.7)
As has been stated above, the q-integral is determined by the ultraviolet cutoff. Finally, we note that in double-layer system the interlayer interaction does give rise to a correction δσ inel to the driving-layer conductivity, which is equal in magnitude and opposite in sign to the transconductivity. This effect is, however, reduced by a factor ∼ (k F ξ) −4 (where ξ is the interlayer distance), as compared to (A.7), see Ref. 61. In this appendix we will derive the general expressions (valid for arbitrary magnetic field) for the kernels B xy in terms of the quasiclassical propagator for a white-noise random potential. This will allow us to reproduce the results of Refs. 18,19, where the interactioninduced corrections to σ xx and ρ xy were studied for a white-noise disorder in the limit B → 0. We will further apply the formalism to calculate the longitudinal MR and the Hall resistivity in a finite magnetic field with ω c ≪ T . The resistivity tensor in yet stronger magnetic field, ω c ≫ T , is studied, in the more general framework of a mixed disorder model in Sec. V.
Using Eqs. (2.38) and (2.40), we get
B (ρ) xx = 1 2τ D 2 − 1 τ Dn x n x D + 1 2 D − n x Dn x − 2 τ n x Dn x D + 2ω c n x Dn y D − 1 − ω 2 c τ 2 τ 2 Dn x Dn x D + 2ω c τ Dn x Dn y D (B.1)
for the kernel describing the longitudinal resistivity, and
B (ρ) xy = ω c 2 DD − 1 τ Dn x n y D − n x Dn y − 2 τ n x Dn y D − 2ω c n x Dn x D − 1 − ω 2 c τ 2 τ 2 Dn x Dn y D − 2ω c τ Dn x Dn x D (B.2)
for the Hall resistivity. The propagator D(φ, φ ′ ) in the case of white-noise disorder can be expressed through the propagator D 0 (φ, φ ′ ), obeying the Liouville-Boltzmann equation with only scattering-out term present in the collision integral,
− iω + iqv F cos(φ − φ q ) + ω c ∂ ∂φ + 1 τ D 0 (φ, φ ′ ) = 2πδ(φ − φ ′ ). (B.3)
As in a zero magnetic field, the total propagator is given by the sum of the ladder-diagrams (thus including the scattering-in processes), yielding
D(φ, φ ′ ) = D 0 (φ, φ ′ ) + dφ 1 2π dφ 2 2π D 0 (φ, φ 1 )D 0 (φ 2 , φ ′ ) τ − D 0 ,
(B.4) which we write symbolically as follows
D = D 0 + D 0 D 0 τ − g 0 . (B.5)
Here we introduced a short-hand notation
g 0 (ω, q) ≡ D 0 = dφ 2π dφ ′ 2π D 0 (ω, q; φ, φ ′ ) (B.6)
for the angle-averaged scattering-out propagator. It turns out that for a white-noise disorder the kernels B xy can be expressed in terms of g 0 (and its derivatives with respect to q and ω). The solution of (B.3) is given by
D 0 (ω, q; φ, φ ′ ) = exp{iqR c [sin(φ ′ − φ q ) − sin(φ − φ q )]} × ∞ n=−∞ exp[in(φ − φ ′ )] −i(ω − nω c ) + 1/τ . (B.7)
It is worth mentioning that in the mixed-disorder model introduced in Sec. V with both, white-noise and smooth disorder present, the solution of the Liouville-Boltzmann equation also has the form (B.4). In that case, the propagator D 0 satisfies the Liouville-Boltzmann equation for a purely smooth disorder (considered in Appendix D) with the replacement ω → ω + i/τ wn , where τ wn is relaxation time due to white-noise potential. Using (B.7) and a series representation for the Bessel functions, we find [see, e.g. Ref. 54]
g 0 (ω, q) = i ω c n J 2 n (qR c ) µ − n = iπ ω c J µ (qR c )J −µ (qR c ) sin πµ , (B.8)
where J µ (z) is the Bessel function and
µ = ω ω c + i ω c τ . (B.9)
In the absence of magnetic field (ω c = 0, R c = v F /ω c = ∞) the propagators D 0 (ω, q; φ, φ ′ ) and g 0 (ω, q) acquire a simple form
D 0 (ω, q; φ, φ ′ ) = 2πδ(φ − φ ′ ) −iω + qv F cos(φ − φ q ) + 1/τ , (B.10) g 0 (ω, q) = 1 q 2 v 2 F + (−iω + 1/τ ) 2 ≡ 1 S(ω, q)
.
(B.11)
To proceed further, we first reduce [using (B.5)] the "matrix elements" appearing in (B.1) and (B.2) to the form containing only the propagators D 0 ,
D = D 0 τ τ − g 0 , (B.12) DD = τ 2 D 0 D 0 (τ − g 0 ) 2 , (B.13) Dn x n β D = τ 2 D 0 n x n β D 0 (τ − g 0 ) 2 , (B.14) n x Dn β = n x D 0 n β + n x D 0 D 0 n β τ − g 0 , (B.15) n x Dn β D = n x D 0 n β D 0 + τ n x D 0 D 0 n β D 0 (τ − g 0 ) 2 , (B.16) Dn x Dn β D = τ 2 D 0 n x D 0 n β D 0 (τ − g 0 ) 2 + τ 2 D 0 n x D 0 D 0 n β D 0 (τ − g 0 ) 3 , (B.17)
where β = x, y. Next, using (B.7) and (B.3) and performing the averaging over φ q , we can express the matrix elements involving D 0 via the propagator g 0 . Introducing the notation W = −iω + 1/τ , we get the following φ q -averaged matrix elements,
D 0 D 0 = −i ∂g 0 ∂ω , (B.18) D 0 n x n x D 0 = − 1 2q 2 v 2 F [1 − Wg 0 ] 2 + ω 2 c 8v 2 F ∂g 0 ∂q 2 , (B.19) n x D 0 n x = ω 2 c 4v 2 F ∂ 2 g 0 ∂q 2 + 1 q ∂g 0 ∂q + g 0 2 ,
(B.20)
n x D 0 n x D 0 = − W 2qv 2 F ∂g 0 ∂q , (B.21) n x D 0 D 0 n x D 0 = [1 − Wg 0 ] 2qv 2 F ∂g 0 ∂q , (B.22) D 0 n x D 0 n x D 0 = − 1 4v 2 F ∂ 2 g 0 ∂q 2 + 1 q ∂g 0 ∂q , (B.23) D 0 n x D 0 2 = − 1 2v 2 F ∂g 0 ∂q 2 , (B.24)
for the "longitudinal correlators", and
D 0 n x n y D 0 = ω c 2qv 2 F [1 − Wg 0 ] ∂g 0 ∂q , (B.25) n x D 0 n y = ω c W 2qv 2 F ∂g 0 ∂q , (B.26) n x D 0 n y D 0 = ω c 4v 2 F ∂ 2 g 0 ∂q 2 + 1 q ∂g 0 ∂q , (B.27) n x D 0 D 0 n y D 0 = ω c 4v 2 F ∂g 0 ∂q 2 , (B.28) D 0 n x D 0 n y D 0 = i 2ω c ∂g 0 ∂ω + iW qv 2 F ∂g 0 ∂q ,B (ρ) xx (ω, q) = τ τ − g 0 2 × 2τ − g 0 2τ 2 g 2 0 + (1 − Wg 0 ) 2 q 2 v 2 F + i τ ∂g 0 ∂ω − 1 qv 2 F τ 2 ∂g 0 ∂q + 1 4v 2 F τ 3 ∂g 0 ∂q 2 × (1 − ω 2 c τ 2 ) 2τ τ − g 0 + ω 2 c τ 2 1 + g 0 2τ + 1 4v 2 F τ 2 ∂ 2 g 0 ∂q 2 + 1 q ∂g 0 ∂q 1 − ω 2 c g 2 0 (B.30) B (ρ) xy (ω, q) = τ τ − g 0 2 × − i 4ω c τ 2 ∂g 0 ∂ω + iW qv 2 F ∂g 0 ∂q + ω c 2qv 2 F τ 1 − Wg 0 − g 0 2τ ∂g 0 ∂q + ω c g 0 4v 2 F τ 2 ∂ 2 g 0 ∂q 2 + 1 q ∂g 0 ∂q + ω c 4v 2 F τ 2 τ + g 0 τ − g 0 ∂g 0 ∂q 2 . (B.31)
In zero magnetic field, we put ω c = 0 and substitute g 0 = 1/S in (B.30). After some algebra, we reduce the obtained expression for the kernel B xx to the form
B (ρ) xx (ω, q) = (qv F ) 2 2τ 3 S 3 (S − 1/τ ) 3 + 3(qv F ) 2 4τ 2 S 3 (S − 1/τ ) 2 + S − W τ S(S − 1/τ ) 2 + (2S − 1/τ )[S − W] 2 2τ (qv F ) 2 S(S − 1/τ ) 2 ,(B (ρ) xx (ω, q) ≃ S 0 + iω τ S 3 0 + [S 0 + iω] 2 τ (qv F ) 2 S 2 0 , = S 0 + iω τ S 2 0 1 S 0 + 1 S 0 − i(ω + i0)
, (2.39) and using (C.1) for exchange interaction, we reproduce the linear-in-T correction to the resistivity in the ballistic regime,
(B.33) where S 0 = [q 2 v 2 F − (ω + i0) 2 ] 1/2 . Substituting (B.33) inδρ F xx ρ 0 = − T E F . (B.34)
Within the approximation of isotropic interaction used in Ref. 18, the Hartree term is determined by the triplet channel and is given by
δρ H xx ρ 0 = − 3 F σ 0 1 + F σ 0 T E F . (B.35)
It is worth noting that one should exercise a certain caution when comparing the experimental data with the results (B.34) and (B.35), even in systems with shortrange impurities. First, the higher angular harmonics F ρ,σ n =0 of the interaction 62 (neglected in the above approximation) may change the numerical coefficient in front of the Hartree term (see discussion in Sec. III E and in Ref. 18). Second, anisotropy of the impurity scattering introduces an extra factor 2πνW (π)τ = 1 (where W (π) is the effective impurity-backscattering probability) in both exchange and Hartree terms (see Sec. II C3 and Appendix C). The anisotropy may arise due to some amount of smooth disorder present in any realistic system, due to a finite range of scatterers, or due to the screening of originally point-like impurities (see Sec. IV). Therefore, the interaction parameter F σ 0 extracted from the measured linear-in-T resistivity with the use of (B.34), (B.35) may differ considerably from that found from a measurement of other quantities (e.g. the resistivity correction in the diffusive limit or the spin susceptibility).
To find the leading contribution to B (ρ) xy in the limit of vanishing magnetic field, we have to expand the propagator g 0 up to the second order in ω c in the first term in curly brackets in (B.31). This can be easily done by treating the term ω c ∂/∂φ in (B.3) as a perturbation, which yields g 0 (B → 0) = g 0 (ω, q; B = 0) + ω 2 c h(ω, q)
= 1 S − ω 2 c q 2 v 2 F (S 2 − 5W 2 ) 8S 7 . (B.36)
After a simple algebra, we find B In arbitrary magnetic field, Eq. (B.30) can be also significantly simplified when the condition of the ballistic regime, T τ ≫ 1, is assumed. Then the leading contribution to the longitudinal MR, ∆ρ xx = ρ xx (B) − ρ xx (0), is determined by the kernel
τ B (ρ) xx (ω, q) ≃ g 2 0 + (1 − Wg 0 ) 2 q 2 v 2 F + i ∂g 0 ∂ω − ω 2 c 4v 2 F ∂g 0 ∂q 2 . (B.39)
The remaining terms in (B.30) yield the contributions to the MR which are smaller at least by an additional factor (T τ ) −1 . Using (B.36) [which tells us that for ω c ≪ T the magnetic-field-induced corrections to the propagator g 0 are small by a factor (ω c /T ) 2 ], we find that the MR for not very strong magnetic fields, ω c ≪ T , is determined by a quadratic in ω c correction to the kernel B In this appendix we calculate the leading ballistic correction to the conductivity at B = 0 for a generic scattering cross-section W (φ− φ ′ ) in the case of the Coulomb interaction. As explained in Sec. II C3, this term (proportional to T τ ) is obtained by substituting the ballistic asymptotics (2.51) of B xx in the general formula (2.35). Likewise, the interaction propagator U (ω, q) entering (2.35) has to be replaced by U (ω, q) = 1 2ν
1 1 + iω D f (C.1)
with the free propagator D f given by Eq. (2.50). Performing the angular integration ... , we get δσ xx ≃ − e 2 2π 2 T τ Im where we introduced the dimensionless variables Ω = ω/T, Q = qv F /T, andW (φ) = 2πντ W (φ). It is convenient to split the interaction propagator as follows 2νU (Ω, Q) = S 0 S 0 + i(Ω + i0) = 1 − iΩ S 0 + i(Ω + i0) , (C.3) where S 0 = [Q 2 − (Ω + i0) 2 ] 1/2 . The first term corresponds to a statically screened interaction and is equivalent to a point-like interaction with V 0 = 1/2ν, the second term results from the dynamical weakening of screening. As discussed in Sec. II C3, the contribution δσ (1) xx of the first (constant) term is proportional to the backscattering probability W (π), see Eq. (2.52). Let us show that this follows also from Eq. (C.2). Performing the variable change Q → S 0 , we find δσ (1) xx ≃ − e 2 2π 2 T τ Im (C.4)
The contour C of integration over S 0 in Eq. (C.4) is shown in Fig. 14. Interchanging the order of integration over φ and S 0 , we see that for any φ = π (i.e. cos φ 2 = 0) the S 0 -integral converges. Furthermore, transforming the integration contour C → C ′ as shown in Fig. 14, it is straightforward to reduceΦ(Ω) to an explicitly real form. Therefore, only the singular point φ = π (where the result of S 0 -integration diverges as 1/| cos φ 2 |, implying that the imaginary part ofΦ(Ω) is determined by a delta-function in φ-integral) contributes to (C.4), so that δσ (1) xx ∝ W (π). To find the corresponding coefficient, one can consider the isotropic scattering W (φ) = const and to integrate over φ first, yielding δσ (1) xx = e 2 πW (π)T τ, (C. 5) in agreement with (2.52). Note that the integral over Ω is formally divergent at the upper limit. It should be cut off at Ω ∼ E F /T yielding a temperature independent contribution ∼ e 2W (π)E F τ which renormalizes the value of the Drude conductivity. We now turn to the contribution δσ (2) xx of the second (dynamical) term in the interaction propagator (C.3), which differs from Eq. (C.4) by an extra factor −iΩ/[S 0 + i(Ω + i0)]. Rotating at φ = π the integration contour as before, we reduce the S 0 -integral to the form (S 0 → −iY )
Ω 2 ∞ Ω dY Y 2 Y + Ω (Y 2 cos 2 φ 2 + Ω 2 sin 2 φ 2 )
, (C. 6) which is again real and thus yields no contribution to δσ (2) xx . Though the point φ = π is singular in this case as well, the singularity is only logarithmic (∼ ln | cos φ 2 |), so that no contribution proportional to W (π) arises. This can be easily checked by assuming W (φ) = const and performing the φ-integration first. Therefore δσ (2) xx = 0, and the linear-in-T term is given by Eq. (C.5).
In the above consideration we have expanded the ballistic propagator D up to terms with one scattering event.
In the case of small-angle scattering this is justified provided T τ s ≫ 1, while in the intermediate temperature range τ −1 ≪ T ≪ τ −1 s processes with many scattering events dominate (though the particle motion is typically close to the straight line). The term δσ (1) xx which is governed by anomalous processes of returns in a time t T −1 ≪ τ is exponentially small in this case, see Sec. II C3. As to the δσ (2) xx contribution to the linear-in-T term, it remains zero in this case as well. To demonstrate this, we use Eq. (2.36). In the first and the third terms we can replace D by the free propagator (2.50), the fourth term gives no T τ contribution, while in the second term we should take into account the angular diffusion (2.9) around the straight trajectory,
1 2 D − n x Dn x → −iω 2τ [q 2 v 2 F − (ω + i0) 2 ] 3/2 . (C.7)
Combining the contributions to B xx of all the three terms, we get In this appendix, we will solve the classical equation for a propagator of a particle moving in a smooth random potential in a magnetic field,
−iω + iqv F cos φ + ω c ∂ ∂φ − 1 τ ∂ 2 ∂φ 2 D(φ, φ ′ ) = 2πδ(φ − φ ′ ). (D.1)
Here the polar angle of the velocity is counted from the angle of q, φ − φ q → φ.
We first consider the diffusive limit, Dq 2 , ω ≪ 1/τ , and solve this equation perturbatively in q for arbitrary magnetic field. Putting q = 0, we obtain the solution in the form D(φ, φ ′ ) = n e in(φ−φ ′ ) −iω + inω c + n 2 /τ , (D.2) which is just a standard expansion in eigenfunctions of the Liouville-Boltzmann operator. We now treat the term δL = iv F q cos φ as a perturbation. The first-order correction to the eigenvalues λ (0) n = −iω + inω c + n 2 /τ vanishes, while the second order correction is
λ (2) n = q 2 v 2 F 2 1 1 − (iω c τ + 2n) 2 (D.3)
and can be neglected along with −iω in all terms except for n = 0 in the diffusive limit. The first order correction to the right eigenfunction for n = 0 reads
Ψ (1) 0,R (φ) = − iqv F τ 1 + ω 2 c τ 2 [cos φ + ω c τ sin φ], (D.4)
while the left eigenfunction differs from (D.4) by a replacement ω c → −ω c . Thus, in the diffusive limit the propagator has the form
D(ω, q; φ, φ ′ ) ≃ 1 Dq 2 − iω × 1 − i qv F τ (cos φ + ω c τ sin φ ) 1 + ω 2 c τ 2 × 1 − i qv F τ (cos φ ′ − ω c τ sin φ ′ ) 1 + ω 2 c τ 2 + n =0 e in(φ−φ ′ ) inω c + n 2 /τ . (D.5)
In a strong magnetic field (ω c τ ≫ 1) one can go beyond the diffusion approximation. In this case one can represent the propagator in the form D(ω, q; φ, φ ′ ) = d(ω, q; φ, φ ′ ) exp[−iqR c (sin φ − sin φ ′ )], (D.6) and solve the equation for d(φ, φ ′ ), −iω − i qv F ω c τ sin φ + ω c + 2i qv F ω c τ cos φ ∂ ∂φ
+ 1 τ qv F ω c 2 cos 2 φ − 1 τ ∂ 2 ∂φ 2 d(φ, φ ′ ) = 2πδ(φ − φ ′ ) (D.7)
perturbatively in q. At q = 0 we have the same solution (D.2) as in the diffusive limit. The first order correction to the eigenvalues is now produced by the q 2 -term in (D.7), λ
(1) n = Dq 2 , with D = R 2 c /2τ the diffusion constant in a strong magnetic field. The second order corrections λ (2) n turn out to be small compared to λ (1) n for (qR c ) 2 ≪ ω c τ . As in the diffusive limit, for calculation of B xx the corrections to the eigenfunctions Ψ n with n = 0 can be neglected. The first-order correction to Ψ 0 is found to be (we drop the term ∝ sin 2φ, since it does not contribute to B xx in the leading order) We thus get the result (6.5) for the singular contribution D s , with Ψ R,L given by (E.3). In a strong magnetic field (ω c τ ≫ 1), we can go beyond the diffusive limit. Proceeding as for an isotropic system, we perform the transformation (D.6). Treating the q-dependent terms in the obtained equation for d(ω, q; φ, φ ′ ) as a perturbation and keeping the singular contribution only, we come to the result (6.10), where χ(φ) can be represented symbolically as
χ(φ) = 1 + iqv F ω 2 c ∂ ∂φ −1Ĉ sin(φ − φ q ). (E.6)
According to (3.3)-(3.5), we only need to calculate averages of the type D and n α D , so that it is sufficient to keep the zero and first harmonics in φ in Eq. (6.10). Using
n αĈ n β = τ −1 x τ −1 y , (E.7)
we then reduce (E.6) to the form (6.11).
FIG. 1: Exchange diagrams for the interaction correction to σ αβ . The wavy (dashed) lines denote the interaction (impurity scattering), the shaded blocks are impurity ladders, and the +/− symbols denote the signs of the Matsubara frequencies. The diagrams obtained by a flip and/or by an exchange + ↔ − should also be included. "Inelastic" part of the diagrams f, g is canceled by a contribution of the Coulomb-drag type, see Appendix A.
FIG. 2 :
2lision integral, determined by the scattering cross-section Diagrams obtained by a flip and/or by an exchange + ↔ − from the diagram d.
the explicit form of the ballistic propagator for the case of white-noise disorder and B → 0 [Eqs. (B.4), (B.6), (B.10), (B.11), and (B.36)] we recover the results for δσ xx and δρ xy obtained in a different way in Refs. 18 and 19, see Appendix B.
(3.1) for calculation of B xx if the identities (3.3), (3.4), and (3.5) are used. (Of course, B xx can also be evaluated directly from Eq. (2.36), but then the second (regular) term D reg in (3.1) has to be included.) Combining all four terms in (2.36), we get
FIG. 4 :
4Functions G0(T τ ) (a) and GF(T τ ) (b) determining the T -dependence of the exchange term for point-like, Eq. (3.12), and Coulomb, Eq. (3.19), interaction, respectively. Diffusive and ballistic asymptotics, Eq. (3.15) and Eq. (3.21), are also shown.
FIG. 5 :
5Hartree diagrams for the interaction correction to σ αβ . The diagrams are labeled in the way as their exchange counterparts inFig. 1. The diagrams obtained by a flip and/or by an exchange + ↔ − should also be included.
FIG. 6 :
6Hartree contribution, GH(T τ ), for (a) weak interaction, κ/kF = 0.1, 0.2, 0.3, 0.5, and (b) strong interaction, F0 = −0.3, −0.4, −0.5 (from bottom to top). Dashed curves represent the exchange contribution.
FIG. 7 :
7The function G σ H (T τ, ǫz), Eq. (3.46), describing the temperature dependence of the triplet contribution is shown for F σ 0 = −0.3 and different values of Zeeman splitting, ǫz = 0.1, 0.3, 0.5, 1.0 (from top to bottom). Dashed curve represents the case ǫz = 0.
F
(x) is obtained by substituting (3.17) and (3.9) in (2.35) [cf. similar calculation for δσ F xx leading to Eqs. (3.18) and (3.20).] The function G ρxy F (x) describing the temperature dependence of the exchange correction to the Hall resistivity is shown in Fig. 8. In the ballistic regime, where G (xy)F (x) dominates, the interaction becomes effectively point-like with νV 0 = 1 2 , so that one can simplify the calculation using G
H
(x), one should use Eqs. (3.9) and (3.41).
F
(T τ ) (upper curve) describing the temperature dependence of the Hall resistivity for point-like and Coulomb interaction, respectively. Diffusive (x ≪ 1) and ballistic (x ≫ 1) asymptotics, Eqs. (3.52) and(3.54), are also shown.
FIG. 9 :
9Schematic plot of the return probability R(t) in a strong magnetic field and smooth disorder. In the ballistic regime, the peaks are separated by the cyclotron period, t1 = 2π/ωc. Dashed curve represents the smoothened return probabilityR(t)
FIG. 10 :
10Schematic illustration of the ballistic dynamics in a strong magnetic field. The thick line shows the particle trajectory (two cyclotron revolutions disturbed by the smooth random potential). The thin line is the diffusive trajectory of the guiding center.
FIG. 11 :
11Functions G mix 0 (T τ ) (a) and G mix F (T τ ) (b) describing the temperature dependence of the resistivity correction due to point-like and Coulomb (exchange) interaction, respectively, in the mixed-disorder model for different values of parameter γ ≡ τsm/τ = 20, 10, 5 (from top to bottom). Dashed curves represent these functions for purely smooth disorder (γ = 1).
point-like interaction, substituting (B.40) in (2.39), we find the following ballistic (T τ ≫ 1) asymptotics of the longitudinal MR,
multiplied by the ballistic asymptotics of the interaction, Eq. (C.1). Substituting this product in Eq. (2.39), we get the Fock contribution to the MR in the form ∆ρ F xx ρ 0 = − (ω c τ ) 2 πk F l 17π 192 T τ , T τ ≫ 1. (5.16)
for δσ yy . In the ballistic regime the time-integrated return probability T −1 dt D(t) scales as (T τ ) −1/2 [see Eqs. (4.5) and (4.
FIG. 12 :
12Magnetoresistivity in a lateral superlattice with modulation wave vector k ex. (a) Quasiclassical Weissoscillations; dashed line shows the resistivity in the absence of modulation. (b) Interaction-induced quantum oscillations in ρyy for three temperatures. The curves correspond to the values of the parameter 2c0(τsm/T τ 2 ) 1/2 = 0.1, 0.3, 0.5 (from top to bottom), assuming mixed disorder. Dashed line represents the resistivity of the non-interacting system. Typical experimental parameters are used: effective mass m = 0.067 × 9.1 × 10 −28 g, electron density ne = 3.16 × 10 11 cm −2 , modulation strength η = 0.05, modulation period a = 260nm, momentum and single-particle relaxation times τ = 100pc and τs = 5pc, respectively.
APPENDIX B: PROPAGATOR AND KERNELSB αβ FOR WHITE-NOISE DISORDER
"Hall correlators". Substituting Eqs. (B.12)-(B.29) in (B.1) and (B.2), we obtain the kernels B (ρ) xx and B (ρ) xy averaged over φ q ,
with Eq. (16a) of Ref. 19. In the ballistic limit, T τ ≫ 1, the leading contribution [O(1/τ 2 )] to B 3iS 0 ω + 5ω 2 ]. (B.38)
2 S 2 0 − 35iω 3 S 0 + 35ω 4 , (B.40) independently of the relation between ω c and τ −1 . Similarly, using (B.36), one can find the correction to correction is independent of the relation between ω c and τ −1 . The results (B.40) and (B.41) are used for calculation of the interaction-induced corrections to ρ xx and ρ xy for the white-noise disorder and ω c ≪ T in Sec. V B. APPENDIX C: LINEAR-IN-T TERM IN THE BALLISTIC LIMIT AT B = 0
2 − (Ω + i0) 2 ] 1/2 [Q 2 − (Ω + i0) 2 ] 1/2 + i(Ω + i0) × dφ 2π (−iΩ)W (φ)(1 − cos φ) [Q 2 cos 2 φ 2 − (Ω + i0) 2 ][Q 2 − (Ω + i0) 2 ]
FIG. 14 :
14The contours C and C ′ of integration over S0 in Eq.
the coefficient of the T τ -term indeed vanishes. APPENDIX D: SOLUTION OF LIOUVILLE-BOLTZMANN EQUATION FOR A SMOOTH DISORDER
0
(φ) ≃ − iqv F τ cos φ (ω c τ ) 2 ,
dominates for ω c τ 1, so that the results of Ref.19 are applicable in sufficiently broad range of magnetic fields.2
330
11π
96 T τ
(5.18)
for the Coulomb interaction. The result (5.18) reduces
in the limit B → 0 to that obtained in Ref. 19 from the
quantum kinetic equation. We see that in view of a rela-
tively small value of the numerical coefficient 49/330, the
first (B-independent) term in square brackets in (5.18)
5 )
5This expression is identical, up to a sign, to the result of Ref.61 for Coulomb drag. This demonstrates that two contributions indeed cancel each other,
B.32) which agrees with Eq. (16b) of Ref. 19 up to an overall factor 1/2τ related to different normalization. In the ballistic limit, T τ ≫ 1, expanding (B.32) in τ −1 , one finds the leading contribution [O(1/τ )] given by the last two terms in (B.32),
leading to Eq.(3.1).To calculate the kernel B xy , we need a more accurate form of the propagator. Therefore, we should analyze the corrections to the eigenvalues and eigenfunctions of the Liouville-Boltzmann operator to the next order in (qR c ) 2 /ω c τ . To do this, it is convenient to perform the transformationand introduce the dimensionless variables β = ω c τ,Q = qR c β/(1 + β 2 ) 1/2 , Ω = 2ωτ . The equation for d(ω, q; φ, φ ′ ) takes then the formwhere we performed a rotationφ = φ+φ β , φ β = arccotβ.as a perturbation to the operatorwe find the unperturbed solutioñand the first-order correction to the eigenvaluesλ (1) n = 0. Calculating the n = 0 eigenfunctions and eigenvalue up to the second order in the perturbation (D.11) we finally obtain the singular part of the propagator for β ≫ 1 with required accuracy,where Q = qR c and the functions χ R,L (φ, Q) are given byAs to the regular part of the propagator, for n = 0 it is sufficient to calculate the eigenfunctions to the first order in the perturbation, which yieldsThe results (D.14)-(D.17) allow us to calculate the kernel B xy (ω, q) in the first non-vanishing order in β −1 , see Sec. III G.APPENDIX E: PROPAGATOR FOR ANISOTROPIC SYSTEMSIn this Appendix, we assume that the collision integral C induces a transport anisotropy, i.e. that the scattering cross-section W (φ, φ ′ ) is not a function of φ − φ ′ . The propagator D(ω, q; φ, φ ′ ) satisfies the equationWe first consider the diffusive limit and concentrate on the leading contribution D s governed by the diffusion mode. This requires finding a lowest eigenvalue Λ 0 of the operator in the l.h.s. of (E.1) and the corresponding left and right eigenfunctions. Treating the term iqv F cos(φ − φ ′ ) perturbatively as in Appendix E, we find Ψ R,L (φ) = 1 − iqv F ±ω c ∂ ∂φ +Ĉ
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[] |
[
"The Inverse Problem of Reconstructing Reaction-Diffusion Systems",
"The Inverse Problem of Reconstructing Reaction-Diffusion Systems"
] |
[
"Barbara Kaltenbacher ",
"William Rundell "
] |
[] |
[] |
This paper considers the inverse problem of recovering state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed to recover these sources. This leads to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.
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10.1088/1361-6420/ab8483
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[
"https://arxiv.org/pdf/2003.00489v1.pdf"
] | 211,677,834 |
2003.00489
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8505163286a402877f215ff297235a005014b405
|
The Inverse Problem of Reconstructing Reaction-Diffusion Systems
Mar 2020 March 3, 2020
Barbara Kaltenbacher
William Rundell
The Inverse Problem of Reconstructing Reaction-Diffusion Systems
Mar 2020 March 3, 2020
This paper considers the inverse problem of recovering state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed to recover these sources. This leads to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.
Introduction
Reaction diffusion equations have a rich history in the building of mathematical models for physical processes. They are descendants of nonlinear ordinary differential equations in time with an added spatial component making for a partial differential equation of parabolic type. These early models dating from the first decades of the twentieth century include that of Fisher in considering the Verhulst logistic equation together with a spatial diffusion or migration, u t − ku xx = f (u) = bu(1 − cu) to take into account migration of species and that of Kolmogorov, Petrovskii and Piskunov in similar models which are now collectively referred to as the Fisher-KPP theory of population modeling, see, [21]. There is also work in combustion theory due to Zeldovich and Frank-Kamenetskii that utilize higher order polynomials in the state variable u and where the diffusion term acts as a balance to the chemical reactions, [8].
The use of systems of reaction diffusion models followed quickly; adding a spatial component to traditional population dynamic models such as predator-prey and competitive species as well as the interaction of multiple species or chemicals. By the early 1950's it was recognized by Alan Turing that solutions to such equations can, under the correct balance of terms, be used to simulate natural pattern formations such as stripes and spots that may arise naturally out of a homogeneous, uniform state [30]. This theory, which can be called a reaction-diffusion theory of morphogenesis, has been a major recurrent theme across many application areas.
These models use the underlying physics to infer assumptions about the specific form of the reaction term f (u). The few constants appearing, if not exactly known, are easily determined in a straightforward way by a least squares fit to data measurements. We envision a more complex situation where the function f (u) (or multiple such functions as we will be considering systems of equations) cannot be assumed to have a specific known form, or to be analytic so that knowing it over a limited range gives a global extension, and therefore must be treated as an undetermined coefficient problem for a nonlinear partial differential equation.
Let Ω be a bounded, simply-connected region in R d with smooth boundary ∂ Ω, and let L := −∇ · (a(x)∇·) + q(x). be a uniformly elliptic operator of second order with L ∞ coef-
ficients u t (x,t) − Lu(x,t) = f 1 (u) + φ 1 (w) + r 1 (x,t, u, v), v t (x,t) − Lv(x,t) = f 2 (v) + φ 2 (w) + r 2 (x,t, u, v), w = w(u, v), (x,t) ∈ Ω × (0, T )(1)
for some fixed time T and subject to the prescribed initial and boundary conditions ∂ u ∂ ν (x,t) + γ 1 (x)u(x,t) = θ 1 (x,t)
∂ v ∂ ν (x,t) + γ 2 (x)v(x,t) = θ 2 (x,t) (x,t) ∈ ∂ Ω × (0, T ) u(x, 0) = u 0 (x), v(x, 0) = v 0 (x) x ∈ Ω(2)
where ν is the outer unit normal and γ i (x), θ i (x,t) are in C β (∂ Ω), where C β is the Schauder space with Hölder exponent β , 0 < β < 1.
In equation (1) we assume r 1 (x,t, u, v) and r 2 (x,t, u, v) are known and the interaction variable w = w(u, v) is also known but either the pair { f 1 , f 2 } or the pair {φ 1 , φ 2 } is unknown and the inverse problems posed are to determine these quantities.
Thus, there are two distinct inverse problems. The first is when we assume the interaction coupling φ i (w) between u and v is known, but both f 1 (u) and f 2 (v) have to be determined.
The second is when we assume the growth rate couplings f i for both u and v are known, but the interaction terms φ 1 (w) and φ 2 (w) have to be determined.
In order to perform these recoveries we must prescribe additional data and we shall again consider two possibilities: the values of u(x, T ) and v(x, T ) taken at a later, fixed time T u(x, T ) = g u (x), v(x, T ) = g v (x), x ∈ Ω;
or the time traces u(x 0 ,t) and v(x 0 ,t) measured at a fixed point x 0 ∈ ∂ Ω and for all t ∈ (0, T ).
u(x 0 ,t) = h u (t), v(x 0 ,t) = h v (t), x 0 ∈ ∂ Ω.(4)
If the boundary conditions at the measurement point x 0 are of Dirichlet type (γ = ∞) then instead we measure the flux a(x 0 )∂ ν at x 0 in (4). Note that final time data corresponds to census data taken a fixed time T for the species involved. The time trace data involves monitoring the population (or of chemical concentrations) at a fixed spatial point as a function of time. Both of these data measurements are quite standard in applications. In (3), the observation domain can be restricted to a subdomain ω of Ω; in view of the fact that the functions to be discovered are univariate, even an appropriately chosen curve in Ω could possibly suffice.
The interaction coupling w of u and v, which we assume known, can take on several forms. The near universal choice in ecological modeling is to take w = uv. This is also common in other applications but other more complex possibilities are in use.
For example, the Gray-Scott model of reaction diffusion, [23], takes w = u 2 v and is a coupling term that often leads to pattern formation. This coupling occurs, for example, in molecular realizations where there is an activator (u) and an inhibitor (v). The antagonistic effect here occurs from the relative depletion of v that is consumed during the production of u. The so-called Brusselator equation (the Walgraef and Aifantis equation) which also leads to the generation of sustained oscillations, instead takes w = bu − cu 2 v and occurs, amongst many other situations, in the dislocation dynamics in materials subjected to cyclic loading, [31].
Other possibilities include w = √ u 2 + v 2 or the nonlocal situation w = Ω (u 2 + v 2 ) dx as well as combinations of all of the above.
The astute reader will already have noted that there are other very similar combinations that might lead to physically-motivated inverse problems. Indeed, this is the case and the analysis from Section 3.1 also applies to other combinations of unknowns than those mentioned above.
In the case of a single equation using time trace data, uniqueness results and the convergence of reconstruction algorithms were shown in [4,27,25] for the recovery of the unknown term f (u). In the more recent paper [16], the authors used final time data as in (4) and showed uniqueness and contractibility results for a fixed point iteration scheme which allowed effective recovery of f (u). We also point to [15] where the reaction term is of the form q(x) f (u) with known f (u) and an unknown space dependent term q(x) that controls the intensity of the reaction.
The reconstruction methods that we consider here are based on a projection of the PDE on the observation manifold, which naturally leads to a fixed point iteration for the unknown terms. As compared to Newton's method, which is a more general approach, this projection principle requires an appropriate form of the given data; in particular, the observation manifold cannot be orthogonal to the dependence direction of the unknown function, i.e., if we aim at reconstructing an x dependent coefficient then time trace observations cannot be used in such a projected approach (and will hardly give good reconstruction results with any other method either). Since here the unknown function does not depend on x or t this problem does not appear, though. Moreover, a convergence proof of regularized Newton type methods can only be carried out under certain conditions on the forward operator, such as the tangential cone condition (see, e.g., [14] and the references therein), which is most probably not satisfied here. Also note that the fixed point approach used here entails a uniqueness result for the inverse problem.
We remark that we have followed standard mathematical practice when writing differential equations by scaling the equation to set parameters not under consideration to unity. Thus in our reconstruction we have taken the volume of Ω to be one and the elliptic operator L to be the Laplacian thereby making the implicit assumption that the diffusion coefficient a(x) in the leading term div(a∇u) is unity. In an actual physical situation this will not be the case; for example the value of a for a molecule diffusing in a gas will be several orders of magnitude smaller.
The outline of the paper is as follows. Section 3 provides an extension of the convergence analysis in [16] for a fixed point iteration as well as its consequences for uniqueness in two directions. Firstly, we "vectorize" this analysis to be applicable for a class of reactiondiffusion equations, characterized by certain conditions. Secondly, we allow for higher space dimensions by carrying out the estimates in Schauder spaces rather than (Hilbert) Sobolev spaces, thus largely avoiding dependence on the space dimension due to Sobolev embeddings. The focus is on the final data case (3) there. In Section 4 we show numerical tests with the discussed fixed point iteration. We provide reconstructions of the pair ( f 1 , f 2 ) for known φ i , r i , w, as well as of the pair (φ 1 , φ 2 ) for known f i , r i , w in (1). Here, both settings of final time data (3) and of time trace data (4) are considered.
Preamble
We start with a short revision of the results in [16], since the present paper builds on these. As a matter of fact, in [16], we considered a scalar problem with only one unknown function f but the setting there is slighty more general than the one described in the introduction (and most of the rest of this paper) in the sense that a subdiffusion equation is considered. More precisely, in [16] we treat the semilinear (sub)diffusion initial boundary value problem D α t u(x,t) + Lu(x,t) = f (u(x,t)) + r(x,t) (x,t) ∈ Ω × (0, T ) ∂ u ∂ ν + γu = 0 on ∂ Ω × (0, T )
u(x, 0) = u 0 (x), x ∈ Ω(5)
where D α t denotes the Djrbashian-Caputo fractional time derivative of order α ∈ (0, 1) which is defined by
D α t u = 1 Γ(1 − α) t 0 u ′ (s) (t − s) α ds(6)
In case α = 1, D α t is the usual first time derivative and (5) becomes a semilinear parabolic equation. For details on fractional differentiation and subdiffusion equations, we refer to, e.g., [1,2,3,20,28,29], see also the tutorial on inverse problems for anomalous diffusion processes [12]; For well-posedness of the forward problem (5) with Lipschitz continuous f we refer to [11,Section 3], see also [16,Section 2].
In [16], the inverse problem of recovering the nonlinearity f from final time data g(x) ≡ u(x, T ),
x ∈ Ω
(assuming that all other coefficients in (5) are known) was approached by a fixed point scheme for the operator T defined by the identity (T f )(g(x)) = D α t u(x, T ; f ) − Lg(x) − r(x, T ) x ∈ Ω . where for given f , the function u(x,t; f ) denotes the solution to (5). This defines T f just on the range of g and therefore it is crucial to assume that all values of u that get inserted into f belong to this range J := g(Ω) ⊇ u(Ω × [0, T )) . (8) More precisely, it suffices to impose this condition at the solution ( f ex , u ex ) of the inverse problem, and to project the iterates onto this range during the fixed point iteration. Obviously, these considerations can be extended to the case of partial observations g(x) ≡ u(x, T ), x ∈ ω on a subset ω ⊂ Ω, under the adapted range condition g(ω) ⊇ u(Ω ×[0, T )).
Since f is an univariate function, measurements on a higher dimensional domain Ω appear to be too much in fact and it can suffice to take observations along a curve ω connecting the points of minimal and maximal values of u(x, T ).
The following example illustrates the fact that this imposes a true constraint on the class of problems that can be expected to exhibit unique identifiability. Example 2. 1. Let ϕ be an eigenfunction of −L with corresponding eigenvalue λ > 0. Take f (u) = cu as well as u 0 = ϕ , r = 0, then u(x,t; f ) = e −(λ −c)t ϕ(x). If ϕ stays positive (which is, e.g., the case is λ is the smallest eigenvalue), and c < λ , then, with ϕ = min x∈Ω ϕ(x) ≥ 0, ϕ = max x∈Ω ϕ(x) > 0, we get for the range of u over all of Ω, and therefore the range condition will be violated.
In [16,Section 3] we proved a self-mapping property of T on a sufficiently small ball in W 1,∞ (J), as well as its contractivity in the parabolic case α = 1 for T large enough provided f is strictly monotonically decreasing, which implies exponential decay of the corresponding solution or actually its time derivative D t u, as long as r vanishes or has exponentially decaying time derivative, see also Lemma 3.1 below. Such a dissipative setting is indeed crucial for proving that the Lipschitz constant of T decreases with increasing final time T , as the following counterexample shows. Example 2. 2. Again, let ϕ(x) be an eigenfunction of −L with corresponding eigenvalue λ > 0. Take f (1) (u) = c 1 u, f (2) (u) = c 2 u as well as u 0 = ϕ , r = 0, and set u (i) (x,t) = u(x,t; f i ) which can be computed explicitely as
u (i) (x,t) = e (c i −λ )t ϕ(x) , u (i) t (x,t) = (c i − λ )e (c i −λ )t ϕ(x) , which yields T f (1) (g(x)
) − T f (2) (g(x)) = u (1) t (x, T ) − u (2) t (x, T ) = (c 1 − λ )e c 1 T − (c 2 − λ )e c 2 T e −λ T ϕ(x) .
Thus, for any combination of norms · X , · Z the contraction factor T f (1) (g)−T f (2) The two examples above clearly show that the two aims (a) range condition and (b) dissipa-tivity (for contractivity) are conflicting, at least as long as we set r = 0 as it was done here. Thus, in order to achieve (a) we need to drive the system by means of r (or alternatively, by inhomogeneos boundary conditions). Luckily, nonvanishing r does not impact the case (b) since these inhomogenieities basically cancel out when taking differences between fixed point iterates for establishing contractivity estimates. Thus it is indeed possible to have range condition and contractivity together. However, for this it is crucial to not only use found data, but to be able to design the experiment in such a way that the data exhibits the desired properties.
(g) Z | f (1) − f (2)
The analysis in [16] is restricted to the case of a single scalar equation and to one space dimension, due to the fact that it is carried out in (Hilbert) Sobolev spaces using embeddings into W 1,∞ . The aim of the analysis here is therefore twofold. First of all, we intend to generalize the approach from [16] towards identification of nonlinearities in certain systems of reaction-diffusion equations, see Section 3. 1. Secondly, we provide an analysis of the above fixed point scheme in Schauder spaces, which allows us to work in higher space dimensions, see Section 3. 2. For the latter case we have to restrict ourselves to the parabolic setting α = 1, as carrying over the regularity theory from, e.g., [6] to the subdiffusion setting, based on results from, e.g., [5], would be a major effort in itself that goes beyond the scope of this paper. This also affects the extension of the analysis from [27,26] for time trace (4) instead of final time data (3), which is why we focus on the final time observation case in the analysis section 3.
A generalization of the scalar diffusion setting to systems of an arbitrary number N of possibly interacting states can be achieved by replacing the unknown function f in (5) by a component wise defined vector valued unknown nonlinearty f = ( f 1 , . . ., f N ), whose action is not only defined on a single state but on a possible combination of these, via known functions w = (w 1 , . . ., w N ). Possible additional known interaction and reaction terms are encapsulated in a set of (now potentially also nonlinear) functions r = (r 1 , . . . , r N ).
We thus consider systems of reaction-(sub)diffusion equations of the form D α t u i (x,t) + (L u) i (x,t) = f i (w i ( u(x,t))) + r i (x,t, u(x,t)) (x,t) ∈ Ω × (0, T ) , i ∈ {1, . . .N} ,
for u = (u 1 , . . ., u N ) subject to the boundary and initial conditions
∂ u i ∂ ν + γ i u i = 0 on ∂ Ω × (0, T ) , i ∈ {1, . . .N}(10)
and
u(x, 0) = u 0 (x), x ∈ Ω .(11)
Well-posedness of this forward problem with Lipschitz continuous nonlinearities f i , r i is a straightforward extension of the results from [11,Section 3], [16,Section 2].
Given the self-adjoint uniformly elliptic operator −L, e.g. − L = diag(−∇ · (A∇·), . . ., −∇ · (A∇·)) + Q(x) (12) with Q : Ω → R N×N , A ∈ R N×N symmetric (uniformly) positive definite, the functions w : I := I 1 × · · · × I N → J := J 1 × · · · × J N , r : Ω × (0, T ) × I → R N , and the data u 0 , γ i , as well as measurements on a subset ω of the domain Ω
g i (x) ≡ u i (x, T ), x ∈ ω , i ∈ {1, . . .N} ,(13)
we wish to determine the unknown functions
f i : J i → R i ∈ {1, . . .N} .
This includes both cases of identifying the reaction terms f i (by setting w i (ξ 1 , ξ 2 ) = ξ i ) and of identifying the interaction terms φ i in (1). We will abbreviate the collection of unknown functions by f = ( f 1 , . . . , f N ), noting that each individual f i might have a different domain of definition J i .
Note that this setting allows for linear and nonlinear coupling among the individual states u i via the known differential operator L (often referred to as cross-diffusion) and the known functions w i as well as r i .
Throughout the analysis we will impose the range condition
I i = [g i , g i ] = [min x∈ω g i (x) , max x∈ω g i (x)] = g i (ω) ⊇ u ex,i (Ω × [0, T )) , i ∈ {1, . . .N} ,(14)
cf, (8), where we assume ω to be compact to guarantee (via Weierstrass' Theorem and continuity of g i ) that I i is indeed a compact interval. Here u ex is the state part of a solution ( f ex , u ex ) of the inverse problem.
Part of the analysis is based on series expansions in terms of the eigenvalues and -functions (λ n , φ n ) n∈N of the self-adjoint elliptic operator L as well as the induced Hilbert spaceṡ H σ (Ω) = {v ∈ L 2 (Ω) :
∞ ∑ n=1 λ σ /2 n v, φ n φ n ∈ L 2 (Ω)} where ·, · denotes the L 2 inner product on Ω, with the norm v Ḣσ (Ω) = ∞ ∑ n=1 λ σ n v, φ n 2 1/2
, that is equivalent to the H σ (Ω) Sobolev norm provided the coefficients of L are sufficiently regular, which we assume to be the case here.
Consider now the spatially one-dimensional setting of Ω ⊆ R 1 being an open interval (0, L), and make the invertibility and smoothness assumptions
g ∈ H 2 (ω; R N ) , w ∈ H 2 (I 1 × · · · × I N ; R N ) N ∑ j=1 ∂ w i ∂ ξ j ( g(x)) g ′ j (x) ≥ β > 0 for all x ∈ ω , i ∈ {1, . . .N} ,(15)
that by the Inverse Function Theorem imply that w i • g : ω → J i is bijective and its inverse is in H 2 (J i ; Ω). Thus we can define the fixed point operator T : X → X , where
X := X 1 × · · · × X N , X i = { f i ∈ W 1,∞ (J i ) : f i (w i ( u 0 )) ∈Ḣ 2 (Ω)} ,(16)
by
(T f ) i (w i ( g(x)) = D α t u i (x, T ; f ) − (L g) i (x) − r i (x, T, g(x)) x ∈ ω , i ∈ {1, . . .N} , (17) where for any j = ( j 1 , . . . , j N ) with j i : J i → R, the function u(x,t) = u(x,t; j) solves D α t u i (x,t) + (L u) i (x,t) = h i (w i (P u(x,t))) + r i (x,t, P u(x,t)) (x,t) ∈ Ω × (0, T ) , i ∈ {1, . . .N} , ∂ u i ∂ ν + γ i u i = 0 on ∂ Ω × (0, T ) , i ∈ {1, . . .N} u(x, 0) = u 0 (x), x ∈ Ω ,(18)
with the projection P : R N → I on the compact cuboid I, i.e., P i ξ = max{g i , min{g i , ξ }}.
The range condition (14) guarantees that any solution ( f ex , u ex ) of the inverse problem (9), (10), (11), (13) is a fixed point of T.
Besides the resonstruction problem (9), (13) with final time data, that will be discussed in detail in the convergence section 3, in the numercial reconstruction section we will also consider an analogous inverse problem of recovering f = ( f 1 , . . .,
f N ) in (9) from time trace data h(t) ≡ u(x 0 ,t), t ∈ (0, T ) ,(19)
for some x 0 ∈ ∂ Ω, cf. Pilant and Rundell [27,26] for the scalar case. Under the invertibility condition
h ∈ C 1,1 (0, T ; R N ) , w ∈ C 2 (I 1 × · · · × I N ; R N ) N ∑ j=1 ∂ w i ∂ ξ j ( h(t)) h ′ j (t) ≥ β > 0 for all t ∈ (0, T ) , i ∈ {1, . . .N} ,(20)
we can, analogously to [27,26], define a fixed point operator T :
X := X 1 × · · · × X N → X 1 × · · · × X N , where X i = C 0,1 (J i ), by (T f ) i (w i ( h(t)) = D α t h i (t) − (L u) i (x 0 ,t; f ) − r i (x 0 ,t, h(t)) t ∈ (0, T ) , i ∈ {1, . . .N} .(21)
The crucial estimates of (L u) i (x 0 ,t; f ) C 0,1 (Ω) required for establishing self-mapping and contraction properties of T on a ball in X as in [27,26] could in principle like there be based on the implicit representation
u(x,t) = ψ(x,t) + t 0 Ω K(x, y,t − τ) f (w( u(y, τ)) + r(y, τ, u(y, τ)) dy
of u by means of the Green's function K and the solution ψ(x,t) = u(x,t; 0) of the linear problem obtained by setting f ≡ 0, and replacing r(x,t, u(x,t)) by r(x,t, 0), together with regularity estimates on the Green's function K. For Green's functions for systems and their regularity in the parabolic case α = 1, see, e.g. [6, Theorem 1, Chapter 9]. In the subdiffusion case α < 1 the Green's function is defined by the Fox H-functions, cf., e.g., [5].
Convergence of a fixed point scheme for final time data
We will now consider convergence of the fixed point scheme defined by the operator T defined by (17) for reconstructing the reaction and interaction functions f i in (9). To some extent we can here build on previous work for the case of one scalar equation and a single function f to be reconstructed in [16]. However, it is also clear that the interaction among several states can complicate the situation considerably and lead to phenomena that would not be possible with single uncoupled equations. Our aim in Section 3.1 is to explore conditions for a scenario that would allow to make some statements on self-mapping and contractivity of T. We are aware of the fact that this is far from capturing the whole multitude of possibilities and interesting cases that can arise in systems. The theoretical results are illustrated by some examples of 2 × 2 systems arising in systems biology.
Another extension made in this section is to get rid of the restriction on the spatially one dimensional setting that had to be imposed in [16] in order to enable certain Sobolev embed-dings, in particular at the transition from H s (Ω) for the space dependent function f (g(x)) to W 1,∞ (J) for the univariate function f (u). We do so by carrying out the analysis in Schauder spaces instead, which basically allows to use the same differentiability order for f (g(x)) and f (u), independently of the dimension of Ω. Since the required regularity results on PDE solutions are so far only available in the literature for α = 1, we restrict ourselves to this case in Section 3. 2. 3.1 Vectorization of the results from [16] in the spatially one-dimensional case
Analogously to the proof of [16, Theorem 3.1] we can establish T as a weakly * continuous self-mapping on a sufficiently large ball in X . (15) hold and assume that κ as well asρ := sup ζ ∈I D t r(·, ·, ζ ) L Q * (0,T ;L 2 (Ω)) are sufficiently small. Then for large enough ρ > 0 the operator T defined by (17) is a self-mapping on the bounded, closed and convex set
Theorem 3.1. Let α ∈ ( 4 5 , 1], σ ∈ ( 3 2 , 2), θ ∈ (0, 2 − 1/α), Q * ≥ 2(2−θ ) 2−σ , Ω ⊆ R 1 an open bounded interval, letB = {h ∈ X : h i W 1,∞ (J i ) ≤ ρ , h i (w i ( u 0 )) + r(·, 0, u 0 ) − L u 0 Ḣσ (Ω) ≤ κ}
and T is weakly* continuous in X as defined in (16). Thus T has a fixed point in B.
Moreover, in the parabolic case α = 1, contractivity of T for sufficiently large final time T follows as in [16,Theorems 3.2,3.3] from the fact that for f (1)
, f (2) ∈ X , the difference T( f (1) ) − T( f (2) ) = ( u (1) − u (2) ) t = z, where D t z i − (L z) i = N ∑ j=1 f (1) i ′ (w i ( u (1) )) ∂ w i ∂ ξ j ( u (1) ) + ∂ r i ∂ ξ j ( u (1) ) )z j + f (1) i ′ (w i ( u (1) )) ∂ w i ∂ ξ j ( u (1) ) + ∂ r i ∂ ξ j ( u (1) ) − f (2) i ′ (w i ( u (2) )) ∂ w i ∂ ξ j ( u (2) ) + ∂ r i ∂ ξ j ( u (2) ) D t u (2) j in Ω × (0, T ) ∂ z i ∂ ν + γ i z i = 0 on ∂ Ω × (0, T ) z i (x, 0) = f (1) i (w i ( u 0 (x))) − f (2) i (w i ( u 0 (x))) x ∈ Ω i ∈ {1, . . .N}.
The factor D t u (2) appearing in the right hand side of this PDE decays exponentially under certain conditions. Lemma 3. 1. Let L be of the form (12) where for some c Q > −λ , with λ > 0 the smallest eigenvalue of −∇ · (A∇·) with homogeneous impedance boundary conditions,
Q(x) − M(x,t, ξ ) − c Q id is nonnegative definite for all x ∈ Ω, t > 0, ξ ∈ I ,(22)
(note that here the definition of nonnegative definiteness of a matrix here does not necessarily include its symmetry) where
M i, j (x,t, ξ ) := f ′ i (w i (ξ )) ∂ w i ∂ ξ j (ξ ) + ∂ r i ∂ ξ j (ξ )(23)
and assume that there exist constants C r , c r > 0 such that |D t r(x,t, ξ )| ≤ C r e −c r t x ∈ Ω , t > 0 , ξ ∈ I , and that D t u i (0) = (L u 0 ) + f i (w i ( u 0 )) + r i (0, u 0 ) ∈ L ∞ (Ω).
Then there exist C 2 , c 2 > 0 (more precisely, c 2 ∈ (0, min{c r , λ +c Q 2 })) such that the solution u of (9), (10), (11) satisfies the exponential decay estimate
|D t u(x,t)| ≤ C 2 e −c 2 t x ∈ Ω , t > 0 .
Proof. Exponential decay of D t u follows from the maximum principle applied to the scalar
function v(x,t) = 1 2 |D t u(x,t)| 2 = 1 2 ∑ N i=1 D t u i (x,t) 2 .
In the case of (12) this satisfies, with c 1 = c Q + λ , the equations
D t v − △v + (c Q − ε)v = −∇D t u T A∇D t u − D t u T (Q(x) − M(x,t, P u(x,t)) − c Q id)D t u − ε 2 |D t u| 2 + D t u T D t r in Ω × (0, T ), ≤ −D t u T (Q(x) − M(x,t, P u(x,t)) − c Q )id)D t u + 1 2ε |D t r| 2 ∂ v ∂ ν + γv = 0 on ∂ Ω × (0, T ) , v(x, 0) = 1 2 |D t u(0)| 2 = 1 2 N ∑ i=1 | − (L u 0 ) + f i (w i ( u 0 )) + r i (0, u 0 )| 2 x ∈ Ω,(24)
where we have used Young's inequality. The right hand side in (24) can be bounded by the exponentially decaying function 1 2ε |D t r(t)| 2 , provided Q(x) − M(x,t, P u(x,t)) − c Q id is positive semidefinite for all (x,t) ∈ Ω × (0, T ), which is guaranteed by (22). By the maximum principle, this implies v ≤v pointwise in Ω × (0, T ), for the solutionv ofv t − △v + (c Q − ε)v = 1 2ε |D t r(t)| 2 with initial datav(x, 0) = 1 2 |D t u(0)| 2 and therefore (cf. [22]) v(x,t) ≤v(x,t) ≤ 1 4ε e − min{2c r ,λ +c Q −ε}t |D t u(0)| 2 for all (x,t) ∈ Ω × (0, T ).
Remark 3.1. A possible way to satisfy condition (22) is by sufficiently strong diffusion in the elliptic operator (12) λ min (Q(x)) + λ > sup
t∈(0,T ),ξ ∈I,ζ ∈R N \{0} 1 |ζ | 2 ζ T ( f ′ ( w(ζ ) d w dξ (ξ ) + d r dξ (x,t, ξ )) ζ
for all x ∈ Ω, t ∈ (0, T ).
We mention in passing that for the proof of Lemma 3.1 it was essential to work with homogeneous boundary conditions.
Along the lines of the proof of [16,Theorem 3.2] we therefore obtain the following contractivity result. Theorem 3.2. Let the assumptions of Lemma 3.1 hold and assume that
f (1) i ′ (w i ( u (1) )) ∂ w i ∂ ξ j ( u (1) ) + ∂ r i ∂ ξ j ( u (1) ) +Q i j L ∞ (0,T ;L ∞ (Ω)) ≤ĉ 1 ,(25)
for some positive semidefinite matrixQ(x) andĉ 1 sufficiently small such that 4ĉ 1 <λ 2 1 , whereλ 1 is the smallest eigenvalue of the operator −L := −L +Q, i.e., 4ĉ 1 <λ 2 1 . Moreover, assume that f
(1) i , f (2) i ∈ W 2,∞ (J i ) and N ∑ i=1 ( f (1) i − f (2) i )(w i ( u (2) (t)) 2 H 1 (Ω) ≤C 1 (g) N ∑ i=1 ( f (1) i − f (2) i )(w i ( g)) H 1 (ω) .(26)
Then there exists a constant C > 0 depending only on f (1)′′ L ∞ (I) , (L u 0 ) + f i (w i ( u 0 )) + r i (0, u 0 ) L ∞ (Ω) and the constantC 1 (g) in (26), such that
(T( f 1 ) − T( f 2 ))(g) Ḣ1 (Ω) ≤ Ce −(λ 1 −4ĉ 1 /λ 1 )T /2 ( f 1 − f 2 )(g) Ḣ1 (Ω) .
withĉ 1 as in (25).
Note that this result does not rely on any Sobolev embeddings and therefore remains valid for higher space dimensions, provided we can make sense of the condition (26), which in one space dimension easily follows even in the general form
N ∑ i=1 h i (w i ( u (2) (t)) 2 H 1 (Ω) ≤C 1 (g) N ∑ i=1 h i (w i ( g)) H 1 (ω) ∀ h ∈ W 2,∞ (J)
from uniform strict monotonicity of the functions g i and x → u (2) i (x,t) for all t ≥ 0 as well as the range condition (14).
Some Examples
We now provide some examples of typical systems to which the analysis above applies.
Example 3.1. D α t u i − △u i = f i (u i ) i ∈ {1, . . . N} ,(27)
where w i (ξ 1 , . . . , ξ N ) = ξ i , and (15) is satisfied for strictly monotone and H 2 smooth g i , with J i = I i = g i (ω). The Jacobian of w is just the identity matrix and M(x,t, (22) is satisfied for monotonically decreasing functions f j , or if the f ′ j have arbitrary sign but their positive values on I j are dominated by λ min (Q(x)) + λ . This clearly extends to
ξ ) = diag( f ′ 1 (ξ 1 ) . . . f ′ N (ξ N )), soD α t u i − △u i = f i (u i ) + r i ( u) i ∈ {1, . . . N}
, with sufficiently small interaction terms r i . A particular case of interest here is the so-called competing species interaction where r i ( u) = −u i ∑ j =i β i j u j with nonnegative coefficients β i j , cf., e.g., [13] and see Example 3.2 below for the case N = 2.
We now turn to some examples of 2 × 2 systems, where condition (22) can be verified by applying the simple criterion
A B C D nonnegative definite ⇐⇒ A ≥ 0 , D ≥ 0 , 4AD ≥ (B +C) 2 .(28)Example 3.2. D α t u − △u = f 1 (u) − β u · v D α t v − △v = f 2 (v) − β u · v (29) with β > 0. Here M(x,t, u, v) = f ′ 1 (u) − β v −β u −β v f ′ 2 (v) − β u
, and we expect both u and v to be nonnegative, thus set
I 1 = [0, u max ] , I 2 = [0, v max ](30)
for some upper bounds u max , v max > 0. Nonnegativity of −M via (28) is equivalent to
− f ′ 1 (u) + β v ≥ 0 , − f ′ 2 (v) + β u ≥ 0 , 4(− f ′ 1 (u) + β v)(− f ′ 2 (v) + β u) ≥ β 2 (u + v) 2 .
A sufficient condition for this to hold is
− f ′ 1 (u) ≥ β 2 u, − f ′ 1 (v) ≥ β 2 v for all u ∈ I 1 , v ∈ I 2 . On the other hand, setting v = 0 implies − f ′ 1 (u) ≥ 0 and − f ′ 1 (u) ≥ β 2 u 2 4(− f ′ 2 (0)+β u) ; likewise we get − f ′ 2 (v) ≥ 0 and − f ′ 2 (v) ≥ β 2 v 2 4(− f ′ 1 (0)+β v) . This implies β ≤ min u∈(0,u max ] 2 u (− f ′ 1 (u) + f ′ 1 (u) 2 − f ′ 2 (0) f ′ 1 (u)) β ≤ min v∈(0,v max ] 2 v (− f ′ 2 (v) + f ′ 2 (v) 2 − f ′ 1 (0) f ′ 2 (v)) ,
i.e., a restriction on the size of the interaction β u.v, in terms of the reactions
{ f 1 , f 2 }. Example 3.3. D α t u − △u = f (u) + r 1 (u, v) D α t v − △v = r 2 (u, v)(31)
see, e.g., [17, equations (1), (2), page 61], where typically r 2 (u, v) = −r 1 (u, v) or at least the individual corresponding terms have opposite sign. More precisely, they are often of the form
r 1 (u, v) = k 1+ u ν 11 v ν 12 − k 1− u µ 11 v µ 12 r 2 (u, v) = k 2+ u ν 21 v ν 22 − k 2− u µ 21 v µ 22(32)
with nonnegative constants k i± , ν i j , µ i j , which would be the characteristic for, e.g., massaction kinetics. In view of the practically relevant setting of nonnegative states, we will again consider the nonlinear functions on
I 1 = [0, u max ] , I 2 = [0, v max ].
In this example,
M(x,t, u, v) = f ′ (u) + r 1,u (u, v) r 1,v (u, v) r 2,u (u, v) r 2,v (u, v)
, so that nonnegativity of −M via (28) is equivalent to
r 2,v (u, v) ≤ 0 ∀ u ∈ [0, u max ] , v ∈ [0, v max ] (33) f ′ (u) ≤ − sup v∈[0,v max ] r 1,u + (r 1,v + r 2,u ) 2 (u, v) −4r 2,v (u, v) ∀ v ∈ [0, v max ] .(34)
In case of the particular form (32), condition (33) is equivalent (by considering the asymptotics as u, v → 0) to ν 21 ≥ µ 21 and ν 22 ≥ µ 22 and ν 22 k 2+ u
ν 21 −µ 21 max v ν 22 −µ 22 max ≤ µ 22 k 2− .(35)
The requirements of f ′ ∈ L ∞ ([0, u max ]), and r 1,
u ∈ L ∞ ([0, u max ] × [0, v max ])
lead to further restrictions on the exponents ν i j , µ i j to avoid singularities at vanishing u, v. Example 3. 4.
D α t u − △u = f (v) + r 1 (u, v) D α t v − △v = r 2 (u, v)(36)
see, e.g., [17, equations (18), (19), page 76].
Here, M(x,t, u, v) =
r 1,u (u, v) f ′ (v) + r 1,v (u, v) r 2,u (u, v) r 2,v (u, v)
, so that nonnegativity of −M via (28) is equivalent to
r 1,u (u, v) ≤ 0 , r 2,v (u, v) ≤ 0 ∀ u ∈ [0, u max ] , v ∈ [0, v max ] (37) | f ′ (v) + r 1,v (u, v) + r 2,u (u, v)| ≤ 2 r 1,u (u, v) r 2,v (u, v) ∀ u ∈ [0, u max ] , v ∈ [0, v max ] .(38)
In the setting of (32), condition (37)
≤ µ 11 k 1− .(39)
Note that f ′ ∈ L ∞ ([0, v max ]) together with (38) does not impose additional constraints on the exponents in (32). Example 3. 5.
D α t u − △u = f 1 (u) + r 1 (u, v) D α t v − △v = f 2 (u) + r 2 (u, v)(40)
see, e.g., [17, page 89].
With M(x,t, u, v) = f ′ 1 (u) + r 1,u (u, v) r 1,v (u, v) f ′ 2 (u) + r 2,u (u, v) r 2,v (u, v)
, nonnegativity of −M via (28) is equiv-
alent to f ′ 1 (u) ≤ − sup v∈[0,v max ] r 1,u ∀ v ∈ [0, v max ] , r 2,v (u, v) ≤ 0 ∀ u ∈ [0, u max ] , v ∈ [0, v max ] (41) | f ′ 2 (u) + r 1,v (u, v) + r 2,u (u, v)| ≤ 2 ( f ′ 1 (u) + r 1,u (u, v)) r 2,v (u, v) ∀ u ∈ [0, u max ] , v ∈ [0, v max ](42)
In the setting of (32), the right hand part of (41) coincides with (35) in Example (31).
Again, f ′ i ∈ L ∞ ([0, v max ])
together with (42) does not impose additional conditions on the exponents in (32).
Analysis in Schauder spaces in the parabolic case and higher space dimensions
In the parabolic case α = 1, the availability of regularity results in Schauder spaces C k,β (Ω× (0, T )), cf., e.g, [6], allows to work in higher space dimensions Ω ⊆ R d , d > 1. Thus these results are new as compared to those in [16] even in case of a single PDE. Note that the Schauder space setting has already been used in [27,26] for the same nonlinearity identification problems in case of time trace (instead of final time) observations.
For simplicity of exposition we here consider the scalar case (abbreviating D t u by u t since there will be no further subscripts here) of recovering f :
J → R in u t (x,t) − Lu(x,t) = f (u(x,t)) + r(x,t) (x,t) ∈ Ω × (0, T ) ∂ u ∂ ν + γu = 0 on ∂ Ω × (0, T ) u(x, 0) = u 0 (x) x ∈ Ω (43) with observations g(x) = u(x, T ) x ∈ ω .(44)
Note that analogously to Section 3.1, this can be extended to the system setting (9)-(13), provided a higher dimensional (with respect to space) version of the condition (15) holds and guarantees invertibility as well as smoothness of the mapping w i • g : ω → J i .
Self-mapping fixed-point operator on spaces of Lipschitz continuous functions
In order to work in function spaces over a fixed interval J := g(ω) = [g, g], like in [16] we project the values of u onto J, which can as well be written by means of a superposition operator (Pu)(x,t) = max{g, min{g, u(x,t)}} = Φ(u(x,t))
with Φ = id on [g, g], Φ ≡ g on (−∞, g], Φ ≡ g on [g, ∞)
. Note that Φ is contained in W 1,∞ (R) = C 0,1 (R) which will be sufficient for the proof of T being a self-mapping on X = C 0,1 (J). Later on, when using higher order Schauder spaces to show contractivity of T, the lack of additional smoothness of Φ will remain an issue.
Moreover, we assume the range condition
J = [g, g] = g(ω) ⊇ u ex (Ω × [0, T ))(45)
to hold for any exact solution ( f ex , u ex ) of the inverse problem (43), (44).
Thus we define the fixed point operator T : X → X by
T f (g(x)) = u t (x, T ; f ) − (Lg)(x) − r(x, T ) x ∈ ω ,(46)
where for some j ∈ C 0,1 (J), the function u(x,t; j) solves
u t − Lu = j(Φ(u)) + r in Ω × (0, T ) ∂ u ∂ ν + γu = 0 on ∂ Ω × (0, T ) u(x, 0) = u 0 (x) x ∈ Ω .(47)
Equation (46) indeed uniquely determines f + := T f on J if, e.g, ω is curve in Ω along which g is strictly monotone. Otherwise, the transition from the multivariate function f + (g) : ω → R, defined on the d-dimensional domain ω , to the real function f : J → R, defined on an interval J ⊆ R, can be carried out by metric projection, cf. [16, Lemma 3.1, Remark 3.1]. We therefore redefine T : X → X by
T f = P g y where y(x) = u t (x, T ; f ) − (Lg)(x) − r(x, T ) x ∈ ω(48)
and assume that the mapping P g : C 0,1 (ω) → C 0,1 (J) satisfies the compatibility condition
f ex = P g (u ex,t (T ) − Lg − r(T ))(49)
for an exact solution ( f ex , u ex ) of the inverse problem (43), (44), and is continuous as a mapping from C 0,1 (ω) to C 0,1 (J), i.e., for all y ∈ C 0,1 (ω), the bound
P g y C 0,1 (J) ≤ C(g) y C 0,1 (ω)(50)
holds. If, e.g, ω is a curve with a regular parametrization ω = {x(τ) : τ ∈ [0, 1]} such that ∇g(x(τ)) ·ẋ(τ) ≥ 1 C(g) |ẋ(τ)| for all τ ∈ [0, 1], then for P g simply defined by (P g w)(g(x)) = w(x), x ∈ ω , the estimate (50) follows from
sup ξ =η∈J |(P g w)(ξ ) − (P g w)(η)| |ξ − η| = sup x =y∈ω |w(x) − w(y)| |g(x) − g(y)| = sup σ =τ∈[0,1] |w(x(σ )) − w(x(τ))| |g(x(σ )) − g(x(τ))| = sup σ =τ∈[0,1] |w(x(σ )) − w(x(τ))| |x(σ ) − x(τ)| |x(σ ) − x(τ)| |g(x(σ )) − g(x(τ))| where |g(x(σ ))−g(x(τ))| = σ τ ∇g(x(ρ)) ·ẋ(ρ) dρ ≥ 1 C(g) σ τ |ẋ(ρ)| dρ ≥ 1 C(g) |x(σ )−x(τ)| .
Assumptions (45) and (49) imply that the f part of any such solution ( f ex , u ex ) is a fixed point of T.
We will now prove that T is a self-mapping on X = C 0,1 (J). To do so, we use the identity
T f (g) − f ex (g) = (u t (T ; f ) − Lg − r(T )) − (u ex,t (T ) − Lu ex (T ) − r(T )) = u t (T ; f ) − u ex,t (T ) ,(51)
to which we apply P g to obtain, using (50),
T f − f ex C 0,1 (J) ≤ C(g) û t (T ) C 0,1 (ω) . whereû(x,t) = u(x,t; f ) − u ex (x,t) with u(x,t) = u(x,t; f ) solving (47) with j = f , and thereforeû = u − u ex solveŝ u t − Lû +cû = f (Φ(u)) − f ex (u) in Ω × (0, T ) ∂û ∂ ν + γû = 0 on ∂ Ω × (0, T ) u(x, 0) = 0 x ∈ Ω . wherec = − f ex (u)− f ex (u ex ) u−u ex ∈ L ∞ ((0, T ) × Ω) since f ex ∈ C 0,1 (J). From [6, Theorem 6, page 65] we obtain that û t (T ) C 0,1 (ω) ≤ û t C 0,1 ((0,T )×Ω) ≤ K f (Φ(u)) − f ex (u) C 0,1 ((0,T )×Ω) ≤ K f • Φ − f ex C 0,1 (R) (1 + u C 0,1 ((0,T )×Ω) )
where u = u ex +û and
û C 0,1 ((0,T )×Ω) ≤ K f (Φ(u)) − f ex (u) C((0,T)×Ω) ≤ f • Φ − f ex C(R) .
Thus, provided f ex ∈ C 0,1 (J) and u ex ∈ C 0,1 ((0, T ) × Ω), we can conclude from f ∈ C 0,1 (J) that also T f ∈ C 0,1 (J). Theorem 3. 3. Under assumptions (45), (49), (50), the operator T defined by (48) is a self-mapping on X = C 0,1 (J).
Contractivity in higher order Schauder spaces
In order to prove contractivity of T, we need to move on to higher order Schauder spaces X ⊆ C 2,β (R) for some fixed β ∈ (0, 1]. The PDE estimates we will use for this purpose will rely on the Schauder space regularity theory for parabolic equations from [6]. Note that in view of the counterexample [7,Problem 4.9], the Hölder exponent β needs to be strictly positive.
A first attempt to circumvent the lack of higher smoothness of the projection operator would be to replace Φ in (47) by a smoothed version Φ ε . However, in order to prove convergence of the resulting approximation as ε → 0, we would need uniform boundedness of Φ ′ ε in C 0,β (R), which -as can be readily checked -is not possible, though. Note that this lack of smoothness of Φ was not an issue in [16], where we worked in Sobolev spaces, since the superposition operator induced by Φ ′ is Lipschitz as an operator from L q to L p for any 1 ≤ p < q < ∞, cf. [9].
We therefore achieve confinement to the observable range J = g(ω) in a different manner, namely by definition of the spaces in which we work as
X = { j ∈ C 2,β (R) : j ′ = 0 on R \ J}.
(52)
Thus f ex ∈ X implies the necessary conditions f ex ′ (g) = f ex ′ (g) = f ex ′′ (g) = f ex ′′ (g) = 0 on the exact solution f ex . To circumvent these conditions, one might, in place of just enforcing constant values outside J, impose C 2,β smooth extrapolation by defining X = { j ∈ C 2,β (R) : j| [g,∞) ∈ Π([g, ∞)) , j| (−∞,g] ∈ Π((−∞, g])} for some low dimensional spaces Π([g, ∞)), Π((−∞, g]) of polynomials or rational functions. However, this would not allow to estimate the global C 2,β (R) norm of f by its corresponding norm on the observable part J. We therefore remain with the space X defined by (52). Note that for all j ∈ X the identity j C k,β (J) = j C k,β (R) k ∈ {0, 1, 2} holds.
In order to map from final time data defined on ω to functions defined on J (or actually on all of R here), we will again use an operator P g : C 2,β (ω) → X and in place of (50), assume bounds with respect to the stronger spaces to hold, i.e., for all y ∈ C 2,β (ω),
P g y C 2,β (R) = P g y C 2,β (J) ≤ C(g) y C 2,β (ω) .(53)
Summarizing, we define T : X → X by (48) with u(x,t, j) denoting the solution of
u t − Lu = j(u) + r in Ω × (0, T ) ∂ u ∂ ν + γu = 0 on ∂ Ω × (0, T ) u(x, 0) = u 0 (x) x ∈ Ω ,(54)
differently from the previous subsection, where we used (47).
Contractivity for small u ex,t :
We will now prove that T is a self-mapping on a C 2,β ball with appropriately chosen radius ρ > 0, and that it contracts the error f − f ex C 2,β (J) . To this end, we use the definition (48) together with (53) and the fact that the function defined by z(x,t) := u t (x,t; f ) − u ex,t (x,t) on Θ := Ω × (0, T ) satisfies the parabolic initial boundary value problem
z t − Lz − f ex ′ (u ex )z = f ′ (u) − f ex ′ (u ex ) u t = 1 0 f ex ′′ (u ex + θû) dθû + ( f − f ex ) ′ (u) u t in Ω × (0, T ) ∂ z ∂ ν + γz = 0 on ∂ Ω × (0, T ) z(x, 0) = ( f − f ex )(u 0 (x)) x ∈ Ω .(55)
Here u = u(·, ·; f ) solves (54) with j = f , andû = u − u ex solveŝ
u t − Lû − 1 0 f ex ′ (u ex + θû) dθû = ( f − f ex )(u) in Ω × (0, T ) ∂û ∂ ν + γû = 0 on ∂ Ω × (0, T ) u(x, 0) = 0 x ∈ Ω .
Using (53) as well as the identity (51) we obtain that
T f − f ex C 2,β (J) ≤ C(g) z(T ) C 2,β (ω) .
To estimate z(T ) C 2,β (ω) , we apply [6, Theorem 6, page 65], together with the fact that for the space-time cylinder Θ = (0, T ) × Ω, the space C 0,β (Θ) with the norm
w C 0,β (Θ) = w C(Θ) + sup (x,t) =(y,s)∈Θ |w(x,t) − w(y, s)| |x − y| 2 + |t − s| β is a Banach algebra i.e., v · w C 0,β (Θ) ≤ v C 0,β (Θ) w C 0,β (Θ) for all v, w ∈ C 0,β (Θ), and, moreover j(w) C 0,β (M) ≤ j C 0,β (R) (1 + w C 0,1 (M) )
holds for all j ∈ C 0,β (R), w ∈ C 0,β (M) and either
M = Θ = (0, T ) × Ω or M = Ω. Assuming f ′ ex (u ex ) ∈ C β (Θ), we get, for Θ t = (0,t) × Ω, z C([0,t];C 2,β (Ω)) ≤ ∑ |m|≤2 D m x z C 0,β (Θ t ) ≤ K 1 0 f ex ′′ (u ex + θû) dθû + ( f − f ex ) ′ (u) C 0,β (Θ) u t C 0,β (Θ t ) + ( f − f ex )(u 0 (x)) 2,β C (Ω) ≤ K f ex ′′ C 0,β (J) (1 + u ex C 0,1 (Θ t ) + û C 0,1 (Θ t ) ) û C 0,β (Θ t ) ) + ( f − f ex ) ′ C 0,β (J) (1 + u ex C 0,1 (Θ t ) + û C 0,1 (Θ t ) ) U Z + f − f ex C 2,β (J) (1 + u 0 C 0,1 (Ω) )(d 2 u 0 2 C 1,β (Ω) + u 0 C 2,β (Ω) + 1) where U Z = ( u ex,t C 0,β (Θ t ) + z C 0,β (Θ t ) ) .
But the latter estimate does not yield contractivity, since for this we would need an estimate by a small multiple of f − f ex C 2,β (J) .
Thus we split z = z r + z 0 into a part z r satisfying the inhomogeneous PDE
z r t − Lz r − f ex ′ (u ex )z r = 1 0 f ex ′′ (u ex + θû) dθû + ( f − f ex ) ′ (u) u t in Ω × (0, T )
with homogeneous initial conditions z r (x, 0) = 0 and a part z 0 satisfying the homogeneous
PDE z 0 t − Lz 0 − f ex ′ (u ex )z 0 = 0 with inhomogeneous initial conditions z 0 (x, 0) = ( f − f ex )(u 0 (x)) x ∈ Ω .
To estimate the z r , we apply [6, Theorem 6, page 65] which yields
z r C([0,t];C 2,β (Ω)) ≤ K f ex ′′ C 0,β (J) (1 + u ex C 0,1 (Θ t ) + û C 0,1 (Θ t ) ) û C 0,β (Θ t ) ) + ( f − f ex ) ′ C 0,β (J) (1 + u ex C 0,1 (Θ t ) + û C 0,1 (Θ t ) ) U Z ,(56)
where U Z is as above and
z C 0,β (Θ t ) ≤ z r C 0,β (Θ t ) + z 0 C 0,β (Θ t ) ≤ ∑ |m|≤2 D m x z r C 0,β (Θ t ) + z 0 C 0,β (Θ t ) .
Here
û C 0,1 (Θ t ) ≤ ∑ |m|≤2 D m xû C 0,β 0 (Θ t )
hence, taking the supremum over τ ∈ [0,t] on both sides
z 0,2 C([0,t];C 2,β (Ω)) ≤ 1 1−c z 0,1 C([0,τ];C 2 (Ω))(60)
i.e.,
z 0 C([0,t];C 2,β (Ω) ≤ 2−c 1−c C Ω H σ ,C 2,β sup τ∈[0,t] Ψ(τ; σ , λ 1 )C L f − f ex C 2 (J) u 0 Ḣ2 (Ω) .(61)
Likewise, for σ 0 > d/2 + β , we have
z 0 C([0,t];C 0,β (Ω)) ≤ C Ω H σ 0 ,C 2,β sup τ∈[0,t] Ψ(τ; σ 0 , λ 1 )C L f − f ex C 2 (J) u 0 Ḣ2 (Ω) ,(62)
where we can avoid the potential singularity of Ψ(t; σ 0 , λ 1 ) at t = 0 in (58) by assuming
2 ≥ σ 0 > d/2 + β(63)
which still admits the three-dimensional space case. (Note that this avoidance is not possible for Ψ(t; σ , λ 1 ) due to the requirement σ > d/2 + 2 + β made above.) However, a problem occurs in (61), since due to the singularity at τ = 0 of Ψ(τ; σ , λ 1 ), the factor sup τ∈[0,t] Ψ(τ; σ , λ 1 ) is not finite. Recall that the time dependence of f ex ′ (u ex )
led to the convolution part z 02 that forced us to take the supremum over τ ∈ [0,t] to arrive at (60). Thus, the part z 0 of z corresponding to the initial condition cannot be controlled in this setting and we need to remove it by assuming the initial condition ( f − f ex )(u 0 ) to vanish. This is in line with existing results on decay of solutions to autonomous equations, see, e.g., [19], that require time periodicity of the coefficient -in our case f ′ ex (u ex ), which is not available here, though.
Doing so, we end up with an estimate of the form
T f − f ex C 2,β (J) ≤ C(K, ρ 0 , ρ 1 , L, Ω)(1 + u ex C 0,1 (Θ) ) u ex,t C 0,β (Θ) f − f ex C 2 (J) .
provided f − f ex C 2,β (J) ≤ ρ small enough. This yields self-mapping and contractivity on a ball of radius ρ around f ex in
X = { j ∈ C 2,β (R) : j ′ = 0 on R \ J , ( j − f ex )(u 0 ) = 0}.(64)
if u ex,t C 0,β (Θ) is sufficiently small. Theorem 3. 4. Let 2 > d/2 + β , and (53) hold, and assume that f ex ∈ X defined as in (64) Then there exist ρ , κ > 0, such that for u ex,t C 0,β (Θ) ≤ κ , the operator T is a self-mapping
on B X ρ ( f ex ) = { j ∈ C 2,β (R) : j ′ = 0 on R \ J , ( j − f ex )(u 0 ) = 0 , j − f ex C 2,β (J) ≤ ρ}. Moreover, the contraction estimates T f 0 − f ex C 2,β (J) ≤ q f 0 − f ex C 2 (J) , T n f 0 − f ex C 2,β (J) ≤ q n f 0 − f ex C 2 (J)
hold for some q ∈ (0, 1) and any f 0 ∈ B X ρ ( f ex ).
Note that assuming Note that this result is valid in space dimensions d ∈ [1, 4 − 2β ), so in particular also for d = 3, as long as β < 1 2 .
u 0 = 0 we get X ⊃ { j ∈ C 2,β (R) : j ′ = 0 on R \ J , ( j − f ex )(0) = 0}.
Contractivity for monotonically decreasing f : To avoid the smallness assumption on u ex,t C(Θ) in Theorem 3.4, we can make use of exponential decay of u t in case of monotonically decreasing f and exponentially decaying r t .
To this end, we assume existence of a nonnegative, only space dependent potentialq, that is sufficiently close to the space and time dependent function f ex
′ (u ex ) f ex ′ (u ex ) +q L ∞ (Ω×(0,T )) ≤č 1 ,(65)
and rewrite (55) as z t − Lz +qz = ( f ex ′ (u ex ) +q)z + y u t =:
r I + r II in Ω × (0, T ) ∂ z ∂ ν + γz = 0 on ∂ Ω × (0, T ) z(x, 0) = ( f − f ex )(u 0 ) x ∈ Ω ,(66)
where
y = 1 0 f ex ′′ (u ex + θû) dθû + ( f − f ex ) ′ (u) .(67)
Moreover, we take advantage of the fact that without the initial data term we gain regularity of f − f ex by iterating, i.e., by (57), we have
T 2 f − f ex C 2,β (J) ≤ C(g)C(K, ρ 0 , ρ 1 )( T f − f ex C 1,β (J) (1 + u ex C 0,1 (Θ) ) u ex,t C 0,β (Θ)(68)
which puts us into the favorable position of only having to estimate the C 1,β norm instead of the C 2,β norm of (T f − f ex )(g) = z(T ). Note that the choice (63) that enabled us to work in higher space dimensions required us to use the H 2 norm of ( f − f ex )(u 0 ), which we estimated by the C 2 norm of f − f ex . To avoid the term f − f ex C 2 (J) in the right hand side of (61), we therefore impose ( f − f ex )(u 0 ) = 0, which can, e.g., be achieved by assuming f (0) = f ex (0) and u 0 ≡ 0.
To achieve contractivity for large enough final time T , we will additionally assume that L is of the form
L = ∇ · (A∇·) with A ∈ R d×d positive definite,(69)
that r t is exponentially decaying, and that f is monotonically decreasing f ′ ≤ 0, which altogether implies that u t appearing in the right hand side of (55) decays exponentially
u t (t) C(Ω) ≤ C 2 e −c 2 t(70)
as in [16,Section 3.3] and Lemma 3.1 above.
In order to make use of dissipativity in estimating the C 1,β (Ω) norm of z(T ) = (T f − T f ex )(g), we take a small deviation via Sobolev spaces. Namely, we use continuity of the embeddings W θ ,p (0,t) → C(0,t) and W 2−2θ ,p (Ω) → C 1,β (Ω) for θ ∈ (0, 1), θ > 1 p , 1 − 2θ > d p + β (which can always be achieved by choosing p ∈ [1, ∞) sufficiently large) and apply interpolation, as well as maximal L p regularity of the operator A = −L +q cf. [18,Proposition 8] to the equation (66). The latter together with causality of the equation yields, for any µ ∈ (0,λ 1 ), whereλ 1 the smallest eigenvalue of A, and the functions defined by z µ (x,t) = e µt z(x,t), r I,µ (x,t) = e µt r I (x,t), r II,µ (x,t) = e µt r II (x,t) that z µ W 1,p (0,t;L p (Ω)) + z µ L p (0,t;W 2,p (Ω)) ≤ C A µ,p r I,µ + r II,µ L p (Ω×(0,t)) (71)
with a constant C A µ,p independent of t . This together with interpolation and the fact that the norm of the embedding W θ ,p (0,t) → C(0,t) is independent of t (by Morrey's inequality) yields
e µt z(t) C 1,β (Ω) ≤ C R + W θ ,p ,C C Ω W 2−2θ ,p ,C 1,β z µ W θ ,p (0,t;W 2−2θ ,p (Ω)) ≤ C R + W θ ,p ,C C Ω W 2−2θ ,p ,C 1 C A µ,p r I,µ + r II,µ L p (Ω×(0,t)) .(72)
For the two terms on the right hand side of (66) we can estimate r I,µ L p (Ω×(0,T )) ≤č 1 |Ω| 1/p t 0 e µs z(s) p C(Ω) ds 1/p ; To estimate r II,µ , we first of all consider the multiplier y, cf. (67),
y C(Ω×(0,t)) ≤ f − f ex C 1 (J)) + f ex ′′ L ∞ (J) û C(Ω×(0,t)) ≤ f − f ex C 1 (J)) + f ex ′′ L ∞ (J) K f − f ex C(J) ≤ (1 + K f ex ′′ L ∞ (J) ) f − f ex C 1 (J)) ,
hence, using (70), we obtain
r II,µ L p (Ω×(0,t)) ≤ (1 + K f ex ′′ L ∞ (J) ) f − f ex C 1 (J)) C 2 t 0 e p(µ−c 2 )s ds 1/p . We therefore choose 0 < µ < min{λ 1 , c 2 } so that t 0 e p(µ−c 2 )s ds ≤ 1 p(c 2 −µ) and get r II,µ L p (Ω×(0,t)) ≤ C 3 f − f ex C 1 (J)) with C 3 = C 2 (1 + K f ex ′′ L ∞ (J) )( 1 p(c 2 −µ) ) 1/p . Altogether, abbreviating C = 2 p−1 C R + W θ ,p ,C C Ω W 2−2θ ,p ,C 1 C A µ,p ,(73)
we end up with the estimate
η(t) ≤ C č p 1 |Ω| t 0 η(s) ds +C p 3 f − f ex p C 1 (J))
for η(t) = e pµt z(t) p C 1,β (Ω) . Now applying Gronwalls's inequality we obtain
η(t) ≤ CC p 3 f − f ex p C 1 (J)) 1 +Cč p 1 |Ω| t 0 e Cč p 1 |Ω|(t−s) ds = CC p 3 e Cč p 1 |Ω|t f − f ex p C 1 (J)) .
Thus withč 1 sufficiently small so that
Cč p 1 |Ω| < pµ < p min{λ 1 , c 2 },(74)
we get
z(T ) p C 1,β (Ω) ≤ CC p 3 e −(pµ−Cč p 1 |Ω|)T f − f ex p C 1 (J))
hence, via (53) and (68), Lipschitz continuity of T 2 with a factor that decays exponentially with T .
Here it is important to note that K in (56), (68) can be chosen as independent of T in the dissipative setting we are considering here, due to the following lemma, whose prove can be found in the appendix. for someq ∈ C 2 (Ω) (depending on x only),q ≥ 0,
c 1 < (epµ) 1/p 2|Ω| 1/p C R + W θ ,p ,C C Ω W 2−2θ ,p ,C 1 C A µ,p ,(76)
for some µ <λ 1 , p ∈ [1, ∞), whereλ 1 > 0 is the smallest eigenvalue of −L +q.
Then there exists a constantK independent of T such that for any j ∈ C 0,β (Θ) the solution z of
z t − Lz + cz = j in Ω × (0, T )
with homogeneous initial and conditions satisfies
∑ |m|≤2 D m x z C 0,β (Θ) ≤K j C 0,β (Θ) .
In particular, the assumptions of this Lemma are satisfied if c = − f ′ ex (u ex ) ≥ 0.
Thus we have proven the following contractivity result. Theorem 3. 5. Let u 0 = 0, f ex (0) = 0, L be of the form (69), f ex ′ (u ex ) ∈ C 2 (Θ), f ex ∈ X , and assume that r t decays exponentially r t (t) C(Ω) ≤ C r e −c r t and that (65) holds with a nonnegative potentialq ∈ C 2 (Ω) and a sufficiently small constantč 1 , cf. Then there exist T > 0 large enough and ρ > 0 small enough, such that the operator T 2 is a self-mapping on B X ρ ( f ex ) = { j ∈ C 2,β (R) : j(0) = 0 , j ′ = 0 on R \ J , j − f ex C 2,β (J) ≤ ρ}. Moreover, the contraction estimates
T 2 f 0 − f ex C 2,β (J) ≤ q f 0 − f ex C 1,β (J) , T 2n f 0 − f ex C 2,β (J) ≤ q n f 0 − f ex C 1,β (J)
hold for some q ∈ (0, 1) and any f 0 ∈ B X ρ ( f ex ). Conclusions on uniqueness analogous to Corollary 3.1 can be drawn.
Reconstructions
We will show the results of numerical experiments using the basic versions of the iterative schemes defined by (17) and (21) for each of the two data types: time trace data consisting of the value of h(t) := u(x 0 ,t) for t ∈ [0, T ]; final time data g(x) := u(x, T ) for some chosen value of T . The numerical results presented will be set in one space dimension although there is no limitation in this regard (other than computational complexity of the direct solvers) as our unknowns are functions of a single variable. Also, in this setting the graphical illustrations are more transparent. Note that in one dimension the curve ω ⊂ Ω becomes Ω itself which we take to be the unit interval. We will also consider only two equations as this case encompasses most of the features of a larger system. For notational convenience we use u and v for the dependent variables in the two equations in the system. As data we took two differing initial values u 0 (x) and v 0 (x) and as boundary conditions we used (homogeneous) Dirichlet at the left endpoint and Neumann at the right; typically different for each of u and v.
We outline below the main steps used to compile the reconstruction examples shown throughout this section. 1. The domain and solvers used. (c) For the spatial case of g meas (x) this filtering used an eigenfunction basis of the elliptic operator to take into account the boundary conditions and projecting onto a basis set of its eigenfunctions using H 2 smoothing of its coefficients to obtain g(x). In the temporal case of h meas (t) where the only constraint is with the initial data function at t = 0, either a H 1 Tikhonov penalty term or a smoothing spline routine was used.
(d) Note that the iteration schemes below themselves contains no specific regularization (although the projection of each iteration onto the range of the data could be considered in this light). For more details on data smoothing and propagation of the noise through the fixed point iteration, we refer to [16, Section 3.5]. 3. Algorithm for reconstructing f 1 , f 2 in section 4.1:
(a) Set f 0 1 , f 0 2 to some initial guess then for k = 0, 1, 2, . . .
(b) compute D t u(T ), D t v(T ) by solving (77) with f 1 = f k 1 , f 2 = f k 2 , and differen- tiating u, v with respect to time (c) update f 1 , f 2 : f k+1 1 (g u (x)) = D t u(x, T ) − △g u (x, T ) − β g u (x, T )g v (x, T ) − r u (x, T ) f k+1 2 (g v (x)) = D t v(x, T ) − △g v (x, T ) − β g u (x, T )g v (x, T ) − r v (x, T )
The iteration for reconstructing f 1 , f 2 from time trace data is defined analogously and involves computation of △u, △v instead of D t u, D t v. 4. Algorithm for reconstructing φ 1 , φ 2 in section 4.3:
(a) Set φ 0 1 , φ 0 2 to some initial guess, then for k = 0, 1, 2, . . .
(b) compute D t u(T ), D t v(T ) by solving (79) with φ 1 = φ k 1 , φ 2 = φ k 2 , and differen- tiating u, v with respect to time (c) update φ 1 , φ 2 : φ k+1 1 (w(g u (x), g v (x))) = 1 β u (D t u(x, T ) − △g u (x, T ) − f 1 (g u (x, T )) − r u (x, T )) φ k+1 2 (w(g u (x), g v (x))) = 1 β v (D t v(x, T ) − △g v (x, T ) − f 1 (g v (x, T )) − r v (x, T ))
Reconstructions of f 1 and f 2
In this first group of reconstructions we seek the recovery of the reaction terms f 1 (u) and f 2 (v) and assume the interaction terms between them are just given by a multiple of φ i (w) = w = uv. Our equations are then
D t u − △u = f 1 (u) + β u · v + r u (x,t, u) D t v − △v = f 2 (v) + β u · v + r v (x,t, v)(77)
representing a "competing species" model if β < 0 and a "symbiotic relationship" if β > 0.
The magniture of β represents the strength of the coupling. The source terms r u (x,t, u) and r v (x,t, v) are assumed known if present. We used a homogeneous Dirichlet boundary conditions at x = 0 and Neumann conditions forcing in flux at x = 1. The initial conditions u 0 and v 0 were different as were r u and r v ,
u 0 (x) = x(1 − 2x + x 2 ), v 0 (x) = sin( π 2 x) , r u (x,t) = 10 sin( π 2 x)t, r v (x,t) = 12(2x − x 2 )t .
The following sample functions to be reconstructed were used:
f 1 (u) = 2u(1 − u)(u − 0.9), f 2 (u) = max{2e −5(v−1) 2 − 0.1v 2 , −2}(78)
The first of these is a version of the Zeldovich combustion model with chosen parameters that are physically relevant, the second is chosen to offer more challenge to the reconstruction process. Depending on the driving boundary conditions and the strengths of the interaction terms φ i the range over which one must recover f i can be considerable. This offers challenges from a computational viewpoint and of course if one "knew" that the correct answer was a polynomial function all the reconstruction process would be nothing other than a least squares fit in some appropriate norm to obtain a small number of constants. We are looking beyond this here and hoping to be able to detect features in these reaction and coupling terms that might drive the model rather than be purely derivative from it.
The iteration schemes (17) and (21) can be implemented pointwise or by representing the unknowns in a set of basis functions. The choice of the latter is important. While many standard models use only low degree polynomials to represent the modelling of f this is clearly a severe limitation. Using high degree polynomials is out of the question due to the severe ill-conditioning recovering Taylor coefficients from data far from the initial point. Rational functions may seem to be a good choice but again their range of accuracy is limited when used over a wide interval. In addition, this is a nonlinear fitting problem -it is also unstable under extrapolation as it is again analytic continuation. There are also other negative effects. Our initial guesses for f i may be quite distant from the actual and in this situation during the iteration process it was frequently found that the denominator of the rational function has zeros sufficiently near to the real axis that reconstructed functions f k of large amplitude resulted at a local point. This had the effect of causing failure of the iteration scheme to converge. Indeed basis functions designed for a fixed and relatively narrow range tend to be suboptimal in this setting.
We found a good choice to be moving Gaussian basis functions. These are known to effectively model many non-linear relationships and since each basis function is non-zero over a small interval this localization is useful in the current situation where f 1 (u) and f 2 (v) are totally locally defined. As a secondary consideration here this locality property results in a sparse matrix that leads to much faster computation of this phase.
As noted earlier, we are initially taking the interaction functions {φ 1 (w), φ 2 (w)} to be a constant β times the identity and also w = uv. If β = 0 then this case is just a complete decoupling of the system and the results of [16] show a unique recovery through the resulting contraction mappings. For β sufficiently small, the analysis of section 3 then shows the same result and the question becomes if this holds true for all β . Our analysis does not cover this case and as we will see below this answer certainly appears to be negative and is illustrated graphically in Figure 1 below.
The norms shown in these figures are discrete L 2 norms of the functions f at the 100 stored values as described previously. We also show reconstructions of f 1 and f 2 achieved after a given number of iterations in Figures 2 and 3. In these figures we show the exact function as a black dashed line and the iterations in bold lines with the ordering: yellow, orange, red, light green and dark green. Typically these correspond to the second, fourth, sixth, eighth and tenth but in he case of the slower converging scheme with β = 1.3 these are at 1, 3, 6, 9, 12. Note for the larger β values the reconstruction progresses by improvements of the values of smaller magnitude first due to the causality inherent in the time trace situation as opposed to one from final time measurements.
1.0 f ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ • • • • • • • • • • • • • • • • • • • • • • • • • 5 10 15 0 1 2 3 4 5 6 7 f (n) 2 − f 2 2 n ⋆ β = −1 ⋄ β = 0.3 • β = 1 • β = 1.3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ • • • • • • • • • • • • • • • • • • • • • • • • • 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 f (n) 1 − f 1 2 n ⋆ β = −1 T = 1 ⋄ β = 0.3 T = 1 • β = 0.5 T = 0.75 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ • • • • • • • • • • • • • • • 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 f (n) 2 − f 2 2 n ⋆ β = −1 T = 1 ⋄ β = 0.3 T = 1 • β = 0.5 T = 0.75 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ • • • • • • • • • • • • • • •0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 f 1 (u) u f act iter 1 iter 2 iter 3 iter 5 iter0 1 2 3 −15 −10 −5 0 f 1 (u) u .1 2 3 4 5 −2 −1 0 1 2 f 2 (u) u .
We should expect in fact that the convergence rate will decrease with increasing β and this is entirely borne out by numerical computations As expected, there is also a difference depending on the sign of β : the case of β < 0 corresponds to a negative interaction and might be expected to be a more stable case and turns out to be the situation. The case β > 0 indicates symbiosis between the quantities represented by u and v and should be expected to be a more unstable situation. This is certainly the outcome in the ordinary differential equation version and sufficiently strong reaction terms will dominate the diffusion effect supplied by the elliptic operator. Thus one of the two of u or v should dominate and lead to blow up of the solution -perhaps in finite time. The conditions for this are quite complex. In the case of an ordinary differential equation the nonlinear right hand side terms must, in general, be uniformly Lipschitz and if they are positive and have asymptotic growth of order u δ for δ > 1 then this occurs. Under the effects of a diffusion operator this is more complex; the power law growth is coupled to, among other things, the dimension of the space, see [10,24,32]. The limitation we must impose in this case is that the measurements taken be up to or at a time T before the onset of the solution failing to exist. Of course, long before this situation the iteration scheme would most likely have failed. This is apparent from Figure 1 However, and we stress this part, we make no prior assumptions on any of the functions f i or φ i of analyticity or indeed any form that would be non-local in nature; the possibility that these reaction terms could have very different properties in different ranges of their arguments is a central hypothesis. Thus simply assuming that small time measurements leading to small values of u and v and the associated reconstruction of f i and φ i over this range extends to a global reconstruction is too restrictive in many physical situations.
Returning to Figure 1 the topmost pair shows the decrease in f (n) i − f i for i = 1, 2 using time trace data on the right boundary point where the solution is being driven. For both f 1 and f 2 the initial approximation was the zero function showing that a good initial approximation isn't critical here. For β greater than about 1.3 the scheme no longer converged. The bottom pair shows the analogous result for final time data. The scheme converged for β less than about 0.3 with the time measurement taken to be T = 1, but if this were reduced, somewhat larger values of β can be used. The case of β = 0.5 and T = 0.75 which approximately corresponds to the maximum value for T with is β , illustrates this situation.
The actual final reconstructions obtained from the above choices of β differ slightly. But as Figure 1 shows the individual iterations show considerable differences. Note also that the range of values of u and v depend quite strongly on β . The reconstructions obtained are shown in Figure 2 for the case of time trace data and in Figure 3 for the case of final time data with parameters as described above.
Reconstructions under added noise
We show below similar reconstructions but under noise in the data. Although from a technical perspective of considering only the norms of the unknowns and the data space these recovery problems appear to be only mildly ill-conditioned this does not take into account the relevant constants that are strongly influenced by the nonlinearities in the model. Thus while good reconstructions can be obtained with a few percent added uniform noise, much more leads to quite poor reconstructions. In Figure 4 we show the same functions f 1 (u) and f 2 (v) reconstructed under noise levels of 0.1% and 1%. These show the quite obvious distinction between the previous reconstructions where a very mild filter was added of the type noted above to offset any inaccuracy in the direct data due to truncation error in the direct solver and filtering to remove active noise.
Reconstructions of the interaction terms φ i
In this section we assume that both f 1 and f 2 are known, but φ 1 (w) and φ 2 (w) in
D t u − △u = f 1 (u) + β u φ 1 (w(u, v)) + r u D t v − △v = f 2 (v) + β v φ 2 (w(u, v)) + r v(79)
are unknown and have to be recovered from either a pair of time trace or a pair of final time data at t = T . The sample values to be reconstructed are where β u and β v are constants that will be used to test how large these functions can be and the iteration scheme still converge. The known functions in the model were set to
φ 1 (w) = (arctan(w) + 2w.e −(w−1). 2 ) φ 2 (w) = 0.1 · (27 − (3 − w) 2 (3 + 2w)) if w < 3 2.7 else(80)f 1 (u) = u(1 − u) , f 2 (v) = v(2 − v) r u (x,t) = 10 sin( π 2 x) · t , r v (x,t) = 12(2x − x 2 ) · t
We show the reconstructions of φ 1 and φ 2 graphically by again displaying four pairs of reconstruction iterates, for each of the values β = −1, 0.1, 1, 10, in Figure 5. Of course different values of β give different solution with differing ranges and this is clearly shown. The exact value is the dotted curve, the curve in yellow is the first iterate, the one in orange is the second, and the one in red the third or fifth. This shows the rapid convergence for a wide range of β values.
In Figure 6 we plot the relative norms of φ (n)
i − φ i,actual (that is, the ratio of this quantity to φ These plots also show the case of β u = −β v = 1 and setting instead w = u 2 v. Such an asymmetrical relationship occurs frequently in chemical reactions. As an example, if φ i (w) = w, and β u , β v have opposite signs this is the classical "Brusselator" model for a Belousov-Zhabotinsky auto-catalytic reaction. It is the theoretical underpinning of the chemistry "magic trick" of a jar of liquid changing colour from red to blue and back again in a repeating cycle. Other colour variations are also possible. The two chemicals used are often potassium bromate and cerium sulphate in an acid base. This is just a specific example of a wider class of periodic cycle solutions of the Turing type, [30]. See also [31,23]. We remark here that from either an analytical or computational perspective other choices of w In these references φ 1 and φ 1 have simple (low degree polynomial) form. The ability to go beyond this and determine a more complex form has clear physical applicability. (x,t) ∈ Q, wherev solvesv t − Lv +qv = 2(č 1 + ε)v + 1 4ε j 2 in Q with homogeneous initial and boundary conditions. The values of v,v can be estimated analogously to (71), (72) as follows e µt v(t) C(Ω) ≤ e µtv (t) C 1 (Ω) ≤ C 2(č 1 + ε) v L p (Ω×(0,t)) + 1 4ε j 2 L p (Ω×(0,t)) ≤ C|Ω| 1/p t 1/p 2(č 1 + ε) v C(Ω×(0,t)) + 1 4ε j 2 C(Ω×(0,t))
with C = C R + W θ ,p ,C C Ω W 2−2θ ,p ,C 1 C A µ , i.e., v(t) C(Ω) ≤ C|Ω| 1/p t 1/p e −µt 2(č 1 + ε) v C(Ω×(0,t)) + 1 4ε j C(Ω×(0,t)) , hence by taking the supremum over t ∈ [0, T ] and using sup t∈R e −µt t 1/p = (epµ) −1/p v C(Q) ≤ C|Ω| 1/p (epµ) −1/p 2(č 1 + ε) v C(Q) + 1 4ε j 2 C(Q) .
Thus withč 1 + ε sufficiently small, more preciselyč 1 + ε < 1 2C |Ω| −1/p (epµ) 1/p ,
w 2 C(Q) ≤ 2 v C(Q) ≤ C|Ω| 1/p (epµ) −1/p 4ε(1 − 2C(č 1 + ε)|Ω| 1/p (epµ) −1/p ) j 2 C(Q) ,
which together with (81) yields the assertion.
Epilogue
It is tempting for authors to wonder how a paper will be received and in this case the answer to the question is likely to depend on the community to which the reader belongs.
The practitioners might feel not enough attention was given for complete answers to specific problems or the range of problems was insufficient; "why was . . . not tackled? Mathematicians might have liked to see theorems containing "sufficently small" conditions replaced by estimates with tangible values. The inverse problems community seeing the complexities arising from the unknown ranges and nonlinearities themselves, might reflect, "but linear inverse problems/equations behave even better."
In some sense these are valid statements. However, reaction diffusion systems are able to model an enormous range of physical problems and coupled systems of nonlinear equations are always going to impose mathematical difficulties. There are indeed easier inverse problems, but it is the above ubiquity and challenges that make them compelling.
We hope to continue this work by expanding the range of questions posed, by looking for better analytic tools and superior computational methods. There is much, much more still to be said.
x,t; f ) = ϕ , whereas for the final time data we have min x∈Ω u(x, T ; f ) = ϕe −(λ −c)T , max x∈Ω u(x, T ; f ) = ϕe −(λ −c)T ,
by the function m(T ) = |(c 1 −λ )e (c 1 −λ )T −(c 2 −λ )e (c 2 −λ )T | |c 1 −c 2 | . Now take c 1 > c 2 > λ ; then m(0) = 1 and m ′ (T ) = (c 1 −λ ) 2 e (c 1 −λ )T −(c 2 −λ ) 2 e (c 2 −λ )T c 1 −c 2 > 0. This makes a contraction for finite time T impossible unless ϕ Z f 0 X (with f 0 (u) = u) is sufficiently small.
Corollary 3 . 1 .
31Under the assumptions of Theorem3.4, there exists at most one C 2 solution f ex of the inverse problem within any ball of radius ρ in X .
(a) The domain was the rectangle [0, L] × [0, T ] where we took L = T = 1. For a direct solver to the reaction diffusion system we used a finite difference scheme based on the Crank-Nicolson integrator and used this in an extrapolation mode resulting in fourth order accuracy in space and time.2. Data assimulation and smoothing(a) To obtain simulated data we used the direct solver at relatively low resolution to obtain either/or a time trace u(x 0 ,t) or final time u(x, T ) values. These values were then sampled at S equally spaced points and uniformly distributed, mean zero, random noise added to form our simulated measurements. The values chosen were S = 20 in the case of spatial data and S = 25 for temporal data.(b) These data values were then interpolated to the entire intervals containing 200 in space and 300 in time using an H 2 or H 1 -filtering scheme thus obtaining the working values of g meas (x) and h meas (t). to be used in the iterative schemes.
Figure 1 :
1Convergence rates of iterations as a function of β . Top: time trace, Bottom: final time.
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Figure 2 :
2Time trace data. Reconstructions of f 1 and f 2 at selected iterations as a function of β : β = −1 (top), β = 1 (middle), β = 1.3 (bottom). T = 1.
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Figure 3 :
3Final time data. Reconstructions of f 1 and f 2 at selected iterations as a function of β .β = −1 (top), β = 0.3 (middle), β = 0.5 (bottom)
which shows the reconstructions for the values β = −1, 0.3, 1, 1.3 and the decreasing rate of convergence with increasing β is evident. The choice of the largest of these values of β closely corresponds to the onset of failure of the iteration scheme in the case of time trace data on 0 ≤ t ≤ T = 1, and to a smaller combination of β and T in the case of final time data.
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Figure 4 :
4Reconstructions of f 1 and f 2 under noise from time trace data. Top: 0.1%, Bottom: 1%.
i
− φ i,actual as a function of the iteration index n, for each of these β -values. Here φ (0) i denotes the initial approximation to the i th function (in this case it was the zero function).
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Figure 5 :
5Reconstructions of φ 1 and φ 2 from time trace data. are possible. For example, we could choose w(u, v) = √ u 2 + v 2 or in the case with more equations as w( u) = u .
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Figure 6 :
6Convergence rates of φ iterations as a function of β . Top: time trace, Bottom: final time.
is equivalent to ν 21 ≥ µ 21 and ν 22 ≥ µ 22 and ν 22 k 2+ u ν 11 ≥ µ 11 and ν 12 ≥ µ 12 and ν 11 k 1+ uν 21 −µ 21
max
v
ν 22 −µ 22
max
≤ µ 22 k 2−
ν 11 −µ 11
max
v
ν 12 −µ 12
max
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1
2
3
−30
−25
−20
−15
−10
−5
0
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2
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Note that in [6, Theorem 1, page 92], | f | 2,α ∼ | f | α on a smooth domain Ω
AcknowledgmentThe work of the first author was supported by the Austrian Science Fund FWF under the grant P30054.The work of the second author was supported in part by the National Science Foundation through award DMS-1620138., that for any real vector ξ and (x,t) ∈ Q = Ω×(0, T) the inequality ∑ d i, j=1 a i, j (x,t)ξ i ξ j ≥ K 2 |ξ | 2 with K 2 > 0, and that j ∈ C β (Q).1Then there exists a constant K 0 depending only on K 1 , K 2 , d such that if w is a solution of w t − Lw = j in Q with w, w t , w x i , w x i ,x j ∈ C 0,β (Q) satisfies the estimateThus it suffices to estimate w C(Q) on the right hand side in terms of j C 0,β (Q) , which we do by means of a maximum principle, similarly to Lemma 3.1. More precisely, since under the assumption (69) v :v(x, 0) = 0 x ∈ Ω . where we have used Young's inequality with some ε > 0 to be chosen below, as well as nonnegativity of w 2 and (75). Hence by the maximum principle 0 ≤ v(x,t) ≤v(x,t) for all
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|
[] |
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"RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization",
"RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization",
"RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization",
"RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization"
] |
[
"Yiqing Shen [email protected] \nSchool of Mathematical Sciences\nShanghai Jiao Tong University\nShanghaiChina\n",
"Yulin Luo \nSchool of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina\n",
"Dinggang Shen [email protected] \nSchool of Biomedical Engineering\nShanghaiTech University\nShanghaiChina\n\nShanghai United Imaging Intelligence Co., Ltd\nShanghaiChina\n",
"Jing Ke [email protected] \nSchool of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina\n\nSchool of Computer Science and Engineering\nUniversity of New South Wales\nSydneyAustralia\n\nBirenTech Research\nShanghaiChina\n",
"Yiqing Shen [email protected] \nSchool of Mathematical Sciences\nShanghai Jiao Tong University\nShanghaiChina\n",
"Yulin Luo \nSchool of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina\n",
"Dinggang Shen [email protected] \nSchool of Biomedical Engineering\nShanghaiTech University\nShanghaiChina\n\nShanghai United Imaging Intelligence Co., Ltd\nShanghaiChina\n",
"Jing Ke [email protected] \nSchool of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina\n\nSchool of Computer Science and Engineering\nUniversity of New South Wales\nSydneyAustralia\n\nBirenTech Research\nShanghaiChina\n"
] |
[
"School of Mathematical Sciences\nShanghai Jiao Tong University\nShanghaiChina",
"School of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina",
"School of Biomedical Engineering\nShanghaiTech University\nShanghaiChina",
"Shanghai United Imaging Intelligence Co., Ltd\nShanghaiChina",
"School of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina",
"School of Computer Science and Engineering\nUniversity of New South Wales\nSydneyAustralia",
"BirenTech Research\nShanghaiChina",
"School of Mathematical Sciences\nShanghai Jiao Tong University\nShanghaiChina",
"School of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina",
"School of Biomedical Engineering\nShanghaiTech University\nShanghaiChina",
"Shanghai United Imaging Intelligence Co., Ltd\nShanghaiChina",
"School of Electronic Information and Electrical Engineering\nShanghai Jiao Tong University\nShanghaiChina",
"School of Computer Science and Engineering\nUniversity of New South Wales\nSydneyAustralia",
"BirenTech Research\nShanghaiChina"
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[] |
Stain variations often decrease the generalization ability of deep learning based approaches in digital histopathology analysis. Two separate proposals, namely stain normalization (SN) and stain augmentation (SA), have been spotlighted to reduce the generalization error, where the former alleviates the stain shift across different medical centers using template image and the latter enriches the accessible stain styles by the simulation of more stain variations. However, their applications are bounded by the selection of template images and the construction of unrealistic styles. To address the problems, we unify SN and SA with a novel RandStainNA scheme, which constrains variable stain styles in a practicable range to train a stain agnostic deep learning model. The RandStainNA is applicable to stain normalization in a collection of color spaces i.e. HED, HSV, LAB. Additionally, we propose a random color space selection scheme to gain extra performance improvement. We evaluate our method by two diagnostic tasks i.e. tissue subtype classification and nuclei segmentation, with various network backbones. The performance superiority over both SA and SN yields that the proposed RandStainNA can consistently improve the generalization ability, that our models can cope with more incoming clinical datasets with unpredicted stain styles. The codes is available at https://github.com/yiqings/RandStainNA.
|
10.1007/978-3-031-16434-7_21
|
[
"https://export.arxiv.org/pdf/2206.12694v1.pdf"
] | 250,073,005 |
2206.12694
|
1370d62a4b49570be9c0137d7b3236ac213f5426
|
RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization
25 Jun 2022
Yiqing Shen [email protected]
School of Mathematical Sciences
Shanghai Jiao Tong University
ShanghaiChina
Yulin Luo
School of Electronic Information and Electrical Engineering
Shanghai Jiao Tong University
ShanghaiChina
Dinggang Shen [email protected]
School of Biomedical Engineering
ShanghaiTech University
ShanghaiChina
Shanghai United Imaging Intelligence Co., Ltd
ShanghaiChina
Jing Ke [email protected]
School of Electronic Information and Electrical Engineering
Shanghai Jiao Tong University
ShanghaiChina
School of Computer Science and Engineering
University of New South Wales
SydneyAustralia
BirenTech Research
ShanghaiChina
RandStainNA: Learning Stain-Agnostic Features from Histology Slides by Bridging Stain Augmentation and Normalization
25 Jun 20222 Y. Shen, Y. Luo, D. Shen, and J. Ke Fig. 1. The overall framework of the proposed RandStainNA.Histology Image · Stain Normalization · Stain Augmenta- tion Equal contributions Correspondence to Jing Ke
Stain variations often decrease the generalization ability of deep learning based approaches in digital histopathology analysis. Two separate proposals, namely stain normalization (SN) and stain augmentation (SA), have been spotlighted to reduce the generalization error, where the former alleviates the stain shift across different medical centers using template image and the latter enriches the accessible stain styles by the simulation of more stain variations. However, their applications are bounded by the selection of template images and the construction of unrealistic styles. To address the problems, we unify SN and SA with a novel RandStainNA scheme, which constrains variable stain styles in a practicable range to train a stain agnostic deep learning model. The RandStainNA is applicable to stain normalization in a collection of color spaces i.e. HED, HSV, LAB. Additionally, we propose a random color space selection scheme to gain extra performance improvement. We evaluate our method by two diagnostic tasks i.e. tissue subtype classification and nuclei segmentation, with various network backbones. The performance superiority over both SA and SN yields that the proposed RandStainNA can consistently improve the generalization ability, that our models can cope with more incoming clinical datasets with unpredicted stain styles. The codes is available at https://github.com/yiqings/RandStainNA.
Fig. 1
. The overall framework of the proposed RandStainNA.
Introduction
Pathology visually exams across a diverse range of tissue types obtained by biopsy or surgical procedure under microscopes [6]. Stains are often applied to reveal underlying patterns to increase the contrast between nuclear components and their surrounding tissues [14]. Nevertheless, the substantial variance in each staining manipulation, e.g. staining protocols, staining scanners, manufacturers, batches of staining may eventually result in a variety of hue [10]. In contrast to pathologists who have adapted themselves to these variations with years' training, deep learning (DL) methods are prone to suffer from performance degradation, with the existence of inter-center stain heterogeneity [2]. Specifically, as color is a salient feature to extract for by deep neural networks, consequently, current successful applications for whole slide images (WSIs) diagnoses are subject to their robustness to color shift among different data centers [5]. There are two primary directions to reduce the generalization error, namely stain normalization and stain augmentation [22].
Stain normalization (SN) aims to reduce the variation by aligning the staincolor distribution of source images to a target template image [11,17,24]. Empirical studies regard stain normalization as an essential prerequisite of downstream applications [2,12,22]. Yet, the capability to pinpoint a representative template image for SN relies heavily on domain prior knowledges. Moreover, in real-world settings such as federated learning, the template-image selection is not feasible due to the privacy regularizations [10], as source images are inaccessible to the central processor as a rule. Some generative adversarial networks (GANs) are proposed recently [17,18] for SN, yet remaining the phenotype recognizability is always problematic. A salient drawback of the sole stain style in SN is the restricted color-correlated features can be mined by deep neural networks.
i.e. F S M = N (M S A , Σ S A ) and F S D = N (M S D , Σ S D )
. The three-step training stage comprises a random selection of color space S, the generated of an associated random stain style template
[M S v , Σ S v ]
, and the normalization of a batch with the generated virtual template. Our approach is downstream task agnostic.
Stain augmentation (SA) seeks a converse direction to SN by simulating stain variations while preserving morphological features intact [21]. Tellz et. al. [20] first tailored data augmentations from RGB color space to H&E color space. Afterward, in parallel with SN approaches, GAN is also widely adopted by stain augmentation applications e.g. HistAuGAN [23].
Previous works have compared the performance between SN and SA without interpretation of their differences [22]. Moreover, we have observed that the mathematical formulations of SN are coincidental with SA, where the transfer of SN depends on a prior Dirichlet distribution [25] and SA distorts images with a uniform distribution [20], depicted in Fig. 1. Hence, we make the first attempt to unify SN and SA for histology image analysis. Two primary contributions are summarized. First, a novel Random Stain Normalization and Augmentation (RandStainNA) method is proposed to bridge stain normalization and stain augmentation, consequently, images can be augmented with more realistic stain styles. Second, a random color space selection scheme is introduced to extend the target scope to various color spaces including HED, HSV, and LAB, to increase flexibility and produce an extra augmentation. The evaluation tasks include tissue classification and nuclei segmentation, and both show our method can consistently improve the performance with a variety of network architectures.
Method Overview. Random Stain Normalization and Augmentation (Rand-StainNA) is a hybrid framework designed to fuse stain normalization and stain augmentation to generate more realistic stain variations. It incorporates randomness to SN by automatically sorting out a random virtual template from pre-estimated stain style distributions. More specifically, from the perception of SN's viewpoint, stain styles 'visible' to the deep neural network are enriched in the training stage. Meanwhile, from the perception from the SA's viewpoint, RandStainNA imposes a restriction on the distortion range and consequently, only a constrained practicable range is 'visible' to CNN. The framework is a general strategy and task agonist, as depicted in Fig. 2.
Stain Style Creation and Characterization. Unlike the formulation of comprehensive color styles of nature images, stain style of histology remains to be a vague concept, which is primarily based on visual examination, restricting to obtain a precise objective for alleviating the stain style variation [16,24]. To narrow this gap, our work first qualitatively defines the stain style covering six parameter, namely the average and standard deviation of each channel in LAB color space [16]. We pick up LAB space for its notable capability to represent heterogeneous styles in medical images [16]. Novelly, we first transfer all histology slides in the training set from RGB space to LAB color space. Then the stain style of im-
age x i are depicted by A i = [a (l) i , a (a) i , a (b) i ] ∈ R 3 and D i = [d (l) i , d (a) i , d (b) i ] ∈ R 3 , where a (c) i , d (c) i
are the average value and standard deviation of each channel c ∈ {l, a, b} in image x i , as shown in the pre-processing stage block in Fig. 2.
Virtual Stain Normalization Template. In routine stain normalization approaches [16,24], a source image is normalized to a pre-selected template image by aligning the average of A s and standard deviation D s of pixel values to the template A t and D t . Thus, it is sufficient to formulate a template image with [A t , D t ]. In the proposed RandStainNA, we expand the uniformly shared one-template mode to a board randomly generated virtual templates [A v , D v ] scheme. To be more specific, iteratively, random A v is sampled from distribution F A , and likewise D v is picked out from the other distribution F D , which are jointly used as the target for every training sample normalization. Empirical results yield that eventual performance are robust to a wide range of distribution types of F A and F D , such as Gaussian and t-distribution. In the rest of this section, we simply leverage Gaussian distribution as the estimation i.e. setting
F A = N (M A , Σ A ), F D = N (M D , Σ D ), where N (M, Σ)
writes for the Gaussian distribution with expectation M and covariance matrix Σ. Notably, due to the orthogonality of channels, Σ is a diagonal matrix i.e. Σ = diag(σ 2 1 , σ 2 2 , σ 2 3 ) for some σ j with j = 1, 2, 3.
Statistics Parameters Estimation For Virtual Template Generation.
The estimation of statistical parameters of M A , Σ A , M D , Σ D are afterwards applied to the formation of stain style discussed above. A proper candidate is attributed to the sample channel mean values of all the training images for M A and M D , as well as the standard deviations of samples for Σ A and Σ D , based on the average value and standard deviation of the whole training set. However, two defects turn out in this discipline that one is the inefficiency to transverse the whole set, and the other is the special cases of infeasibility e.g. federated learning or lifelong learning. Therefore, we provide a more computation-efficient alternative, by randomly curating a small number of patches from the training set and applying their sample mean and standard deviation as
M A , Σ A , M D , Σ D .
The empirical results suggest it can achieve competitive performance.
Image-wise Normalization With Random Virtual Template. After transferring image x i from RGB into LAB space, we write the pixel value as [l, a, b]. We denote the average and standard deviation (std) for each channel of this image as
A i = [a (l) i , a (a) i , a (b) i ] and D i = [d (l) i , d (a) i , d (b)
i ], and the generated random virtual template associated to
x i from F A , F D as A v = [a (l) v , a (a) v , a (b) v ] and D i = [d (l) v , d (a) v , d (b) v ].
Then the image-wise normalization based on random virtual template is formulated as
l = d (l) v d (l) i (l − a (l) i ) + a (l) v a = d (a) v d (a) i (a − a (a) i ) + a (a) v b = d (b) v d (b) i (b − a (b) i ) + a (b) v ,(1)
Then, we transfer [l , a , b ] from LAB back to RGB space. Notably, we generate different virtual templates for images that vary at every epoch during the training stage. Therefore, RandStainNA can largely increase the data variations with the on-the-fly generate virtual templates.
Determine Random Color Space for Augmentation. By the computation of the stain style parameters [M, Σ] of distinct color spaces, we can derive their associated F A and F D . Afterwards, we extend our RandStainNA from LAB to other color spaces e.g. HED, HSV. This extension allows the proposal of a random color space selection scheme, which will further strengthen the regularization effect. The candidate pool comprises three widely-used color spaces in the domain of histology, i.e. HED, HSV, LAB. Training iteratively, an initial color space S is an arbitrary decision with equal probability i.e. p = 1 3 , or with manually-assigned values depending of the performance of each independent space. Subsequently, a virtual template is assigned to associate with S to perform image-wise stain normalization.
Experiments
Dataset and Evaluation Metrics. We evaluate our proposed RandStainNA on two image analysis tasks i.e. classification and segmentation. Regarding the patch-level classification task, we use a widely-used histology public dataset NCT-CRC-HE-100K-NONORM for training and validation, with the addition of the CRC-VAL-HE-7K dataset for external testing [9]. These two sets comprise a number of 100,000 and 7180 histological patches respectively, from colorectal cancer patients of multiple data centers with heterogeneous stain styles. We randomly pick up 80% from NCT-CRC-HE-100K-NONORM for training and the rest 20% for validation. The original dataset covers nine categories, but for the category of background, we can straightforwardly identify them in pre-processing stage with OTSU algorithm [15] and thus it is removed in our experiment for a more reliable result. The top-1 classification accuracy (%) is used as the metric for the 8-category classification task. For the nuclei segmentation task, we use a small public dataset of MoNuSeg [12], with Dice and IoU as the metrics.
Network Architecture and Settings. In the classification task, we employ six backbone architectures to perform the evaluations, namely the ResNet-18 [7], ResNet-50 [7], MobileNetV3-Small [8], EfficientNetB0 [19], ViT-Tiny [3] and SwinTransformer-Tiny [13]. These networks, including CNN and transformers, may represent a wide range of network capabilities, which effectively demonstrate the adaptability of our method in different settings. In the nuclei segmentation task, we use CIA-Net as the backbone [26] for its notable performance in small set processing. We use a consistent training scheme for distinct networks for performance comparison with stain augmentation and stain normalization methods. Detailed training schemes and hyper-parameter settings are shown in the supplementary material. We perform 3 random runs and compute the average for each experiments.
Compared Methods. All models are trained with morphology augmentation, namely the random vertical and horizontal flip. In both evaluation tasks, we compare our method with existing stain normalization [16] and stain augmentation [21,22] approaches performed in the three color spaces i.e. HED, HSV, LAB. Regarding the stain augmentation in HED, we employ a multiplication rule [22] that adds noise to each channel i.e. p = p * ε 1 + ε 2 , where p is the augmented pixel value and p is the original pixel value, and ε 1 and ε 2 are uniform random noises, termed as stain augmentation scheme #1 (SA1). For the SA in HSV, we adopt an addition rule i.e. p = p + p * ε [22], termed as stain augmentation scheme #2 (SA2). We integrate the above two schemes for LAB stain augmentation, due to an absence of literature works for SA in LAB. We also configure two augmentation settings according to different degrees of distortion i.e. range of random noise, denoted as light (L) and strong (S) [21,22]. To fully retain recognizable morphological features, we do not take GAN-related approaches for comparison. Table 1. Test accuracy (%) comparison on the tissue type classification task. We compare our method with stain augmentation (SA) [22] and stain normalization (SN) [16] in three color space i.e. LAB, HSV, HED [22]. In SA, we follow previous work [22] by leveraging two settings, namely light (L) and strong (S), determined by the degree of distortion. The best and second best are marked in boldface and with * respectively.
Method
ResNet18 Results. Our method can consistently improve the baseline performance of the six backbone architectures in terms of test accuracy, with the implementation in three color spaces, as shown in Table 3. Therefore, it yields the effectiveness of RandStainNA. The hybrid architecture can outperform a sole deployment of either SN or SA. With the random color space selection scheme (denoted as 'full'), the RandStainNA achieves further performance improvement. To demonstrate the effects achieved with different approaches straightforwardly, we visualize the original raw images with stain variations, SN images, SA images, and images processed with our RandStainNA in Fig. 3. In the visualization graph, we use the results from SA and SN performed in HSV space as an example, which shows very similar outcomes in LAB and HED spaces. As shown, the SN unifies stain styles into a shared template that may leave out many useful features [22], and the SA may generate unrealistic images. In contrast, our method generates much more realistic images to reorganize by both human and deep learning techniques. The ble 2. For SN and SA in each color space, we pick up one configuration with higher performance in the classification task. Our method also achieves the best performance to demonstrate its effectiveness in various downstream tasks.
Conclusion
The proposed RandStainNA framework aims to cope with the inevitable stain variance problem for clinical pathology image analysis. Leveraging the advantages of both stain normalization and stain augmentation, the proposed framework produces more realistic stain variations to train stain agnostic DL models. Additionally, RandStainNA is straightforward practically and efficient when applied as an on-the-fly augmentation technique, in comparison with most current GANs. Moreover, the result shows the feasibility to train robust downstream classification and segmentation networks on various architectures. One future direction of our current works is the expansion of color spaces, e.g. YUV, YCbCr, YPbPr, YIQ, XYZ [4], to further improve the generalization ability. Table 3. Test F1-score comparison on the tissue type classification task. We follow the same settings as Table 1 50 128 1e-1→1e-3 SGD 0.9 1e-4 Cosine Warm-up w. 3ep ResNet50 50 128 1e-1→1e-3 SGD 0.9 1e-4 Cosine Warm-up w. 3ep MobileNetV3-S 50 128 1e-1→1e-3 SGD 0.9 1e-4 Cosine Warm-up w. 3ep EfficientNet-B0 50 128 1e-1→1e-3 SGD 0.9 1e-4 Cosine Warm-up w. 3ep ViT-T 50 512 1e-3→1e-5 Adamw -5e-2 Cosine Warm-up w. 3ep Swin-T 50 512 1e-3→1e-5 Adamw -5e-2 Cosine Warm-up w. 3ep CIA-Net 30 4 2e-5→4e-6 Adam --
Supplementary Material
Step decay 0.95 per ep Table 6. Patch-level classification performance comparison on BACH in terms of test top-1 accuracy and AUC. The Data, annotations, and description of BACH is freely available at https://iciar2018-challenge.grand-challenge.org/. For all the compared methods, we follow the same settings as Table 2 in the paper. We use ResNet-18 as the backbone.
Fig. 2 .
2The overall pipeline of the proposed RandStainNA that fuses stain normalization and stain augmentation. Prior to the training stage, random virtual template generation functions are defined
Fig. 4 Fig. 3 .
43provides the UMAP embedding of stain style parameters of [M, Σ] in the associated solutions. The nuclei segmentation results are listed in the Ta-The illustrative patch examples of (a) raw images, (b) stain-normalized images, (c) stain-augmented images, (d) images processed with the proposed RandStainNA. We incorporate the results of four random runs into one image patch to demonstrate the different grades of randomness maybe achieved by the stain augmentation methods and our RandStainNA in (c) and (d).
Fig. 4 .
4UMAP[1] embedding of the stain style charactiersitic statistics i.e. [M, Σ] of raw images, stain normalized images, stain augmented images and those augmented with our RandStainNA. As shown, our method can enrich the realistic stain styles in training CNNs.
Fig. 5 .
5T-sne visualization of the penultimate layers trained with baseline (a), stain augmentation (b), stain normalization (c), our proposed RandStainNA (d).
Table 2. Performance comparison on nuclei segmentation in terms of Dice and IoU.ResNet50 MobileNet EfficientNet ViT SwinTransformer
Baseline
84.20
72.07
80.25
79.62
72.85
71.06
SA1-L+LAB 84.62
77.24
84.09
81.71
75.17
69.42
SA1-S+LAB 89.35
87.97
90.79
90.81
84.77
76.76
SA2-L+LAB 93.55
92.81
92.72
93.66
89.90
88.42
SA2-S+LAB 90.77
90.50
89.13
89.64
84.84
75.13
SA1-L+HED 92.47
89.55
90.81
92.57
86.22
81.17
SA1-S+HED 88.77
87.92
86.58
88.67
87.42
76.50
SA2-L+HSV 91.39
88.97
88.97
91.10
82.80
78.70
SA2-S+HSV 91.93
90.06
90.22
91.76
85.81
77.83
SN+LAB
93.01
91.40
92.23
91.92
89.77
88.80
SN+HED
91.38
90.57
89.54
91.34
88.29
86.60
SN+HSV
93.85
93.86
92.38
93.90
90.63
86.21
Ours (LAB) 94.44*
93.97
93.94
93.54
90.30
91.01
Ours (HED)
93.28
93.61
91.69
92.67
91.41
90.03
Ours (HSV)
94.04
94.12*
94.06*
94.81
93.27*
92.75
Ours (Full)
94.66
94.45
94.53
94.62*
93.34
92.39*
Metrics Baseline SA+LAB SA+HED SA+HSV SN+LAB SN+HED SN+HSV Ours
Dice
0.7270 0.7297
0.7354
0.7349
0.7792
0.7668
0.7780 0.7802
IoU
0.5564 0.5665
0.5758
0.5712
0.6302
0.6119
0.6291 0.6335
in paper. The best performance is marked in boldface. Our proposed consistently achieve the best performance.Method
ResNet18 ResNet50 MobileNet EfficientNet ViT SwinTransformer
Baseline
0.785
0.628
0.737
0.738
0.673
0.624
SA1-L+LAB 0.784
0.689
0.788
0.761
0.699
0.594
SA1-S+LAB 0.853
0.815
0.870
0.869
0.795
0.691
SA2-L+LAB 0.913
0.904
0.901
0.914
0.863
0.845
SA2-S+LAB 0.877
0.875
0.859
0.864
0.817
0.690
SA1-L+HED 0.890
0.858
0.875
0.897
0.831
0.764
SA1-S+HED 0.853
0.839
0.831
0.855
0.845
0.699
SA2-L+HSV 0.883
0.854
0.855
0.880
0.786
0.718
SA2-S+HSV 0.896
0.866
0.876
0.892
0.825
0.705
SN+LAB
0.899
0.870
0.883
0.878
0.855
0.835
SN+HED
0.890
0.860
0.852
0.879
0.838
0.800
SN+HSV
0.920
0.922
0.900
0.921
0.885
0.805
Ours (LAB)
0.927
0.912
0.922
0.914
0.857
0.873
Ours (HED)
0.905
0.915
0.881
0.904
0.898
0.866
Ours (HSV)
0.920
0.922
0.920
0.930
0.913
0.899
Ours (Full)
0.926
0.925
0.927
0.926
0.915
0.896
(a)
(b)
(c)
(d)
Table 4 .
4Training configurations. Notably, for all the compared methods i.e. stain normalization and stain augmentation, we use a consistent training settings. E, BS, LR, Opt., M and WD represents the epoch, batch size, learning rate, optimization, momentum and weight decay ratio respectively.Networks
E BS
LR
Opt. M WD Scheduler Other Strategy
ResNet18
Table 5 .
5Model complexity comparison in terms of number of trainable parameters and floating point operations per second (FLOPS).Networks
Number of Parameters (×10 6 ) FLOPS (×10 8 )
ResNet18
11.69
1.82
ResNet50
25.56
4.11
MobileNetV3-Small
2.54
59.66
EfficientNetB0
5.29
401.68
ViT-Tiny
5.67
1.08
SwinTransformer-Tiny
27.40
4.35
CIA-Net
15.67
331.27
Table 7 .
7Metrics Baseline SA+LAB SA+HED SA+HSV SN+LAB SN+HED SN+HSV Ours Classification comparison on BACH with MobileNetV3-Small.Accuracy 72.68
74.18
74.22
73.43
74.98
62.50
74.01 79.11
AUC
0.914
0.931
0.929
0.918
0.933
0.870
0.923 0.951
Metrics Baseline SA+LAB SA+HED SA+HSV SN+LAB SN+HED SN+HSV Ours
Accuracy 74.22
77.66
75.99
77.33
78.34
68.72
74.69 79.08
AUC
0.925
0.937
0.937
0.938
0.944
0.884
0.924 0.949
Acknowledgements. This work has been supported by NSFC grants 62102247.
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